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A Review of Trimming in Isogeometric Analysis: Challenges, Data Exchange and Simulation Aspects

June 2017
Archives of Computational Methods in Engineering 25(8):1-69

DOI:10.1007/s11831-017-9220-9

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CC BY 4.0

Authors:

Benjamin Marussig

Graz University of Technology

Thomas J.R. Hughes

University of Texas at Austin

We review the treatment of trimmed geometries in the context of design, data exchange, and computational simulation. Such models are omnipresent in current engineering modeling and play a key role for the integration of design and analysis. The problems induced by trimming are often underestimated due to the conceptional simplicity of the procedure. In this work, several challenges and pitfalls are described.

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Arch Computat Methods Eng

DOI 10.1007/s11831-017-9220-9

ORIGINAL PAPER

A Review ofTrimming inIsogeometric Analysis: Challenges, Data

Exchange andSimulation Aspects

BenjaminMarussig1 · ThomasJ.R.Hughes1

Received: 16 February 2017 / Accepted: 17 March 2017

geometric objects. On the other hand, computational analy-

sis has focused on the problem-solving part of engineering.

Thus, the main attention has been drawn to the develop-

ment of mathematical models governing physical phenom-

ena as well as the reliability and eﬃciency of their numeri-

cal treatment. Still, design models are usually the starting

point of the analysis process in order to deﬁne the domain

of interest. In current engineering design, however, they are

subsequently approximated by ﬁnite element meshes for

computation. Since this is a fundamental step in conven-

tional simulations, there is a substantial body of literature

on meshing, see e.g.,[28, 94, 98, 281, 322] and the refer-

ences cited therein. Finite element analysis (FEA) was a

widely used commercially available procedure in engineer-

ing prior to the advent of commercial CAD. Nevertheless,

FEA ﬁnds itself separated from design by its own represen-

tation of geometrical objects, which is diﬀerent from CAD.

The given situation has contributed to a loss of communi-

cation between these ﬁelds, both of which are essential in

the process of addressing practical engineering problems.

Isogeometric analysis [66, 140] provides an alternative

to the conventional analysis methodology that converts

CAD models for use in FEA. The key idea is to perform

numerical simulations based on CAGD technologies.

Besides the fact that this synthesis oﬀers several computa-

tional beneﬁts, such as high continuity [67, 68, 187], the

long term goal of isogeometric analysis is to enhance the

overall engineering product development process by clos-

ing the gap between design and analysis. An invaluable

byproduct of this eﬀort is the initiation of a dialog between

these two communities which had drifted apart.

Gaining insight into each others’ ﬁelds is an essen-

tial component to tackle the problem of interoperability

between analysis and design representations. It is easily

overlooked that each discipline has its open challenges

Abstract We review the treatment of trimmed geometries

in the context of design, data exchange, and computational

simulation. Such models are omnipresent in current engi-

neering modeling and play a key role for the integration of

design and analysis. The problems induced by trimming are

often underestimated due to the conceptional simplicity of

the procedure. In this work, several challenges and pitfalls

are described.

1 Introduction

Trimming is much more complicated than most peo-

ple think. It is one of the most fundamental procedures in

Computer Aided Geometric Design (CAGD) that allows

the construction of complex geometries. Unfortunately, it

is also the source of one of the most serious impediments

to interoperability between Computer Aided Design (CAD)

systems and downstream applications like numerical simu-

lation [268]. This work aims to increase awareness of this

issue by providing a broad overview of trimmed geometries,

addressing design, data exchange, and analysis aspects.

Once upon a time, the original vision of CAD was the

holistic treatment of the engineering design process [247].

However, it has emerged as an autonomous discipline

which seeks to optimize the modeling and visualization of

The original version of this article was revised as OpenChoice

Grants were missing.

* Benjamin Marussig

marussig@utexas.edu

1 Institute forComputational Engineering andSciences, The

University ofTexas atAustin, 201 East 24th St, Stop C0200,

Austin, TX78712, USA

B.Marussig, T.J.R.Hughes

1 3

and limitations. Due to the capability of current CAD sys-

tems it may seem that design models are ideal creations,

but this notion is simply not true. There are several open

issues that need to be solved. Compact overviews are given

in the independent compilations of Kasik etal. [151] and

Piegl [228]. In these papers, robustness and interoperabil-

ity issues are identiﬁed as crucial CAD problems. Although

trimmed geometries are not explicitly mentioned in these

papers, they play a central role in both cases.

The most common description of CAGD models is the

boundary representation (B-Rep) where an object is repre-

sented by its boundary surfaces rather than a volume dis-

cretization. These surfaces are usually constructed indepen-

dently from each other and often only certain regions of a

surface are supposed to be part of the actual object. Trim-

ming allows a modeler to cut away the superﬂuous sur-

face areas. To be precise, the visualization of the surfaces

is adapted while their parameterization and mathematical

description remain unchanged. This procedure is very con-

venient and inevitable in many operations such as surface-

to-surface intersection. However, the main problem is that

trimming cannot practically be performed exactly within

CAGD applications. Thus, the ﬁnal object possesses small

gaps and overlaps between its surfaces. Figure1 illustrates

some inaccuracies of a model deﬁned by a torus inter-

sected by a plane. Note that the discrepancy between the

computed intersection

CTrimmed

and the related exact solu-

tion

CTorus

is scarcely visible. The imperfections of trimmed

geometries are usually very well hidden from the user, but

they surface as soon as a design model is applied to down-

stream applications. To use the words of Piegl [228]:

While one can cheat the eye in computer graphics and

animation, the milling machine is not as forgiving.

Numerical simulation of practical trimmed models is

more than the analysis of a speciﬁc type of a CAGD repre-

sentation. It rather addresses the core issue of the interoper-

ability between design and analysis, namely the appropriate

treatment of the deﬁciencies of design models. To be clear,

this problem is not restricted to isogeometric analysis, but

manifests itself as complications during the meshing pro-

cess in the case of conventional analysis methodology. In

fact, geometry repair and corrections of design models are

mandatory tasks, before actual mesh generation can be

applied [86, 114]. Isogeometric analysis of trimmed geom-

etries tackles these issues directly at the source, i.e., the

design model. Thus, many pitfalls that may occur in a

meshing process can be circumvented [61].

(a) (b)

Figure 1(c)

Figure1(d) C

Torus

Trimmed

Originalobjects Trimmed geometry

Approximate visualization Approximate intersection

Fig. 1 Model of a half of a torus: a initial non-trimmed torus and

surface deﬁned in the xy-plane, b resulting trimmed object, c closeup

showing the deviation from the visualization mesh (blue background)

and the computed intersection curve

CTrimmed

(yellow) of the objects,

and d closeup illustrating the diﬀerence of the original inner circle of

the torus

CTorus

(red) to the intersection curve computed. The images

c and d are captured in top view. (Color ﬁgure online)

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

It is important to note that the CAD community is also

inﬂuenced by the ongoing dialog. An increasing number

of researchers propose new modeling concepts that take

the needs of downstream applications into account, see

e.g.[61, 203, 247]. We believe that the aligned eﬀorts of

both communities are the keys to unite design and analysis,

resulting in a holistic treatment of the engineering design

process.

This paper intends to encourage the interaction of

these ﬁelds by providing an overview of various aspects

related to trimmed geometries. Section2 begins by review-

ing some basic concepts frequently used in CAGD. It is

focused on non-uniform rational B-spline (NURBS) based

B-Rep models since they are the most popular representa-

tion in engineering design. Based on this, Sect.3 addresses

the role of trimming in the context of design. A critical

assessment of exchanging data between diﬀerent software

packages is provided in Sect. 4. Finally, various strate-

gies to deal with trimmed geometries in an isogeometric

analysis process are outlined in Sect.5. Each of these three

review sections closes with a brief summary of the main

points and their discussion. Section6 moves on to focus on

a particular aspect, namely the stabilization of a trimmed

basis. In the concluding section, the main ﬁndings are sum-

marized and some open research questions are listed.

2 CAGD Fundamentals

B-splines and their rational counterpart NURBS provide

the basis for the geometric modeling of most engineering

models. This section gives a brief overview of this CAGD

technology focusing on aspects which are crucial for the

subsequent discussion. For further information related to

spline theory the interested reader is referred to [34, 62,

83]. Detailed descriptions of eﬃcient algorithms can be

found in[230]. In the present paper, the terms B-spline and

NURBS are used to refer to basis functions. The geomet-

ric objects described using these functions, i.e.,curves and

surfaces, may be generally denoted as patches.

2.1 Basis Functions

B-splines

Bi,p

consist of piecewise polynomial segments

which are connected by a certain smoothness. They are

deﬁned recursively for a ﬁxed polynomial degree

by a

strictly convex combination of B-splines of the previous

degree,

p−1,

given by

(1)

i,p(u)=

u−u

ui+p−ui

Bi,p−1(u)

+ui+p+1−u

i+p+1

−u

i+1

Bi+1,p−1(u)

with

The essential element for this construction is the knot vec-

tor

𝛯

characterized as a non-decreasing sequence of coor-

dinates

ui⩽ui+1

. The parameters

are termed knots and

the half-open interval

[ui,ui+1)

is called

th knot span. Each

knot span has

p+1

non-vanishing B-splines as illustrated

inFig.2. Each basis function is entirely deﬁned by

p+2

knots and its support,

supp{Bi,p}={ui,…,ui+p+1},

is local.

Within each non-zero knot span

s,us<us+1,

of its support,

Bi,p

is described by a polynomial segment

s

Each knot

value indicates a location within the parameter space which

is not

C∞

-continuous, i.e.,where two adjacent

s

join. Suc-

cessive knots may share the same value, which is indicated

by the knot multiplicity

i.e.,

ui=ui+1=⋯=ui+m−1.

general, the continuity between adjacent segments is

Cp−m

This control of continuity is demonstrated for a quadratic

B-spline in Fig. 3. If the multiplicity of the ﬁrst and last

knot is equal to

p+1,

the knot vector is denoted an open

knot vector. The knot sequence

is a special from of such a knot vector since it yields the

classical

th-degree Bernstein polynomials. To be precise,

Bernstein polynomials are usually deﬁned over the interval

(2)

i,0(u)=

{

1if ui

≤

u<ui+1

0otherwise.

(3)

𝛯

{

=⋯=u

p+1

=⋯=u

2p+1},

s+1

Bs,0

s+1

Bs−1,1

Bs,1

s+1

Bs−2,2

Bs−1,2

Bs,2

Fig. 2 Non-vanishing B-splines

of knot span

for diﬀerent

degrees

p={0, 1, 2}

which are based on a knot vector with equally

spaced knots

1 2 3 4

(a)

1 2

5 4

(b)

Ξ=

{

1,2,3,4

}

Ξ=

{

1,2.5,2.5,4

}

Fig. 3 Polynomial segments

s

of a quadratic B-spline due to diﬀer-

ent knot vectors

𝛯.

Note the diﬀerent continuity C between the seg-

ments based on the knot multiplicity

B.Marussig, T.J.R.Hughes

1 3

[0, 1].

If necessary, the restriction to this interval can be

easily accomplished by a coordinate transformation.

As a whole, B-splines based on a common knot vector

𝛯

form a partition of unity, i.e.,

and they are linearly independent, i.e.,

is satisﬁed if and only if

ci=0, i=0, …,I−1.

Due to the

latter property, every piecewise polynomial

fp,𝛯

of degree

over a knot sequence

𝛯

can be uniquely described by a

linear combination of the corresponding

Bi,p.

Hence, they

form a basis of the space

𝕊p,𝛯

collecting all such functions

An example of a cubic B-spline basis deﬁned by an open

knot vector is shown in Fig.4.

The ﬁrst derivative of B-splines are computed by a lin-

ear combination of B-splines of the previous degree

For the computation of the kth derivative, this is general-

ized to

with

(4)

I−1

∑

i=0

Bi,p(u)=1, u∈

[

u0,uI+p

(5)

I−1

∑

i=0

Bi,p(u)ci=

(6)

𝕊

p,𝛯=

I−1

∑

i=0

Bi,pci,ci∈ℝ

(7)

�

i,p(u)=

ui+p−ui

Bi,p−1(u)

−p

i+p+1

−u

i+1

Bi+1,p−1(u)

(8)

(k)

i,p(u)= p!

(p−k)!

∑

𝓁=0

ak,𝓁Bi+𝓁,p−k(u)

Remark 1 The knot diﬀerences of the denominators

involved in the recursive formulae (1), (7) and (8) can

become zero. In such a case the quotient is deﬁned to be

zero.

2.2 Curves

B-spline curves of degree

are deﬁned by basis functions

Bi,p

due to a knot vector

𝛯

with corresponding coeﬃcients

in model space1

which denote control points. The geo-

metrical mapping



from parameter space to model space

is given by

with

representing the total number of basis functions. The

derivative is

In general, control points

are not interpolatory,

i.e.,they do not lie on the curve. The connection of

straight lines is called the control polygon and it provides an

approximation of the actual curve. An important property

of a B-spline curve is that it is contained within the convex

hull of its control polygon. In particular, a polynomial seg-

ment related to a non-zero knot span

i.e.,

u∈[us,us+1),

is in the convex hull of the control points

cs−p,…,cs.

The

continuity of the whole piecewise polynomial curve

C(u)

is inherited from its underlying basis functions, i.e., the

continuity at knots is determined by the knot multiplicity,

and the position of its control points. These relationships

are illustrated inFig.5. Note that the interpolatory B-spline

B4,2

ofFig.5a corresponds to the kink at

inFig.5b and

that the second polynomial segment lies within the con-

vex hull of

c3.

If the curve consist of a single poly-

nomial segment, i.e., the associated

𝛯

is of form (3), the

curve is referred to as Bézier curve. A polynomial segment

of a B-spline curve is termed a Bézier segment, if it can be

0,0

=1,

k,0 =

ak−1,0

ui+p−k+1−ui

k,𝓁=

ak−1,𝓁−ak−1,𝓁−1

ui+p+𝓁−k+1−ui+𝓁

𝓁=1, …,k−

k,k=

−ak−1,k−1

i+p+1

−u

i+k

(9)



(u):=C(u)=

I−1

∑

i=0

Bi,p(u)ci

(10)

𝐉

(u):=

I−1

∑

i=0

B�

i,p(u)ci

0 1 2 3 4

B0,3

B1,3B2,3B3,3B4,3B5,3

B6,3

Fig. 4 B-spline basis speciﬁed by an open knot vector,

i.e.,

𝛯={0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4}

1 The model space is also referred to as physical space.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

represented by a Bézier curve. In Fig.5b, this is the case

for the segment

u∈[3, 4]

deﬁned by the control points

c6.

B-spline curves can be generalized to represent rational

functions such as conic sections. For this purpose, weights

are associated with the control points such that

where

denotes the spatial dimension of the model space.

The homogeneous coordinates

specify a B-spline curve

Ch(u)

in a projective space

ℝd+1.

In order to obtain a curve

ℝd

the geometrical mapping (9) is extended by a per-

spective mapping



with the center at the origin of

ℝd+1.

This projection is given by

where

=(C

,…,C

)

⊺

are the homogeneous vector

components of the curve and the weighting function is

determined by

The application of Eq.(12) is illustrated inFig.6. The pro-

jection

C(u)

is denoted as a non-uniform rational B-spline

(NURBS) curve. The term rational indicates that the result-

ing curves are piecewise rational polynomials, whereas the

term non-uniform emphasizes that the knot values can be

distributed arbitrarily.

(11)

(

i)⊺

(

i)⊺

∈ℝ

d+1,

(12)

(u)=



(Ch(u)) =

Cw(

)

w(u),

(13)

(u)=

I−1

∑

i=0

Bi,p(u)wi

The derivative of the NURBS geometrical mapping is

deﬁned by

with

Another way to represent NURBS curves is

with

The weighting function

w(u)

is the same as in Eq.(13) and

Ri,p

denotes a NURBS basis function. Since the weights

are now associated with B-splines

Bi,p

the mapping (17)

employs control points

of the model space. In gen-

eral, NURBS curves degenerate to B-spline curves, if all

weights are equal. Hence, they are a generalization of them.

(14)

𝐉

(u):=

w(u)

𝜕Cw(u)

𝜕u−

𝜕w(u)

𝜕u

Cw(u)

(w(u))

(15)

𝜕

w(u)

𝜕u=

I−1

∑

i=0

B�

i,p(u)wi

(16)

𝜕

Cw(u)

𝜕u=

I−1

∑

i=0

B�

i,p(u)cw

(17)

(u)=

I−1

∑

i=0

Ri,p(u)ci

(18)

i,p(u)=

i,p

(u)

w(u).

0 1 2 3 4

B0,2

B1,2B2,2B3,2

B4,2

B5,2

B6,2

(a)

(b)

Basis functions

Quadratic B-spline curve

Fig. 5 Example of a B-spline curve: a B-splines based on

𝛯={0, 0, 0, 1, 2, 3, 3, 4, 4, 4}

and b a corresponding piecewise

polynomial curve. In b, circles denote control points and the dotted

lines indicate the convex hull of the dashed curve segment

u∈[1, 2)

Ch(u)

C(u)

c1c2

Fig. 6 Perspective mapping



of a quadratic B-spline curve

Ch(u)

homogeneous form

ℝ3

to a circular arc

C(u)

in model space

ℝ2.

The

mapping is indicated by dashed lines

B.Marussig, T.J.R.Hughes

1 3

The properties of B-spline curves apply to their rational

counterpart as well, if the weights are non-negative, which

is usually the case.

2.3 Spline Interpolation

In case of a spline interpolation problem, a given function f

shall be approximated by a B-spline patch

f:=

∑I−1

i=0

i,p

They agree at

data sites

̄u

if and only if

The corresponding system of equations consists of

the unknown coeﬃcients

and the spline collocation

matrix

𝐀u

which is deﬁned by

The Schoenberg–Whitney theorem [34, 83] states that the

matrix

𝐀u

is invertible if and only if

Since condition(21) guarantees that

𝐀u

does not become

singular, it is expected that the corresponding condition

number gets large if

̄u

approaches the limits of its allowed

range. Non-uniformity of

̄u

is another reason for an increas-

ing condition number. In fact, it gets arbitrary large if two

interpolation sites approach each other, while the others are

ﬁxed. Several authors [11, 34, 184] recommend to interpo-

late at the Greville abscissae

which are obtained by the

following knot average

These abscissae are well known in CAGD and used for dif-

ferent purposes, e.g.,to generate a linear geometrical map-

ping [83]. The most important feature of this approach is

that it induces a stable interpolation scheme for moderate

degrees

2.4 Tensor Product Surfaces

Tensor product surfaces allow an extremely eﬃcient evalu-

ation of patches. They play an important role in CAGD. In

particular, B-spline and NURBS patches are very common.

Bivariate basis functions for B-spline patches are obtained

by the tensor product of univariate B-splines which are

deﬁned by separate knot vectors

𝛯I

and

𝛯J.

These knot vec-

tors determine the parameterization in the directions

and

respectively. Moreover, they span the bivariate basis of a

(19)

(

̄uj

)

I−1

∑

i=0

Bi,p

(

̄uj

)

ci,j=0, …,I−

(20)

𝐀u

[j,i]=B

i,p(

̄u

,i,j=0, …,I−

(21)

Bi,p(

̄u

i)≠

0, i=0, …,I−

(22)

i+1

i+2

+⋯+u

i+p

patch. Combined with a bidirectional grid of control points

ci,j

the geometrical mapping



is determined by

The polynomial degrees are denoted by

and

respec-

tively for each parametric direction. The Jacobian of the

mapping (23) is computed by substituting the occurring

univariate B-splines by their ﬁrst derivatives, alternately for

each direction. In general, derivatives of B-spline patches

are speciﬁed by

The eﬃciency of tensor product surfaces stems from the

fact that their evaluation can be performed by a successive

evaluation of curves [62]. Suppose the parametric value

viso

is ﬁxed, the surface equation yields

(23)



(u,v):=S(u,v)=

I−1

∑

i=0

J−1

∑

j=0

Bi,p(u)Bj,q(v)ci,j

(24)

𝜕k+l

𝜕

ku𝜕lv

S(u,v)=

I−1

∑

i=0

J−1

∑

j=0

B(k)

i,p(u)B(l)

j,q(v)ci,j

(a)

(b)

2.5

B2,2(v)B2,2(u)

Parameter space

Basis function

Fig. 7 A bivariate basis determined by

𝛯I={1, 1, 1, 2, 3, 4, 4, 4}

and

𝛯J={1, 1, 1, 2.5, 2.5, 4, 4, 4}

for

and

respectively: a shows

the bivariate basis spanned by

𝛯I

and

𝛯J,

whereas b illustrates the

construction of a corresponding bivariate B-spline

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

with

Ciso(u)

denoting an isocurve of the surface deﬁned by

new control points

ci.

Hence, a surface can be evaluated by

I+1

J+1

curve evaluations, depending which paramet-

ric direction is evaluated ﬁrst.

The tensor product nature of the patches is illustrated

inFig.7 by means of a bivariate basis. Note that the uni-

variate knot values propagate through the whole param-

eter space. If both knot vectors of the resulting patch are

of form(3), it is referred to as Bézier surface. NURBS sur-

faces are derived analogous to curves by the introduction of

weights.

2.5 Constructing Patches byBoundary Curves

The most basic surface construction scheme is to connect

two curves

with

i=1, 2

by a linear interpolation. The

resulting surfaces are termed ruled surfaces and they are

deﬁned as

where

u,v∈[0, 1].

have the same degree and knot

vector, it is straightforward to represent

as a single ten-

sor product surface. In this case the connection lines on

associate points of equal parameter value. Alternatively, the

rulling (26) could also be performed according to relative

arc length. This yields a diﬀerent geometry which cannot

be converted to a NURBS patch [230].

The construction of Coons patches is another very com-

mon procedure. Thereby, a surface

is sought to ﬁt four

boundary curves

Ci(u)

and

Cj(v)

with

i=1, 2

and

j=3, 4.

The parameter range is again

u,v∈[0, 1].

The curves have

to satisfy the following compatibility conditions at the cor-

ners of the surface

(25)

(u,viso)=

I−1

∑

i=0

J−1

∑

j=0

Bi,p(u)Bj,q(viso)ci,j

I−1

∑

i=0

Bi,p(u)(J−1

∑

j=0

Bj,q(viso)ci,j

)

I−1

∑

i=0

Bi,p(u)̃

ci=Ciso(u),

(26)

(u,v)=(1−v)C1(u)+vC2(u)

=(1−v)S

(u,0)+vS

(u,1),

(27)

Sc(0, 0)=C1(

=0)=C3(

=0),

(28)

Sc(1, 0)=C1(u=1)=C4(v=0),

(29)

Sc(0, 1)=C2(u=0)=C3(v=1),

(30)

Sc(1, 1)=C2(u=1)=C4(v=1).

C1(u)

C2(u)

C4(v)

C3(v)

Sc(0,0)

Sc(1,0)

Sc(1,1)

Sc(0,1)

(a)

(b)

(c)

(d)

Ruled surface Sr

u(u, v)

Ruled surfaceSr

v(u, v)

Bilinearinterpolation Sc(u, v)

Coons patch S(u, v)

Fig. 8 Components of a bilinear Coons patch due to the boundary

curves

Ci(u)

and

Cj(v)

highlighted by thick lines

B.Marussig, T.J.R.Hughes

1 3

Using a bilinear interpolation a Coons patch is given by

where

and

are ruled surfaces based on

Ci(u)

and

Cj(v),

respectively, and

is the bilinear interpolant to the four

corner points

These various parts of a Coons patch are visualized in

Fig. 8. Equation (31) can be generalized by using two

arbitrary smooth interpolation functions

f0(s)

and

f1(s)

fulﬁlling

and

The corresponding Coons patch can be expressed in matrix

form as

with

0∈ℝd

denoting the zero vector. Various functions

may be used to specify

such as Hermite polynomials or

trigonometric functions. In case of Bernstein polynomials,

the surfaces

and

are in Bézier or B-spline form

and the resulting Coons patch can be represented as a sin-

gle NURBS surface.

Finally, Gordon surfaces are a further generalization of

Coons patches, where the surface

and

interpolate sets

of isocurves rather than boundary curves. Gordon surfaces

are also referred to as transﬁnite interpolation [103]. The

term indicates that these surfaces interpolate an inﬁnite

number of points, i.e.,the boundary curves and isocurves.

(31)

Sc(

)−

(32)

c(u,v)=

[

]⊺[

(0, 0)S

(0, 1)

Sc(1, 0)Sc(1, 1)

][

(33)

fk(𝓁)=𝛿k𝓁,k,𝓁=0, 1,

(34)

f0(s)+f1(s)=1, s∈[0, 1],s=u,v.

(35)

u,v

−

⎡

⎢

⎣

−1

f0(u)

f1(u)

⎤

⎥

⎦

⊺

⎡

⎢

⎣

0Sc(u,0)Sc(u,1)

Sc(0, v)Sc(0, 0)Sc(0, 1)

Sc(1, v)Sc(1, 0)Sc(1, 1)

⎤

⎥

⎦

⎡

⎢

⎣

−1

f0(v)

f1(v)

⎤

⎥

⎦

Based on this deﬁnition, ruled surfaces, Coons patches, and

Gordon surfaces may be generally referred to as transﬁnite

interpolations.

2.6 Representation ofTriangles

Triangular patches may be represented by tensor product

surfaces despite their four-sided nature. Therefore, either

a side or a point is degenerated as shown in Fig.9. Such

degenerated patches are often used since it is convenient to

use only one surface representation. However, it is apparent

that this can lead to a distorted parameterization. In addi-

tion, the enforcement of continuity between adjacent sur-

faces is diﬃcult in this case.

An alternative is to use triangular patches. A point

on such surfaces is deﬁned by barycentric coordinates,

i.e.,

(r,s,t)

with

r+s+t=1.

We will focus on triangular

Bézier patches

S△

which are speciﬁed as

with

representing linearly independent bivariate Bernstein

polynomials of degree

The related control points

ci,j,k

form a triangular array as shown in Fig.10 for the cubic

case. The resulting patch fulﬁlls the convex hull property

and its boundaries are Bézier curves. Rational triangular

Bézier patches may be deﬁned again by the introduction of

weights. Despite the potential of triangular patches, there

are currently no commercial CAD applications that admit

the use of splines on triangulations.

(36)

S△

(r,s,t)=

∑

i+j+k=p

i,j,k⩾0

Bi,j,k,p(r,s,t)ci,j,k

(37)

i,j,k,p(r,s,t)=

i!j!k!

risjtk

(a) (b)

Degenerated side Degeneratedpoint

Fig. 9 Tensor product representation of triangular patches: a an

angle between adjacent edges has 180

◦

and b a side shrinks to a

point. Circles mark the corner points of the resulting patch

c3,0,0c0,0,3

c0,3,0

c2,0,1c1,0,2

c2,1,0

c1,2,0

c0,1,2

c0,2,1

c1,1,1

(a) (b)

Control grid structureTriangularpatch

Fig. 10 Triangular Bézier patch of degree

p=3:

a the general struc-

ture of the control grid and b a corresponding surface

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

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2.7 Trimmed Surfaces

In order to represent arbitrary surface boundaries when

using tensor product surfaces, patches can be modiﬁed by

trimming procedures. For this purpose, curves are deﬁned

within the parameter space of a surface

S(u,v).

These trim-

ming curves

Ct(̃u)

are usually B-spline or NURBS curves.

They are given by

where

∈ℝ

are the control points of the trimming curve

given in the parameter space of the trimmed surface. Con-

nected trimming curves are ordered such that they form a

closed directed loop. Loops also include the boundary of

the original patch if it is intersected by trimming curves.

These loops divide the resulting trimmed patch into distinct

parts where the curves’ directions determine which parts

are visible. In other words, trimming procedures are used

to deﬁne visible areas

v

over surfaces independent of the

underlying parameter space.

As a result, surfaces with non-rectangular topologies

can be represented in a very simple way. An example of a

trimmed patch is shown in Fig.11. It is emphasized that

the mathematical description, i.e., the tensor product basis

and the related control grid, of the original patch does not

change and is never updated to reﬂect the trimmed bound-

ary represented by the independent trimming curves.

Trimmed surfaces should be considered as an “engineer-

ing” extension of tensor product patches [83]. On the one

hand, they permit a convenient way to deﬁne arbitrary sur-

face topologies and provide a means for visually display-

ing them in graphics systems. On the other hand, they do

not oﬀer a canonical solution to related problems such as

a smooth connection of two adjacent patches along a trim-

ming curve, although the graphics system leads the user

to incorrectly believe so. In fact, enormous eﬀort has been

and is still devoted to resolve the shortcomings of trimming

procedures as discussed later on in Sect.3.

2.8 Solid Models

Most CAGD objects are geometrically represented by their

boundary only. In other words, these models consist of sev-

eral boundary patches

𝛾

where

denotes the entire boundary of the object. If

is a curve, several patches may be needed to represent

(38)

t(̃u)=

[

u(̃u)

v(̃u)

]

I−1

∑

i=0

Ri,p(̃u)ct

(39)

Γ=

⋃

i=1

𝛾i

distinct sections with diﬀerent polynomial degrees. This is

not a critical issue since curves can be joined rather eas-

ily, even with a certain continuity. However, a problem

arises as soon as surfaces are considered, because tensor

(a)

(b)

(c)

RegularB-splinepatch

Trimmed parameter space

Trimmed patch

Fig. 11 Trimmed tensor product surface: a regular surface deﬁned

by a tensor product basis, b trimmed parameter space where a loop

of trimming curves (thick line) speciﬁes the visible part

v

of c the

resulting trimmed surface as displayed by a CAD system’s graphics

display. The arrow in b denotes the direction of the trimming curves

B.Marussig, T.J.R.Hughes

1 3

product surfaces are four-sided by deﬁnition. A single reg-

ular NURBS surface may be closed equivalently to a cyl-

inder or a torus. Spherical objects may be represented as

well, if degenerated edges are introduced. Yet, more com-

plicated objects such as a double torus require a partition

into multiple NURBS patches. The connection of two adja-

cent surfaces is complicated, especially if a certain conti-

nuity is desired. In general, non-conforming parameteriza-

tions along surface boundaries need to be expected.

In addition to the geometric representation of the bound-

ary patches, the topology of the object has to be described.

It addresses the connectivity of the various components,

and the corresponding entities are termed

– vertices relating to points,

– edges relating to curves,

– faces relating to surface.

It should be noted that the descriptions of an object’s shape,

i.e.,geometry, and its structure, i.e.,topology, are separated

[290]. By deﬁnition, a B-Rep model always consists of a

data structure of both topological and geometric objects.

Regarding isogeometric analysis, the parameterization of

the geometry is a further important issue that should be

taken into account. Figure 12 summarizes these various

perspectives of a CAGD model representing a simple solid.

The corresponding object consists of three trimmed sur-

faces and an untrimmed, or regular one. It is apparent that

even simple CAGD models rely on multiple non-conform-

ing surfaces.

Finally, it should be mentioned that B-Reps are also

used to describe dimensionally reduced objects, i.e.,shell

structures. In this case, the patches

𝛾

specify the object

itself rather than its boundary. It is important to note that

the terms surface model and solid model do not refer to the

dimension of an object. In CAGD, they rather indicate if a

model contains topology information (solid model) or not

(surface model). Based on the brief outline given here, the

discussion on the representation of CAGD models will be

continued in Sect.3.2.

3 Trimming inComputer Aided Geometric

Design

There is a large body of literature on trimmed B-spline

and NURBS geometries in CAGD. Trimming is addressed

in the context of surface intersection, the development for

appropriate data structures for solid modeling, the visu-

alization of objects which is referred to as rendering, and

remodeling approaches. The following outline of these

topics is meant to be comprehensive, but it is by no means

complete. Further, some auxiliary techniques are presented.

The motivation for this section is twofold: ﬁrst of all, it

provides an overview of the historical development of trim-

ming approaches in the ﬁeld of CAGD. Apart from being

interesting in its own right, this insight exposes a num-

ber of general challenges, techniques, and ideas regarding

trimmed geometries. Hence, it is hoped that the subsequent

sections also give insight to further strategies dealing with

trimming in the context of isogeometric analysis.

3.1 Surface Intersection

Trimming is closely related to the problem of surface-to-

surface intersection. In general, the intersection of two par-

ametric surfaces

face4

edge 1

edge 2

edge 3

edge 4

edge 5

edge 6

vertex 1

vertex 2

vertex 3

vertex 4

face 1

face 2

face3

(a)

(b) (c)

Visualizationand topologicalentities

ComponentsParameterization

Fig. 12 Diﬀerent perspectives of a CAGD solid model: a visible part

of the object and its topological entities (to be precise, the related

geometric objects, i.e., points, curves, and surfaces, are displayed),

b the geometric segments of the B-Rep and c the underlying math-

ematical parameterization of each surface. In c, dashed lines mark the

boundary of the visible area and gray lines indicate the underlying

tensor product basis. Note that the parameterization along common

edges does not match

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

leads to a system of three nonlinear equations, i.e.,the three

coordinate diﬀerences of

and

, with four unknowns

u,v,s,t

[62]. If surfaces intersect, the solution usually

yields curves, but also subsurfaces or points may occur.

The computation of intersections is one of many “geomet-

ric interrogation” techniques, or processes, employed in

all types of modeling. The development of a good surface

intersection scheme is far from trivial since the method has

to balance three contradictory goals: accuracy, eﬃciency,

and robustness. The surveys [219, 220] and the text-

books[4, 130, 221] provide detailed information on various

approaches. Surface intersection algorithms can be broadly

classiﬁed into four main categories: (i) analytic methods,

(ii) lattice evaluation, (iii) subdivision methods, and (iv)

marching methods.

3.1.1 Analytic Methods

The intersection of two surfaces may be solved analytically,

i.e., an explicit representation of the intersection curve is

obtained. Early solid modeling systems used analytic meth-

ods to obtain exact parametric representations of the inter-

section of quadratic surfaces[40]. The intersection problem

always has a simple solution when both surfaces are given

as functions in implicit form[130]. The good news is that

parametric surfaces can always be represented implicitly

[265], but the main problem is that the algebraic complex-

ity of the intersection increases rapidly with the degree of

the surfaces. This is often illustrated by a popular exam-

ple of the intersection of two bicubic patches which has an

algebraic degree of 324 as shown by Sederberg [265, 266].

In addition, the intersection of two bicubic patches has a

(40)

(u,v)=

(

(u,v),y

(u,v),z

(u,v)

(41)

(s,t)=

(

(s,t),y

(s,t),z

(s,t)

genus2 of 433 and only curves of genus 0, i.e.,all degree

two curves, cubic curves with one double point, quartic

curves with three double points or one triple point, etc.,

can be expressed parametrically using rational polynomi-

als [152]. Figure 13 illustrates two examples of implicit

cubic curves with diﬀerent genus. The complexity of sur-

face intersection curves has also been discussed in the

study of Farouki and Hinds[88]. It is argued that the deri-

vation of an implicit representation is not practical and an

approximation scheme may be preferred. In general, ana-

lytic methods have been restricted to low degree intersec-

tions, which yield exact results very fast.

3.1.2 Lattice Evaluation

The basic idea of this technique is to reduce the dimension-

ality of surface intersections by computing intersections of

a number of isoparametric curves [186, 250] (see Fig.14).

Once the discrete intersection points are obtained, they are

sorted and connected by an interpolation scheme. In order

to deﬁne an intersection curve, lattice evaluation involves

an initial choice of a proper grid resolution. This is cru-

cial for both the robustness and eﬃciency of the method.

Unfortunately, determination of an appropriate discrete step

size is not straightforward and, if too coarse, may lead to a

failure in identifying critical features [170].

Curve intersection schemes are also useful in the con-

text of ray tracing for visualization and point classiﬁca-

tion in solid modeling [220]. In these applications a patch

is intersected by a straight line, as discussed later on in

Sects.3.3.2 and3.5.2.

10−1

−1

Genus 1

y−x3=0

10−1

−1

Genus 0

y2+x3

−2xy =0

Fig. 13 Cubic curves with diﬀerent genus. Note the double point in

case of the genus 0 curve

Fig. 14 Intersection points based on line-to-surface computations

2 The genus g of a plane algebraic curve is speciﬁed by the degree

of the curve and the number

and multiplicity

of its singular

points:



p2−3p+2−



i=1mi(mi−1)

.

B.Marussig, T.J.R.Hughes

1 3

3.1.3 Subdivision Methods

The key idea of subdivision approaches is to compute an

intersection using approximations of the patches involved,

rather than the actual objects themselves. These approxi-

mations are often deﬁned by piecewise linear elements.

Consequently, the intersection problem is subdivided into

many, but signiﬁcantly simpler, problems. The ﬁnal inter-

section curve is obtained by merging the individual inter-

section results.

A good approximation of the original objects is of

course essential for the accuracy of such techniques. How-

ever, subdivision algorithms become ineﬃcient for high-

precision evaluation, especially if the decomposition is

performed uniformly. A considerable improvement can be

achieved if the region of the intersection, i.e.,the aﬀected

elements, is estimated in a preprocessing step. This can

be carried out by bounding boxes that completely enclose

the corresponding element. Various construction schemes

of such bounding boxes are outlined in Sect.3.5.1. Their

common aim is to allow an eﬃcient determination if two

objects are clearly separated or not. In particular, ele-

ments are recursively reﬁned if their bounding boxes over-

lap, which leads to a non-uniform, adaptive subdivision

algorithm as shown in Fig.15. This process of successive

reﬁnement and removing of separable boxes is referred to

as a divide-and-conquer principle[130].

An important advantage of subdivision methods is that

they do not require starting points. On the other hand, the

drawbacks can be summarized according to Patrikalakis

and Maekawa [221] as follows: (i) they are only able to

isolate zero-dimensional solutions, (ii)there is no certainty

that each root has been extracted, (iii) the number of roots

in the remaining subdomains is typically not provided, and

(iv) there is no explicit information about root multiplicities

without additional computations. Last but not least, (v) the

method is not eﬃcient in case of high-precision or higher

order evaluations[181, 219].

3.1.4 Marching Methods

Marching methods3 derive an intersection curve by step-

ping piecewise along the curve, e.g., [14, 20, 84]. Such

methods usually consist of a search, a marching, and a

sorting phase. The ﬁrst phase detects an appropriate start-

ing point on the intersecting curve. Often, this is performed

by a subdivision or lattice approach. In the marching phase,

a point sequence along the intersection curve is developed

starting from the points determined in the previous phase.

The direction and the length of the next step are deﬁned

by the local diﬀerential geometry. Finally, the individual

points and segments of the intersection curve are sorted

and merged to disjoint pieces and curve loops.

According to Hoschek and Lasser [130], all marching

methods share some common problems: (i)determination

of good starting points, (ii)detection of all branches of the

intersecting curve, (iii)avoiding of multiple detections of

a intersection segment, (iv) correct behavior at self-inter-

sections and singularities, (v)proper choice of the direction

and length of the subsequent step, and (vi) a robust auto-

matic stopping criterion.

Despite all these issues, marching methods are by far

the most widely used approaches due to their generality

and ease of implementation [170]. In addition, accuracy

improvement can be easily achieved by decreasing the step

size, and they are also very eﬃcient, especially in combina-

tion with subdivision methods.

At this point, it should be emphasized that the problems

related to topology detection of the intersection curve,

i.e.,ﬁnding all its branches and singular points, apply to all

intersection methods and several authors have addressed

them, e.g.,[6, 105, 168, 222, 267, 283].

3.1.5 Hybrid Methods

Every intersection method type has its beneﬁts and draw-

backs, hence a number of authors have established hybrid

methods that combine features of the diﬀerent categories.

One of the elementary surface intersection schemes has

been proposed by Houghton et al. [133]. The algorithm

combines a divide-and-conquer approach with a New-

ton–Raphson procedure: ﬁrstly, the surface is subdivided

into ﬂat sub-pieces. Then, each sub-piece is approximated

by two triangles and the intersection of these triangles is

computed. In the next step, the resulting linear segments

are sorted and connected using the information provided

by the subdivision tree. Finally, the intersection points are

reﬁned by the Newton-Raphson scheme. The main advan-

tage of this method is that it is very general and can be

Fig. 15 Determination of an intersection region of two curves by

means of a divide-and-conquer scheme that uses axis-aligned bound-

ing boxes

3 Marching methods are also referred to as tracing methods.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

applied to any surface representation, in contrast to earlier

techniques that utilize properties of certain surface types,

e.g.,[46, 111, 165, 180, 224, 256].

Barnhill etal.[19] presented another general procedure

to compute the intersection of two rectangular

patches.

It relies on a combination of subdivision and a marching

scheme. It does not assume a special structure of the inter-

secting surfaces and special cases are considered, e.g.,inﬁ-

nite plane intersections, creases, and self-intersection. The

algorithm has been enhanced in [20], including the uti-

lization of the divide-and-conquer concept presented by

Houghton etal.[133].

Another combination of a divide-and-conquer subdivi-

sion with an iterative marching approach has been devel-

oped by Kriezis et al. [169]. The method enables inter-

secting algebraic surfaces of any degree with rational

biquadratic and bicubic patches.

Krishnan and Manocha[170] developed an approach for

NURBS surfaces that combines marching methods with the

algebraic formulation. The starting points on the intersec-

tion curve are computed by Bézier curve–surface intersec-

tions that are obtained by eigenvalue computations. More-

over, they introduced a technique that allows detection of

singularities during the tracing process.

3.1.6 Representation oftheIntersection Curve

Various techniques for the computation of approximate

solutions to the surface-to-surface intersection problem

have been outlined so far. It remains to discuss the actual

representation of the result.

In general, three distinct representations of an intersec-

tion are obtained. On the one hand, the intersection curve

in model space is computed. This may seem to be the main

objective of the whole procedure at ﬁrst glance, yet it is

just a part of the overall solution process. The intersection

curve has to be represented in each parameter space of the

trimmed patches. These curves are referred to as trimming

curves in the following and are needed to determine which

surface points are visible.

Intersection and trimming curves can be deﬁned by any

kind of representation, but usually low-degree B-splines

are used. They are constructed based on a set of sampling

points that result from the surface-to-surface intersec-

tion algorithm applied [207]. Subsequently, an interpola-

tion scheme or another curve-ﬁtting technique is used to

generate a continuous approximation of the intersection

in model space

In general, this curve does not lie on

either of the intersecting surfaces. A trimming curve

obtained based on the sampling points given in the corre-

sponding parameter space [240]. The related curve ̃

in the

model space is obtained by evaluating the equation of the

surface

along its

Alternatively, ̃

may be represented

explicitly. DeRose et al. [71] presented an eﬃcient and

stable algorithm based on blossoming4 that can be use to

exactly compute the control points of ̃

Such an expan-

sion of

S◦Ct

into an explicit representation

may be used

to join another patch to a trimmed surface, but there is no

computational beneﬁt [189]. Curves on surfaces have a

degree of

p(m+n)

with

denoting the degree of the trim-

ming curve and m and n correspond to the degrees of the

trimmed surface [83]. Renner and Weiß [240] compared

exact and approximate representations of

and con-

cluded that high degree is the main reason for preferring an

approximation scheme. Furthermore, they formulated the

following requirements for such a scheme: (i) low degree,

(ii) fast and stable generation, (iii) full control over devia-

tion between exact curve and approximation, and (iv) con-

sideration of the speciﬁc surface geometry. According to

them, these requirements are often not satisﬁed in current

CAD systems. Besides the schemes presented in [240], sev-

eral other approaches have been proposed to compute good

approximate curves on surfaces, see e.g., [90, 119, 317].

It is emphasized that ̃

does not coincide with the inter-

section curve in model space

regardless of its representa-

tion. In addition, all procedures related to trimming curves

are performed for each patch separately. Hence, the images

of these curves ̃

do not coincide, neither with each other,

nor with

As a consequence, gaps and overlaps may

occur between intersecting patches. There is no connec-

tion between these three representations of the intersection;

although the sample points provide some information dur-

ing the construction, this data is only stored temporarily

during the approximation procedure and never retained in

memory for further use. These various approximations of a

surface-to-surface intersection are summarized in Fig.16.

Currently, the most common geometric modeling ker-

nels are ACIS, C3D, and Parasolid. They provide

software components for the representation and manipula-

tion of objects, and form the geometric core of many CAD

applications. All of them use splines for the description of

trimming curves [43, 65, 282]. Yet, the representation of

the intersection curve in model space varies: ACIS deﬁnes

it by a three-dimensional B-spline curve, Parasolid uses

a set of sorted intersection points that can be interpreted as

a linear approximation, and in C3D the intersection curve

is not stored at all. In C3D, trimming curves are computed

such that they have the same radius and derivatives at the

same parametric values. However, this is only satisﬁed at

the intersection point used for the construction, for more

details see[102].

4 A multi-aﬃne and totally symmetric mapping is called a blossom

(or polar form). Blossoms can be used to deﬁne spline algorithms in

an elegant way. For details see [236].

B.Marussig, T.J.R.Hughes

1 3

3.2 Solid Modeling

Solid modeling is concerned with the use of unambiguous

representations of three-dimensional objects. It is based on

a consistent set of principles for mathematical and com-

puter modeling, and focuses on informational complete-

ness, physical ﬁdelity, and universality [276]. An essential

aspect is the topology of complex models. The consid-

eration of topology is indeed the fundamental diﬀerence

between solid models and surface models, because the lat-

ter describes only the geometry of an object [5]. Topologi-

cal properties are not metrical, but address connectivity and

dimensional continuity of a model [207]. There are several

textbooks on solid modeling [120, 194, 207, 290] and for

an elaboration of the historical development of this ﬁeld of

research the interested reader is referred to the landmark

paper of Requicha [242] and the subsequent surveys by him

and his co-authors [241, 243, 244, 251]. In the following,

the progress towards a trimmed solid model as well as the

related challenges are outlined.

3.2.1 Formulation ofTrimmed Solid Models

Pierre Bézier outlined the idea of trimming already in the

1970s. In his paper[29], it is proposed to perform the seg-

menting of a model by curves deﬁned on a square patch

as illustrated in Fig. 17. These curves are termed “trans-

posant”, the present participle of the French word for trans-

pose. The concept reduces the amount of data and enables

an easier blend with other patches. However, this idea was

presented with little theoretical support and solid modeling

requires an adequate mathematical theory as emphasized in

a survey by Requicha and Voelcker[243].

It took some time to develop a rigorous way to represent

trimmed free-form models. There are three broad catego-

ries for representing geometric objects: (i)decomposition,

(ii)boundary, and (iii)constructive representations. Popu-

lar examples of decomposition representations are voxel

models where a solid is approximated by identical cubic

cells. Advantages and limitations of this approach are dis-

cussed in [153]. A B-Rep5 (B-Rep) deﬁnes an object by

its bounded geometry, along with an associated topologi-

cal structure of corresponding entities, such as faces, edges,

and vertices. The beneﬁts of storing an object’s shape by

means of its boundary were already elaborated in the

seminal work of Braid[36]. Most B-Reps consist of sev-

eral surface patches and additional information is stored to

eﬃciently identify the various components and their rela-

tion to each other[241]. Various data structures for B-Reps

have been used, e.g.,[21, 73, 106], to ﬁnd a compromise

between storage requirements and response to topological

questions. The best known constructive representation is

so-called constructive solid geometry (CSG)[241]. Simple

S1(u, v)

S2(s, t)

Modelspace

Parameter spaceofS1(u, v)

Parameter space of S2(s, t)

(a)

(b)

(c)

Fig. 16 Independent curve interpolation of an ordered point set to

obtain approximations of the intersection of two patches

S1(u,v)

and

S2(s,t).

The set of sampling points depends on the surface-to-surface

intersection algorithm applied. The subsequent interpolation of these

points is performed in a the model space and the parameter space of b

S1(u,v)

and c

S2(s,t)

leading to the curves

and

respectively.

The point data is usually discarded once the curves are constructed

5 B-Rep models with free-form surfaces are also referred to as sculp-

tured models.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

primitives are combined by means of rigid motions and

regularized Boolean operations—union, intersection, and

diﬀerence. The resulting object is represented by a binary

tree where the internal nodes correspond to the Boolean

operations and the primitive solids (or half spaces) are

given in the leaves. An example of such a tree is given in

Fig.18.

Remark 2 The developments of isogeometric analysis and

additive manufacturing using heterogeneous materials yield

to a growing interest in another representation of three-

dimensional geometric models, namely volumetric repre-

sentations (V-Reps). As a matter of fact, several researchers

in the CAGD community have started addressing this issue,

see e.g.,[31, 197, 321], including the deﬁnition of trimmed

V-Reps [203].

Trimmed solid models combine concepts of B-Rep and

constructive representation, i.e., they consist of free-form

B-Reps merged by Boolean operations[245]. A uniﬁcation

of CSG and free-form surfaces was presented in the early

1980s [56], perhaps for the ﬁrst time. It was proposed to

use models with straight edges but free-form surface inter-

polation in between. In the context of Boolean operations,

however, the merging of B-Reps and CSG is more involved.

Gossard etal.[104] developed a polyhedral modeler which

combines the two representations by means of a graph

structure. In particular, two relative position operators have

been implemented. For manifold polyhedral objects, the

implementation of Boolean operations is well understood

[194]. However, the deﬁnition of a convenient representa-

tion for trimmed NURBS is a major challenge since the

topology of patches becomes quite complicated if Boolean

operations are performed [120]. B-Rep solid modeling uti-

lizes surface-to-surface intersection schemes to create arbi-

trarily deﬁned free-form geometric entities, but the corre-

sponding algorithms require more than just computing the

intersection curve. Weiler’s thesis[310] provides a study

on topological data structures. He summarized the essential

attributes of geometric modeling operators as follows:

(i) Determination of the topological descriptions,

(ii) Determination of the geometric surface descriptions,

(iii) Guarantee that the geometry corresponds unambigu-

ously to the topology.

Setting up a topology requires the classiﬁcation of the

neighborhood of various entities (faces, edges, and verti-

ces) involved in the intersections[120]. The correlation of

topology and geometry becomes particularly complicated if

intersection curves have singularities or self-intersections.

In addition, various forms of set membership classiﬁcation,

i.e.,the determination if parts are inside, outside, or on the

boundary of a domain, are used to compute B-Reps through

Boolean operations. In order to determine if a surface point

is inside or outside of the surface, the trimming curve must

be deﬁned in the parameter space as noted before. If the

trimming curve would only be deﬁned in the model space,

the problem would be in fact ill-deﬁned[205].

Fig. 17 An early sketch of a

trimmed surface (reprinted from

[29], with permission from

Elsevier)

−

Fig. 18 Representation of the object shown in Fig.12a by means of

a CSG tree: the object is speciﬁed by a composition of simple solids

using Boolean operations, i.e.,union (U) and diﬀerence (−)

B.Marussig, T.J.R.Hughes

1 3

The ﬁrst formulation of a trimmed patch representation

that supports Boolean operations and free-form geometry

was presented in the late 1980s by Casale and Bobrow[47,

49]. The domain of trimmed patches is speciﬁed by the

two-dimensional equivalent of a CSG tree. Hence, a

B-Rep is obtained that contains topology information of its

trimmed components. Patches are intersected similar to the

divide-and-conquer procedure of [133], but the trimming

curve is then also transformed into the parameter space in

order to perform Boolean operations and set membership

classiﬁcations. At the same time, a rigorous trimmed sur-

face deﬁnition has been formulated by Farouki [85]. The

formulation is based on Boolean operation deﬁnitions. In

particular, a trimmed patch is given by its parametric and

implicit surface equations together with a trimming bound-

ary that is deﬁned as a tree structure of non-intersecting

and nested piecewise-algebraic loops. These loops consist

of monotonic branches. The integral over the trimmed sur-

face is determined by a proper tessellation of the patch. It

should, however, be pointed out that all these approaches

fail to guaranty exact topological consistency since the

images of the trimming curves do not match in general, as

noted by Farouki etal.[87]. This leads to gaps and overlaps

of the solid model, which can introduce failure of down-

stream applications such as numerical simulations.

3.2.2 Robustness Issues

Several robustness issues arise in case of imprecise geo-

metric operations. As a matter of fact, the numerical out-

put from simple geometric operations can already be quite

inaccurate. Complications may occur even for linear ele-

ments as discussed by Hoﬀmann [120–123] or the compu-

tation of the convex hull of a set of points [16], for instance.

These problems are induced by propagation of numerical

conversion, roundoﬀ, and digit-cancellation errors of ﬂoat-

ing point representation. The issue of rounding errors of

numerical computations is known for a long time, at least

since an early study by Forsythe [92]. Investigating the

eﬀects of ﬂoating point arithmetic on intersection algo-

rithms is an important area of research [220]. Since inter-

section problems can be expressed as a nonlinear poly-

nomial system of equations, the robustness issue maybe

addressed from a computational point of view. Troubles

arise if the problem is ill-conditioned which is for example

the case for tangential intersections and surface overlaps

[135, 195].

The key issue is that numerical errors may cause mis-

judgment as pointed out in [291]. Since the geometrical

decisions are based on approximate data and arithme-

tic operations of limited precision, there is an interval of

uncertainty in which the numerical data cannot yield fur-

ther information [122]. Of course, the situation gets even

more delicate if trimmed free-form surfaces are involved

where approximation errors are quite apparent due to the

gaps and overlaps between intersecting patches. In case of

topological decisions, the accumulation of approximation

errors is especially crucial since inaccuracy leads to incon-

sistency of the output as indicated in Fig.19.

There is a large amount of research that addresses the

issue of accurate and robust solid modeling. The various

concepts are outlined in the following subsections. The

approaches are based on tolerances, interval arithmetic, and

exact arithmetic.

3.2.2.1 Tolerances Often, tolerances are used to assess

the quality of operations like the computation of an inter-

section [19, 130]. Several authors have suggested to use

adaptive tolerances where each element of the model is

associated with its own tolerance, e.g.,[143, 271]. In addi-

tion, tolerances may be dynamically updated [82]. Robust-

ness of topology decisions may be improved by choosing

the related precision higher than the one for the input data

[291]. Another strategy is to adjust the data in order to obtain

topologically consistent functions [204]. There are various

other approaches that improve the application of tolerance

and the interested reader is referred to the review of Hong

and Chang[128] for a comprehensive discussion. In fact, all

common CAD software tools are based on a user-deﬁned

tolerance that determines the accuracy of the geometrical

operations performed. For example, the default tolerance

values of ACIS are

10−6

for the comparison of points and

10−3

for the diﬀerence of an approximate curve or surface to

its exact counterpart [65]. Unfortunately, tolerances cannot

guarantee robust algorithms since they do not deal with the

inherent problem of limited-precision arithmetic.

(a)

(b)

Close intersections

Top ological error

Fig. 19 Example of an incorrect topology: a two intersection points

are close together which may lead to b an incorrect topological

placement along the vertical line due to numerical approximation

errors (re-execution of the original example [204])

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

3.2.2.2 Interval Arithmetic In case of interval arithme-

tic, e.g.,[76, 208], numerical errors are taken into account

by associating an interval of possible values to a variable.

This approach is correct in the sense that result intervals

are guaranteed to contain the real number that is the value

of the expression. The interval size indicates the reliability

of ﬂoating point computations. In particular, a narrow inter-

val is obtained in case of a successful operation whereas

a wide interval reveals a risk [117]. This concept may be

modiﬁed to rounded intervals to assure that the computed

endpoints always contain the exact interval, see e.g.,[3, 57,

193]. Interval arithmetic may also be combined with back-

ward error analysis [255].

In the context of solid modeling, Hu et al. [134–136]

suggested to use interval NURBS, i.e., NURBS patches

with interval arithmetic. The control points are described

by interval numbers rather than real numbers. Conse-

quently, they are replaced by control boxes and thus, curves

and surfaces are represented by slender tubes and thin

shells, respectively. A conceptional sketch is illustrated in

Fig.20. The object is deﬁned by a graph with nodes rep-

resenting the topological entities. Each node has two lists:

one for higher dimensional nodes and another for lower

dimensional nodes that are arranged in counterclockwise

order. This data structure has been applied to Boolean oper-

ations [136] and various intersection problems including

ill-conditioned cases [134]. Gaps between actual intersect-

ing objects are avoided and no intersection point is missed.

However, objects that do not intersect each other originally,

may do after several geometric processing steps using

rounded interval arithmetic. Furthermore, interval arithme-

tic approaches cannot achieve very high precision in rea-

sonable computation time [220].

3.2.2.3 Exact Arithmetic In order to achieve robustness,

algorithms have been developed that are based on exact

arithmetic, which is the standard in symbolic computation

[318]. Most of these approaches focus on polyhedral objects,

see e.g., [26, 93, 291]. For linear geometries like planes

and their intersections, exact rational arithmetic is enough

to handle all necessary numbers. However, more involved

objects rely on real algebraic numbers and therefore, they

require more complicated data structures and algorithms.

Keyser etal.[157–159] presented such a scheme for curve-

to-curve intersection in a plane and the theoretical frame-

work for exact computation based on algebraic numbers

has been discussed by Yap [318]. A combination of exact

approach and ﬂoating point calculation may also be used as

suggested by Hoﬀmann etal.[123]. In their paper, symbolic

reasoning is used when ﬂoating point calculation yields

ambiguous results. Krishnan etal.[171] demonstrated that

exact arithmetic can be applied to large industrial models.

They presented a B-Rep modeling system dealing with

models using over 50,000 trimmed Bézier patches.

Still, the main drawback of such approaches is their eﬃ-

ciency. Exact computation can be several orders of mag-

nitude slower than a corresponding ﬂoating point imple-

mentation [156]. According to Patrikalakis and Maekawa

[220], much research remains to be done in bringing such

methods to practice. In particular, more eﬃcient algorithms

should be explored that are generally applicable in low and

high degree problems.

3.2.2.4 Concluding Remarks Overall, the formulation of

robust solid models with trimmed patches is still an open

issue. Tolerance based approaches are usually preferred

since they are faster than the more precise ones. Hence, there

is again a tradeoﬀ between eﬃciency, accuracy, and robust-

ness as discussed in the context of intersection schemes. Of

course, the importance of these properties to the object rep-

resentation strategy depends on the application context [45].

In fact, the problem of topologically correct merging

of trimmed surfaces is such a challenge that more recent

research in CAGD tries to circumvent this issue by employ-

ing other surface descriptions like T-splines and subdivi-

sion surfaces. These representations inherently possess a

consistent topology and corresponding models are water-

tight, i.e.,they do not have unwanted gaps or holes. How-

ever, they also have some drawbacks and the transforma-

tion of the original object usually leads to approximations,

at least in the vicinity of the intersection curve as discussed

later on in Sect.3.4.

3.3 Rendering

In computer graphics rendering refers to the process of

generating images of a CAGD model. There are two dif-

ferent ways to approach this goal. On the one hand, indirect

schemes ﬁrst tessellate the surfaces of the object and the

actual visualization is based on this render-mesh. On the

other hand, rendering may be performed directly on free-

form surfaces by ray tracing. For a general introduction

(a) (b) (c)

Ideal Actual Interval

Fig. 20 Four splines representing a quadrilateral: a ideal mathemati-

cal object, b ﬂoating point model with approximation errors, and c

interval arithmetic based representation

B.Marussig, T.J.R.Hughes

1 3

to the creation of realistic images, the interested reader is

referred to the textbook of Glassner[99].

3.3.1 Tessellation

All commercial rendering systems tessellate free-form sur-

faces before rendering, because it is more eﬃcient to opti-

mize the code for a single type of primitive [27].

Early on, trimmed surfaces had been rendered using the

de Boor [33], Oslo [60], or Boehm’s knot insertion [32]

algorithm. In addition to the subdivision, the regions must

be sorted to ﬁnd out which ones are hidden and have to be

removed for rendering, see e.g.,[52, 179, 292]. In general,

subdivision approaches are expensive if they are performed

to pixel level.

The rendering of trimmed NURBS surfaces can also be

carried out using a combination of subdivision and adap-

tive forward diﬀerencing[185, 275]. This method allows

fast sampling of a large number of points, but suﬀers from

error propagation. The main drawback in rendering trans-

parent objects is the redundant pixel painting in adaptive

forward diﬀerencing. Furthermore, the overall performance

of the algorithm obtained is rather slow[191].

Rockwood etal.[249] presented a scheme enabling ren-

dering of trimmed surfaces in real-time. Firstly, the surface

is tessellated, i.e.,approximated by linear triangles or other

polygons. Therefore, all surfaces are subdivided into indi-

vidual Bézier patches. A trimmed Bézier patch may be sub-

divided further to obtain monotone regions that have con-

vex boundaries in the parameter space [165]. Each patch

is tessellated into a grid of rectangles which are connected

to the region boundaries by triangles. The actual rendering

is performed on the approximate mesh. This idea has been

adapted and enhanced by several other authors, e.g., [2,

176, 191].

The triangulation of trimmed surfaces by a restricted

Delaunay triangulation has been proposed by Sheng and

Hirsch[279]. The basic idea of this technique is to compute

the approximation mesh in the parameter space. Although it

has been developed for stereolithography6 applications, the

suitability for rendering is emphasized. Stereolithography

was also the motivation in[74] where trimmed surfaces are

triangulated by an adaptive subdivision scheme. In contrast

to the approach by Rockwood etal.[249], both algorithms

contain strategies to avoid cracks between patches. A gen-

eral discussion on how to avoid edge gaps in case of an

adaptive subdivision is given by Dehaemer and Zyda [69].

According to Vigo and Brunet[300], the main drawback

of the approaches previously mentioned [249, 279] is that

the resulting elements may be odd-shaped, especially near

the boundary. They suggested to overcome this issues by a

piecewise linear approximation of trimmed surfaces using

a triangular mesh that is based on a max-min angle crite-

rion. The algorithm is designed so that the resulting mesh

can be used for stereolithography, FEA, and rendering. The

mesh obtained consists of shape-regular elements and has

no cracks between patches.

The determination of a proper step size of a tessellation

is of course an important issue. The elements should not be

too small in order to avoid oversampling of the surface, nor

too big, since this would decrease the quality of the ren-

dering [1]. Lane and Carpenter [178] presented a formula

for calculating the upper bound of the distance between a

right triangle interpolating a surface. Later, the bound was

improved by Filip et al. [89]. Based on this work, Sheng

and Hirsch [279] derived the following formula for arbi-

trary triangles: the approximation error can be estimated

by the diﬀerence of a parametric surface

S(u,v)

to a linear

triangle

T(u,v)

where T is the correspond region in the parameter space,

𝜆

denotes the maximal edge length of

T(u,v),

and

are

speciﬁed by

Hence, the upper bounds of second derivatives of the sur-

face are required. Once these bounds are determined,

𝜆

can

be computed for a given tolerance

𝜀

The bounds on the second derivatives for a B-spline sur-

face (43)–(45) can be computed by constrained optimiza-

tion [89] or conversion to a Chebyshev basis [279]. Piegl

and Richard [229] use the fact that the derivative of a

B-spline is again a B-spline to deﬁne the upper approxima-

tion bounds by computing the maxima of the control points

of the diﬀerentiated surfaces. They address the treatment

(42)

sup

(u,v)∈T

S(u,v)−T(u,v)

≤2

𝜆2



M1+2M2+M3

,

(43)

1=sup

(u,v)∈T

‖

𝜕2S(u,v)

𝜕2u

‖

(44)

2=sup

(u,v)∈T

‖

𝜕2S(u,v)

𝜕u𝜕v

‖

(45)

3=sup

(u,v)∈T

‖

𝜕2S(u,v)

𝜕2v

‖

(46)

𝜆

(

𝜀

2(M1+2M2+M3))1∕2

6 Stereolithography is an early and widely used 3D printing tech-

nique.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

of rational surfaces by means of homogeneous coordinates

and adjustment of the tolerance due to the perspective map-

ping (12).

A few years later, Piegl and Tiller[231] proposed a tri-

angulation scheme which is geometry-based, i.e.,the pro-

cedure is based on the geometry rather than the param-

eterization. The trimming curves are polygonized in the

model space by cubic Bézier curves and the surface itself

is subdivided by its control net. The main advantage of

this approach is that the trimmed NURBS surface is not

required to have more than

-continuity, in contrast to

the previous methods that assumed that the surfaces are

-continuous in order to estimate a step length in the

parameter space [89]. Elber [79] proposed two alternative

approaches that are also independent of the parameteriza-

tion: one based on an intermediate linear surface ﬁt and

another based on global normal curvature. In general, tes-

sellations do not require an element connectivity or shape

regular elements. However, several authors have presented

the construction of conforming meshes for trimmed patches

that yield triangles with good aspect ratios [55, 57, 58].

Irregular meshes are also an issue regarding hardware

implementation on the graphical processing unit (GPU).

Moreton [206] presented tessellation of polynomial sur-

faces for hardware rendering using forward diﬀerences and

dividing the work of tessellation between CPU and GPU.

To avoid gaps along shared boundaries of patches due to

the diﬀerent ﬂoating point engines, all boundary curves of

the patches are calculated on the GPU. In order to enable

GPU based tessellation, Guthe etal.[108] presented a trim

texture scheme which can be parallelized. In this approach,

the visible domain is speciﬁed based on a texture-map of

black and white pixels, hence the trimming task is per-

formed on pixel-level.

3.3.2 Ray Tracing

In contrast to tessellation, ray tracing tries to compute an

image one pixel at a time [76]. Every object in a scene is

tested if it intersects with rays spawned from an eye-point

as indicated in Fig.21. The result must return at least the

closest intersection point and the corresponding normal of

the surface for each ray. Hence, the heart of any ray tracing

package is the set of ray intersection routines [99].

Ray tracing is a powerful, yet simple approach to image

generation [144]. Already early attempts of this technique

have been successfully applied for automatic shading of

objects [8], modeling of global lightning eﬀects [311], and

the visualization of fuzzy reﬂections and blurred phenom-

ena [64]. In the context of parametric surfaces, the pio-

neering works focus on diﬀerent ray-surface intersection

methods [144, 146, 297]. Various intersection algorithms

have been developed. One of the ﬁrst algorithms used a lat-

tice approach where the problem is reduced to ﬁnding the

root of univariate polynomials [146]. Several numerical

methods based on Newton schemes have been employed,

e.g.,[198, 293, 297]. Nishita etal. [214] introduced a ray

tracing technique for trimmed patches based on Bézier clip-

ping. This concept has been adapted and enhance by a large

number of researchers, e.g.,[44, 78, 97, 217, 303]. Bézier

clipping is discussed in more detail in Sect.3.5.2.

Ray tracing of trimmed patches has also been addressed.

Usually, the untrimmed surface is intersected ﬁrst and the

determination if the intersection point lies inside or out-

side of the trimmed domain is performed in a subsequent

step. This point classiﬁcation task can be employed by ray-

tests, e.g.,[198, 214, 261]. The regions that require trim-

ming may be identiﬁed in a preprocessing step in order to

improve the performance [97]. Section3.5.2 provides more

information on the ray-test concept. An alternative way of

point classiﬁcation is to generate a trim texture that returns

whether a point is inside or not [108]. This approach is very

eﬃcient since it requires only a single texture look-up to

classify a domain point. However, the trim texture has to be

updated every time the view changes [313].

One of the greatest challenges of ray tracing is eﬃcient

execution [99]. Hence, many researchers have focused

on this issue. Early attempts improved the performance

by means of bounding box trees, e.g.,[198, 316]. Havran

[116] compared a number of such schemes and concluded

that the kd-tree is the best general-purpose acceleration

structures for CPU. Kd-trees deﬁne a binary space partition

that always employs axis-aligned splitting planes. Pharr

etal.[226] have shown that coherence can be exploited to

improve ray tracing. Their rendering algorithms improve

locality of data storage and data reference. Further improve-

ment can be obtained by simplifying and streamlining the

basic algorithms in order to exploit performance features of

Eye

˜x1

Image plane,

or screen

Fig. 21 Rays spawned from an eye-point in order to get a pixel-wise

image of an object

B.Marussig, T.J.R.Hughes

1 3

processors like single instruction, multiple data extensions

[27, 97, 302]. Purcell et al. [235] demonstrated that the

entire ray tracing process can be performed on the GPU.

Since then, several other GPU based approaches emerged,

e.g.,[91, 172, 217, 261].

Despite the eﬃciency deﬁcit compared to tessellation

schemes, direct rendering of surfaces has several advan-

tages. The memory requirements and preprocessing costs

are reduced since fewer primitives are used and geometric

precision and image quality are improved by eliminating

artifacts [27].

3.4 Remodeling ofTrimmed Models

Solid models with trimmed surfaces suﬀer from robust-

ness issues that may lead to inconsistencies as previously

discussed in Sect.3.2.2. In order to obtain an unambiguous

and watertight description of a solid model, several authors

considered replacing trimmed objects by other surface rep-

resentations. In particular, it has been suggested to remodel

trimmed surfaces by means of a set of regular patches, sub-

division surfaces, or T-splines.

3.4.1 Regular Patches

The treatment of trimmed surfaces in the early automo-

tive industry was discussed by Sarraga and Waters [257],

in which a repatching method is proposed. To be precise,

the intersection curves are used as edges of new regular

patches approximating the original surface. As pointed

out by Sarraga and Waters, repatching has several distinct

disadvantages for modeling, but it is applied as a compro-

mise between the complexity of free-form surfaces and

the requirements of solid modeling. The common aim of

the subsequent approaches is to improve this compromise.

Besides the desire for an unambiguous and robust solid

model, exchange of geometric data between dissimilar

CAD software has been a motivation for this remodeling

concept. Various constructions for the repatching procedure

have been proposed. Hoschek and Schneider [131] con-

vert trimmed rational Bézier patches into a set of bicubic

and biquintic Bézier patches. The segmentation is based

on arguments related to the curvature of the surface and

conditions on the geometrical continuity. The procedure

combines some of Hoschek’s previous works, i.e., [129,

132], and consists of four steps: (i) determination of new

geometrically oriented boundary curves, (ii) approxima-

tion of these curves, (iii) ﬁtting of the interior of each patch

using geometric continuity conditions for the boundary

and corner points, and (iv) approximation of the intersec-

tion curves of trimmed surfaces. The use of ruled sur-

faces [110], Coons patches [41, 301], and Clough–Tocher

splines7 [167] have also been suggested to remodel trimmed

surfaces. Another concept is based on clipping isoparamet-

ric curves of a B-spline surface [9]. Later, this approach has

been adapted for the design of aircraft fuselages and wings

[304, 320].

Generalized Voronoi diagrams may be used to obtain a

proper decomposition of the trimmed domain with multiple

trimming curves [110, 142]. Thereby, the parameter space

is partitioned into convex polygons such that each poly-

gon contains exactly one trimming curve as illustrated in

Fig.22. Details on Voronoi diagrams can be found in the

survey of Aurenhammer [10].

Another strategy to remodel trimmed models is local

perturbation. In contrast to repatching, the control points

of the original surfaces are modiﬁed in order to obtain

an unambiguous conﬁgurations along the intersection

curves. Hu and Sun[137] proposed to close gaps between

trimmed B-spline surface by an algorithm that moves one

of the patches towards the trimming curve deﬁned by the

other one. This approach modiﬁes the control point of the

patch near the trimming curve using singular value decom-

position. It can be used to improve the accuracy of small

gaps, but yields bad-shaped surfaces if the gaps are too

large. Moreover, this approach does not produce an exact

topological consistency. Song etal. [285] deﬁnes the dif-

ferences of corresponding trimming curves by means of a

so-called error curve in model space. It is speciﬁed so that

its coeﬃcients depend linearly upon the control points of

the intersecting surfaces. The perturbation is carried out by

setting all coeﬃcients of this curve to zero. This is found

by solving a linear system of equations and results in an

adaptation of the control points. A complement to this

work was presented by Farouki etal. [87]. They propose

Fig. 22 Generalized Voronoi diagram for ﬁve trimming curves (re-

execution of the original ﬁgure of [110])

7 Clough–Tocher is a splitting scheme to construct

-continuous

splines over triangulations.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

to remodel trimmed surface by a hybrid collection of ten-

sor product patches and triangular patches. In particular,

they demonstrated the approximation of trimmed bicubic

patches by quintic triangular patches such that the intersec-

tion curves are explicitly deﬁned by one side of a triangular

Bézier patch. The approach considers pairs of rectangular

patches that intersect along a single diagonal arc. This can

be achieved by a preprocessing step as described in the fol-

low-up paper[115].

3.4.2 Subdivision Surfaces

The basic concept of subdivision approaches goes back to

the 1970s. Chaikin developed an elegant algorithm to draw

a curve by cutting the corners of a linear polygon [54]. The

basic steps of the procedure are shown in Fig.23. Later,

it was shown that this cutting algorithm converges to a

quadratic B-spline curve and the initial polygon is equiv-

alent to its control polygon [246]. This idea of sequential

subdivision of a control polygon was generalized by Doo

and Sabin [75] as well as Catmull and Clark [53] to com-

pute bi-quadratic and bi-cubic B-spline surfaces, respec-

tively. Since then, a vast number of diﬀerent subdivision

schemes emerged for various surface types, such as trian-

gular splines [190] and NURBS patches [51], for instance.

The ﬁnal objects of subdivision schemes are referred to as

limit curves or surfaces. The distinguishing feature of these

approaches is that they can be applied to arbitrary control

polygons which are not restricted to a regular grid struc-

ture. The smoothness between the resulting surfaces is con-

trolled by the subdivision scheme.

In 2001, the issue of Boolean operation for subdivi-

sion surfaces was addressed. Litke etal.[188] presented a

trim operator, but do not address surface-to-surface inter-

sections. An algorithm for approximate intersections was

developed by Biermann etal. [30]. High accurate results

can be achieved at additional computational expense. Both

approaches employ subdivision based on triangular splines.

Shen et al. [278] convert trimmed NURBS surfaces to

untrimmed subdivision surfaces using Bézier edge condi-

tions. The limit surface ﬁts the original object to a speci-

ﬁed tolerance. The resulting Catmull–Clark models are

watertight and smooth along the intersection. Recently,

Shen etal.[277] presented a generalization of the approach

that converts B-Rep models of regular and trimmed bicu-

bic NURBS patches to a single NURBS-compatible sub-

division surface. During this process, a quadrilateral

mesh topology is constructed in the parameter space of

each patch and the corresponding control points are com-

puted by solving a ﬁtting problem. Finally, the individual

parts are merged into a single subdivision mesh. In order

to obtain gap-free joints, the preserved boundary curves in

model space are used as target curves of the subdivision

surface.

Subdivision models possess a greater ﬂexibility due to

their inherent topological consistence while conventional

NURBS models have greater control of an objects shape.

This attribute of subdivision attracted considerable atten-

tion, especially in the ﬁeld of computer animation [70]. For

a detailed discussion on subdivision schemes the interested

reader is referred to the textbook [308].

3.4.3 T-splines

T-splines were introduced by Sederberg et al. [270] in

2003. They are generalizations of B-splines that allow

T-junctions in the parameter space and the control net of

(a) (b)

Level 1 Level 2

Level 3 Limit curve

Fig. 23 Chaikin’s corner-cutting algorithm: construction of a quad-

ratic B-spline curve by a subdivision of the control polygon

Fig. 24 An example of a parameter space with T-junctions which are

highlighted by circles

B.Marussig, T.J.R.Hughes

1 3

a surface as illustrated in Fig. 24. In a subsequent paper

[268], the related ability of local reﬁnement is used to close

gaps between trimmed surfaces by converting them to a

single watertight T-spline model. The resulting T-spline

representation can be converted to a collection of NURBS

surfaces again, without introducing an approximation

error. On the other hand, conversion to the T-spline repre-

sentation includes some perturbation in the vicinity of the

intersection. It is argued that the approximation error can

be made arbitrarily small, and the perturbation can be con-

ﬁned to an arbitrarily narrow neighborhood of the trimming

curve. The conversion is performed such that

-continuity

is obtained between the intersecting surfaces. These papers

are focused on cubic splines since they are the most impor-

tant ones in CAGD. However, the T-spline concept is not

restricted to the cubic case.

3.5 Auxiliary Techniques

Techniques and strategies frequently used in the context of

trimming are outlined in this section. They may be useful

for researchers dealing with trimmed models in isogeomet-

ric analysis.

3.5.1 Bounding Boxes

Bounding boxes are often applied to signiﬁcantly accelerate

geometrical computations. The basic idea is to use rough

approximations of objects in order to get a fast indicator

if two regions are well separated or not. Hence, involved

operations have to be carried out only if necessary. These

approximations may be reﬁned adaptively as in divide-and-

conquer based surface intersection approaches introduced

in Sect.3.1.3.

The simplest and perhaps most common approach is

to embed objects into min-max boxes where the corner

points of the object deﬁne an axis-parallel box. The axis

aligned setting is not mandatory but allows the most eﬃ-

cient evaluation of the distance between two boxes [116].

Some authors suggest to use oriented bounding boxes to

improve the geometry approximation, e.g.,[15, 20, 133]. In

this case, the bounding box is rotated such that it is aligned

with the connection of the corner points of the surface it

encloses. An object can also be bounded by a combination

of slaps, also known as fat lines, with diﬀerent orientations

[154]. Slaps denote regions between two parallel planes

which are speciﬁed by their normal vector. This concept

includes conventional bounding boxes, simply by using

two orthogonal slaps. Figure25 summarizes these various

bounding box types.

Bounding boxes constructed by corner points do not

guarantee the enclosing of the whole spline, especially

if a spline is highly curved. The convex hull property

of the control points can be used in order to get a proper

approximation. Consequently, the area of the bounding box

increases since it is computed based on the control polygon

rather than the actual geometry, as illustrated in Fig.26.

Sederberg and Nishita [269] proposed an optimized bound

for planar quadratic and cubic Bézier curves. They sug-

gested deﬁning the bounding region by lines parallel to

the connection

𝓁

of the ﬁrst and last control point. They

are determined by the minimal and maximal distance

the other control points

perpendicular to

𝓁.

The tighter

bound is determined in the quadratic case by

while for the cubic splines it is

with the scaling factor

𝛼

given by

Figure 27 illustrates the improvement of the bounding

boxes due to these bounds. Note that the bounding boxes

(47)

min =min

{

}

and dmax =max

{

(48)

dmin

=𝛼

⋅

min

{

0, d

2},

(49)

dmax

=𝛼

⋅

max

{

0, d

2},

(50)

𝛼

{

4if d1d2>

otherwise.

(a) (b) (c)



Axis-parallel Oriented Slaps

Fig. 25 Various types of bounding boxes for the same curve. The

orientation of the enclosing region is indicated by arrows

(a) (b)

Fig. 26 Construction of axis-parallel bounding boxes by a the end-

points and b the control points of a spline

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

are oriented according to the locations of the ﬁrst and last

control point.

Another way to assure that a curve lies within its

bounding box is to subdivide it into monotonic regions.

The essential idea is that if a domain of any continuously

diﬀerentiable function f is subdivided at its characteris-

tic values, the range of f on each of the subintervals can

be simply found by evaluating f at the endpoints of that

subinterval [165, 208]. The set of characteristic points may

include zeros of the ﬁrst or second derivatives of f, start

and end points of open curves, and singular points such as

cusps or self-intersections. Figure 28 shows an example

of a B-spline curve that has been divided into monotonic

regions and the corresponding bounding boxes. In order

to detect these points, a preprocessing step is required.

Despite this additional eﬀort, monotonic regions have

been used in several application like intersecting planer

curves [156, 157] and surfaces [84], tessellation of trimmed

NURBS [249], and ray tracing [261].

3.5.2 Point Classiﬁcation

One of the most fundamental operations in the context of

trimmed surfaces is the determination if a point

of a patch

is inside or outside the visible domain. This can be done

by counting the number of intersections of a ray emanat-

ing from

with the trimming curves and the boundary of

the patch. If the number is odd

is inside and otherwise it

is outside of the visible area. The direction of the ray can

be chosen arbitrary. This rule is based on the Jordan curve

theorem, that is, every simple closed planar curve sepa-

rates the plane into a bounded interior and an unbounded

exterior region [109]. Hence, the intersection is determined

in the parameter space of the patch, in contrast to the ray

tracing approach for rendering outlined in Sect.3.3.2. Fur-

thermore, if a trimming curve is not closed, it is associate

to the visible part of the patch boundary to obtain a closed

loop as illustrated in Fig.29. Another possibility is to con-

nect open trimming curves with the non-visible boundary

of the patch and intersect only with the trimming curves.

It should be noted that in the latter case, the even-odd rule

turns upside down, i.e.,

is inside the visible domain if the

number of intersections is even.

Despite its conceptional simplicity, the implementa-

tion of the corresponding algorithm is not trivial [77].

For example, ambiguous cases may occur like tangency

between the ray and the curve. Nishita et al. [214] pro-

posed the following procedure: the ray is chosen such that

it intersects perpendicularly with the closest boundary of

the patch. As a consequence, the parameter space is divided

into four quadrants which meet at the origin of the ray as

9d2

9d1

4d1

(a)

(b)

(c)

Fig. 27 Tighter bounds for bounding boxes for quadratic and cubic

B-spline curves. The original bounding boxes are shown by dotted

lines whereas dashed lines mark the improved ones

Fig. 28 Deﬁnition of axis-parallel bounding boxes based on mono-

tonic regions. The white points mark the characteristic points consid-

ered

Fig. 29 Classiﬁcation of interior and exterior points by counting

intersections of the trimming curve with a ray

B.Marussig, T.J.R.Hughes

1 3

shown in Fig.30. They are labeled counter-clockwise such

that the quadrants I and IV are adjacent to the ray. If the

trimming curve is speciﬁed as a set of Bézier curves the

following cases may be considered:

(i) There are no intersections with the ray if all control

points of the trimming curve are within the quadrants

I and II, or II and III, or III and IV.

(ii) All control points of a trimming curve are within the

quadrants I and IV. The number of intersections is

even if the endpoints of curve are in the same quad-

rant; otherwise it is odd.

Tangency between the ray and the trimming curve do

not pose any problem for these exclusion criteria. How-

ever, it should be pointed out that an intersection may be

counted twice if the ray goes through an endpoint which is

shared by two trimming curves. For the other cases where

the intersection has to be computed, Nishita et al. [214]

suggested to employ Bézier clipping. This concept has been

introduced by Sederberg and Nishita [269] in the context

of curve-to-curve intersection and locating points of tan-

gency between two planar Bézier curves. The basic idea is

to use the convex hull property of Bézier curves to iden-

tify regions of the curves which do not include the solu-

tion. The bounding regions are deﬁned by fat lines parallel

and perpendicular to the line through the endpoints of the

Bézier curve. By iteratively clipping away such regions, the

algorithm converges to the solution at a quadratic rate and

with a guarantee of robustness.

In particular, the ray is deﬁned implicitly by

The coordinates are denoted by x and y in order to

emphasize that this approach is applicable for any plane

(51)

ax +by +c=0 with a2+b2=1.

coordinate system. The distance

d(u)

of a point on the

Bézier curve

C(u)

to the ray

𝓁

is given by

The coeﬃcients

are the distances of the control points

of the Bézier curve to the ray and

Bi,p

are Bernstein polyno-

mials of degree

Equation(52) can be represented as a

non-parametric Bézier curve

C(u,d(u))

where the values

are related to their corresponding Greville abscissae,

i.e.,

The relationship of the original and non-para-

metric Bézier curve are depicted in Fig.31.

The roots of

C(u,d(u))

are equivalent to the paramet-

ric values

at which

𝓁

intersects

C(u).

Hence, the convex

hull of

C(u,d(u))

can be used to identify regions where

the objects do not intersect. To be precise, the minimal

and maximal parametric values, i.e.,

umin

and

umax,

of the

intersections of the convex hull with the

-axis splits the

parameter space into three regions of which only one,

(52)

(u)=

∑

i=0

diBi,pwith di=axi+byi+c

III

IIIIV

Fig. 30 Speciﬁcation of quadrants for the point classiﬁcation proce-

dure of Nishita etal.[214]



C(u)

d(u)

(0,d

3,d

1

3,d

2

(1,d

umin umax

(a)

(b)

Intersection

Non-parametric curve

Fig. 31 Bézier clipping: a intersection of a ray

𝓁

with a Bézier curve

C(u)

and b the corresponding non-parametric Bézier curve which is

used to determine the parameter range

[umax,umin ]

that contains the

intersection of

C(u)

and

𝓁

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

i.e.,

umin ⩽u⩽umax,

has to be considered for the intersec-

tion with the ray. This region is extracted as a Bézier curve

by means of knot insertion and the procedure is repeated

until a certain tolerance is reached.

This technique can also be utilized to determine the

intersection of two Bézier curves by iteratively clipping

both objects [269]. Sherbrooke and Patrikalakis [280]

developed a generalization of Bézier clipping that allows

computing the roots of an n-dimensional system. The so-

called Projected-Polyhedron method subdivides an object

into Bézier segments and generates each side of its bound-

ing boxes by projecting the control points onto diﬀerent

planes. Thus, only the convex hull of two-dimensional

point sets has to be computed.

3.6 Summary andDiscussion

The previous parts of this section shed light on various

aspects of trimmed NURBS in the context of CAD. On the

basis of the discussion of surface intersection in Sect.3.1,

it can be concluded that there is no canonical way to derive

trimming curves, but a wide range of diﬀerent techniques

to address this problem. Further, an exact representation of

an intersection of two patches is not feasible in most cases.

In fact, several distinct curves are usually used to specify a

single intersection: a curve in model space and trimming

curves in the parameter spaces of all patches involved.

These curves are independent approximations of the actual

intersection and there is no connection between them. This

missing link makes it very diﬃcult to transfer information

from a trimmed patch to an adjacent one.

The necessity of approximation yields gaps and over-

laps between intersecting patches. As a result, robustness

problems arise for solid modeling as outlined in Sect.3.2.

Considerable eﬀort has been devoted to derive consistent

trimmed models. Still, the problem is unresolved and the

absence of truly robust representations poses a demanding

challenge, especially for downstream applications. Even

within the ﬁeld of CAGD, the replacement of trimmed sur-

faces by other representations may be needed. Tessellations

are used in the context of rendering, for instance. In this

particular case, the main reason is eﬃciency as discussed

in Sect.3.3. However, all other remodeling approaches pre-

sented in Sect.3.4 are motivated by the ﬂaws of trimmed

models and the limitation of tensor product patches due to

their four-sided nature. It should be emphasized that these

schemes may yield watertight models, but there are cer-

tain tradeoﬀs. First of all, approximations are introduced

at least in the vicinity of trimming curves. The number of

control variables increases particularly if regular tensor

product patches are used for the remodeling. Subdivision

surfaces and T-splines are promising techniques, but may

induce new problems like extraordinary vertices8 and lin-

ear dependence of the basis functions. In addition, they

are designed for a speciﬁc surface type which may be an

issue if a model consist of parts with diﬀerent polynomial

degree. Overall, it is apparent that there is no simple solu-

tion to the trimming problem.

Despite their diﬃculties, trimmed NURBS are the stand-

ard in engineering design and for the exchange of geo-

metrical information in general. On the one hand, trimmed

tensor product surfaces persist for historical reasons since

they are a well-established technology, integrated in cur-

rent CAD software. On the other hand, this representation

distinguishes itself by its eﬃciency, precision, and simplic-

ity. Trimming problems are hidden from the user who usu-

ally designs a model with the help of black box algorithms.

Isogeometric analysis and adaptive manufacturing may

lead to new developments in CAGD, but trimmed models

are the state of the art and changes will certainly take time.

It is not clear to the authors whether trimmed NURBS or

other techniques like T-splines and subdivision surfaces

will triumph in the future, but it is good to see the competi-

tion. At this juncture, however, trimmed NURBS seem to

be the dominant technology of engineering design.

4 Exchange Standards

At the beginning of this section, general considerations for

exchanging data between diﬀerent computer software sys-

tems is discussed. Next, the most popular neutral exchange

standards, i.e., the Initial Graphics Exchange Speciﬁca-

tion (IGES) and the Standard for the Exchange of Product

Model Data (STEP), are brieﬂy introduced and compared.

Finally, this section closes with some concluding remarks.

4.1 General Considerations

In modern CAD systems, parameters and constraints gov-

ern the design of a model, rather than the deﬁnition of

speciﬁc control points. Further essential components are

local features and the construction history. All these vari-

ous factors are referred to as design intent [162]. Each soft-

ware has its own native data structure to keep track of the

geometry, the topology, and the design intent of its models.

Thus, a translation process is required when information

is exchanged between systems with diﬀerent native struc-

tures. The conversion of formats may seem like an easy

task, but it is in fact very complicated. Usually, there is no

8 Regular internal vertices have four incident edges, also referred to

as valency

k=4.

All other settings, i.e.,

k≠4,

are denoted as extraor-

dinary vertices.

B.Marussig, T.J.R.Hughes

1 3

direct mapping from one format to another. This holds true

in particular for information related to the design intent

since there are no canonical guidelines for its representa-

tion. Consequently, the exchange of the complete data of a

CAD model between diﬀerent systems is scarcely possible,

especially when the systems are designed for diﬀerent pur-

poses. In most cases, only the geometric information of the

ﬁnal object is transferred.

This interoperability issue has been investigated in a

study focusing on the USautomotive supply chain [294].

The following possible solutions have been discussed: (i)

standardization on a single system, (ii) point-to-point trans-

lation, and (iii) neutral format translation.

In case of a single system standardization the same

native format is used for all processes, e.g., design and

analysis. The main advantage is that the compatibility of

the model data is assured since no translation is required.

However, this approach implies the restriction to a single

system. Consequently, every part has to be adjusted to the

developments of the dominant application of the software.

Most importantly, translation problems can arise even

within one system due to diﬀerent software versions—just

imagine you would like to open a PowerPoint presentation

created 10 years ago.

The basic idea of point-to-point translation is to convert

a native format of a system directly to a native format of

another one. This concept works reasonably well for unam-

biguous data exchange tasks. Unfortunately, it is not always

clear how a given information should be translated so that

it is properly interpreted in another native format. In addi-

tion, a high degree of vendor cooperation is necessary in

order to develop a direct translator. Similar to the previ-

ous strategy, direct translators have to be rewritten for each

new system or perhaps even for new versions of the same

software.

Neutral format translation is based on a common neutral

format for the exchange of (geometric) data. This approach

enables an independent development of various tools work-

ing on the same model. The minimization of dependencies

simpliﬁes the maintenance of each software and eventu-

ally leads to robust implementations since a clean code

is designed to do one thing well, as noted by Stroustrup9

[196]. Further, vendors are more willing to develop transla-

tors for neutral formats since it does not require the disclo-

sure of proprietary code. This is beneﬁcial since interpreta-

tion errors of the native format are most likely minimized

when the conversion is provided by vendors themselves.

An additional advantage of neutral formats is that they are

ideally suited for long term storage of data. However, there

are also a number of weaknesses. First of all, it is not pos-

sible to capture the design intent and thus, translation to a

neutral format provides only a snapshot of the current geo-

metric model. In general, every translation leads to loss of

information and the quality of an exchanged model depends

on the capability of the neutral format used.

In the context of isogeometric analysis, the minimal

requirement is the accurate exchange of geometrical infor-

mation of the ﬁnal model. Topology is also essential to

assess the connectivity between patches. The reconstruc-

tion of topological data based on edge comparison or

related strategies is very cumbersome and extremely error-

prone, especially in cases of trimmed geometries where

edges only coincide within a certain tolerance as elaborated

on in Sect.3. The following two approaches are suggested:

(i) direct extraction of topological data from CAD software

by a point-to-point translation or (ii) using a neutral format

that is able to cope with topological data. The former may

be preferred if there is a cooperation with a CAD vendor

and the developments focus on the speciﬁc product. How-

ever, neutral exchange formats will be discussed in the fol-

lowing because they are the most general and independent

approaches. Despite their deﬁciencies, native formats seem

to be the most sustainable solution.

4.2 Neutral Format Translators

Concepts for a common data exchange format emerged

in the 1970s. These attempts were borne by a variety of

partners from industry, academia, and government [101].

Based on the initiative of the CAD user community, in par-

ticular General Electric and Boeing, vendors agreed to cre-

ate an American national standard for CAD data exchange.

The ﬁnal result was the ﬁrst version of IGES [209]. IGES

provided the technical groundwork to a more involved

exchange format, namely STEP.

4.2.1 IGES

The name of this neutral exchange format already reveals

its original purpose [101]:

– Initial10 to suggest that it would not replace the work of

the American National Standards Institute.

– Graphics not geometry, to acknowledge that academics

may come up with superior mathematical descriptions.

– Exchange to suggest that it would not dictate how ven-

dors must implement their native database.

– Speciﬁcation to indicate that it is not imposed to be a

standard.

10 The word “interim” was used in the ﬁrst draft.

9 Bjarne Stroustrup is the inventor of the programming language

C++.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

IGES provided an important and very practical ﬁrst solu-

tion to the exchange problem, resulting in a ﬁle format that

is implemented in almost every CAD system. Regarding

current literature on isogeometric analysis, it seems that

IGES is still the preferred choice when it comes to the

extraction of geometric information. Here, we will try to

disprove this notion.

According to Goldstein etal.[101] and the studies cited

within, the shortcomings of IGES can by summarized as

follows: (i) it contains several ways to capture the same

information leading to ambiguous interpretation, (ii) loss

of information during exchange, (iii) development without

rigorous technical discipline, (iv) restriction of exchange

capabilities due to the compliance with earlier IGES ver-

sions, (v) it was developed as a method to exchange engi-

neering drawings, but not designed for more sophisticated

product data, (vi) vendors implemented only portions of

IGES, and (vii) there is no mechanism for testing the trans-

lators. In addition, IGES is a national standard which may

lead to translation problems if other than US software is

used. Most importantly, IGES is a stagnant exchange for-

mat. The last oﬃcial version of IGES, i.e., version 5.3

[155], was published more than 20 years ago in 1996.

Although IGES continues to be deployed in industry, its

main legacy is the disclosure of several weaknesses of the

neutral exchange concept, thereby enhancing new emerging

standards. The most notable one is STEP, which provides

a broader, more robust standard for the exchange of data

[101].

4.2.2 STEP

Since 1984 the International Organization for Standardiza-

tion (ISO) has been working on a standard for the exchange

of product data and its ﬁrst parts were published in 1994

[232]. The objective of this development eﬀort—one of the

largest ever undertaken by ISO—is the complete and unam-

biguous deﬁnition of a product throughout its entire life

cycle, which is independent of any computer system [264].

Hence, the corresponding standard includes the exchange

of CAD data, yet its scope is much broader.

STEP is the informal term for the standard oﬃcially

denoted as ISO 10303. It is organized by an accumulation

of various parts uniﬁed by a set of fundamental principals.

These parts are referred to as ISO 10303-xxx, where xxx is

determined by the part number. Each of them is separately

published and has to pass several development phases sum-

marized in Table1. Each part is associated with one of the

following series: (i) description methods, (ii) implementa-

tion speciﬁcations, (iii) conformance testing, (iv) generic

integrated resources, (v) application integrated resources,

(vi) application protocols. Figure 32 gives an overview

of these various components of STEP. The description

is given by the common formal speciﬁcation language

EXPRESS (Series 10) deﬁning data types, entities, rules,

functions, and so on [274]. It is not a programming lan-

guage, but has an object-oriented ﬂavor. The transfer of

data is deﬁned by the implementation speciﬁcations (Series

20). The exchange by a neutral ACSII ﬁle is addressed in

Part 21, “clear text encoding the exchange structure.” This

STEP-ﬁle transfer is the most widely used data exchange

form of STEP [264]. However, other approaches, like

shared memory access, are covered by the series as well,

see e.g., Part 22, “standard data access interface.” Con-

formance tests provide the veriﬁcation requirements (Series

30).

The most fundamental components of STEP are the inte-

grated resources. They contain generic information such

(Parts 40–199)

Integrated resources

–Fundamentalsof

product description

–Representation of

geometry andtopology

–Representation

speciﬁcation

(Series 10)

EXPRESS

(Series 30)

Conformance

testing

–General

concepts

–Test labre-

quirements

(Series 200)

Application protocols

–Conﬁguration controlleddesign

–Explicit draughting

–Associative draughting

Implementationspeciﬁcation

–Physicalﬁle

–STEP data access interface

(Series 20)

Fig. 32 Structure of STEP (re-execution of the original diagram

[177])

Table 1 STEP stages

Stages Names

PWI Preliminary work item

NWI New work item

AWI Approved work item

WD Working draft

CD Committee draft

FCD Final committee draft

DIS Draft international standard

FDIS Final draft international standard

PRF Proof of new international standard

IS International standard

B.Marussig, T.J.R.Hughes

1 3

as geometric data and display attributes (Series 40) as well

as further elementary units that are specialized for certain

application areas (Series 100). For example, Part 42, “geo-

metric and topological representations,” focuses on the

deﬁnition of geometric models in general, while Part 104,

“ﬁnite element analysis,” is devoted to applications in the

context of FEA. These parts provide the entities needed to

build application protocols denoted as APs (Series 200).

They are the link to the needs of industry and other users.

Their purpose is to interpret the STEP data in the con-

text of a speciﬁc application which may be part of one or

more stages of a life cycle of a particular product. Part 209,

“multidisciplinary analysis and design,” addresses engi-

neering analysis. Each STEP application protocol is further

subdivided into a set of conformation classes (CCs). These

subsets must be completely implemented if a translator

claims to be conform with the standard [233]. Hence, it is

important to know what conformance classes are supported

by a software system. This modular structure, with several

APs and their CCs, may seem complex and daunting, but

it gives users the necessary transparency of what can be

expected of the data exchanged. Moreover, the complexity

of the overall concept of STEP does not imply that it is dif-

ﬁcult to use.

The downside of the broad scope of STEP is the large

amount of detailed information which may seem over-

whelming at ﬁrst glance. In addition, ISO documents are

not available for free, but can be purchased through the ISO

homepage.11 There are, however, helpful resources to start

with: the US Consortium called Product Data Exchange

using STEP (PDES, Inc.) provides several resources12 like

a handbook for a general introduction to STEP [264]. Fur-

ther background articles are also released by the developers

of STEP tools, Inc.13 ISO permits EXPRESS listings to be

distributed without copyright restrictions and several exam-

ples are given in the software’s archive.14

One of the advantages of STEP is that it is more than

just a speciﬁcation for exchanging geometric informa-

tion. It provides a complete product data format allowing

the integration of business and technical data of an object,

from design to analysis, manufacturing, sales, and service

[294]. STEP is perfectly aligned with the spirit of isogeo-

metric analysis, i.e., unifying ﬁelds. The most important

feature of STEP is its extensibility. Eﬀorts have been made

to include the design intent into STEP [162, 233, 234]. Par-

ticularly interesting for isogeometric analysis is the speci-

ﬁcation of volumetric NURBS and local reﬁnement in the

next versions of Part 42 and other parts [284].

4.2.3 Comparative Example

In order to demonstrate the representation of trimmed

geometries in IGES and STEP, an example of two inter-

secting planes is considered. A square

[0, 5]2

within the xy-

plane is perpendicularly intersected along its diagonal by

another plane surface as illustrated in Fig.33. Thereby, the

perpendicular patch is also trimmed into two halves by the

square.

The model investigated has been constructed using the

software Rhinoceros and the intersection has been

computed in two diﬀerent ways: using (i) the trim-com-

mand and (ii) the Boolean-command, respectively. Both

schemes lead to the same geometry, yet the topology varies

as indicated by the diﬀerent highlighting of Fig.33a and

b. To be precise, the trim-command produces a surface

model that consists of two independent trimmed surfaces,

while the Boolean-command results in a solid model

where the patches are connected. Both models have been

exported to neutral exchange formats. The correspond-

ing IGES and STEP ﬁles are provided in the Appendix.

In the following, certain aspects of the exported ﬁles are

discussed.

Fig. 33 Design model with the same geometry but diﬀerent topol-

ogy: a two independent trimmed surfaces and b connected surfaces

by a Boolean operation. In a, the isocurves of the separated surfaces

are displayed in black and red, respectively. (Color ﬁgure online)

Table 2 Entity types of the IGES example

Numbers Names Information

126 Rational B-spline curve Geometry

128 Rational B-spline surface Geometry

141 Boundary entity Topology

143 Bounded surface entity Topology

11 http://www.iso.org, September 2016.

12 https://www.pdesinc.org/ResourceIndex.html, September 2016.

13 http://www.steptools.com/library/standard/, September 2016

14 http://www.steptools.com/sc4/archive/, September 2016.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

Remark 3 The default setting of the Rhinoceros export

has been chosen, i.e., AP213AutomotiveDesign, for the

STEP examples. However, the elements discussed in this

section are not aﬀected by this choice.

4.2.3.1 File Structure The ﬁxed ASCII ﬁle format15 of

IGES is structured by the subsequent sections: Start (S),

Global (G), Directory Entry (D), Parameter Data (P), and

Termination (T). The letters in the brackets label these dis-

tinct parts and they are shown in column 73 of every ﬁle.

The Directory Entry and the Parameter Data is speciﬁed by

entities which are associated with a unique type number.

Table2 lists the entities used in this example. The Direc-

tory Entries provide attribute information for each entity in

an IGES ﬁle. Each entry is ﬁxed in size and is speciﬁed by

20 ﬁelds. The ﬁrst ﬁeld contains the entity type and the sec-

ond one points to the ﬁrst line of the related Parameter Data

record. This connection is shown for a rational B-spline sur-

face in Fig.34a. The Parameter Data, on the other hand, is

free-formatted and it consists of a sequence of integer and

real numbers starting with the entity type number.

STEP ﬁles are easy to read since the language used is

based on an English-like syntax [274]. In general, an accu-

mulation of entities pointing to each other shapes the struc-

ture of the exchange data. Lines specifying entities begin

with the symbol #, followed by the unique identiﬁer of

the corresponding object. This identiﬁer is used to con-

nect various entities with each other as shown in Fig.34b.

Besides pointers, an entity may consists of integers, real

numbers, Booleans (.F./.T.), and enumeration ﬂags (e.g.,.

UNSPECIFIED.).

4.2.3.2 Surfaces Representation Both exchange formats

provide the fundamental informations of B-spline patches,

i.e., degree, knot vectors, and control points, together with

auxiliary information. In case of IGES, a sequence of num-

bers separated by commas is used, while STEP additionally

groups associated components using brackets. In Fig.35, the

representations of the regular square patch are compared.

Note that knot vectors are speciﬁed by knot values with their

multiplicity and that coordinates of control points are stored

128,1,1,1,1 ,0,0,1,0 ,0,0.0D0,0.0 D0,7.071067811865475D0, 0000005P 3

7.071067811865475D0,0.0D0,0.0D0,9.999999999999998D0, 0000005P 4

9.999999999999998D0,1.0D0,1.0D0,1.0D0,1.0D0,5.0D0,0.0 D0,-5.0D0,0000005P5

8.881784197001252D-16,4.999999999999999D0,-5.0D0 ,5.0D0,0.0 D0,0000005P6

4.999999999999998D0,8.881784197001252D-16,4.999999999999999D0,0000005P7

4.999999999999998D0,0.0D0,7.071067811865475D0,0.0D0 ,0000005P 8

P5000000;0D899999999999999.9 9

128300100000010000D5

1280-17800TrimSrf0D6

Type Number Sequence Number

#80=B_SPLINE_SURFACE_WITH_KNOTS(’’,1,1,((#108,#109),(#110,#111)),

.UNSPECIFIED.,.F.,.F.,.F.,(2,2),(2 ,2),(0. ,7.07106781186547),(0. ,10.),

.UNSPECIFIED.);

#108=CARTESIAN_POINT(’’,(5.,0.,-5.));

#109=CARTESIAN_POINT(’’,(5.,0.,5.));

#110=CARTESIAN_POINT(’’,(8.88178419700125E-16,5.,-5.));

#111=CARTESIAN_POINT(’’,(8.88178419700125E-16,5.,5.));

Identiﬁer

Type

(a) (b)

IGES STEP

Fig. 34 Entity connection of the exchange formats. Pointers are indicated by arrows. The examples have been extracted from Files1 and 3 of

the Appendix, respectively

128,1,1,1,1,0,0,1,0,0,0.0D0,0.0D0,5.0D0,5.0D0,0.0D0,0.0D0,5.0D0 ,0000027P47

5.0D0,1.0D0,1.0D0,1.0D0,1.0D0,0.0D0,0.0D0,0.0D0,0.0D0,5.0D0,0000027P48

0.0D0,5.0D0,0.0D0,0.0D0,5.0D0,5.0D0,0.0D0,0.0D0,5.0D0,0.0D0, 0000027P49

5.0D0;0000027P 50

Upper index I−1 and J−1

Degree pand q

Not closed

Polynomial

Non periodic

Knot vector ΞI

Knot vector ΞJ

Weightsw0,...

Control pointcoordinates

x0,y

0,z

0,x

1,...

Start and end

of u-direction

Start and end

of v-direction

#81=B_SPLINE_SURFACE_WITH_KNOTS(’’,1,1,((#132,#133),(#134,#135)),

.UNSPECIFIED.,.F.,.F.,.F.,(2,2),(2,2),(0.,5.),(0.,5.) ,.UNSPECIFIED.);

Degree pand q

Controlpoints

Knot vector

type

Knot values

of ΞIand ΞJ

Knot

multiplicity

Not self-

intersecting

Not

closed

B-spline

surface form

(a) (b)

IGES STEP

Fig. 35 Descriptions of the surface model’s regular square patch. The B-spline surface data has been extracted from Files1 and 3 of the Appen-

dix, respectively

15 There exist also a compressed format for details see [155].

B.Marussig, T.J.R.Hughes

1 3

within its own entity in case of STEP. Hence, error-prone

comparisons of ﬂoating point numbers are avoided.

Regarding trimmed surfaces, the following informa-

tion is provided in both exchange formats: (i) the regular

surface, (ii) the related loops of trimming curves, and (iii)

their counterparts in model space. All this information can

be found in a single IGES entity, i.e., 141. In particular, the

fourth and ﬁfth number within the sequence of this entity

deﬁne the reference to the regular surface and the number

of related curves, respectively. These numbers are followed

by arrays of the size 4. The ﬁrst value of an array refers

to model space curves while the last value points to trim-

ming curves. The total number of arrays is determined by

the number of related curves.

In case of STEP, the trimmed surface data is not coa-

lesced in a single object, but it is embedded in a graph

structure. Hence, the information is represented by vari-

ous diﬀerent entities which are linked together. The

ADVANCED_FACE entity may be viewed as the start-

ing point of the graph structure that speciﬁes the trimmed

patch. Figure36 illustrates such a collection of entities. For

the sake of clarity, some intermediate entities have been

neglected as indicated by the dashed lines.

4.2.3.3 Topology So far, the speciﬁcation of certain

parts of a model has been addressed. Here, the diﬀerences

between the exchange formats regarding an object’s topo-

logical information is examined by comparing the output

for the surface model and solid model shown in Fig.33. The

former is deﬁned by two independent surfaces, while the

latter is a single coherent manifold.

In the following the square patch in the xy-plane is

denoted by

S□

and the perpendicular patch is referred to as

⟂

The corresponding edges of the model are labeled

e□

with

i={1, …,3}

and

e⟂

with

j={1, …,4},

respectively.

The topology due to the STEP and IGES formats is com-

pared in Fig.37. To be precise, the provided edge loop data

is shown. Further details are neglected for the sake of brev-

ity, but the entire ﬁles can be found in the Appendix.

Comparing Fig. 37a and b shows that IGES yields the

same output for both models. In other words, the diﬀerent

topologies of them are not recognized. Note that the only

values that diﬀer are the sequence numbers of the entities

which are completely independent from the actual object.

In fact, the topological connection of

S□

and

S⟂

is lost in

case of the solid model, despite the simplicity of the exam-

ple. That the solid model has been properly constructed is

proven by the STEP output shown in Fig.37d where the

edges

e⟂

and

e□

are joined in a single reference, i.e.,

#48.

ADVANCED FA CE

EDGE LOOP B SPLINE SURFACE WITH KNOTS

SURFACE CURV E

PCURVE

B SPLINE CURVE WITH KNOTS

(Model space curve)

B SPLINE CURVE WITH KNOTS

(Trimming curve)

Fig. 36 Graph related to a trimmed surface in STEP. Entities that

provide geometrical information are highlighted in gray. Intermediate

nodes may be skipped which is indicated by dashed lines

(a)

(b)

(c)

(d)

IGES: surface model

IGES: solidmodel

STEP: surface model

STEP: solid model

141,1,3,5,4 ,7,1,1,9,11,1,1,13,15,1,1 ,17,19,1,1,21 ;0000023P 45

141,1,3,27 ,3,29,1,1 ,31,33,1,1 ,35,37,1,1 ,39 P1400000; 68

S⊥e⊥

1e⊥

2e⊥

3e⊥

Se

1e

2e

141,1,3,5,4 ,7,1,1,9,11,1,1,13,15,1,1 ,17,19,1,1 ,21;0000023P 42

141,1,3,27 ,3,29,1,1 ,31,33,1,1 ,35,37,1,1 ,39 P1400000; 65

S⊥e⊥

1e⊥

2e⊥

3e⊥

Se

1e

2e

#22=EDGE_LOOP(’’,(#24 ,#25,#26,#27));

#23=EDGE_LOOP(’’,(#28 ,#29,#30));

#24=ORIENTED_EDGE(’’,*,*,#52,. T.);

#25=ORIENTED_EDGE(’’,*,*,#53,. T.);

#26=ORIENTED_EDGE(’’,*,*,#54,. T.);

#27=ORIENTED_EDGE(’’,*,*,#55,. T.);

#28=ORIENTED_EDGE(’’,*,*,#56,. T.);

#29=ORIENTED_EDGE(’’,*,*,#57,. T.);

#30=ORIENTED_EDGE(’’,*,*,#58,. T.);

e⊥

1,...,e⊥

e

1,e

2,e

#19=EDGE_LOOP(’’,(#21 ,#22 ,#23,#24));

#20=EDGE_LOOP(’’,(#25 ,#26,#27));

#21=ORIENTED_EDGE(’’,* ,*,#48 ,.T.);

#22=ORIENTED_EDGE(’’,* ,*,#49,.T.);

#23=ORIENTED_EDGE(’’,* ,*,#50,.T.);

#24=ORIENTED_EDGE(’’,* ,*,#51,.T.);

#25=ORIENTED_EDGE(’’,* ,*,#52,.T.);

#26=ORIENTED_EDGE(’’,* ,*,#53,.T.);

#27=ORIENTED_EDGE(’’,* ,*,#48 ,.F.);

e⊥

2,e⊥

3,e⊥

e

1,e

e⊥

1,e

e⊥

1,e

Fig. 37 The STEP and IGES loop data of the model illustrated in

Fig.33. Entries referring to edges are labeled by

e□

and

e⟂

The index

numbers are not consistent and thus, may diﬀer from sub-ﬁgure to

sub-ﬁgure. In d, the highlighted pointer, i.e.,

#48,

refers to the edge

where the two surfaces

S□

and

S⟂

intersect

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

The STEP data related to the surface model is illustrated in

Fig.37c.

4.3 Summary andDiscussion

In Sect.4.1, the problem of exchanging data between dif-

ferent systems is outlined from a general point of view. It

is argued that the use of neutral exchange standards is the

most comprehensive way for this task. Nevertheless, every

mapping from one system to another may cause problems,

especially with respect to the design intent of a model

where no canonical representation exists. There is often no

one-to-one translation from one format to another, which

leaves room for (mis)interpretation. As a result, exchange is

usually restricted to snapshots of an object’s geometry. The

transfer of topology data is also possible if (i) the design

model is properly constructed as a coherent solid model

and (ii) the neutral format is able to capture this informa-

tion. It is apparent that the capability of the neutral stand-

ard applied is essential for the quality and success of the

exchange. This has been demonstrated by a simple example

given in Sect.4.2.3 where IGES does not export the topol-

ogy correctly.

STEP should generally be preferred as a neutral

exchange format. According to Tassey etal.[294], STEP is

superior to other translators because it

– addresses many types of data,

– incorporates a superset of elements common to all sys-

tems,

– supports special application needs, and

– provides for international exchanges.

In their paper, several studies are discussed in which STEP

excels with respect to the quality of exchanging data of

industrial examples. In addition, this standard is constantly

developed and improved, e.g., by its enhancements for

isogeometric analysis [284].

Theoretically, the broad scope and modular structure of

STEP provides coverage of various application domains

which are indicated by the application protocols and their

conformance classes. However, this functionality has to be

supported by the CAD vendors. Most vendors have chosen

to implement only certain parts of STEP, i.e.,some con-

formance classes of AP 203 and AP 214 [264]. It is not sur-

prising that vendors seem to show little interest in neutral

exchange formats, since their implementation slows down

the development of the actual software and users become

more independent from their products. Hence, it is likely

that neutral ﬁle formats will always provide less informa-

tion than the original model. Translation errors may be

avoided if the needed data is extracted directly from the

native format, but this requires vendor interaction and the

restriction to a single software. This alternative is not very

sustainable since a native format may become obsolete

after a new software version is released.

5 Isogeometric Analysis ofTrimmed Geometries

Isogeometric analysis of trimmed NURBS is an impor-

tant research area, simply due to the omnipresence of such

geometry representations. Integration of design and analy-

sis can only be achieved if the simulation is able to cope

with CAGD models that are actually used in the design

process. Moreover, sound treatment of trimmed solid mod-

els is also an essential step for the derivation of volumetric

representations.

Current attempts to integrate trimmed geometries into

isogeometric analysis may be classiﬁed as global and

local approaches. The latter uses the parameter space of

the trimmed patch as background parameterization and

the trimming curves determine the domain of interest, i.e.,

v,

for the analysis. Knot spans that are cut by trimming

curves require special attention during the simulation. In

that sense, local approaches are closely related to ﬁctitious

domain methods,16 see e.g.,[124, 239, 252, 259]. Conse-

quently, similar tasks have to be undertaken: (i) detection

of trimmed elements, (ii) application of special integra-

tion schemes in these elements, and (iii) stabilization of

the trimmed basis. CAGD models are not modiﬁed but the

analysis has to deal with all the related robustness issues

pointed out in Sect. 3.2.2. Global reconstruction, on the

other hand, substitutes a trimmed surface by one or sev-

eral regular patches which can be analyzed with regular

integration rules. In other words, it is endeavored to ﬁx the

design model, before it is used in the downstream applica-

tion, e.g.,the simulation. These approaches are similar to

remodeling schemes in CAGD presented in Sect. 3.4.1.

Isogeometric analysis of subdivision surfaces, e.g., [59,

248, 309], and T-splines, e.g.,[22, 262, 263, 321], may be

included into the class of global reconstruction techniques.

However, the discussion of the analysis of these representa-

tions is beyond the scope of this review.

Coupling of multiple patches is another issue that

has to be addressed. Adjacent patches usually have non-

matching parameterizations and a robust treatment of tol-

erances is required to link the degrees of freedom along

an intersection due to the gaps between trimmed surfaces

and the missing link between their trimming curves. Local

16 There are various names for ﬁctitious domain methods such as

embedded domain methods, ﬁnite cell methods, WEB-spline meth-

ods, and immersed boundary methods. The principle idea, however,

is the same.

B.Marussig, T.J.R.Hughes

1 3

approach have to deal with the issue directly during the

analysis, while global approach apply this crucial step

beforehand during the remodeling phase. The coupling

itself is usually performed by a weak coupling technique.

Alternatively, some global schemes try to establish match-

ing parameterizations during the reconstruction procedure.

This allows an explicit coupling of patches and a better

control of the continuity between adjacent patches [149].

It is also noteworthy that the coupling procedure may be

neglected in certain simulation methods. For example, the

boundary element method and the Nyström method do not

require certain continuity between elements or patches, see

e.g.,[218, 223, 225, 319].

The following approaches for analyzing trimmed geom-

etries have been applied to ﬁnite element and boundary ele-

ment methods. The former focuses on shell analysis while

the latter is used for volumetric B-Rep models. However,

the basic concepts are not restricted to a speciﬁc simulation

type since in both cases the treatment of trimmed surfaces

is in the focus. It will be highlighted if a certain part explic-

itly applies for a speciﬁc simulation method.

The overview begins with a short historical note, which,

to the best of the authors’ knowledge is the ﬁrst direct sim-

ulation with trimmed patches. Afterwards, the current state

of research is reviewed in the Sects. 5.2 and 5.3 addressing

global and local approaches, respectively.

5.1 The First Analysis ofTrimmed Models

It is fascinating that the analysis of trimmed patches goes

back to the genesis of trimmed patch formulations. In fact,

Casale etal.[48, 50] presented an analysis of such geom-

etries a few years after they had suggested one of the ﬁrst

trimmed solid model formulations [47, 49]. In particular,

trimmed patch boundary elements had been proposed.

The basic idea of their approach is to employ the

trimmed patch for the geometrical representation and to

deﬁne an independent Lagrange interpolation over the

tensor product surface for the description of the physical

variables. This additional basis does not take the trimming

curves into account. Thus, the nodes of the Lagrange inter-

polation may lie inside or outside the trimmed domain. This

is emphasized by using the term virtual nodes. The analy-

sis is performed by means of a collocated boundary ele-

ment formulation, see e.g.,[96], where all Lagrange nodes

contribute to the system matrix. If a node is outside of the

trimmed domain, the jump term coeﬃcient17 of the bound-

ary integral equation is set to 1 since the node is not part of

the object’s boundary. Numerical integration is performed

over a triangulation of the trimmed domain. These triangles

are used to deﬁne integration regions only and do not con-

tribute any degrees of freedom.

This concept has various deﬁciencies, but it consists of

features that can be found in current approaches as well.

For example, deﬁning the geometrical mapping by the

trimmed parameter space, but the physical ﬁelds by a dif-

ferent (spline) basis is employed in some global techniques

presented later. There are also similarities to local schemes

since the trimmed domain is treated like a background

parameterization leading to special considerations regard-

ing numerical integration and points that are not within the

domain. Furthermore, the motivation for the application

of the boundary element method was the same as today in

isogeometric analysis, i.e.,the potential of a direct analysis

of B-Rep models without the need of generating a volumet-

ric discretization.

5.2 Global Approaches

Global reconstruction schemes decompose trimmed sur-

faces into regular patches. The general concept is the same

as presented in Sect. 3.4.1 in the context of CAGD. The

distinguishing aspect is that the following strategies are

aimed to provide analysis-suitable models.

5.2.1 Reconstruction byRuled Surfaces

Trimming curves

may be used to deﬁne a mapping

t

such that a regular tensor product basis speciﬁes the valid

area

v

of the corresponding trimmed patch, as proposed

by Beer etal.[25]. To be precise,

t

is given by a linear

interpolation between two opposing

,i={1, 2}

The

geometrical mapping to the model space is performed by

the original trimmed patch, hence the approach is also

referred to as double mapping method.

The following assumptions are made for the sake of

notational simplicity. Firstly, the regular basis functions are

deﬁned over a unit square, i.e.,

s,t∈[0, 1].

In addition, it

is assumed that both trimming curves are speciﬁed within

the same parameter range

̃u∈[a,b].

Based on that, the

intrinsic coordinate

̃u

can be linked to the boundaries of the

regular basis at

t=0

and

t=1

by the coordinate transfor-

mations

f(s)

and

g(s).

They are given by

and

These equations traverse the interval of

̃u

in opposite direc-

tions, e.g.,

f(0)=g(1)=a,

since one of the trimming

curves has to be evaluated in reverse order. Finally,

t

determined by

(53)

̃u=f(s)=a+s(b−a),

(54)

̃u=g(s)=b+s(a−b).

17 Usually, the jump term coeﬃcient depends on the geometric angle

of the boundary at the point considered, see e.g.,[96].

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

From a CAGD point of view, the mapping (55) is equiva-

lent to the one of a ruled surface (26). The main diﬀerence

is that the ruled surface is deﬁned in the parameter space in

this case. The geometric mapping

,

however, is still per-

formed by the trimmed patch. Figure38 summarizes the

concept of the double mapping approach.

The main advantage of this approach is its simplicity and

ease of implementation. However, there are various restric-

tions: ﬁrst of all, the assumption that

v

is governed by two

opposing trimming curves limits the application to very

speciﬁc trimming situations. Furthermore, the four-sided

nature of

v

is implied. Consequently, trimmed patches

with more complex topology have to be decomposed by an

additional preprocessing step. There is no control over the

quality of the parameterization due to the mapping

t.

Ele-

ments may become very distorted depending on the posi-

tion of the trimming curves

Such a situation occurs for

a triangular-shaped

v,

see Fig. 9. Since the parameteri-

zation is completely independent of the basis functions of

the trimmed parameter space, the double mapping method

works well for Bézier patches. An integration issue arises

as soon as B-spline patches are considered. The problem is

depict in Fig.39. Note that the parameter lines of the geom-

etry representation propagate through the elements deﬁned

by the mapped regular parameterization. Thus, integration

(55)

Sv

(s,t)=(1−t)C

(f(s)) + tC

(g(s))

of the regular elements is not performed over a

C∞

-continu-

ous region. In order to get a proper distribution of quad-

rature points, the elements must be subdivided along the

non-smooth edges. The speciﬁcation of such regions is not

straightforward. To conclude, the double mapping method

is a simple solution for (Bézier) patches which had been

trimmed during the design process, at least for ones that

can be represented by a regular patch.

5.2.2 Reconstruction byCoons Patches

A natural extension of the previous method is to deﬁne

the mapping

t

to a trimmed parameter space by means of

Coons patches. In contrast to the ruled surface interpola-

tion (55),

t

takes four boundary curves into account. Ran-

drianarivony [237, 238] developed such an approach, which

has been applied to wavelet Galerkin BEM in collabora-

tion with Harbrecht [113]. Although they do not focus on

isogeometric analysis per se, most of their techniques can

be directly utilized: (i) decomposition of

v

into several

four-sided patches, (ii) identiﬁcation if

t

is a diﬀeomor-

phism,18 and (iii) construction of matching parameteriza-

tions of adjacent patches.

The ﬁrst step of the decomposition procedure is to sub-

stitute the trimming curves

of each patch by a linear

approximation

The vertices

are located along

as illustrated in Fig.40a.

should be as coarse as pos-

sible since the number of vertices determines the number

of patches that decompose

v.

As initial approximation,

the endpoints of the trimming curves may be used. How-

ever,

has to be ﬁne enough to resolve the topology of the

trimmed patch, e.g.,lines of exterior loops may not inter-

sect ones of interior loops. In order to get a single polygon

representing

v,

interior loops are connected to exterior

Av≡SAv(s, t)

Fig. 38 Double mapping scheme to ﬁt a regular tensor product sur-

face to a trimmed patch. The ﬁrst mapping

t

speciﬁes the transfor-

mation to the valid area

v

of the trimmed parameter space, while

the geometric mapping is denoted by

.

The trimming curves

(̃u),i={1, 2}

are illustrated by thick lines

Cp−m

Fig. 39 Double mapping method for a B-spline patch. The dotted

lines indicate parameter curves that are not

C∞

-continuity within

v

18 A diﬀeomorphism is a

C∞

mapping with a

C∞

inverse.

B.Marussig, T.J.R.Hughes

1 3

loops by so-called double edges. The vertices of the result-

ing polygon are basis for the decomposition of

v

into a set

of quadrilaterals

.

Therefore, it is important that the total

number of vertices

is even. In the next step, the straight

boundary curves of

are replaced by the complementary

portions of

An example of a decomposition is shown

in Fig.40b. Due to this procedure, the following problems

may arise. The most obvious one is that the curved bound-

ary may intersect an internal edge. In addition, sharp cor-

ners become degenerated points if the corresponding

smoother than

C0.

As a result, no diﬀeomorphism for this

region can be found [237]. Finally, it is not assured that a

Coons patch interpolation is regular. Such problems arise

particularly in case of non-convex domains. An example of

a non-regular Coons patch where the parametric lines of the

surface overspill is shown in Fig.41. A remedy to the men-

tioned issues is local reﬁnement of

or the aﬀected

i.

The detection of the ﬁrst two problems is straightforward,

but determination of a Coons patch’s regularity requires a

more detailed discussion.

The following identiﬁcation procedure assumes that the

Coons patch interpolation (35) is planar and described by

boundary curves given in Bézier form, i.e.,

(56)

i(u)=

∑

k=0

Bk,p(u)ci

k,i=1, 2, u∈[0, 1]

The interpolation function

is also described as a Bézier

polynomial

For the determination of the regularity, the factors

𝜇,𝜏,

and

𝛼

are speciﬁed by

The indices

and

are deﬁned as above and the factors in

Eqs. (60) and (61) are determined by

with ̂

=(c

−c

)

Finally, the constant

𝛽

deﬁned such that

(57)

j(v)=

∑

k=0

Bk,p(v)cj

k,j=3, 4, v∈[0, 1]

(58)

f1(s)=

∑

k=0

Bk,p(s)𝜓k=1−f0(s),s∈[0, 1]

(59)

𝜇

:=max

�

(s)

:s∈[0, 1]

(60)

𝜏

:=min

{

𝜏ij

k𝓁

}

,k,𝓁=0, …,p

(61)

𝛼

:=max

{

𝛼

,𝛼

2}.

(62)

𝜏

k𝓁:=p2det

[

k+1−ci

k,cj

𝓁+1−cj

𝓁

(63)

𝛼

1:=max

k=0,…,p

𝜇



k−c3



+𝜓k̂c+



0−c1

,

(64)

𝛼

2:=max

k=0,…,p{

𝜇

||(

k−c

)

+𝜓k̂c+

(

0−c

)||},

(65)

p‖

‖

𝜓k

(

𝓁+1−c2

𝓁+c1

𝓁−c1

𝓁+1

)

(

𝓁+1−c1

𝓁

)‖

‖

‖≤

𝛽

R1R2

(a)

(b)

Approximation of the trimmed domain

Decomposition

Fig. 40 Decomposing of a trimmed domain

v

into b regular four-

sided patches

i.

In a, the trimming curves are continuous whereas

the linear approximation is illustrated by dashed lines. Further, vertex

and

are connected by a double edge

C1(u)

C4(v)

C2(u)

C3(v)

Fig. 41 Example of a planar non-regular Coons patch where iso-

curves overlap

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

for all

𝓁=0, …,p−1

and

k=0, …,p.

Based on these def-

initions, the Coons patch mapping is regular if

These are only suﬃcient conditions. If they are not ful-

ﬁlled a subdivision procedure is employed. Therefore,

another suﬃcient condition is derived. Assuming that a

Coons patch is represented as Bézier surface with control

points

i,j,

the Jacobian can also be described as a Bézier

function of degree

with the control points

with

m,n=0, …,2p

and the coeﬃcients

where

If the coeﬃcients

m,n

have the same sign the Coons patch

mapping is a diﬀeomorphism. In case of unequal signs, the

patch is adaptively subdivided and for each sub-patch the

coeﬃcients

Jsub

m,n

are computed. The procedure stops as soon

as the signs of

Jsub

m,n

do not change within every sub-patch.

(66)

p‖

‖

𝜓k

(

𝓁+1−c4

𝓁+c3

𝓁−c3

𝓁+1

)

(

𝓁+1−c3

𝓁

)‖

‖

‖≤

𝛽

(67)

2𝛼𝛽 +𝛼2<𝜏 and 𝜏>0.

(68)

m,n=∑

i+k=m

j+𝓁=n

C(i,j,k,𝓁)

(

)(

)

(

i+k

)

(

)(

𝓁

)

(

j+𝓁

(69)

(i,j,k,𝓁):=𝓁

[

pD(i−1, j,k,𝓁−1)

+(1−i

p)D(i,j,k,𝓁−1)]

+(1−𝓁

p)[i

pD(i−1, j,k,𝓁

)

(

1−i

D(i,j,k,𝓁)

]

(70)

(i,j,k,𝓁):=p2det

[

i+1,j−cC

i,j,cC

k,𝓁+1−cC

k,𝓁

The overall Coons patch is not regular, if it consists of sub-

patches with diﬀerent signs. It is emphasized that the sub-

division is only performed to determine the regularity of

the mapping. The corresponding proofs and more informa-

tion can be found in [113, 237].

In case of multiple patches, a matching parameterization

between adjacent surfaces is sought. The connectivity is

described by a graph. During the decomposition procedure,

the polygon vertices of the adjacent patches

and

are

computed so that

where

and x

𝓁

are the vertices along the common edge

and



denotes the geometrical mapping (23). If a vertex x

is added to obtain an even-numbered approximation

corresponding x

𝓁

needs to be added in the adjacent patch.

Thus, odd faces can be converted to even ones only in pairs

as illustrated in Fig.42. It should be noted that the inserted

vertices propagate through faces which are even already.

In order to minimize the aﬀected faces, the shortest path

connecting two faces is computed by the application of

Dijkstra’s algorithm [72] to the connectivity graph. In addi-

tion to matching vertices, trimming curves

Ct∈[a,b]

are

parameterized by means of the chord length of the corre-

sponding intersection curve in model space. The trimming

curve segment of quadrilaterals

i

is initially deﬁned by

(t⋅a

+(1−t)b

)∈[a

]⊂[a,b]

The new represen-

tation ̂

is given by

with

𝜙i

denoting the inverse of the length function

Hence, the images of the trimming curve

of adjacent

patches match at the same parametric values, i.e.,the same

chord length. This procedure is applied before the Coons

patch construction. For details on the computation of the

reparameterization the interested reader is referred to [238].

In contrast to the approach described in the previous sub-

section, this reconstruction scheme addresses the partition-

ing of trimmed domains into several four-sided regions, the

regularity of these regions, and the connection of adjacent

patches. Yet, some aspects are unresolved. For instance,

the compatibility condition (71) implies that trimming

curves coincide which is usually not the case as discussed

in Sect.3. Since the trimming curves do not describe the

same curve in model space, the chord length parameteri-

zation may lead to diverse results. Thus, a robust imple-

mentation is required that treats the tolerances involved. As

(71)

i

(xi

j

𝓁

)

(72)

:=̄

◦𝜙

,𝜙

(

𝜆

i)−1,

(73)

𝜆

i(t):=�t

‖



◦̄

dt (𝜃)

‖

d𝜃

Fig. 42 Converting a pair of odd faces to even ones. Circles indicate

the initial vertices of the faces and crosses mark those vertices that

had been added to obtain faces with an even number of vertices

B.Marussig, T.J.R.Hughes

1 3

pointed out in Randrianarivony’s thesis [237], the recon-

struction algorithm might fail if the inaccuracies of CAGD

models are too large. User interaction is required to ﬁnd an

adequate tolerance threshold. IGES data has been used as

exchange format in the related publications, which makes

the tolerance treatment even more delicate since the topol-

ogy of the objects has to be reconstructed as elaborated in

Sect.4. Finally, the problem of integrating over

C∞

-contin-

uous regions in case of B-spline patches is not addressed.

5.2.3 Reconstruction byIsocurves

Recently, Urick [298] presented a reconstruction approach

based on isocurves (25). In contrast to the previous

schemes, the trimmed patch is replaced by a new param-

eterization and a new set of control points. The overall

procedure consists of several steps including (i) topology

detection, (ii) parameter space analysis and determination

of knot vector superset, (iii) reparameterization of trimmed

parameter spaces, (iv) computation of corresponding con-

trol points, and (v) the treatment of multiple trimming

curves.

In order to identify the topology of the trimmed domain

v,

characteristic points

of the trimming curves

are

determined. The points considered are summarized in

Table 3. They represent characteristic points commonly

used in surface-to-surface intersection schemes (i.e.,types

0, 2, and 3) [221], along with an additional point previously

not utilized (type 1). The main purpose of this classiﬁcation

is to detect portions

𝛾

of a trimming curve that are asso-

ciated to either the

-direction, i.e.,

𝛾u

or the

-direction,

i.e.,

𝛾v

of the parameter space. With this in mind, the most

signiﬁcant points are those of types 0 and 1, because they

indicate a possible transition from

𝛾u

𝛾v

Each

𝛾

together

with its opposing edge of the parameter space speciﬁes a

four-sided regions

.

An example of a segmentation of

v

is illustrated in Fig.43. Note that not every characteristic

point of type 1, i.e.,

x1,

yields a new portion. Hence, the

sequence of characteristic points has to be examined rather

than the classiﬁcation of individual points.

Once all reconstruction regions



are detected, the

parameterization of adjacent patches is aligned. The fol-

lowing knot cross-seeding procedure establishes a one-

to-one relation of points along the intersection curve

C(s)

in model space and the related trimming curves of the

surfaces

Si(u,v).

Firstly,

C(s)

is reﬁned so that it is deﬁned

by Bézier segments, i.e., the multiplicity of all knots is

equal to the curve’s degree. Furthermore, each

Si(u,v)

subjected to knot insertion in the

-direction and

-direction

at their characteristic points

𝛾u

and

𝛾v

portions, respec-

tively. Thus, the knot vectors of the surfaces incorporate

the locations of all

. In the next step, the knot information

is exchanged across the diﬀerent objects involved in order

to deﬁne a knot vector superset. During this process, new

Bézier segments are introduced to

C(s).

In particular, knots

are added at the parametric values

̂s

that correspond to the

knots of

Si(u,v).

The values

̂s

are determined by minimiz-

ing the distance of

C(s)

to isocurves

Ciso(u)

and

Ciso(v)

each

Si(u,v)

and the ﬁxed parameters of these isocurves are

determined by the knot values of the surfaces. As a result,

the reﬁned intersection curve and its superset knot vector

reﬂect the knots and topological characteristics of itself,

the related surfaces, and their trimming curves. Finally, this

information is passed on to the surfaces, i.e., all

Si(u,v)

are reﬁned at the interior knots of

C(s)

including the knots

of the adjacent

Sj(u,v),j≠i.

This is done by minimizing

the distance between the points of

C(s)

and

Si(u,v)

. The

exchange of knot data is necessary in order to guarantee

that patches are connected along their intersection after the

reconstruction.

Reparameterization is required to obtain a conforming

basis for each four-sided region

.

Suppose



is related to

𝛾v

-portion19 of a trimming curve, then it is described by a

set of isocurves

{

iso

(u)}K

k=1

along ﬁxed parameter values

Table 3 Characteristic points

of a trimming curve

Types Description

0 End points, kinks, and cusps

1 Slope relative to

is 1 or −1

2 Slope relative to

is 0

3 Slope relative to

∞

x0x1

x1x1

γu

γv

Fig. 43 Determination of the trimming curve portions

𝛾u

and

𝛾v

which are associated to the parametric direction

and

respectively.

The boundary of the related regions

i

within the trimmed domain

are indicated by dashed lines. Characteristic points

are marked by

crosses which correspond to the sloped of the curve. The point type is

denoted by the related superscript

19 Regions related to a

𝛾u

-portion are treated in an analogous manner.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

siso

Note that the values

siso

correspond to the parameteri-

zation of the intersection curve

C(s)

rather than the v-direc-

tion of the trimmed surface. Thus, the reparameterized

region



will be conformal with

C(s)

and the reparameter-

ized counterpart of an adjacent surface will be conformal

as well. Due to the cross-seeding process,

𝛾v

is linked to

at least one Bézier segment of

C(s).

The positions

siso

the isocurves

Ciso

)

are determined by the endpoints and

Greville abscissae of these Bézier segments. The corre-

sponding parametric values in the

-direction are labeled

̂uk

and represent the locations where the distance of

Ciso

)

minimal to

C(s).

Knot insertion at

̂uk

is applied to extract

the part of

Ciso

)

that is within the domain of interest,

i.e.,the current four-sided region

.

The values

̂uk

vary for

each

Ciso

)

since they are distributed close to the trimming

curves. In other words, parameter intervals of the isocurves

within



do not match in general. To overcome this issue,

all

Ciso

)

are reparameterized to be speciﬁed by the same

basis.

The simplest way to establish the reparameterization

is to use a linear coordinate transformation so that all iso-

curves are deﬁned over a common range, combined with

a subsequent accumulation of the shifted interior knots of

each knot vector. However, this technique yields a large

number of basis function since interior knots of isocurves

with diﬀerent initial intervals do not coincide after the

transformation. Furthermore, the size of the resulting knot

spans may vary excessively because the alteration of shifted

interior knots can be arbitrarily small.

Hence, a nonlinear reparameterization is preferred. A set

of functions

{

(̃u)}

k=1

is sought that maps the correspond-

ing

Ciso

(u)∈[a

to a common range

Ciso

(̃u)∈[c,d

]

without modifying interior knots. These functions can be

represented by univariate splines with scalar coeﬃcients

i.e.,

Note that the linear coordinate transformation is a special

case of this formulation where the degree of the function

is set to

q=1.

The composite of an isocurve

Ciso

)

and its

fk(̃u)

determines the reparameterization

The degree of the resulting curve is given by

̃p=pq

where

refers to the original degree of the isocurve. If the long-

est isocurve is taken as target parameterization, it can be

adjusted by using conventional degree elevation to

̃p.

The

other curves are subjected to a nonlinear reparameteriza-

tion based on their

fk(̃u).

For technical details on nonlin-

ear reparameterization of curves the interested reader is

referred to the textbook [230].

(74)

fk(̃u)=

I−1

∑

i=0

Bi,q(̃u)ck

i,q>1, k=1, …,K

(75)

Ciso

(̃u)=C

iso

(̃u)),̃u∈[c,d]

It remains to ﬁnd a way to coordinate the individual

fk(̃u)

to obtain a global reparameterization for the whole

reconstruction domain



that yields a new valid tensor

product parameter space

.

The key idea is to represent the

global transformation as a spline surface

f(̃u,s).

This sur-

face includes all

fk(̃u)

as isocurves, i.e., f(̃u,s

iso

)=f

(̃u)

The bivariate reparameterization is given by

with the degree

of the intersection curve segment and a

grid of scalar control coeﬃcients

ci,j.

If the degree in the

-direction of the trimmed surface varies from

ps,

the degree

of the segment may be adjusted by means of degree eleva-

tion. Equation(76) can be represented as a non-parametric

(76)

f(̃u,s)=

I−1

∑

i=0

J−1

∑

j=0

Bi,̃p(̃u)Bj,ps(s)ci,j

(a)

(b)

Reparameterization surface

Isocurves

siso

0siso

1siso

2siso

s˜u

f(˜u, s)

310

˜u

Fig. 44 Reparameterization of a trimmed parameter space of a bicu-

bic Bézier patch: a surface

f(̃u,s)

that reparameterizes the trimmed

parameter space

to a regular one

̃u

indicated by the vertical and

plane grid, respectively. Lines on the surface mark the associated iso-

curve along

siso

bProﬁle of the isocurves

fk(̃u)

B.Marussig, T.J.R.Hughes

1 3

surface by linking the coeﬃcients

ci,j

to their Greville

abscissae (22).

The corresponding control points are deﬁned as

The coeﬃcients

ci,j

can be deﬁned by the user as long as the

following restrictions are met:

(i) The function

f(̃u,s)

must be strictly monotonic in

the

̃u

-direction so that intervals do not overlap.

(ii) The spline surface must employ the same target knot

vector

𝛯

in the

̃u

-direction.

(iii) Each knot value

of the initial knot vectors must

be mapped to a distinct ̃

∈

𝛯

for all isocurves,

i.e.,

=f(̃u

iso

),k=1, …,K

An illustration of such a bivariate reparameterization func-

tion

f(̃u,s)

is provided in Fig.44a and the corresponding

isocurves are shown in Fig.44b. It should be noted that the

parameter space spanned by the

̃u

-axis and

-axis is deﬁned

by straight parameter lines only, in contrast to the original

basis spanned by the

-axis and

-axis. It is emphasized that

(77)

i,j=

⎡

⎢

⎣

(̃ui+1+̃ui+2+⋯+̃ui+̃p)∕ ̃p

(sj+1+sj+2+⋯+sj+ps)∕ps

i,j

⎤

⎥

⎦

the graphs in Fig. 44b intersect at the common interior

knots

̃u

i=ui=

{

3}.

The ﬁnal step of the reconstruction scheme is to deter-

mine the control points of the reparameterized regions

.

Therefore, we recall the speciﬁcation of the control points

of isocurves

Ciso

)

as a weighted combination of surface

control points

ci,j

Isocurves have been introduced at the Greville abscissae

of the Bézier segments along the reconstruction boundary

𝛾v

Hence, the number of isocurves is equal to the number

of unknowns, i.e.,

J=K,

and the control points

ci,j

can be

computed based on the known isocurve control points

inverting the system of equations (78). The control points

along the boundary

𝛾v

are already known beforehand,

i.e.,the control points of

C(s),

and do not need to be com-

puted explicitly.

It is quite astonishing that the procedure described

remains the same when multiple trimming curves are

involved. Instead of assessing the topology of all trimming

curves at once, the trimming curve or more precisely each

𝛾

is processed successively and independently of each other.

In fact, it does not matter if the portions

𝛾

originate from

one or several trimming curves. After each reparameteri-

zation the parameter space is updated and the next region

is addressed. The iterative evolution of the reconstructed

regions



is displayed in Fig.45. To be clear, the regions

0

and

1

shown in Fig.45b and the regions

2

and

3

displayed in Fig.45c are not constructed at the same time.

The ﬁnal outcome of the reconstruction is a new set of

patches with aligned parameter spaces that share the control

point of their intersection curves. Thus, the reconstructed

object is watertight. It is emphasized that this holds true

even for non-manifolds. These beneﬁts come at the price

of an alteration of the initial geometry and an increase of

the degrees of freedom. Since the concept has been pre-

sented just recently, there are several open research topics

to explore. For instance, an estimation of the geometrical

error introduced with respect to the degree of the reparam-

eterization function and the number of isocurves would be

of great interest. This could be the basis for an optimization

procedure for the deﬁnition of the reparameterization func-

tion. Another topic might be the quality of the resulting ele-

ments in model space, especially at the transitions from

𝛾u

𝛾v

regions.

We close the discussion of the isocurve reconstruction

approach with some application remarks. Firstly, the con-

cept can also be applied locally. In this paper it is focused

on the tensor product case where reﬁnement propagates

(78)

J−1

∑

j=0

Bj,q(siso

k)ci,j,k=1, …,K

γu

γv

γu

γv

γu

(a) (b)

Topology detection and

characteristic points

First two reconstruction

regions

Further reconstruction

regions after basis update

Reparameterized pa-

rameter spaces

Fig. 45 Successive evolution of the isocurve reconstruction of a

quadratic trimmed patch using a quadratic reconstruction function

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

through the whole domain for the sake of simplicity. A

locally reconstructed parameter space may be represented

by any local basis like T-splines, hierarchical B-splines, or

LR-splines. Secondly, the degree of the resulting patches

may become large, depending on the degree of the repa-

rameterization function. It might be beneﬁcial to apply

a degree reduction technique after the reconstruction, but

this introduces additional approximation errors. Finally, the

intersection curves should have a good parameterization

since they play an essential role during the reconstruction.

Therefore, it might be advisable to reparameterize the inter-

section curve, e.g.,by its chord length, at the beginning of

the overall procedure.

5.2.4 Reconstruction byTriangular Bézier Splines

Another recent attempt has been proposed by Xia and

Qian[314]. They employ triangular Bézier patches(36) to

convert trimmed models to watertight representations. The

convergence behavior of these splines has been assessed

by these authors and co-workers in [315]. The conversion

involves the following steps: (i) subdivision of all surfaces

into tensor product Bézier patches, (ii) exact representation

of non-trimmed patches by two Bézier triangles, (iii) knot

cross-seeding between adjacent patches, (iv) approximation

of the region along the trimming curve using Bézier trian-

gles, and (v) substitution of the resulting control points of

the approximate trimming curve by corresponding control

points of the intersection curve in model space.

The ﬁrst step can be easily accomplished by means of

knot insertion. The second one is performed following

Goldman and Filip [100]. In particular, a non-trimmed ten-

sor product patch with control points

c□

m,n

can be converted

to two triangular Bézier patches by

where

i+j+k=p+q

and

(

𝛼

𝛽

)

are binomial coeﬃcients

deﬁned as

Equation(79) yields the control points

c△

i,j,k

of one triangu-

lar patch using

c□

m,n

:0 ⩽m⩽p;0 ⩽n⩽q

The control

points of the other triangular patch are obtained by revers-

ing the order of the original control points,

(79)

△

i,j,k=1

(p+q

∑

m=0

min {j,q−i+m}

∑

n=max {0,j−p+m}

c□

m,n

(

)(

)

(

p+q−i−j

p+n−m−j

)

(80)

(

𝛼

𝛽

)

:=𝛼!

(𝛼−𝛽)!𝛽!

i.e.,

c□

p−m,q−n

:0 ⩽m⩽p;0 ⩽n⩽q

The degree of the

resulting patches is determined by

p+q.

It is emphasized

that this transformation does not introduce an approxima-

tion error.

Next, the relationship of adjacent patches along the

intersection is established. This is done similar to the knot

cross-seeding procedure described in the previous subsec-

tion. Hence, we adopt this term here as well. Figure 46

summarizes the basic procedure. Firstly, the intersection

curve

C(̃x)

in model space and the trimming curves

(̃u

)

1(˜u)

2(˜

C(˜x)

X◦Ct

1(˜u)

X◦Ct

2(˜s)

C(˜x)

X◦Ct

1(˜u)

X◦Ct

2(˜s)

(a)

(b)

(c)

Initial setting

Knot cross-seeding

Final triangulation

Fig. 46 Generation of conforming triangulations along the intersec-

tion of two patches: a deﬁnition of Bézier segments of the intersec-

tion curve

C(̃x)

and the trimming curves

(̃u

)

and

(̃s)

b Closest

point projection to ﬁnd corresponding points on the other curves. c

Addition of Bézier segments due to the exchanged points and speci-

ﬁcation of associated triangular regions. Segments are marked by

crosses, black points, and white points based on their origin. The oﬀ-

set between

C(̃x)

and the images



◦C

(̃u

)

and



◦C

(̃s

)

shall empha-

size that they do not coincide in model space. In b, arrows indicate

the projections performed

B.Marussig, T.J.R.Hughes

1 3

and

(̃s

)

are subdivided into Bézier segments at their knot

values and intersections with the trimmed parameter space.

Then, the endpoints of these segments are projected to the

other curves and the corresponding parametric values are

computed. In other words, the trimming curve are reﬁned

based on the knot information of the other trimming curve

and the intersection curve in model space. Consequently,

the resulting Bézier segments of a curve have correspond-

ing counterparts in the other curves. However, the distinct

segments do not coincide in model space. At this point, the

purpose of the knot cross-seeding is to obtain an aligned

triangulation along the intersection. Triangular patches are

speciﬁed within each trimmed surface so that one of their

boundaries represents a Bézier segment.

The construction of these triangular Bézier patches

which are arbitrarily located within the trimmed surface is

performed accordingly to Lasser [182]. In general, a Bézier

triangle T of degree

̃p

in a tensor product basis of a sur-

face

R(u,v)

of degrees

and

yields a triangular patch

S△(r,s,t)

of degree

̃p(p+q).

Xia and Qian[314] focus on

the linear case, i.e.,

̃p=1,

meaning that the trimming curve

is approximated by linear segments. Thus, the composition

S△(r,s,t)=R(T(r,s,t))

is given by

where

𝐈

,p+q

refer to the Bernstein basis (37) of the triangu-

lar patch,

𝐈

is an index triplet

(i,j,k),

and

|𝐈|=i+j+k.

remains to determine the corresponding control points

c△

𝐈.

Therefore, the construction points

Rp,q

0,0

𝐈

u,v

𝐈

)

of the blos-

som of

R(u,v)

are needed. Regarding the

-direction, for

instance, these points are recursively deﬁned by

where the control points of the surface

R(u,v)

are used as

initial values R

0,0

i,j.

The construction in the

-direction is

performed in an analogous manner. The superscripts a and

b denote distinct steps of the recurrence relation in the

-direction and

-direction, respectively. Note that Eq. (82)

is an adaptation of the de Casteljau algorithm that allows

employing new parameter values

𝐈

in every iteration.

Likewise, the index tuples are given by

𝐈v=𝐈v

1+⋯+𝐈v

and

𝐈u=𝐈u

1+⋯+𝐈u

𝛼

with

𝛼

referring to the related super-

script, i.e., a or

a−1.

Using this recursion, the control

points of the triangular patch

S△(r,s,t)

are obtained by

(81)

S△

(r,s,t)=

∑

𝐈

|=p+q

B𝐈,p+q(r,s,t)c

△

𝐈

(82)

a,b

i,j(ua

𝐈u,vb

𝐈v)=

(

1−u𝐈u

)

Ra−1,b

i,j(ua−1

𝐈u,vb

𝐈v

)

+u𝐈uaRa−1,b

i+1,j

(ua−1

𝐈u,vb

𝐈v),

(83)

△

𝐈=

∑

𝐈u+𝐈v=𝐈

(

p+q

𝐈

)

Rp,q

0,0(up

𝐈u,vq

𝐈v)

with

where each of these index triples consists of

and further

This procedure is applied to cover the valid area of

every trimmed Bézier surface by a set of triangular Bézier

patches. Each Bézier segment of a trimming curve is repre-

sented by an edge of such a triangular patch. Finally, those

edges are replaced by the corresponding Bézier segment of

the intersection curve in model space. Since this substitu-

tion is carried out for all patches, a seamless join between

adjacent surfaces is obtained. The approximation error

introduced may be controlled by reﬁning the patches along

the trimming curves.

It is worth noting that Xia and Qian [314] use their

reconstruction procedure as an intermediate step in order to

set up a volumetric parameterization of B-Rep models. The

watertight triangular Bézier surface representation provides

the starting point for a construction of volumetric Bézier

tetrahedra.

(84)

𝐈u=

𝐈

1+⋯+

𝐈

and 𝐈

𝐈

1+⋯+

𝐈

(85)

𝐈

𝛼

𝛽=

(

i𝛼

𝛽,j𝛼

𝛽,k𝛼

𝛽

)

with i𝛼

𝛽,j𝛼

𝛽,k𝛼

𝛽∈{0, 1}

(86)



𝐈u





𝐈u



+⋯+



𝐈u



(87)



𝐈v





𝐈v



+⋯+



𝐈v



(88)

𝐈

|=|

𝐈

u|+|

𝐈

v|=p+q.

(a) (b)

Fig. 47 Illustration of the most common valid cutting patterns of a

single knot span. The actual element type is determined by the direc-

tion of the trimming curve. The crosses highlight the intersection

points of the trimming curve with the element

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

5.3 Local Approaches

Local techniques employ a completely diﬀerent philosophy

than their global counterparts, that is, the geometry model

is not modiﬁed but the analysis has to deal with all deﬁ-

ciencies of trimmed solid models. Thereby, the trimmed

parameter space is used as background parameterization

for the simulation while the trimming curves determine the

domain of interest. Hence, the analyzed area is embedded

in a regular grid of knot spans which consists of interior,

exterior, and cut elements. The following subsections dis-

cuss (i) the detection of theses distinct element sets, (ii) the

integration of cut elements, (iii) the treatment of multipatch

geometries, and (iv) the stability of a trimmed basis.

5.3.1 Element Detection

Before the actual analysis can be performed, the various

element types and their position within the trimmed basis

need to be identiﬁed. Interior elements are deﬁned by non-

zero knot spans that are completely within the valid domain

and can be treated as in regular isogeometric analysis.

Exterior ones, on the other hand, can be ignored since their

entire support is outside of the domain of interest. Cut ele-

ments require special attention. One of the advantages of

local approaches is that the cutting patterns of these ele-

ments are relatively simple compared to the complexity of

the overall trimming curve. Figure47 depicts topological

cases of cut elements that are usually considered, e.g.,[160,

161, 199, 202, 260]. It should be pointed out that other

cases may exits as well, e.g.,an element containing more

than one trimming curve. These situations occur especially

when the basis is very coarse. In general, the complexity of

a trimming curve’s topology within an element decreases

as the ﬁneness of the parameter space increases. Hence,

(local) reﬁnement is a common way to resolve invalid cut-

ting patterns. This reﬁnement may be performed for inte-

gration purpose only. Thus, no new knots are introduced,

but the invalid element is subdivided in several valid inte-

gration regions. An alternative is to extend the valid cutting

patterns as suggested by Wang etal.[307] or the construc-

tion of tailored integration rules for each cut element as

proposed by Nagy and Benson [211]. However, the beneﬁt

of a restricted number of trimming cases facilitates the sub-

sequent integration process.

Considering the situations shown in Fig. 47, cut ele-

ments have either 3, 4, or 5 edges, where one of them is

a portion of the trimming curve. In this paper, we adapt

the notation of Schmidt etal. [260] and label the type of

cut elements by their number of edges. Interior and exte-

rior elements are referred to as elements of type 1 and −1,

respectively. Figure 48 illustrates a trimmed parameter

space and the related element types. Note that the knot

span in the upper right corner is an example of an invalid

case since smooth element edges are usually assumed for

the numerical integration. Possible strategies to deal with

this element include subdivision into several integration

regions, treatment as a type 4 element with two curved

edges, or knot reﬁnement through the kink of the curve. In

general, kinks and straight trimming curves that are aligned

with parameter lines are usual suspects for introducing spe-

cial cases.

The portions of the trimming curve which are within

each element have to be determined in order to get a proper

description of cut knot spans. In particular, the intersec-

tions of the parameter grid with the trimming curve

Ct(̃u)

are required, together with the corresponding parametric

values

̃u+

The overall element detection task consists of

the classiﬁcation of knot span with respect to the trimming

-1 -1 -1 33-1

344513

415454

343-1 3 ?

Fig. 48 Trimmed parameter space and corresponding element types:

1 labels untrimmed knot spans whereas −1 denotes knot spans which

are outside of the computational domain. In case of trimmed knot

spans the element type indicates the number of interior edges, i.e.,3,

4, or 5. The question mark indicates a special case. The intersections

of the trimming curve with the parameter lines are highlighted by

crosses

rout

|d|

(a)

−

(b)

Fig. 49 Detection of cut elements according to Kim etal.[160, 161]:

a ﬁrst assessment based on the inscribed and circumscribed circles

of the element and if necessary, b further comparison of the signed

distance of the element corners

B.Marussig, T.J.R.Hughes

1 3

curves and the determination of trimming curve portions

related to cut elements.

Kim etal.[160, 161] and Schmidt etal.[260] presented

two diﬀerent algorithmic solutions for the element detec-

tion problem. The procedure suggested by the former can

be summarized by:

(i) Compute the minimal signed distance

di,j

of the

center of each non-zero knot span to all trimming

curves to separate interior and exterior elements.

(ii) Identify cut elements by comparing

di,j

with the

radii

rin

i,j

and

rout

i,j

of the inscribed and circumscribed

circles of the element. If

i,j⩽

di,j

<rout

i,j

the signed

distance of the element corner nodes to the trimming

curve are computed and compared as well.

(iii) Compute intersection points for each element cut by

the trimming curve.

Both cases of the second step which specify cut elements

are illustrated in Fig.49. In Fig. 49a, the distance of the

element’s center to the trimming curve is smaller than the

radius of the inscribed circle, whereas in Fig.49b, the cut

element is identiﬁed since the signed distances of its corner

nodes are positive and negative.

On the other hand, Schmidt etal.[260] recommend to

label all non-zero knot spans as interior elements as start-

ing point for the following procedure:

(i) Determine all intersection points of the trimming

curve and the grid produced by the tensor product

of the knot vectors and sort them in a nondecreasing

order with respect to the related values

̃u+

(ii) Assign the element type of cut elements based on the

position of successive intersection points.

(iii) Detect exterior elements based on their position rela-

tive to the cut elements.

It should be noted that successive intersection points mark

start and end of trimming curve portions within an element.

For the last task, the exterior nodes of cut elements can be

used to initialize an incremental algorithm setting adjacent

elements which are not labeled as cut elements to −1 [286].

Figure50 illustrates the situation described. The nodes of

these exterior elements are then used to determine further

exterior elements. The procedure stops as soon as there are

any adjacent elements of type 1 left.

On this basis, it may be concluded that the present

approaches tackle the problem from two diﬀerent direc-

tions. The former puts the element type in the focus with

a subsequent calculation of the intersections of cut ele-

ments with the trimming curve, whereas the latter com-

putes all intersections and derives the type of the elements

afterwards. In general, the most important property of

an element detection algorithm is its robustness since it

hardly eﬀects the overall eﬃciency of the simulation. Both

approaches require a robust implementation of the curve-

to-grid intersection computation. The Bézier clipping tech-

nique described in Sect.3.5.2 could be used. The treatment

of invalid cutting patterns applies also to both algorithms

and depends on the subsequent integration procedure. The

main diﬀerence between the approaches is that the for-

mer relies on a robust implementation of a point projec-

tion algorithm in order to determine the signed distance of

a point to the trimming curve, while the latter requires a

robust technique for detecting exterior elements based on

the intersection information. There is perhaps no objective

way to prefer one scheme to the other, but we would like

to share our experiences with both schemes by mentioning

some possible pitfalls in the following paragraphs.

The ﬁrst point addresses the detection of exterior ele-

ments based on their relative position to cut elements. The

starting point is illustrated in Fig.50. Elements of type 1

are changed to type −1 if they are adjacent to the exterior

nodes of cut and exterior elements. The search for adjacent

elements can be performed in an incremental manner as

it is done in “ﬂood ﬁll” algorithms, which are commonly

used in graphics software [286]. It should, however, be

emphasized that intersected grid nodes need special atten-

tion since the search for adjacent elements might propagate

at these points to the valid domain. Furthermore, a proper

treatment of zero knot spans is required.

The other note is concerned with the calculation of the

signed distance to trimming curves. The shortest distance

of a test point

to a trimming curve

Ct(̃u)

is deﬁned as

443

Intersection Intersected gridnodeExterior node

Fig. 50 Starting point for the separation of interior and exterior ele-

ments following the procedure of Schmidt etal. [260]. White knot

spans are not classiﬁed yet. The arrow indicates the direction of the

trimming curve

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

A Newton–Raphson iteration scheme is employed to deter-

mine the parametric values

̃u∗

[192, 230]. The correspond-

ing signs indicates on which side of

Ct(̃u)

the point

located. It can be computed by the cross product of the tan-

gent vector

t=(tu,tv)⊺

at the projected point x

(̃u

∗)

and the direction vector

d=(du,dv)⊺

from

xt.

In two

dimensions, the analog to the cross product is given by the

determinant, hence the sign is calculated by

In case of non-smooth trimming curves, more than one

minimum might exist as pointed out in [287]. From a prac-

tical point of view this is only relevant if these minima

have diﬀerent signs. Such cases appear in the vicinity of

sharp corners as shown in Fig.51. The correct sign can be

determined by the projected distance calculated by the dot

product

If the angle

𝛼=0,

i.e.,

e1=e2,

the curvatures of the curves

may be compared

where

𝜅1

denotes the curvature of the curve that ends at the

corner.

5.3.2 Integration

Various strategies to integrate cut elements

̃𝜏 ∈v

are out-

lined in this subsection. In general, numerical integration is

performed using conventional Gauss–Legendre quadrature.

The integral over each

̃𝜏

(89)

=min

{‖

‖

(̃u)−x

t‖

‖}

‖

(̃u

∗

)−x

t‖

‖.

(90)

s=tudv−tvdu.

(91)

ei=v⋅ti,i=1, 2,

(92)

=sign

{

min

{

2}}.

(93)

{

1if 𝜅1>𝜅

−1otherwise

is substituted by a weighted sum of point evaluations

The related quadrature points

and the correspond-

ing weights

are speciﬁed in the reference element

̀𝜏 = [−1, 1]2.

The coordinates for the pointwise evalu-

ation of the integrand f are determined by the inte-

gral transformation

r

(𝜉,𝜂):ℝ

↦ℝ

from

̀𝜏

̃𝜏

and the geometrical mapping

(u,v):ℝ2

↦

ℝ3,

i.e.,

yg=(ug,vg)=(r(𝜉g,𝜂g))

as illustrated in

Fig. 52. In order to take these mappings into account,

the quadrature weights

are multiplied by the Gram’s

determinant

given by the Jacobian matrix

𝐉

of the geometrical map-

ping. The Jacobian determinant of

r

is evaluated with respect to the reference coordinates

𝜉g

and

𝜂g

of the integration point

yg.

The deﬁnition of the inte-

gral transformation

r

and the related

J̀𝜏

is straightforward

in case of regular elements. However, the domain of cut

(94)

̃𝜏 =

∫

̃𝜏

f(x)dΩ̃𝜏

(95)

̃𝜏 ≈

∑

g=1

(

)

G(ug,vg)J̀𝜏 (𝜉g,𝜂g)wg

(96)

(u,v):=

√

det

(

𝐉⊺



(u,v)𝐉(u,v)

(97)

̀𝜏 (𝜉

,𝜂

)=det

(

𝐉(𝜉

,𝜂

)

1(˜u)

(˜

)

(˜

)

e1=e2

(a) (b)

Fig. 51 Determination of the correct sign in case of multiple trim-

ming curves

(̃u

)

which describe an acute angle. The area which

returns ambiguous signs is indicated in gray. (Courtesy of Jakob W.

Steidl)

120

0.5

Fig. 52 Distribution of quadrature points indicated by black points

over a regular patch

B.Marussig, T.J.R.Hughes

1 3

elements is more complex and thus, the deﬁnition of

r

more involved.

Numerical integration of cut elements is required in var-

ious simulation schemes. Besides the analysis of trimmed

geometries, it is also needed in the context of ﬁctitious

domain methods and the extended ﬁnite element method.

There are numerous approaches and a vast body of litera-

ture proposing strategies to specify a proper integration

̃𝜏 .

It may be performed so that the trimming curve is

taken into account in an exact or approximate manner. In

this paper, the main focus is on techniques presented in the

context of trimmed NURBS objects. They can be broadly

classiﬁed into the following categories: (i) local reconstruc-

tion, (ii) approximate treatment, and (iii) exact treatment.

The former is performed in model space, while the others

operate in the parameter space in general.

5.3.2.1 Local Reconstruction Schmidt et al. [260] sug-

gested to perform the adjustment of the integration region

by a local reconstruction of the trimmed patch. Therefore,

each cut element in the model space

𝜏

is remodeled as a

single reconstruction patch

̂𝜏 .

In particular,

̂𝜏

is speciﬁed as

a Bézier patch with degrees

̂p⩾p

and

̂q⩾q,

where

and

refer to the degrees of the origin surface. The key idea is

to represent

̂𝜏

in terms of the original control points of

𝜏.

transformation matrix

𝐓

provides the relationship between

the control points of the original and reconstructed patch. It

can be computed by means of a least squares approximation

where the system of equations is given by

This equation consists of

representing the (unknown) control points of

̂𝜏 ,

the (known)

control points of

𝜏,

and a set of sampling points

interpo-

lated by both patches. The total number of control points

involved is determined by the number of non-zero basis

functions, i.e.,

̂n=(̂p+1)( ̂q+1)

and

n=(p+1)(q+1).

The number of sampling points, i.e.,

l+1,

is arbitrary but

larger than

̂n.

The basis function values of

̂𝜏

are pro-

vided by

and the corresponding values of

𝜏

are given by

(98)

𝐁

𝐏=𝐒=𝐁𝐏.

(99)

𝐏

⎡

⎢

⎣

⋮

ĉn−1

⎤

⎥

⎦

,𝐏=

⎡

⎢

⎣

⋮

cn−1

⎤

⎥

⎦

, and 𝐒=

⎡

⎢

⎣

⋮

⎤

⎥

⎦

(100)

𝐁

⎡

⎢

⎣

B0,̂p(̂u

B0,̂q(̂v

0)⋯

B̂p,̂p(̂u

B̂q,̂q(̂v

⋮⋱⋮

0,̂p

(̂us

)

0,̂q

(̂vs

)⋯

̂p,̂p

(̂us

)

̂q,̂q

(̂vs

)

⎤

⎥

⎦

Note that a correlation between the parametric values has to

be established so that

i=𝜏(us

i,vs

i)= ̂𝜏 (̂us

i,̂vs

i),i=0, …,l.

The system of equations (98) is overdetermined consisting

l+1

equations and

̂n

unknowns. In general, it cannot be

solved exactly, but a good approximation of the solution

can be found by forming

which yields the deﬁnition of the transformation matrix

and the relation between the control points

As a result, numerical integration can be performed

based on the regular reconstruction patch

̂𝜏

and the sim-

ple mapping of the regular integration can be applied. The

values obtained are distributed to the control points of the

original patch using the transformation matrix

𝐓.

For more

details, the interested reader is referred to [260].

This procedure can be directly applied to cut elements

of types 3 and 4 as speciﬁed in Sect.5.3.1. Type 5 elements

may be subdivided into two four-sided regions. A drawback

of the local reconstruction scheme is that it introduces an

additional approximation error since the system of equa-

tions (102) cannot be solved exactly. Moreover, the stability

of the computation of the transformation matrix might be

aﬀected if only a very small region of a cut element needs

to be reconstructed.

5.3.2.2 Approximated Trimming Curve The following

two schemes approximate the trimming curve

in order to

deﬁne proper integration points within the parameter space.

(101)

𝐁

⎡

⎢

⎣

B0,p(us

0)B0,q(vs

0)⋯Bp,p(us

0)Bq,q(vs

⋮⋱⋮

0,p

(us

0,q

(vs

)⋯B

p,p

(us

q,q

(vs

)

⎤

⎥

⎦

(102)

𝐁⊺

𝐁

𝐏

𝐁

⊺

𝐒

𝐁

⊺

𝐁𝐏

(103)

𝐓

(

𝐁⊺

𝐁

)−1

𝐁⊺𝐁

(104)

𝐏=𝐓𝐏.

≈

˜ρ

Fig. 53 Approximation of the cut element by a polytope

̃𝜌 .

The con-

trol points of the trimming curve are marked by circles

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

One uses a linear approximation of

to set up a tailored

integration rule, whereas the other applies an adaptive sub-

division to approximate the domain of the cut element.

A tailored integration rule can be established for each

cut element

̃𝜏

as proposed in [211, 305, 306]. The control

polygon20 of

is used to represent

̃𝜏

by a polytope

̃𝜌

shown in Fig.53. The integral over

̃𝜌

can be reduced to a

sum of line integrals over the edges of

̃𝜌

using Lasserre’s

theorems [183]. Therefore, the integration domain

Ω̃𝜌

has

to be convex. Thus, a preprocessing step is applied to rep-

resent non-convex regions by a combination of convex

ones. The line integrals provide reference solutions for the

right-hand side of a set of moment-ﬁtting equations given

They are used to computed a tailored quadrature rule,

i.e.,points

yi=(ui,vi)⊺

and weights

wi,

for all functions

of the desired functions space, e.g.,monomials up to a cer-

tain degree. The goal is to ﬁnd the lowest number of

that the Eq. (105) are satisﬁed up to a certain tolerance, for

each

̃𝜏

or rather

̃𝜌 .

The algorithm proposed in [211] starts

with an initial set of

and successively eliminates one

superﬂuous point after another. In each step, the reduced

set of points is used to solve the system of equations (105)

in the least squares sense.

The construction of a tailored quadrature has two main

beneﬁts: (i) the number of integration points per

̃𝜏

is opti-

mized and (ii) all cutting patterns are covered by a sin-

gle technique, including cases which had been labeled as

invalid in Sect.5.3.1. Of course, this comes at the price of

a more involved preprocessing phase since every cut ele-

ment has to be treated individually. Furthermore, an error

is introduced due to the approximation of

̃𝜏

̃𝜌 .

This error

can be reduced by reﬁnement of the trimming curve since

the control polygon converges to it. Still, the smooth higher

degree representation is removed by a linear one. Finally,

it should be pointed out that the reduction of integra-

tion points does not take the smoothness of the basis into

account, as it is done in case of optimized quadrature rules

for regular splines, see e.g.,[118].

A completely diﬀerent strategy for the integration of cut

elements in based on adaptive subdivision. Researchers

who developed the ﬁnite cell method applied this technique

to trimmed geometries [107, 239, 253, 254]. The basic idea

is to use a composed Gauss quadrature that aggregates inte-

gration points along the trimming curve. A cut element

̃𝜏

decomposed into axis-aligned sub-cells

̃𝜏 ⊞

based on a tree-

structure, i.e., a quadtree in two dimensions. Starting from

the initial cut element, each sub-cell is further subdivided

into equally spaced sub-cells if it contains the trimming

curve as displayed in Fig.54a. This recursive procedure is

performed up to a user-deﬁned maximal depth. Following

the spirit of ﬁctitious domain methods the integral

over

the complete element is deﬁned as

The factors

̃𝜏

⊞

and

I−

̃𝜏

⊞

are the integrals over the valid

domain

v

and the complementary exterior domain

−

respectively. Integration points in the interior of

v

are

(105)

∑

i=1

fj(ui,vi)wi=∫Ω

̃𝜌

fj(u,v)dΩ̃𝜌 ,j=1, …,n

(106)

∑

i=1

̃𝜏 ⊞

(𝛼v)+

∑

j=1

I−

̃𝜏 ⊞

(𝛼−)

(a)

(b)

Conventional

Reduced

Fig. 54 Sub-cell structure of a single cut element: a conventional

approach with quadrature points distributed within the valid (black

points) and exterior (green points) domain and b reduced approach

integrating over the whole element (orange points) and the valid

domain (black points). The sub-cells are indicated by dashed lines.

(Color ﬁgure online)

20 To be precise, it is suggested to use the control points of the inter-

section curves in the model space and to apply point inversion to

determine their location in the parameter space. However, there is no

particular reason why the control polygon of the trimming curve can-

not be used directly.

B.Marussig, T.J.R.Hughes

1 3

multiplied by

𝛼v=1,

whereas exterior integration points

are multiplied by a value that is almost zero,

e.g.,

𝛼−

=10−14

as suggested in [253]. The integration pro-

cedure can be improved with respect to the number of

quadrature points by

where

I−

̃𝜏 (𝛼−)

represents the integral over the whole cut ele-

ment without taken the trimming curve into account. The

integration over the valid domain is performed as before by

the composite quadrature, yet with another weighting fac-

tor, i.e.,

(𝛼v−𝛼−).

Such an improved sub-cell integration is

illustrated in Fig.54b.

The key features of this approach are its simplicity and

generality. The deﬁnition of integral transformation

r

and its Jacobian is straightforward, due to the axis-aligned

shape of the sub-cells. Again, all cutting patterns (includ-

ing invalid ones) can be addressed with a single algorithm.

Moreover, the algorithm can be easily extended to higher

dimensions. The downside is that the trimming curve is

only approximated. Consequently, the integration region

is not represented exactly and an additional approxima-

tion error is introduced. In fact, the accuracy of the integral

ceases at a certain threshold [173, 175]. This threshold may

be improved by the subdivision depth, but a ﬁne resolution

of sub-cells results in a vast number of quadrature points.

Further, reﬁned sub-cells do not converge to the trimming

curve in contrast to the previous approach. One of the great

successes of the ﬁnite cell method was the demonstrated

ability to achieve higher rates of convergence for higher-

order elements and splines, and even exponential rates in

the context of the p-method.

5.3.2.3 Exact Trimming Curve The following techniques

focus on deﬁning a proper mapping

r

from the reference

element

̀𝜏

to the cut element

̃𝜏 ∈v

so that the trimming

curve is exactly represented. Depending on the cutting pat-

tern,

̃𝜏

may be represented by a disjointed set of integration

regions

̃𝜏 ⊡

such that

In contrast to the sub-cells of the previous scheme, the

regions

̃𝜏 ⊡

are not aligned with the axes of the parameter

space and at least one

̃𝜏 ⊡

has an edge which is described by

the portion of the trimming curve

within

̃𝜏 .

There are various ways to specify

r.

Ruled surface (26)

and Coons patch (35) interpolation may be applied, where

the portion of the trimming curve within

̃𝜏

is considered

(107)

c=I−

̃𝜏 (𝛼−)+

∑

i=1

̃𝜏 ⊞

(𝛼v−𝛼−)

(108)

̃𝜏

⋃

i=1

̃𝜏 ⊡

for the construction [199, 307]. An example of local ruled

surface mappings for various element types are shown in

Fig.55a. These methods may be interpreted as local coun-

terparts of the global reconstruction schemes presented in

Sects.5.2.1 and 5.2.2. It is worth noting that approaches

based on the blending function method [95, 173, 174] can

be included into this category, because this method also

employs a transﬁnite mapping [103]. In the nested Jaco-

bian approach, integral transformation is also deﬁned by a

local NURBS surface combined with a nested subdivision

[38, 227]. Thus,

r

consists of the local surface mapping

and an additional transformation to the subregion. A cor-

responding distribution of quadrature points is shown in

Fig.55b. In contrast to both previous references, i.e.,[199,

307], type 5 elements are not decomposed into three trian-

gular ones, but a bisection of the knot span is performed.

Recently, an adaptive Gaussian integration procedure has

been proposed [37]. This variation of the nested Jacobian

approach deﬁnes the local surface parameterization within

the reference space instead of the trimmed parameter space

as illustrated in Fig. 55c. Therefore, the trimming curve

is transformed to the reference space by scaling and rota-

tion. The integration points and their weights are adapted

by scaling the

𝜂

-direction such that the points are located

within the region described by the transformed trimming

curve. The motivation for the adaptive Gaussian integration

procedure is to treat the various cutting patterns by a single

approach.

Another very common strategy is to adopt the inte-

gration scheme developed in the context of the NURBS-

enhanced ﬁnite element method [147, 148, 160, 161, 272,

273]. Using this scheme, every cut element is subdivided

into a set of triangles. Those triangles that only consist of

straight edges are subjected to conventional integration

rules for linear triangles. The other triangles are treated by

a series of mappings that take the curved edge into account

Figure55d displays the components of this series. Suppose

the corner nodes of the triangle in the trimmed parameter

space are labeled

x▵

where the beginning and the end

of the trimming curve portion within the considered trian-

gle are denoted by

x▵

and

x▵

respectively. The transforma-

tion

s,t:x(s,t)

↦

x(u,v)

describes the mapping of a linear

three node element

In order to address the curved edge, the trimming curve is

transformed into the

s,t

-coordinate system by

(109)

r:=s,t(̃u,𝜁(𝜉,𝜂(𝜉,𝜂))).

(110)

s,t

:=x(u,v)=tx

▵

+(1−s−t)x

▵

+sx

▵

(111)

𝜙

(̃u)=

−1

s,t

⋅C

(̃u)

[

xΔ

−xΔ

xΔ

−xΔ

−1

(

Ct(̃u)−xΔ

2).

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

The next mapping

̃u,𝜁:x(̃u,𝜁)

↦

x(s,t)

converts the tri-

angular domain into a rectangular one which possesses

straight edges only. It is given by

Finally, the transformation

𝜉,𝜂:x(𝜉,𝜂)

↦

x(̃u,𝜁)

of the

reference space

[−1, 1]2

to the rectangular region is per-

formed by

(112)



̃u,𝜁:=

{

s=𝜙s(̃u)(1−𝜁),

t=𝜙

(̃u)(1−𝜁)+𝜁

(113)



𝜉,𝜂:=

{

̃u=𝜉

2(̃ue−̃ub)+ 1

2(̃ue+̃ub

)

𝜁=𝜂

where

̃ub

and

̃ue

are the parametric values of the beginning

and the end of the trimming curve portion within the trian-

gle, i.e.,

(̃u

)=x

▵

and

(̃u

)=x

▵

The Jacobian deter-

minant of the overall mapping

r

is determined by

with

(114)

J̀𝜏

=det

(

𝐉

s,t)

det

(

𝐉

̃u,𝜁)

det

(

𝐉

𝜉,𝜂),

(115)

𝐉

s,t=

[

3−u

3−v

uΔ

−uΔ

vΔ

−vΔ

2],

(116)

𝐉

̃u,𝜍=

[

𝜕𝜙s(̃u)

𝜕̃u(1−𝜍)𝜕𝜙t(̃u)

𝜕̃u(1−𝜍)

−𝜙s(̃u)1−𝜙t(̃u)

Rotation

and

scaling

of Ct

˜u

˜ub˜ue

φ(˜u)=X−1

s,t ·Ct(˜u)

Xs,t

X˜

u,ζ

Xξ,η

(c)

(d)

(a) Local ruled surface mapping (b) Nested Jacobian approach

Adaptive Gaussianintegration procedure

NURBS enhanced FEM integration

Fig. 55 Distribution of quadrature points due to various approaches which represent the trimming curve exactly. Dashed lines indicate a subdi-

vision of a cut element into integration regions

B.Marussig, T.J.R.Hughes

1 3

The coeﬃcients

u▵

and

v▵

refer to the coordinates of the

corner nodes

x▵

and the derivative of the transformed trim-

ming curve is calculated by

The various integration schemes are summarized in

Fig.55. Their common and most essential feature is that

the integration region is exactly represented. The main dif-

ference between the strategies is the partitioning of a cut

element

̃𝜏

into integration regions

̃𝜏 ⊡

In fact, the series of

mappings (109) shown in Fig.55d yields the same distri-

bution of quadrature points over a triangular element as a

ruled surface interpolation (26) illustrated in Fig. 55a, if

the trimming curve is a B-spline curve. In case of NURBS

curves, on the other hand, diﬀerent distributions are

obtained. These two cases are compared in Fig.56.

In general, it seems that good results can be obtained

with either of these concepts, especially for moderate

degrees. However, it has been demonstrated that the prop-

erties of coordinate mappings and the corresponding place-

ment of interior nodes is crucial for the convergence behav-

ior of conventional higher degree (

p>3

) ﬁnite elements

[216]. With this in mind, additional research might be use-

ful to assess the quality of the mapping schemes presented

with respect to their performance for higher degree.

5.3.3 Multipatch Geometries

A robust treatment of multiple patches is the most chal-

lenging part of analyzing trimmed geometries. While

single patch analysis can exploit the beneﬁts of trimmed

(117)

𝐉

𝜉,𝜂=

[

2(̃ue−̃ub)0

2].

(118)

𝜕𝜙

(̃u)

𝜕̃u=

[

xΔ

3−xΔ

2xΔ

1−xΔ

]

−1

(

𝜕C

(̃u)

𝜕̃u

representations, all the deﬁciencies elaborated in Sect. 3

surface as soon as solid models described by several

trimmed patches are considered. In case of ﬁnite element

methods, the main ingredients to cope with this situation

are: (i) a weak coupling formulation and (ii) a robust pro-

cedure linking the degrees of freedom of adjacent patches

to each other. This subsection closes with some general

statements regarding the continuity of trimmed multipatch

geometries.

5.3.3.1 Weak Coupling Weak enforcement of constraints

is a common problem in computational mechanics, see e.g.,

[138, 166, 312] and the references cited therein. Such tech-

niques are required in several contexts like mesh-independ-

ent imposing of essential boundary conditions and domain

decomposition methods. The latter covers a versatile ﬁeld

of applications including contact problems, parallelization,

and coupling of subdomains described by diﬀerent physics

or non-conforming discretizations. Numerous approaches

have been developed and each one possesses diﬀerent bene-

ﬁts and disadvantages. The most popular schemes are based

on Lagrange multipliers [12], the penalty method [13, 139],

or Nitsche’s method [112, 215]. These methods are sepa-

rated by a ﬁne line: the penalty method may be viewed as

an approximation of the Lagrange multiplier method [139].

Furthermore, the Nitsche method may be referred to as a

consistent penalty method [252]. In addition, the close rela-

tionship of the Nitsche method to the stabilized Lagrange

multiplier method [17, 18] has been outlined in [288].

The use of Lagrange multipliers is a very general way

to enforce constraints to a system of equations which is

applicable to all kinds of problems. Following Huerta

et al. [138] the main disadvantages are: (i) the system of

equations increases due to the Lagrange multipliers which

are incorporated as additional degrees of freedom, (ii) the

resulting system is not positive deﬁnite, and (iii) the intro-

duction of a separate ﬁeld for the Lagrange multipliers

yields a saddle-point problem which must satisfy a stability

condition known as the inf–sup or Babuška–Brezzi condi-

tion. In order to fulﬁll the last point, the interpolation ﬁelds

of the unknowns and the Lagrange multipliers must be

coordinated, which is not a trivial task, examples of choices

for the interpolation functions can be found in [139].

The penalty method is easy to implement and avoids the

problems mentioned above. However, uniform convergence

to the solution can only be guaranteed if the applied penalty

parameter increases as the mesh is reﬁned [7]. This is cru-

cial since the system matrix becomes ill-conditioned when

the penalty parameter gets large. Usually, a ﬁxed parameter

value is chosen and as a result, the quality of the approxi-

mation cannot be improved below a certain error.

Nitsche’s method introduces a penalty term too, but

it is considerably smaller than in the penalty method [80,

(a) (b)

B-spline curve NURBScurve

Fig. 56 Comparison of the distribution of Gauss points within a cut

element of type 3 based on the NURBS enhanced FEM mapping

(circles) and ruled surface parameterization (crosses). The trimming

curve is described by either a a B-spline curve or b a NURBS curve

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

254]. According to Huerta etal.[138] the only problem of

Nitsche’s method is that it is not as general as the other pro-

cedure. Thus, it is not straightforward to provide an imple-

mentation for some problem types.

These techniques have been successfully applied to vari-

ous isogeometric analysis applications, e.g. [23, 24, 39,

80, 81, 163, 213]. A comparison of the three schemes can

be found in [7]. Also in the context of coupling trimmed

patches, the Lagrange multiplier method [307], the penalty

method [37, 38], and the Nitsche method [107, 164, 254]

have been successfully applied already. In none of these

publications, the surface type motivated the choice of the

weak coupling strategy. In other words, trimmed patches do

not introduce additional arguments to prefer one approach

to the other. Nevertheless, it is important to emphasize

again that trimming curves of adjacent patches do not

describe the same curve in model space. Thus, adjacent

patches have non-conforming parameterizations as well as

gaps and overlaps along their intersection.

5.3.3.2 Linking of Degrees of Freedom Breitenberger

etal. [38] presented a procedure that is able to deal with

complex design models and it has been discussed in more

detail in the related thesis [37]. In addition to a weak cou-

pling formulation, trimming curves of adjacent patches

are connected by so-called edge elements that contain the

required topological information. To be precise, the trim-

ming curves are treated by a master–slave concept where

points of the slave curve are mapped to the master curve.

These points are the intersections of the slave trimming

curve with the grid lines of its own parameter space. The

mapping to the master curve is performed in model space by

means of a point inversion algorithm [192, 230]. The algo-

rithm is usually carried out by a Newton–Raphson scheme

and provides the closest projection of a point to a curve as

shown in Fig.57. In addition, the related parametric val-

ues of the master curve are provided by the point inversion

scheme. The accumulation of these values and the original

grid intersections of the master curve deﬁne a set of inte-

gration regions. Within each region, quadrature points are

speciﬁed and the corresponding points of the slave curve

can again be computed by the point inversion algorithm. To

sum up, the relation of two related trimming curves is estab-

lish by an iterative procedure in model space which com-

putes the shortest distance of a point deﬁned by one curve to

the other curve. This is indeed the same concept as for the

knot cross-seeding procedures presented in Sect.5.2 in the

context of global approaches. In theory, this is a straightfor-

ward task, but its robust implementation is challenging and

crucial for the overall performance of an analysis.

In the following we would like to highlight the impor-

tance of a robust association of adjacent patches by show-

ing an example presented in [37, 38]. The basic setting of

the problem is shown in Fig.58a. This benchmark for geo-

metric nonlinear shell analysis describes a cantilever that

is subjected to an end moment. If the maximal moment

Mmax

is applied, the cantilever deforms to a closed circular

ring. Figure 58b illustrates the numerical solution of this

problem for various parameterizations. Note the diﬀerent

level of complexity along the edges of adjacent patches. It

clearly demonstrates the vast diversity of situations that my

occur in case of multipatch geometries even if they repre-

sent the same geometry.

Another important aspect studied by this example is the

inﬂuence of the gap and overlap size between patches. Con-

sider the geometrical discretization illustrated in Fig. 59.

A gap–overlap function f is introduced to specify a user-

deﬁned inaccuracy along the curved intersection. Posi-

tive and negative values of f represent gaps and overlaps,

respectively. The trimming curves are linked by the point

inversion algorithm as described before. The resulting ver-

tical displacements at the cantilever’s end

uTip

of represen-

tations with diﬀerent f are related to a reference solution

uref

obtained with

f=0.

The diﬀerence is calculated by

and the related results are summarized in Fig. 60. Based

on the corresponding graph, it can be concluded that small

gaps which are within CAD tolerance, i.e., 0.001 units,

barely inﬂuence the quality of the simulation. The diﬀerent

behavior of gaps and overlaps can be explained by the min-

imal distance computation: in contrast to gaps, the assign-

ment of points of the slave curve to the master curve is not

unique in case of overlaps.

(119)

uTip(f)−ure f

master

slave

Fig. 57 Closest point projections of a slave patch to a master patch.

The lines on the surfaces represent the grid of the underlying param-

eter space. The intersections of these lines with the trimming curves

are illustrated by white and black dots for the slave and master patch,

respectively. The projections themselves are indicated by arrows

B.Marussig, T.J.R.Hughes

1 3

5.3.3.3 Continuity Considerations The continuity along

the intersection of two trimmed patches is usually not higher

than

C0.

The construction of corresponding

isogeometric

spaces with optimal approximation properties is well under-

stood for conforming parameterizations [299]. Brivadis

etal.[39] showed this also for weakly imposed

condi-

tions. However, their isogeometric mortar method focuses

on regular patches and a modiﬁcation of the basis functions

at the boundary is required to obtain stability, if the same

degree is used for the primal and dual spaces. Although the

inﬂuence of non-matching interfaces is discussed as well,

the application in the context of trimmed surfaces has yet to

be investigated in more detail.

The construction of smooth isogeometric spaces for

trimmed models is an even more complicated open topic.

In fact, smooth isogeometric spaces on unstructured geom-

etries are a challenging and open problem in general [141,

296]. Locking eﬀects may occur even for regular planar

multipatch conﬁgurations [63, 150]. At this point, it should

be noted that T-splines or subdivision surfaces provide

geometric models which are globally smooth almost eve-

rywhere. Nevertheless, these representations seem to lack

x, u

−utip

wtip

M=Mmax

Initial mesh

0.2

0.4

0.6

0.8

0 5 10 15 20

End moment(xπ50/3)

Tip deﬂections

exact

M=0.7Mmax

M=0.35Mmax

E=1.2·106

ν=0,L=12

b=1,h=0.1

Mmax =2πEI/L=50π/3

Single patch Matching

patches

Non-matching

patches

Straight-trimmed

patches

Curve-trimmed

patches

Patches

with gap

Arbitrarily-trimmed

patches

(a)

(b)

Test setting

Results for various parameterizations withM=Mmax

Fig. 58 Diﬀerent geometry models for analyzing a cantilever sub-

jected to an end moment: a deﬁnition of the problem and b result-

ing solutions. In b, diﬀerent gray scales indicate the distinct patches.

Note the various complexities of the connection of adjacent patches.

(Courtesy of Breitenberger [37, 38])

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

optimal approximation properties due to the existence of

extraordinary vertices [145, 212].

5.3.4 Stabilization

A trimmed basis contains basis functions which are cut

by the trimming curve and exist only partially within the

valid area

v.

In order to clarify the problem statement,

Fig.61 illustrates a trimmed univariate basis. It should be

noted that the Greville abscissae of cut basis functions may

be located outside of

v.

In the example given, this is the

case for

B4,2.

These points cannot be used for collocation

or spline interpolation problems, despite the fact that they

are the preferred choice for setting up a stable system of

equations (see Sect.2.3). Furthermore, the support of cut

basis functions may be arbitrary small, e.g.,this would be

the case for

B4,2

as the trimming location

approaches the

knot value 2. Thus, the condition number of the resulting

system matrices can become very large. In other words, a

trimmed basis is not guaranteed to be stable.

In order to emphasize this stability issue an interpolation

problem is examined: a given function

shall be approximated by a B-spline surface

Sh.

They agree

at k interpolation points

xi,j=(̄ui,̄vj)⊺

where k represents

the total number of bivariate basis functions involved. The

further components of the corresponding system of equa-

tions are the unknown coeﬃcients

ci,j

and the bivariate

spline collocation matrix

𝐀.

The matrix is deﬁned by

(120)

f(u,v)=

√(−1.2 −u)

+(−1.2 −v)

(121)

𝐀

[i+j

⋅

J,m+n

⋅

J]=B

i,p(

̄u

j,q(

̄v

n),

1.37

L=1.37 +10.63 +

10.63

Closest points

gap

overlap

Master

Slave

gap situation

overlap situation

Closest points

Fig. 59 Geometry representation and deﬁnition of the gap–overlap

parameter f for the investigation of the eﬀect of non-watertight geom-

etries on numerical results. (Courtesy of Michael Breitenberger [37,

38])

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-6 -4 -2 0 2 4 6

Diﬀerencefor z-displacement

103

CAD tolerance (0.001)

Fig. 60 Comparison of the relative vertical displacement

related to

the gap–overlap parameter f. Gaps and overlaps are indicated by posi-

tive and negative values, respectively. The gray area of the diagram

indicates the default tolerance of the CAD software used. (Courtesy

of Breitenberger [37, 38])

000 1 2 3 444

0,2

B1,2B2,2B3,2B4,2

5,2

Fig. 61 Univariate basis trimmed at a parameter

There are basis

functions which are fully inside (green), partially inside (blue), and

completely outside (dotted) of

v.

The Greville abscissae of the con-

sidered basis functions are marked by circles. (Color ﬁgure online)

B.Marussig, T.J.R.Hughes

1 3

with

i,m=0, …,I

and

j,n=0, …,J,

where

and

are

the number of basis functions in each parametric directions.

The initial parameter space is given by an open knot vector

with a uniform discretization from −1 to 1 in both direc-

tions, i.e.,

u,v∈ [−1, 1],

and the knot span size is speciﬁed

h=0.125.

A trimming parameter

t∈[0.5, 1)

deter-

mines the square domain

v∈ [−1, t]2

considered for the

interpolation problem. The interpolation points

of cut

basis functions may have to be shifted into

v.

Exterior

basis functions that are completely outside of

v

are not

involved in the interpolation process. The quality and sta-

bility of the approximation

are speciﬁed by the relative

interpolation error measured in the

-norm

‖

𝜖rel

‖

‖L2

as well

as the condition number of the spline collocation

matrix

𝜅(𝐀).

The results are summarized in Fig.62 for vari-

ous degree with

p=q.

It can be observed that the condition number of

𝐀

considerably inﬂuenced by the trimming parameter

particular, a peak is reached as soon as

approaches a knot

value, i.e.,a support of cut basis functions becomes very

small. Furthermore, the approximation quality is aﬀected.

The peaks of the relative error

‖

𝜖rel

‖

‖L2

near knot values are

in fact disastrous. Hence, it is evident that the straightfor-

ward application of a trimmed basis negatively aﬀects the

condition number and subsequently the quality of the

approximation.

The stability aspect of local approaches for the analysis

of trimmed geometries has scarcely been considered in pre-

vious works. It is worth noting that Nitsche formulations

may incorporate parameters which take cut elements into

account, see e.g., [42, 80, 289]. A method-independent

alternative that exploit the properties of B-splines is out-

lined in Sect.6.

5.4 Summary andDiscussion

Various approaches to incorporate trimmed geometries

into an analysis have been described in this review. While

Sect. 5.1 addresses an early attempt which combines

trimmed patches with Lagrange interpolation, recent

research is the focus of the subsequent Sects. (5.2) and

(5.3). To recapitulate the ﬁndings of the current approaches:

there are two fundamentally diﬀerent philosophies to deal

with trimmed models. One seeks to resolve the deﬁciencies

of trimmed models by a reconstruction of the geometric

representation. This is performed as a preprocessing step

before the actual analysis. Since these procedures aﬀect

entire patches and their connection, they are referred to as

global approaches in this work. The other philosophy is to

accept the ﬂaws of trimmed models, implying that the anal-

ysis has to be adaptable enough to cope with them. This

capability is accomplished by treating the occurring trim-

ming situations on the knot span level. Hence, we classify

such techniques as local approaches.

Global approaches address the core of the problem and

aim to solve it at its origin. In fact, they are similar to the

remodeling schemes of CAGD outlined in Sect.3.4. They

share the same shortcomings such as an increased num-

ber of control points and the dependence on a four-sided

domain if regular tensor product surfaces are used for the

reconstruction. It can be argued that global approaches are

more related to CAGD than analysis. Consequently, their

success is also determined by the acceptance in the design

community. However, a compelling global scheme could

eventually lead to design models which can be directly

applied not only to analysis but all downstream applica-

tions, which is the holy grail of the trimming problem.

0.50.625 0.75 0.875 1

101

108

1015

1022

1029

κ(A)

Conditionnumber

0.50.625 0.75 0.875 1

10−6

100

106

1012

1018

relL2

Relative error

p=2 p=3 p=4

Fig. 62 Condition number

𝜅(𝐀)

and relative interpolation error

‖

𝜖rel

‖

‖L

of the bivariate basis for several degrees

in both parametric

directions related to the trimming parameter

The subdivision of the

horizontal axis corresponds to the knot values of the trimmed basis

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

Local approaches focus on enhancing the analysis and

thus, may seem more feasible for researchers in the ﬁeld

of computational mechanics. In fact, the majority of the

publications on isogeometric analysis of trimmed geom-

etries employ such concepts. There is a close relation to

ﬁctitious domain, or immersed, methods since the trimmed

parameter space is used as a background parameterization.

Hence, similar challenges have to be addressed: (i) detec-

tion of elements cut be the trimming curve, (ii) special

integration schemes for these elements, (iii) weak coupling

of adjacent patches, and (iv) the stability issue induced by

the trimmed basis. The main diﬀerence is that additional

eﬀort has to be made to associate the degrees of freedom

of adjacent patches, keeping in mind that their intersections

possess non-matching parameterizations, gaps, and over-

laps. These distinct tasks are clearly separated from each

other. For example, weak coupling is mandatory for ﬁnite

element methods but may be neglected if a boundary ele-

ment method is applied. Most researchers have drawn their

attention to the integration of cut elements. The application

of weak formulations has also been addressed by several

authors. On the other hand, the stability of a trimmed basis

and the robust association of adjacent patches are barely

discussed in the literature, despite the fact that the latter

task is crucial for the analysis of practical design models.

Another issue of using a trimmed basis for the analysis is

that the Greville abscissae of cut basis functions are not

guaranteed to be located inside of the domain of interest.

Consequently, an application to interpolation and colloca-

tion methods requires further considerations. However, the

modular structure of local approaches is indeed a beneﬁt

compared to global approaches which require a self-con-

tained concept which becomes more and more sophisti-

cated with its capabilities.

6 Stabilization ofaTrimmed Basis

There are two reasons for presenting a distinct section on

the stabilization of trimmed parameter spaces: ﬁrst and

foremost, we want to draw attention to this issue which has

been scarcely discussed so far, and, in addition, some of

our recent research is focused on this topic allowing a more

detailed observation of it. The general problem statement

has already been given in Sect. 5.3.4, where it has been

demonstrated that basis functions cut by a trimming curve

can yield ill-conditioned system matrices. Further, Greville

abscissae of such basis functions may be outside of the

valid domain and thus, they cannot be applied to methods

which employ these points like isogeometric collocation

[11, 258]. In order to identify the troublesome components,

we classify the basis functions of a trimmed parameter

space as stable, degenerate, or exterior. The support of the

latter is completely outside of the valid domain

v

and

hence, it can be neglected for the analysis. The distinguish-

ing feature of the other types is that the Greville abscis-

sae of stable B-splines are within

v

whereas the Greville

abscissae of degenerate ones are outside of

v.

The following stabilization scheme resolves the issues

induced by degenerate basis functions in a simple and

ﬂexible manner. The concept is referred to as extended

B-splines. Originally, these splines have been developed

by Höllig and co-workers in the context of a B-spline

based ﬁctitious domain method [124–127]. Here, the main

aspects of extended B-splines are outlined based on the

ﬁndings provided in [199, 202].

6.1 Deﬁnition ofExtended B‑splines

We start the description of extended B-splines by recalling

two fundamental properties of conventional B-spline: (i)

B-splines

Bi,p

are represented by a set of polynomial seg-

ments

s

and (ii) B-splines form a basis of a space

𝕊p,𝛯

which contains every piecewise polynomial

fp,𝛯

of degree

over a knot sequence

𝛯.

The former property is illustrated

in Fig. 63. It should be noted that each polynomial seg-

ment

s

may be extended beyond its associated knot span

With this in mind, it is straightforward to grasp the essen-

tial idea of extended B-splines, namely to re-established the

stability of a trimmed basis by substituting degenerate, and

therefore potentially unstable, B-splines by extensions of

stable ones. These extensions can be exactly represented by

the basis since they are within

𝕊p,𝛯

by deﬁnition.

The overall construction procedure of extended

B-splines is summarized in Fig. 64. Firstly, it is deter-

mined if the Greville abscissae of non-exterior B-splines

are located inside or outside of

v.

In the latter case the

basis function is labeled as degenerate and the corre-

sponding index is stored in the index-set

𝕁.

Secondly, the

polynomial segments of trimmed knot spans are replaced

by the extensions of the polynomial segments of the clos-

est non-trimmed knot span that contains stable B-splines

only. These extensions together with the polynomial seg-

ments of the non-trimmed knot spans form the extended

B0=1

2(u−1)2

1=3

−

22

B2=1

2(4−u)2

1 2 3 4

Fig. 63 Polynomial segments

s

of a B-spline. The extensions of the

segments are indicated by dashed lines

B.Marussig, T.J.R.Hughes

1 3

B-spline basis. The ﬁnal step is to represent the extended

B-splines by a linear combination of the original B-splines.

An extended B-spline is deﬁned by

(122)

i,p=Bi,p+

∑

j∈𝕁i

ei,jBj,p

where

Bi,p

is the stable B-spline that provides the extension

and

𝕁i

is the index-set of all degenerate B-splines related to

the current

i,p.

The extrapolation weights

ei,j

can be deter-

mined by solving an interpolation problem. To be precise,

the given polynomial function f of the extension over the

trimmed knot span shall be represented by means of the

basis functions

Bj,p.

The coeﬃcient

ei,i

is trivial since

i,p

must be equal to

Bi,p

within the non-trimmed knot spans,

thus

ei,i=1.

Spline interpolation as described in Sect.2.3 is not opti-

mal to compute

ei,j

because the Greville abscissae of

Bj,p

are not located within the trimmed knot span in general.

Hence, a quasi interpolation scheme is preferred which

allows an explicit computation of B-spline coeﬃcients. In

particular, the so-called de Boor–Fix or dual functional

𝜆j,p

[34, 35] is used: for any piecewise polynomial

f∈𝕊p,𝛯,

with

(123)

J−1

∑

j=0

𝜆j,p(f)Bj,p

(124)

𝜆

j,p(f)= 1

∑

k=0

(−1)k𝜓(p−k)

j,p(𝜇j)f(k)(𝜇j)

B0B1B2B3

111 2 3 444

(a)

(b)

(c)

(d)

111 2 3 444

¯u0/∈A

v⇒={0}

111 2 3 444

1Be

2Be

111 2 3 444

Fig. 64 Basic procedure to get from a conventional to d extended

B-splines: b determination of degenerate B-splines and substitution

of trimmed polynomial segments by c extensions of non-trimmed

ones

0.25

−0.75

1.0

−0.75

2.25

−3.0

1.0

−3.0

4.0

−

1.5

0.5

0.5−1.52

−

0.5

−1.5

Fig. 65 The construction of bivariate extrapolation weights

ei,j

for

a biquadratic basis. Stable B-splines are marked by black and green

circles. The shown values of

ei,j

are related to the degenerate basis

function marked by the blue circle in the upper right corner of the

parameter space. B-splines of the closest non-trimmed knot span are

indicated by green circles. (Color ﬁgure online)

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

The evaluation point

𝜇j

can be chosen arbitrarily within

[uj,uj+p+1].

Substituting f of Eq. (124) by

s

yields the

extrapolation weights

(125)

𝜓

j,p(u)=

∏

m=1

(

u−uj+m

(126)

i,j=1

∑

k=0

(−1)k𝜓(p−k)

j,p(𝜇j)



s(k)

i(𝜇j)

When the polynomials

𝜓j,p

and

s

are expressed in power

basis form

expression (126) simpliﬁes to

The interested reader is referred to [202] for details

on the conversion to power basis form and further details

regarding the evaluation of the dual functional. In case of

a uniform knot vector, a simpliﬁed formula can be derived

which solely relies on the indices of the B-splines involved,

see e.g.,[124].

Bivariate extrapolation weights are simply obtained

by the tensor product of their univariate counterparts

calculated for each parametric direction as illustrated

inFig.65. Note that the degenerate B-spline is distributed

(p+1)(q+1)

stable ones.

6.2 Properties ofExtended B‑splines

Extended B-splines inherit most essential properties of

conventional B-splines [124, 125, 127]. They are linearly

independent and polynomial precision is guaranteed. Thus,

they form a basis for a spline space. Each knot span has

exactly

p+1

non-vanishing basis functions which span the

space of all polynomials of degree

⩽p

over

v.

Further-

more, approximation estimates have the same convergence

order as conventional B-splines. Extended B-splines have

local support in the sense that only B-splines near the trim-

ming curve are subjected to the extension procedure. The

actual size of the aﬀected region depends on the ﬁneness

of the parameter space, the degree of its basis functions,

and the number of degenerate

Bj,p

related to the stable

Bi,p.

The latter is given by the cardinality of the corresponding

index-set

#𝕁i.

Figure 66 illustrates various examples of

extended B-splines. The basis function shown in Fig.66a is

in fact a conventional B-spline since it is far away from the

trimming curve.

However, there are also some diﬀerences. It is impor-

tant to note that the extrapolation weights may be nega-

tive, hence the evaluation of extended B-splines may lead

to negative values. Conventional B-splines, on the other

hand, are strictly non-negative. This property is exploited

in some contact formulations[295] and structural optimi-

zation [210], for instance. In such cases, the application

(127)

𝜓

j,p(u)=

∑

k=0

𝛽kukand



i(u)=

∑

k=0

𝛼kuk

(128)

i,j=1

∑

k=0

(−1)k(p−k)!𝛽p−kk!𝛼k

Fig. 66 Bivariate extended B-splines

with various cardinali-

ties of the index-set

𝕁i

which indicates the number of related degen-

erate B-splines. Note that a is in fact a conventional B-spline,

i.e.,

p≡B

i,p

since

𝕁i

is empty

B.Marussig, T.J.R.Hughes

1 3

of extended B-splines requires further considerations. The

main diﬀerence in favor of extended B-splines is the stabil-

ity of the corresponding basis. The condition number of a

system is independent of the location of the trimming curve

due to the substitution of B-splines with small support.

Another beneﬁt is that all Greville abscissae are located

within

v

by construction.

6.3 Assembling

Extended B-splines can be applied to an analysis in a very

convenient manner. Suppose we have a linear system of n

equations, one for each stable B-spline, set up by all basis

functions m which are at least partially inside

v.

This

yields

with

m>n.

Upon this point, only conventional B-splines

have been used to compute the system matrix

𝐊.

In other

words,

𝐊

is set up as usual but equations related to degen-

erate B-splines are neglected. In order to obtain a square

matrix an extension matrix

𝐄∈ℝm×n

is introduced [124].

This sparse matrix

𝐄

contains all extrapolation weights

ei,j

including the trivial ones, i.e.,

ei,i=1.

The transformation

of the original to the stable extended B-spline basis is per-

formed by multiplying the extension matrix to the system

matrix. The resulting stable system

is subsequently solved and the obtained solution

𝐮

cor-

responds to the extended B-splines of the unknown ﬁeld.

In case of multi-patch geometries, the extrapolation

weights

ei,j

of each patch have to be assembled to

𝐄

with

respect to the global degrees of freedom. The applica-

tion of the extension operator is particularly convenient, if

extended B-splines are added to an existing code.

6.4 Application toNURBS

The stabilization described is tailored to B-spline func-

tions where it is exploited that the extensions of any poly-

nomial segment

s

can be exactly represented by a linear

combination of basis functions of the trimmed knot span.

In case of NURBS this property is not guaranteed due

to the local inﬂuence of the weights. In order to apply

extended B-splines to a trimmed NURBS basis, two dif-

ferent approaches may be used: (i) conversion of the

CAGD model to a B-spline representation or (ii)applica-

tion of an independent ﬁeld approximation [199–201].

The beneﬁt of the latter is that it allows performing the

analysis based on the original NURBS model without any

geometrical approximations. The key idea of independ-

ent ﬁeld approximation is to use diﬀerent basis functions

(129)

𝐊𝐮 =𝐟where 𝐮,𝐟∈ℝnand 𝐊∈ℝn×m,

(130)

𝐊e

𝐮

𝐟

with

𝐊

𝐊𝐄

𝐊

e∈

ℝ

n×n,

for the representation of the geometry and the approxima-

tions of the physical ﬁelds. Hence, conventional B-splines

can be used for the discretization of the ﬁeld variables over

NURBS patches. This allows the straightforward applica-

tion of extended B-splines. In addition, the combination of

NURBS for the geometry description and B-splines for the

approximations has been shown to be more eﬃcient [199]

and does not lead to a loss of accuracy [184, 201, 299].

There is one caveat: independent ﬁelds are inconsistent

with the isoparametric concept in mechanics and can upset

the precise representation of constant strain states and rigid

body motions [139].

6.5 Assessment ofStability

In order to assess the approximation quality and stability of

extended B-splines the same interpolation problem as in

Sect.5.3.4 is considered. Again, the relative interpolation

0.50.625 0.75 0.875 1

101

108

1015

1022

1029

κ(A)

Condition number

0.50.625 0.75 0.875 1

10−6

100

106

1012

1018

relL2

Relative error

p=2 p=3 p=4

Fig. 67 Spline interpolation problem with extended B-splines for

several degrees

The condition number

𝜅(𝐀)

and the relative inter-

polation error

‖

𝜖rel

‖

‖L

are related to the trimming parameter

The

labels of the horizontal axis indicate knots of the trimmed basis

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

error measured in the

-norm

‖

𝜖rel

‖

and the condition

number of the spline collocation matrix

𝜅(𝐀)

are examined.

The results are summarized in Fig.67.

Comparing Figs. 62 and  67 shows the signiﬁcant

improvement of extended B-splines. If extended B-splines

are used,

𝜅(𝐀)

hardly changes and is independent of the

trimming parameter

In other words, the extended B-spline

basis is stable. Consequently, the approximation qual-

ity is signiﬁcantly improved. In Fig. 67, the reduction of

the approximation accuracy occurs only due to the reduc-

tion of the degrees of freedom

i.e.,number of extended

B-spline, as the trimming parameter

t→0.5.

6.6 Summary andDiscussion

The concept of extended B-splines substitutes unstable

basis functions by extensions of stable ones. It is estab-

lished in a very ﬂexible manner and requires only the pres-

ence of a suﬃcient number of stable basis functions. In

general, this requirement is non-restrictive and can be ful-

ﬁlled by reﬁnement of the basis. Still, it may be an issue if

the design object contains very small ﬁllets. Only B-splines

close to the trimming curve are aﬀected by the stabiliza-

tion procedure. The number of B-splines depends on the

distance of the trimming curve to the knot span which pro-

vides the stable B-splines. This correlates with the degree

of the basis function since the size of its support extends

over

p+1

knot spans.

7 Final Remarks andConclusions

The present work accumulates several topics related to the

treatment of trimmed models and the interoperability prob-

lem between CAD and downstream applications in general.

It is apparent that trimming is a fundamental technique

for geometric design. Most importantly, it enables the

computation of intersections between free-form surfaces.

However, intersection curves cannot be determined exactly,

which leads to various problems. As a result, an intersec-

tion is usually approximated by several independent curves,

one in model space and one in the parameter space of each

surface involved. Their images in model space do not coin-

cide and there is no link between these curves. The result-

ing gaps and overlaps between the surfaces yield robustness

issues due to a lack of exact topological consistency. These

problems are still unresolved, despite the fact that they have

been the focus of an enormous amount of research.

Since the robustness issues of trimmed models are par-

ticularly crucial for downstream applications, the exchange

of CAD data is examined as well. Neutral exchange stand-

ards seem to be the most comprehensive strategy, but it is

important to note that all translations lead to loss of infor-

mation. Moreover, the capabilities of the various exchange

formats are not equivalent. It is demonstrated that STEP is

superior to IGES. STEP should be preferred in general and

especially if the topology of a model needs to be extracted.

In the context of analysis, the current approaches can be

divided into two diﬀerent philosophies. On the one hand,

global approaches aim to ﬁx the problems of trimmed mod-

els before the simulation by a reconstruction of the geo-

metric representation. Using local approaches, on the other

hand, trimmed models are directly employed, but the analy-

sis has to be enhanced in order to deal with all the ﬂaws of

the geometric representation. It may be argued that the for-

mer addresses the issue from a CAD point of view, whereas

the latter utilizes an analysis perspective. The fact that the

problem can be tackled by these diverse directions empha-

sizes the central role of trimmed models for the integration

of design and analysis.

The main conclusions of the present review can be sum-

marized as follows:

– Trimming seems to be a simple and benign procedure at

ﬁrst glance, but its consequences are profound.

– Robustness issues are the price for the ﬂexibility of

trimmed models.

– Flaws of trimmed models are usually hidden from the

user, but surface as soon as they are applied to a down-

stream application.

– To overcome these issues is a crucial aspect regarding

the integration of design and analysis.

– The success of CAD data exchange depends on the

quality of the design model and the capability of the

exchange format.

– There is no canonical way to deal with trimmed models,

neither in analysis nor in design, at least so far.

It is hoped that this review provides a helpful introduction

to the topic and an impetus for further research activities.

There are indeed several open issues worth exploring: opti-

mization of the reparameterization of the global approach

B.Marussig, T.J.R.Hughes

1 3

based on isocurves, assessment of the aﬀect of gaps on the

analysis in case of local schemes, and application to isoge-

ometric collocation, just to name a few. In general, the step

from academic examples to practical multipatch models

is perhaps the most challenging task. Robust algorithms

should be able to take the tolerances of a design model into

account. On the other hand, it may be an unrealistic aim to

ﬁnd a solution that can deal with every possible trimmed

geometry. Similar to the quality of conventional meshes,

an isogeometric simulation is eﬀected by the quality of the

design model. Hence, the speciﬁcation of distinct prop-

erties that classify a design model to be analysis-suitable

are needed so that a designer can get a direct feedback if a

model requires an improvement—providing the right infor-

mation to the right person at the right time. We close this

review by emphasizing that a holistic treatment of the engi-

neering design process requires the aligned eﬀorts of both

the design and the analysis communities.

Acknowledgements Open access funding provided by Austrian Sci-

ence Fund (FWF). We thank Michael Breitenberger and Ben Urick

for their useful advise and fruitful discussions. This research was sup-

ported by the Austrian Science Fund (FWF): J 3884-N32, and the

Oﬃce of Naval Research (ONR): N00014-17-1-2039. This support is

gratefully acknowledged.

Compliance with Ethical Standards

Conﬂict of interest The authors declare that they have no conﬂict

of interest.

Appendix: Exchange Data File Examples

The source ﬁles of the neutral exchange format example

presented in Sect.4.2.3 are given in this section. The mod-

els have been constructed with the commercial CAD soft-

ware Rhinoceros 5.

It should be pointed out that the IGES examples, i.e.,

Files1 and 2, provide the same information although the

topological data of the two models was diﬀerent before the

extraction procedure. In particular, these ﬁles diﬀer only in

the representation of some ﬂoating point values, e.g.,0.0D0

and 8.881D-16, and the sequence of the numbering of a few

parametric data entities, e.g.,0000029P of File 1 is equal

to 0000037P of File2. The corresponding STEP examples

with the correct topology data are given in Files3 and 4.

The interested reader is referred to the homepage of

STEP Tools, Inc.21 for further examples of STEP

ﬁles covering various application protocols.

File 1: IGES trimming example–surface model

,,1H;,,

HC:\Users\Marussig\Desktop \LinuxSync\simpleTrimmingExampleTrim.igs,G 2

HRhinoceros (Sep 27 2012),31HTrout Lake IGES 012 Sep 27 2012,

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5.0D0; 0000027P50

21 http://www.steptools.com, 8 2016.

A Review ofTrimming inIsogeometric Analysis: Challenges, Data Exchange andSimulation Aspects

1 3

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File

2: IGES trimming example – solid model

,,1H;,,G1

HC:\Users\Marussig\Desktop\LinuxSync \simpleTrimmingExampleBooleanIGES.G 2

igs

,G3

HRhinoceros(Sep 27 2012 ),31HTrout Lake IGES 012 Sep 27 2012,G4

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File 3: STEP trimming example–surface model

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B.Marussig, T.J.R.Hughes

1 3

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Scalable field-aligned reparameterization for trimmed NURBS

Article

Full-text available

May 2025
ENG COMPUT-GERMANY

In engineering design, one of the most daunting problems in the design-through-analysis workflow is to deal with trimmed Non-Uniform Rational B-Splines (NURBS), which often involve topological/geometric issues and lead to inevitable gaps and overlaps in the model. Given the dominance of the trimming technology in CAD systems, reconstructing such a model as a watertight representation is highly desired. While remarkable progress has been made in recent years, especially with the advancement of isogeometric analysis (IGA), there still lack a fully automatic and scalable tool to achieve this reconstruction goal. To address this issue, we present a semi-automatic and scalable reparameterization pipeline based on a scalable and feature-aligned meshing tool, QuadriFlow (Huang et al., in: Eurographics symposium on geometry processing, vol 37, no 5, pp 147–160, 2018). On top of it, we provide support for open surfaces to deal with engineering shell structures, and perform sophisticated patch simplification to remove undesired tiny/slender patches. As a result, we obtain a watertight spline surface (multi-patch NURBS or unstructured splines) with a simple quadrilateral layout. Through several challenging models from industry applications, we demonstrate the efficacy and efficiency of the proposed pipeline as well as its integration with IGA. Our source code is publicly available on GitHub (ScaleUntrim, 2024. https://doi.org/10.5281/zenodo.14979911).

Robust Containment Queries over Collections of Trimmed NURBS Surfaces via Generalized Winding Numbers

Preprint

Apr 2025

Efficient and accurate evaluation of containment queries for regions bound by trimmed NURBS surfaces is important in many graphics and engineering applications. However, the algebraic complexity of surface-surface intersections makes gaps and overlaps between surfaces difficult to avoid for in-the-wild surface models. By considering this problem through the lens of the generalized winding number (GWN), a mathematical construction that is indifferent to the arrangement of surfaces in the shape, we can define a containment query that is robust to model watertightness. Applying contemporary techniques for the 3D GWN on arbitrary curved surfaces would require some form of geometric discretization, potentially inducing containment misclassifications near boundary components. In contrast, our proposed method computes an accurate GWN directly on the curved geometry of the input model. We accomplish this using a novel reformulation of the relevant surface integral using Stokes' theorem, which in turn permits an efficient adaptive quadrature calculation on the boundary and trimming curves of the model. While this is sufficient for "far-field" query points that are distant from the surface, we augment this approach for "near-field" query points (i.e., within a bounding box) and even those coincident to the surface patches via a strategy that directly identifies and accounts for the jump discontinuity in the scalar field. We demonstrate that our method of evaluating the GWN field is robust to complex trimming geometry in a CAD model, and is accurate up to arbitrary precision at arbitrary distances from the surface. Furthermore, the derived containment query is robust to non-watertightness while respecting all curved features of the input shape.

Higher-order accurate mass lumping for explicit isogeometric methods based on approximate dual basis functions

Article

Full-text available

Jan 2025
COMPUT MECH

This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and row-sum mass lumping. We demonstrate that the Petrov–Galerkin method developed in our previous work—utilizing row-sum mass lumping—can be interpreted as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions, while preserving optimal spatial accuracy. The mathematical analysis is substantiated by spectral analysis, two-dimensional linear benchmarks illustrating robustness under mesh distortion and a three-dimensional geometrically non-linear forced vibration analysis. The obtained results reveal that our approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.

Cross-talk effects in trimmed isogeometric shells and the control point duplication approach

Article

Apr 2025
COMPUT METHOD APPL M

A hybrid isogeometric collocation method on implicitly trimmed domains

Article

Apr 2025
COMPUT METHOD APPL M

THB++ splines with Applications to Isogeometric Analysis

Preprint

Mar 2025

Robust multi-curve trimming IGA method for layerwise composite plates with arbitrary cutouts

Article

Mar 2025
COMPOS STRUCT

The isogeometric boundary element method: A alternative approach to the analysis of trimmed geometrical models

Article

Mar 2025
COMPUT METHOD APPL M

Gernot Beer

Static Bending Behavior of Two-Directional Functionally Graded Porous Microshell

Chapter

Feb 2025

Deep-neural-network-based framework for the accelerating uncertainty quantification of a structural–acoustic fully coupled system in a shallow sea

Article

Feb 2025
ENG ANAL BOUND ELEM

AUTOMATIC CONFORMAL DECOMPOSITION OF ELEMENTS CUT BY NURBS

Conference Paper

Full-text available

Jan 2016

INTEGRATION OF DESIGN AND ANALYSIS THROUGH BOUNDARY INTEGRAL EQUATIONS

Conference Paper

Full-text available

Jan 2016

Mathematics of Isogeometric Analysis: A Conspectus

Chapter

Dec 2017

Isogeometric analysis (IGA) is a successful generalization of finite element analysis. The use of smooth splines simplifies the interaction with geometric design, but also opens new frontiers to the numerical solution of partial differential equations. In this chapter we present an overview of the main mathematical properties and results that explain the advantages of IGA. We describe the construction of isogeometric scalar and vector spaces, and their approximation and spectral properties.

Geometric processing of CAD data and meshes as input of integral equation solvers

Thesis

Nov 2006

Maharavo Randrianarivony

Among the presently known numerical solvers of integral equations, two main categories of approaches can be traced: mesh-free approaches, mesh-based approaches. We will propose some techniques to process geometric data so that they can be efficiently used in subsequent numerical treatments of integral equations. In order to prepare geometric information so that the above two approaches can be automatically applied, we need the following items: (1) Splitting a given surface into several four-sided patches, (2) Generating a diffeomorphism from the unit square to a foursided patch, (3) Generating a mesh M on a given surface, (4) Patching of a given triangulation. In order to have a splitting, we need to approximate the surfaces first by polygonal regions. We use afterwards quadrangulation techniques by removing quadrilaterals repeatedly. We will generate the diffeomorphisms by means of transfinite interpolations of Coons and Gordon types. The generation of a mesh M from a piecewise Riemannian surface will use some generalized Delaunay techniques in which the mesh size will be determined with the help of the Laplace-Beltrami operator. We will describe our experiences with the IGES format because of two reasons. First, most of our implementations have been done with it. Next, some of the proposed methodologies assume that the curve and surface representations are similar to those of IGES. Patching a mesh consists in approximating or interpolating it by a set of practical surfaces such as B-spline patches. That approach proves useful when we want to utilize a mesh-free integral equation solver but the input geometry is represented as a mesh.

Geometric Modeling and Mesh Generation from Scanned Images

Book

Apr 2016

Yongjie Jessica Zhang

Cutting-Edge Techniques to Better Analyze and Predict Complex Physical Phenomena Geometric Modeling and Mesh Generation from Scanned Images shows how to integrate image processing, geometric modeling, and mesh generation with the finite element method (FEM) to solve problems in computational biology, medicine, materials science, and engineering. Based on the author's recent research and course at Carnegie Mellon University, the text explains the fundamentals of medical imaging, image processing, computational geometry, mesh generation, visualization, and finite element analysis. It also explores novel and advanced applications in computational biology, medicine, materials science, and other engineering areas. One of the first to cover this emerging interdisciplinary field, the book addresses biomedical/material imaging, image processing, geometric modeling and visualization, FEM, and biomedical and engineering applications. It introduces image-mesh-simulation pipelines, reviews numerical methods used in various modules of the pipelines, and discusses several scanning techniques, including ones to probe polycrystalline materials. The book next presents the fundamentals of geometric modeling and computer graphics, geometric objects and transformations, and curves and surfaces as well as two isocontouring methods: marching cubes and dual contouring. It then describes various triangular/tetrahedral and quadrilateral/hexahedral mesh generation techniques. The book also discusses volumetric T-spline modeling for isogeometric analysis (IGA) and introduces some new developments of FEM in recent years with applications.

Pitfalls in Computation, or Why a Math Book isn't Enough

Article

Nov 1970

George E. Forsythe

Isogeometric Analysis: Toward Integration of CAD and FEA

Book

Sep 2009

The authors are the originators of isogeometric analysis, are excellent scientists and good educators. It is very original. There is no other book on this topic. Ren de Borst, Eindhoven University of Technology Written by leading experts in the field and featuring fully integrated colour throughout, Isogeometric Analysis provides a groundbreaking solution for the integration of CAD and FEA technologies. Tom Hughes and his researchers, Austin Cottrell and Yuri Bazilevs, present their pioneering isogeometric approach, which aims to integrate the two techniques of CAD and FEA using precise NURBS geometry in the FEA application. This technology offers the potential to revolutionise automobile, ship and airplane design and analysis by allowing models to be designed, tested and adjusted in one integrative stage. Providing a systematic approach to the topic, the authors begin with a tutorial introducing the foundations of Isogeometric Analysis, before advancing to a comprehensive coverage of the most recent developments in the technique. The authors offer a clear explanation as to how to add isogeometric capabilities to existing finite element computer programs, demonstrating how to implement and use the technology. Detailed programming examples and datasets are included to impart a thorough knowledge and understanding of the material. Provides examples of different applications, showing the reader how to implement isogeometric models Addresses readers on both sides of the CAD/FEA divide Describes Non-Uniform Rational B-Splines (NURBS) basis functions

Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations

Article

Dec 2017

We present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes. Acknowledging the differing requirements posed by design (e.g., the convenience of an intuitive control net) and analysis (e.g., good approximation behavior), we propose the construction of a separate, smooth spline space for each while ensuring isogeometric compatibility - requiring the geometric models to be members of the analysis-suitable spaces. The methodology is simple and is presented for bi-cubic splines; extensions to higher degrees are possible, and are briefly discussed. The presentation has been structured to show compatibility with T-splines - a state-of-the-art CAD technology - but the approach should extend to other locally refinable spline technologies (based on local tensor-product structures). An instantiation of the framework is presented, and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework, and show significantly higher convergence rates than attained previously in the considered setting.

Boundary Element Methods for Engineers and Scientists

Book

Jan 2003

Optimized triangulation of parametric surfaces

Article

Jan 1991

A Review of Trimming in Isogeometric Analysis: Challenges, Data Exchange and Simulation Aspects

Abstract

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