# A non-conforming monolithic finite element method for problems of coupled mechanics.

# Figures

A non-conforming monolithic ﬁnite element method for problems of

coupled mechanics

D. Nordsletten

*

, D. Kay, N. Smith

Oxford Computing Laboratory, University of Oxford, UK

article info

Article history:

Received 19 February 2009

Received in revised form 26 January 2010

Accepted 25 May 2010

Available online 16 June 2010

Keywords:

Finite element method

Mortar/domain decomposition

Navier–Stokes

Arbitrary Lagrangian Eulerian

Finite elasticity mechanics

Non-matching grids

abstract

In this study, a Lagrange multiplier technique is developed to solve problems of coupled

mechanics and is applied to the case of a Newtonian ﬂuid coupled to a quasi-static hyper-

elastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier

is used to weakly impose displacement/velocity continuity as well as equal, but opposite,

force. Through this approach, both mesh conformity and kinematic variable interpolation

may be selected independently within each mechanical body, allowing for the selection

of grid size and interpolation most appropriate for the underlying physics. In addition,

the transfer of mechanical energy in the coupled system is proven to be conserved. The

ﬁdelity of the technique for coupled ﬂuid–solid mechanics is demonstrated through a ser-

ies of numerical experiments which examine the construction of the Lagrange multiplier

space, stability of the scheme, and show optimal convergence rates. The beneﬁts of non-

conformity in multi-physics problems is also highlighted. Finally, the method is applied

to a simpliﬁed elliptical model of the cardiac left ventricle.

Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction

Coupled mechanical systems are prevalent in a wide-range of disciplines, where conservation principles lay the foundation

for multi-physics simulations. Examples in engineering range from aerodynamics [26,25,62,63] through to biomechanics

[59,34,69,17]. In these applications, analysis of the underlying physics requires a comprehensive characterization of the

mechanical interactions between bodies. As a result, a number of algorithms addressing coupled mechanics problems have

been proposed in the literature; falling broadly into three classes (with some exceptions, cf. [20,2]): monolithic, partitioned

and immersion.

Monolithic methods, see [30,21,66,39], denote those schemes in which each subproblem is assembled into a single global

system. Because of this, monolithic schemes typically rely on equivalent numerical discretizations and methods between

bodies. The requirement of equal reﬁnement has the potential to signiﬁcantly impact either the scheme’s computational

efﬁciency (if one body is excessively reﬁned) or accuracy (if one body is under-reﬁned).

1

However, these schemes are gener-

ally numerically stable for a wide-range of physical parameters [9,8].

In contrast, partitioned schemes, see [25], are those in which the subproblems are dealt with independently. This approach

avoids assembly of a single global system and solves the global system iteratively (in some sense [16,20,26,25]). As a result,

these schemes tend to be more computationally efﬁcient than their monolithic counterparts, capitalizing on efﬁcient

0021-9991/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.jcp.2010.05.043

* Corresponding author. Tel.: +44 1865 282257.

E-mail address: david.nordsletten@gmail.com (D. Nordsletten).

1

The level of under/over reﬁnement seen is relative to the governing physics and how one chooses to best approximate them.

Journal of Computational Physics 229 (2010) 7571–7593

Contents lists available at ScienceDirect

Journal of Computational Physics

journal homepage: www.elsevier.com/locate/jcp

subproblem solution techniques and reduced system size. This efﬁciency is limited by the stability and accuracy of the meth-

od, which is restricted (particularly when the physics of one body do not dominate that governing the other [30]).

Immersion techniques are also commonly used for coupled problems, allowing boundary constraints to be imposed along

interfaces which are embedded in a host domain. Two such techniques are the ﬁctitious domain [12,5,27,7,29,61] and

immersed boundary [47,48,24,43,67,46,44,71] methods. Both methods are particularly attractive for problems where one

body is interior to another (e.g. ﬂuid ﬂow through a ﬁberous network or aortic valve movement [34]). Immersion techniques

are well-suited for these problems as they allow complex interfaces as well as dramatic interior movement, avoiding intri-

cate grid construction and adaptation. However, for problems where both bodies are independent or large deformations are

experienced on the exterior of a body, these methods can become inefﬁcient (as a larger host domain is required). Further,

due to the arbitrary insertion of constraints, the coupling conditions are not smooth [27], weakening the exactness of the

coupled solution.

The techniques discussed have advantages and disadvantages, and their appropriate usage is largely application depen-

dent. However, an alternative to these methods can be found in the domain decomposition [4,3,37,18] and mortar domain

[11,10] techniques. These methods, developed primarily for parallel implementations on high performance computers, break

a single ﬁnite element system into a number of subproblems subject to some form of coupling constraint. Using these foun-

dations, we have detailed a linear mechanical theory for coupling using an additional Lagrange Multiplier. Through this ap-

proach, the beneﬁts of both monolithic and partitioned schemes (i.e. stability and non-conformity) may be effectively

combined.

In this paper, we proceed to extend and examine this approach for the case of more complex coupled systems, focusing on

the coupling of a Navier–Stokes ﬂuid and a quasi-static hyperelastic solid. Here, we outline the solution procedure, empha-

sizing the selection of function spaces for the discrete problem. A priori stability bounds are shown for the coupled problem,

demonstrating the energy preservation of the method. Finally, the method is tested by a series of numerical experiments,

showing both convergence and stability for complex non-linear coupled mechanical systems.

1.1. Model problem

In this paper, we focus on the coupling of a Navier–Poisson ﬂuid and a quasi-static hyperelastic solid which satisfy

Problems 1 and 2, respectively. Though the paper focuses on these models, the scheme may be generalized to other coupled

mechanical systems. The linking of these problems is enforced via Problem 3, ensuring kinematic continuity and equal and

opposite traction.

The ﬂuid and solid will be represented geometrically by the domains

X

1

and

X

2

, respectively. In both cases,

X

i

R

d

I; i ¼ 1; 2 is a moving domain which alters shape through the time interval I = [0,T] (note, d = dim

X

). The bound-

ary of each domain,

C

i

, is treated to be at least Lipshitz continuous and is partitioned so that C

i

¼ C

N

i

C

D

i

C

C

i

(where N, D,

and C refer to the Neumann, Dirichlet and Coupling subdomains of the boundary, respectively). In this case, the two domains

are coupled about

C

C

:¼ C

C

1

¼ C

C

2

.

Problem 1 (Navier–Stokes Equations). Consider ﬂow over

X

1

. Let v and p be the velocity and pressure state variables, which

satisfy,

q

@

v

@t

þ

r

x

q

vv

l

r

x

v

þ pIðÞ¼f

1

in X

1

; ð1aÞ

r

x

v

¼ 0inX

1

; ð1bÞ

v

¼ g

D

1

on

C

D

1

; ð1cÞ

l

r

x

v

pIðÞn ¼ g

N

1

on

C

N

1

; ð1dÞ

v

ð; 0Þ¼

v

0

in X

1

ð0Þ; ð1eÞ

where

l

the viscosity,

q

the density, (I)

jk

:=d

jk

, n the outward boundary normal, v

0

the initial velocity, r

x

the Eulerian

gradient operator, f

1

is the contribution to momentum of body forces, and g

D

1

¼ g

D

1

ðx; tÞ and g

N

1

¼ g

N

1

ðx; tÞ the given Dirichlet

and Neumann boundary data.

Problem 2 (Quasi-Static Finite Elasticity). Consider ﬁnite elasticity mechanics over

X

2

. Let u and u be the displacement and

pressure state variables, which satisfy,

r

x

r

ðuÞ

u

IðÞ¼f

2

in X

2

; ð2aÞ

@

t

JðuÞ¼0inX

2

; ð2bÞ

u ¼ g

D

2

on

C

D

2

; ð2cÞ

r

ðuÞ

u

IðÞn ¼ g

N

2

on

C

N

2

; ð2dÞ

uð; 0Þ¼u

0

in X

2

ð0Þ; ð2eÞ

7572 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

where @

t

is the Lagrangian/arbitrary Lagrangian–Eulerian time derivative (refer to Section 3),

r

ðuÞ2R

dd

is a symmetric

stress tensor, JðuÞ¼det j

r

0

uj the mapping Jacobian, r

0

is the gradient operator with respect to undeformed coordinates,

n the outward boundary normal, u

0

the initial displacement, f

2

is the contribution to momentum of body forces, and

g

D

2

¼ g

D

2

ðx; tÞ and g

N

2

¼ g

N

2

ðx; tÞ the given Dirichlet and Neumann boundary data.

The above problems are common models for detailing ﬂuid ﬂow and solid deformation. Problem 1, the Navier–Stokes sys-

tem [45,31–33], is the application of Cauchy’s ﬁrst law and mass conservation to a linear, isothermal, incompressible ﬂuid.

The so-called arbitrary Lagrangian–Eulerian, or ALE, form of the Navier–Stokes system, a generalization of the governing

equations for moving domains, will be used in this work [38,68,53,55,58]. Problem 2 is the quasi-static ﬁnite elasticity sys-

tem [45,13,50,51], governing the displacement of an isothermal, incompressible solid with negligible inertia (relative to that

of the ﬂuid). Though the quasi-static solid model was selected – as it is commonly applied for the study of biological tissues

[50,51] – transient ﬁnite elasticity is not precluded from the presented method.

Problem 3 (Coupling). Let t

1

=(

l

r

x

v pI) n and t

2

=(

r

(u) uI) n, then

X

1

and

X

2

are coupled by the following

conditions,

t

1

þ t

2

¼ 0on

C

C

; ð3aÞ

@

t

u

v

¼ 0on

C

C

: ð3bÞ

The coupling of subproblems into a single multi-physics system enforces equal, but opposite, traction and kinematic

equivalence across the common interface,

C

C

. The force constraint of Eq. (3a) ensures each subproblem provides equal,

but opposite, tractions, point-wise at the interface. Further, the kinematic condition of equal velocity, seen in Eq. (3b), sub-

jects each subproblem to a Lagrangian adherence of boundary points (no slip).

The coupled system is assembled through the combination of Problems 1–3. Using standard Galerkin ﬁnite elements, this

system is reformulated in weak form. As stated, the emphasis of this approach is to allow non-conforming domains while main-

taining stability. We note, however, that non-conforming domains invalidate the very premise of Problem 3 as

C

C

1

is not neces-

sarily equal to

C

C

2

or

C

C

. To circumventthese issues, while still maintaining optimality and stability, theconstraints of Problem 3

are transformed to a third computational domain,

N

, and a Lagrange multiplier, k, is introduced (see Sections 3 and 4).

As we will demonstrate, the mechanical systems given in Problems 1 and 2 may be coupled using an added Lagrange

multiplier, resulting in the familiar weak saddle point system shown in Problem 4.

Problem 4 (Weakly Coupled Fluid–Solid System). The weakly coupled ﬂuid–solid system may be expressed by the following

saddle point problem.

Find ðX; ZÞ2Y

I;D

Z

I

such that,

A

K

ðX; YÞþB

K

ðZ; YÞ¼F

I

ðYÞ

8

Y 2 Y

0

; ð4aÞ

B

K

ðQ; LXÞ¼0

8

Q 2 Z

I

; ð4bÞ

where

X ¼ð

v

; uÞ; Y ¼ðy

1

; y

2

Þ; Z ¼ðp;

u

; kÞ; Q ¼ðq

1

; q

2

; qÞ:

K # I ¼½0; T; A

I

and B

I

are operators on the kinematic/Lagrange multiplier variables and test spaces, F

I

a functional on the

test space Y

0

; L being the linear operator LU =(v,@

t

u), and Y

I

and Z

I

the appropriate test function spaces.

1.2. Overview

The remainder of this paper is structured as follows. We begin in Section 2 by deﬁning initial terminology and notation.

The continuous weak form of both ﬂuid and solid problems is then introduced in Section 3, allowing the natural introduction

of the added Lagrange multiplier variable and construction of Problem 4. The system is then discretized and the solution pro-

cedure outlined in Section 4. From the linear theory [57], important aspects – both for stability and well-posedness – are

noted. In both continuous and discrete systems, a priori stability is shown under certain restrictions on the hyperelastic

law. Finally, the method is applied to a series of numerical experiments in Section 6, demonstrating its stability and conver-

gence for a range of test problems.

2. Preliminary notation

As we consider the Lagrangian and arbitrary Lagrangian–Eulerian (ALE) reference frames (see Fig. 1) we introduce the

following terms and notations. For the transient physical domain,

X

, we deﬁne a bijective mapping, denoted P

K

X

:

X

!

K

,

to a static reference domain,

K

. The deﬁned mapping then provides a relation between coordinates

g

2

K

and coordinates

x 2

X

(

s

) for all

s

2 I, i.e.

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7573

P

K

X

ð

s

Þ : Xð

s

Þ!

K

;

g

¼P

K

X

ðx;

s

Þ;

8

x 2 Xð

s

Þ;

8

g

2

K

; ð5Þ

As the mapping is bijective, we may deﬁne its inverse, P

X

K

. Further, P

K

X

is deﬁned in so that regions of the physical boundary,

C

, map to regions of the reference boundary, . In this case we think of

X

as the transient Lagrangian or ALE domain and

K

its static reference frame. Using this mapping between the reference and physical domains, a function, f :

K

! R

d

, described

on the static reference frame

K

, may be interpreted on

X

as

^

f ,

^

f ðx; tÞ :¼ f ð

g

; tÞ; x ¼P

X

K

ð

g

; tÞ;

8

g

2

K

; t 2 I: ð6Þ

The temporal rate-of-change of f on

K

represents the change in the ﬁeld f along some trajectory in

X

. This time derivative

[53,58] can be deﬁned as,

@

t

f ð

g

; tÞ :¼ lim

jdj!0

f ð

g

; t þ dÞf ð

g

; tÞ

d

;

g

2

K

; ð7Þ

which is related to the Lagrangian (D/Dt) and Eulerian time (@/@t) derivatives by,

D

Dt

f ¼

@

@t

^

f þ

v

r

x

^

f ¼ @

t

f þð

v

wÞ

r

x

^

f ; ð8Þ

where

f represents f projected into the Lagrangian frame, w ¼ @

t

P

X

K

is the domain velocity and v the Lagrangian frame velocity

(equivalent to the ﬂuid/solid material velocity). Note that when the domain velocity is equivalently the Lagrangian frame

velocity, @

t

is simply the Lagrangian time derivative, D/Dt. Similarly, the gradients, r

x

and r

g

,on

X

and

K

, respectively,

are related by Eq. (9), where F

t

is the deformation gradient tensor of the mapping (at some time t 2 I),

G

t

¼ F

T

t

r

g

¼

r

x

; F

t

¼

r

g

P

X

K

ðtÞ; ð9Þ

so that, the gradient of

^

f on

X

is,

2

r

x

^

f ¼G

t

f ¼ F

T

t

r

g

f ¼

r

g

f

F

1

t

: ð10Þ

As the projection of f to

^

f may distort the spatial interpretation of

^

f , the gradients r

g

and r

x

are generally not equivalent.

Further, as

r

x

acts on functions on

X

; G

t

is introduced to represent the action of the r

x

-gradient on functions of

K

. To avoid

cluttering notation, the transformation gradient and deformation gradient operators, G

t

and F

t

, will not be distinguished be-

tween ﬂuid and solid domains – though they will differ, in general – as the appropriate operator and underlying mapping are

implicit to each domain.

Lastly, the integral on

X

(t) at some time t may be detailed on

K

in terms of the weighted integral given by Eq. (11).

Z

X

ðtÞ

^

f dx ¼

Z

K

f J

X

;t

d

g

; J

X

;t

¼ detjF

t

j: ð11Þ

Fig. 1. Example of ALE (light grey,

X

1

) and Lagrangian ( dark grey,

X

2

) moving domains at time points 0 and t. Here the reference domains,

K

1

and

K

2

, are

mapped by there respective projections ðPÞ. Arrows show the position of points

g

1

and

g

2

in the reference domain in

X

1

and

X

2

. Regions around these

points are also mapped, showing the relative dilation of volume characterized by the mapping Jacobian, J . The coupling interface, is mapped to the third

domain

N

.

2

For clarity, in this notation we denote the dyadic product of two vectors, y and z,asyz. The tensor components of yz are given as (yz)

jk

= y

k

z

j

.

7574 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

3. Continuous weak formulation

In this section, the classical Navier–Stokes system, seen in Problem 1, is transformed into its ALE weak form to enable the

solution of ﬂuid ﬂow on the dynamic domain,

X

1

. Similarly, the weak form of the quasi-static ﬁnite elasticity system, shown

in Problem 2, is also derived. Subsequently, a weak formulation of the coupling constraint is derived, using the linear coupled

theory developed in [57] to guide formulation construction. Assembling all weak form equations, the coupled system is

constructed as shown in Problem 4.

3.1. Problem 1 weak formulation

We begin with the ALE form of the Navier–Stokes equations (refer to [68,49,72,40,55,58] for derivations), a generalization

of the classic Navier–Stokes equations to moving domains. In the ALE form, the ﬂuid problem is posed on a static reference

geometry,

K

1

, which is bijectively related to

X

1

via the mapping, P

X

1

K

1

[53,58] (see Section 2). Derivation of the mapping is

done as detailed in [58].

To write the ALE weak form, consider the velocity and pressure, (v,p), on the reference domain,

K

1

. The state variable

spaces are deﬁned as [64,65,54],

V :¼ H

1

ð

K

1

Þ; Wð

K

1

Þ :¼ L

2

ð

K

1

Þ; ð12Þ

and

V

I

:¼ L

1

ðI; L

2

ð

K

1

ÞÞ \ L

2

ðI; H

1

ð

K

1

ÞÞ; W

I

ð

K

1

Þ :¼ L

1

ðI; L

2

ð

K

1

ÞÞ: ð13Þ

These spaces may be tailored for the weak form solutions to Problem 1 by selecting only those functions which match g

D

1

or 0

on the Dirichlet boundaries, i.e.

V

I;D

¼fy 2 V

I

jy ¼ g

D

1

on

D

1

g; V

0

¼fy 2 Vjy ¼ 0on

D

1

g:

The weak ALE form of Problem 1 may then be written as: Find ð

v

; pÞ2V

I;D

W

I

ð

K

1

Þ, such that for any ðy; qÞ2V

0

Wð

K

1

Þ

and any [a,b] # I,

q

m

b

ð

v

; yÞ

q

m

a

ð

v

; yÞþ

q

Z

b

a

c

s

ð

v

w;

v

; yÞd

s

þ

l

Z

b

a

a

s

ð

v

; yÞd

s

Z

b

a

b

s

ðp; yÞd

s

Z

b

a

Z

C

1

t

1

ð

s

ÞyJ

C

1

;

s

d

g

d

s

¼

Z

b

a

m

s

ðf

1

; yÞ; ð14aÞ

Z

b

a

b

s

ðq;

v

Þd

s

¼ 0; ð14bÞ

where t

1

is the traction on the coupling interface, f

1

2 L

2

(I;L

2

(

K

1

)) represents body forces, and

m

s

ð

v

; yÞ :¼

Z

K

1

v

ð

s

ÞyJ

X

1

;

s

d

g

; ð15Þ

a

s

ð

v

; yÞ :¼

Z

K

1

G

s

v

ð

s

Þ : G

s

yJ

X

1

;

s

d

g

; ð16Þ

b

s

ðp; yÞ :¼

Z

K

1

pð

s

ÞG

s

yJ

X

1

;

s

d

g

; ð17Þ

c

s

ðz w;

v

; yÞ :¼

Z

K

1

G

s

zð

s

Þwð

s

ÞðÞ

v

ð

s

Þ½y

1

2

G

s

zð

s

ÞðÞ

v

ð

s

Þy

J

X

1

;

s

d

g

; ð18Þ

are operators, where m represents the integrated acceleration term, a the linear viscous term, b the pressure term, and c the

convective term (where the second component – which is zero in the continuous form – is added for stability [53]). For ease,

we have assumed g

N

1

:¼ 0 in Eq. (14a).

Remark 1. Note, by deﬁnition, the test function y is deﬁned to be constant over time on the reference domain.

3.2. Problem 2 weak formulation

In Problem 2, a solid mechanical system was introduced. Here, we follow the common approach of considering the hyper-

elastic body in the Lagrangian reference/coordinate frame [13,54]. Similar to the ALE case, reference and physical geometries

(denoted

K

2

and

X

2

, respectively) are deﬁned. Both domains are related by the mapping in Eq. (5) however, in this case

P

X

2

K

2

ð

s

Þ :¼ uð

g

;

s

Þþ

g

, where u is the displacement ﬁeld and

g

a point in

K

2

(i.e.

K

2

represents the unstressed body). The

displacement and pressure, (u,u), of the solid – also considered on the reference domain – are sought in spaces,

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7575

U :¼ H

1

ð

K

2

Þ; W :¼ L

2

ð

K

2

Þ; ð19Þ

and

U

I

:¼ L

1

ðI; H

1

ð

K

2

ÞÞ; W

I

ð

K

2

Þ :¼ L

1

ðI; L

2

ð

K

2

ÞÞ: ð20Þ

Similarly, these spaces may be tailored for the weak form solutions to Problem 2 by selecting functions which match g

D

2

or 0

on the Dirichlet boundaries, i.e.

U

I;D

¼fy 2 V

I

jy ¼ g

D

2

on

D

2

g; U

0

¼fy 2 Vjy ¼ 0 on

D

2

g:

Remark 2. The solution u(

s

), at a time

s

2 I, satisﬁes,

uð

s

Þ2U

H

U;

where,

U

H

:¼ y 2 UjLð

g

Þ¼yð

g

Þþ

g

;

g

2

K

2

where L is a bijective mappingfg:

The weak form of Problem 2 may then be written as (refer to [45,13,50] for derivations): Find ðu; pÞ2U

I;D

W

I

ð

K

2

Þ, such

that for any ðy; qÞ2U

0

Wð

K

2

Þ and [a,b] # I,

Z

b

a

^

s

s

ðu; yÞb

s

ð

u

; yÞ d

s

Z

b

a

Z

C

2

t

2

ð

s

ÞyJ

C

2

;

s

d

g

d

s

¼

Z

b

a

m

s

ðf

2

; yÞd

s

; ð21aÞ

m

b

ðq; 1Þm

a

ðq; 1Þ¼0; ð21bÞ

where m

s

ðf

2

; yÞ¼

R

X

2

ð

s

Þ

f

2

ð

s

Þyd

g

; t

2

is the traction on the coupling interface, and f

2

2 L

2

(I;L

2

(

K

2

)) represents the body

forces, b is the operator seen in Eq. (17) set on

K

2

, and

^

s

s

ðu; yÞ :¼

Z

K

2

r

s

ðuÞ : G

s

yJ

X

2

;

s

d

g

: ð22Þ

For ease, we have assumed g

N

2

:¼ 0 in Eq. (21a). Eq. (21b), which preserves the mass/incompressibility of the solid body, re-

quires that the integral measure of any L

2

-function is equivalent. That is (by selecting a = 0 and b =

s

),

Z

K

2

J

X

2

;

s

qd

g

¼

Z

K

2

J

X

2

;0

qd

g

;

8

q 2 L

2

ð

K

2

Þ: ð23Þ

Assuming the solid law is sufﬁciently smooth so that

r

ðuÞ : G

s

y 2 L

2

ð

K

2

Þ (at almost every point in time),

s

s

ðu; yÞ :¼

Z

K

2

r

s

ðuÞ : G

s

yd

g

¼

^

s

s

ðu; yÞ

8

y 2 U; ð24Þ

and by the smoothness of f

2

(and deﬁning M

s

ðf

2

; yÞ¼

R

K

2

f

2

ð

s

Þyd

g

),

m

s

ðf

2

; yÞ¼M

s

ðf

2

; yÞ;

8

y 2 U: ð25Þ

As a result, the weak solid mechanical system of (21) may be recast as: Find ðu; pÞ2U

I;D

W

I

ð

K

2

Þ, such that for any

ðy; qÞ2U

0

Wð

K

2

Þ and [a,b] # I,

Z

b

a

s

s

ðu; yÞd

s

Z

b

a

b

s

ð

u

; yÞd

s

Z

b

a

Z

C

2

t

2

ð

s

ÞyJ

C

2

;

s

d

g

d

s

¼

Z

b

a

M

s

ðf

2

; yÞd

s

; ð26aÞ

m

b

ðq; 1ÞM

0

ðq; 1Þ¼0; ð26bÞ

where the stress and forcing terms are no longer weighted by the mapping Jacobian (which is, itself, a function u).

3.3. Introduction of the Lagrange multiplier

Along the coupling interface, the traction forces in both ﬂuid and solid problems, denoted t

1

and t

2

, arise naturally

through integration by parts [32,33]. The result is a duality pairing appearing in the weak formulation along all

Neumann-type boundaries. Considering this pairing on the coupled interface, noting Eq. (3a), and summing the weak form

Eqs. (14a) and (26a), we may deﬁne the added Lagrange multiplier, k.

l

s

ðk; y

2

y

1

Þ :¼

Z

C

kð

s

Þðy

2

y

1

Þd

g

¼

Z

C

ðt

1

ð

s

Þy

1

þ t

2

ð

s

Þy

2

ÞJ

s

d

g

;

for any test functions, ðy

1

; y

2

Þ2V

0

U

0

, where

s

2 I. This form suggests that the weighted traction k(

s

) 2 H

1/2

(

C

), which

is consistent with the fact that the tractions are restrictions to the boundary of the normal gradients of v and u, respectively.

Similarly, considering the weak imposition of the kinematic condition (3b), multiplying by a test function q 2 H

1/2

(

C

),

provides the sensible weak constraint,

7576 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

l

s

ðq;@

t

u

v

Þ¼0: ð27Þ

The substitution of tractions by k, and the incorporation of the weak kinematic constraint in Eq. (27) introduces an additional

saddle point condition. As such, the selection of the test space for k is subject to the inf–sup, or LBB, condition [42,6,14,15,52],

ensuring the uniqueness of k. Though k(

s

) 2 H

1/2

(

C

) is a natural space, due to Dirichlet conditions incorporated in V

0

and

U

0

, this space may contain components in the nullspace (for instance, if ð

C

\

D

1

Þ\ð

C

\

D

2

Þ – ;). However, a space of

Lagrange multipliers, denoted M , may be formed by removing the nullspace of l [57], i.e.

H

1=2

ð

C

Þ¼M N

l

;

where,

N

l

¼fq 2 H

1=2

ð

C

Þj lðq; y

2

y

1

Þ¼0;

8

ðy

1

; y

2

Þ2V

0

U

0

g:

With this selection, the introduced saddle point problem can be shown to satisfy the LBB condition [57].

3.4. Coupled system

In the previous sections, the weak form ALE Navier–Stokes equations, quasi-static ﬁnite elasticity equations, and coupling

conditions were introduced. Combining these results, a global system seen in Problem 4 may be formulated. Identifying the

state variables (velocity, displacement, ﬂuid/solid pressure, and the Lagrange multiplier, k) as either kinematic X =(v,u)or

Lagrange multipliers Z =(p,u,k), we may deﬁne the spaces and operators of Problem 4 (where K =[a,b] # I).

Y

0

:¼ V

0

U

0

; Y

I

:¼ V

I

U

I

; ð28Þ

Z :¼Wð

K

1

ÞWð

K

2

ÞM ð

C

Þ; Z

I

:¼W

I

ð

K

1

ÞW

I

ð

K

2

ÞM

I

ð

C

Þ; ð29Þ

A

K

ðU; YÞ :¼

q

m

b

ð

v

; y

1

Þþ

Z

K

a

s

ð

v

; y

1

Þþc

s

ð

v

w;

v

; y

1

Þþs

s

ðu; y

2

Þd

s

; ð30Þ

B

K

ðZ; YÞ :¼

Z

K

l

s

ðk; y

2

y

1

Þb

s

ðp; y

1

Þb

s

ð

u

; y

2

Þd

s

; ð31Þ

F

K

ðYÞ :¼

Z

K

m

s

ðf

1

; y

1

ÞþM

s

ðf

2

; y

2

Þ½d

s

þ m

a

ð

v

; y

1

Þ: ð32Þ

Assembling these deﬁnitions, the coupled mechanical system satisﬁes the ﬂuid, solid and coupling problems introduced in

Eqs. (14), (26) and (27).

4. Discrete formulation

The previous section detailed the translation of the classical Problems 1–3 into their weak forms, showing that the

coupled system follows the classic saddle point structure of Problem 4. In this section, the discretization of the coupled sys-

tem is considered using Galerkin ﬁnite elements and implicit Euler time stepping. The discrete weak forms are then linear-

ized and broken down into a general block matrix system.

4.1. Discretization

4.1.1. Spatiotemporal discretization

The reference domains,

K

1

and

K

2

, are split into ﬁnitely many, non-overlapping elements, e, which assemble to form the

mesh T

h

ð

K

i

Þ,

3

i.e.

T

h

ð

K

i

Þ¼ e

1

; ...e

N

i

; h ¼ max

e2T

h

ðK

i

Þ

diamðeÞ i ¼ 1; 2:

The time domain, I, is also divided into N

I

non-overlapping intervals (t

n1

,t

n

), t

n1

< t

n

, t

0

= 0 and t

N

= T. Over each time inter-

val t 2 (t

n1

,t

n

], the velocity, pressure, Lagrange multiplier and domain velocity are taken as constants in time, while the solid

displacement is necessarily piecewise linear in time to satisfy kinematic equality at the coupled interface.

4.1.2. Finite dimensional subspaces

The discretizations T

h

ð

K

1

Þ and T

h

ð

K

2

Þ provide a foundation for forming ﬁnite dimensional polynomial subspaces. Let e

M

be a given reference, or master, element and P

j

ðe

M

Þ be the polynomial space containing all polynomials of degree

4

no more

than

j

,

j

P 1. Then, for every e 2T

h

ðK

i

Þ, i = 1, 2, we require there be a well-deﬁned bijective, linear or curvilinear map,

3

In this paper, we consider tetrahedral, hexahedral and curvilinear elements.

4

For tensor product basis, the degree in each direction, while for those basis on triangles or tetrahedra, the total degree (i.e. x

1

x

2

is degree 2, x

2

1

x

2

is degree 3,

etc.).

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7577

P

e

e

M

: e

M

! e; P

e

e

M

2 P

j

i

ðKÞ

ðe

M

Þ

d

: ð33Þ

where

j

i

(

K

) P 1 is the polynomial degree of the element mapping for

K

i

, and P

e

e

M

a map which transforms the master ele-

ment, e

M

, to the mesh element, e 2T

h

ð

K

i

Þ. With a mapping deﬁned between elements of the mesh and the master element,

any set of polynomials on e

M

may be projected onto e.

Assembling these locally deﬁned sets of polynomials, a continuous polynomial space, S

h,

j

(

K

i

), may be constructed,

S

h;

j

ð

K

i

; T

h

Þ :¼ y : T

h

ð

K

i

Þ!Rjy 2C T

h

ð

K

i

Þ

; yj

e

2 P

j

ðeÞ;

8

e 2Tð

K

i

Þ

;

which deﬁnes all

j

th-order piecewise-continuous polynomial functions on the mesh elements of T

h

ð

K

i

Þ. The discrete set,

S

h,

j

, provides the foundation for constructing the ﬁnite dimensional velocity, displacement, and pressure spaces, i.e.

V

h

:¼ S

h;

j

ð

v

Þ

ð

K

1

; T

h

Þ

hi

d

; U

h

:¼ S

h;

j

ðuÞ

ð

K

2

; T

h

Þ

hi

d

;

W

h

ð

K

1

Þ :¼ S

h;

j

ðpÞ

ð

K

1

; T

h

Þ; W

h

ð

K

2

Þ :¼ S

h;

j

ð

u

Þ

ð

K

2

; T

h

Þ;

where

j

(v) is the order of interpolation of the velocity and similarly for the displacement/pressures. Though the power of the

kinematic variables is arbitrary, in this paper we use the LBB-stable general Taylor–Hood elements, see [28,15]. Each discrete

subspace has ﬁnite dimension which is given by,

N

v

¼ spanV

h

; N

u

¼ spanU

h

; N

p

¼ spanW

h

ð

K

1

Þ; N

u

¼ spanW

h

ð

K

2

Þ:

By deﬁnition, the dimension of each discrete subspace is related to the underlying nodal Lagrange space. For example, denot-

ing the dimension of S

h;

j

ð

v

Þ

ð

K

1

; T

h

Þ as N

j

(v)

, the dimension of V

h

is N

v

= dN

j

(v)

(and similarly for the solid displacement);

while in the case of the pressure, N

p

= N

j

(p)

(and similarly for the solid pressure). Due to the ﬁnite dimension, all of the dis-

crete approximations may be written as a weighted sum of basis functions [60,32,33], denoted

w

and

w

for scalar and vector

functions, respectively. Here subscripts will be used to indicate the state variable, and superscripts the basis function. Thus,

at any time t

n

, n 2 [0,N],

v

h;n

¼ V

n

W

v

; u

h;n

¼ U

n

W

u

; p

h;n

¼ P

n

W

p

;

u

h;n

¼

u

n

W

u

; ð34Þ

where

W

v

¼

w

1

v

.

.

.

w

N

v

v

0

B

B

@

1

C

C

A

; W

u

¼

w

1

u

.

.

.

w

N

u

u

0

B

B

@

1

C

C

A

; W

p

¼

w

1

p

.

.

.

w

N

p

p

0

B

B

B

@

1

C

C

C

A

; W

u

¼

w

1

u

.

.

.

w

N

u

u

0

B

B

B

@

1

C

C

C

A

; ð35Þ

and, for example, V

n

is a constant vector with N

v

components (i.e. V

n

2 R

N

v

). These weightings – V

n

, U

n

, etc. – are then selected

in order to satisfy the discrete coupled system.

4.1.3. Finite dimensional Lagrange multiplier subspace

In the ﬂuid–solid system, coupling conditions are weakly upheld on

C

. While not signiﬁcant for the continuous problem,

in the discrete setting two problems arise. Due to the variability between mesh size and geometric interpolation (linear ver-

sus curvilinear), the domain on the ﬂuid side need not be equivalent to that on the solid side, i.e. T

h

ð

C

1

Þ and T

h

ð

C

2

Þ. In order

to pose a sensible weak form constraint in this setting, it is clear that some mapping between each approximation of

C

must

be constructed.

Beyond the complications introduced by the discrete physical interfaces, the use of different meshes and state variable

interpolations generally results in different trace spaces on each side (note, in the continuous setting both were

H

1/2

(

C

), refer to Section 3.3). As stated in Section 3.3, a key component to the existence and uniqueness of k is the estab-

lishment of an inf–sup stable space, M . The ability of this approach to succeed in the discrete setting is then linked to the

inf–sup stability of l on V

h

0

; U

h

0

and M

h

.

To overcome the complications due to different spatial meshes of the coupling boundary, a third domain,

N

R

d1

,is

introduced which is bijectively mapped to T

h

ð

C

1

Þ and T

h

ð

C

2

Þ by P

C

i

N

(see Eq. (5) and Fig. 2). Mapping the constraints to this

domain, the Lagrange multiplier, k, is redeﬁned as the Jacobian weighted traction on

N

. As a result, the discrete k

h

and

operator, l, is redeﬁned as,

l

s

ðk

h

; y

h

2

y

h

1

Þ :¼

Z

N

k

h

ð

s

Þð

^

y

h

2

^

y

h

1

Þdn ¼

Z

C

ðt

h

1

ð

s

Þy

h

1

þ t

h

2

ð

s

Þy

h

2

ÞJ

s

d

g

;

where t

h

1

and t

h

2

are the tractions resulting from the discrete approximations. Note that hats are added to y

h

1

and y

h

2

as they

must be projected to

N

by the mapping, i.e. following Eq. (36),

^

y

h

2

ðnÞ :¼ y

h

2

ð

g

Þ; n ¼P

N

C

2

ð

g

Þ;

8

g

2

C

2

: ð36Þ

7578 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

The key to executing this computation is the creation of the third domain and the mapping used to relate it to each discrete

boundary. In general, the domain boundaries can be complex – as seen in Fig. 2 – leading to non-trivial maps. However, each

discrete boundary is a parametric mapping of a surface of the reference element, e

M

, see Section 4.1.2. Deﬁning the mapping

from these simple shapes, we can use simple linear transformations, denoted A, for each element boundary regardless of the

complexity of the mapping between the master element and its shape in the mesh. Such parametric mappings are automat-

ically generated in some tessellation algorithms, such as that of [56], which build surface meshes of another type on a para-

metric surface mesh.

This choice of mapping shapes the selection of

N

. A convenient choice for this mapping is selecting

N

2 R

d1

to be a

disjoint set of open regions as seen in Fig. 2, i.e.

N ¼

1

N

fg

:

In this case, h

i

is some region over which the ﬂuid and solid regions overlap. Using the embedding of meshes, a natural

choice is for h

i

to be a square or triangle from which the other mesh was derived. More complex selections may be chosen

using a collection of elements. This process is detailed in Fig. 2, where a curvilinear hexahedral solid mesh is coupled to a

tetrahedral ﬂuid mesh along the coupled boundary.

Focusing now on the construction of M

h

, an approximation space for the Lagrange Multiplier, k

h

, must be deﬁned which

satisﬁes the inf–sup condition. This may be accomplished by noting Remark 3.

Remark 3. As the operator, l, is linear, inf–sup stability may be guaranteed by selecting M

h

as the set (or subset) of basis

functions seen in

N

from the ﬂuid OR solid [57].

As the Lagrange multiplier is deﬁned on

N

, a mesh may be easily created (as done for the other state variables). In this

case, T

h

ð

N

Þ is deﬁned by,

T

h

ðNÞ¼ T

h

ð

1

Þ; ...; T

h

ð

N

Þfg; T

h

ð

j

Þ¼ #

j1

; ...;#

jM

j

no

; j ¼ 1; ...; N

;

denote the discretization of each region, h 2

N

, where N

6 N

#

:¼

P

N

j¼1

M

j

. We note that # 2T

h

ð

N

Þ is a triangle or square in

d = 3, or a line segment in d = 2. Following Remark 3, we will choose the tessellation T

h

ð

N

Þ to be the ﬂuid or solid boundary.

In Fig. 2, for example, T

h

ð

N

Þ may be selected as the set of square elements or set of triangles. M

h

may then be deﬁned as,

Fig. 2. Shown are the meshes resulting from each approximation of the boundary

C

; T

h

ð

C

1

Þ and T

h

ð

C

2

Þ, for a curvilinear hexahedral solid mesh coupled

to a tetrahedral ﬂuid mesh. Each surface element in the meshes are deﬁned in relation to a set of simple squares or triangles (which compose boundaries of

the master element for each mesh). Denoting the surface element

c

c;j

2T

h

ð

C

1

Þ, the corresponding surface in the master element is given as

^

c

c;j

(and

similarly for the solid). Deﬁning the ﬂuid mesh nested within the parametric solid mesh, simple afﬁne mappings may be used to deﬁne the relationship

between these meshes, denoted A

c

,j

for the surface element

^

c

c;j

. As the ﬂuid mesh is deﬁned relative to the master element surfaces of the solid mesh, a

natural choice for f

1

N

g are simply the master elements themselves; however, this is not essential (as seen for h

1

). In this case, three conforming

regions are selected – denoted R(h

1

) for h

1

, etc. – where R(h

1

) is composed of the top two elements of the solid, while R(h

2

) and R(h

3

) are simply the

master surface elements of the solid. Each ﬂuid element is then mapped into these regions, respecting continuity within each region. That is, the afﬁne

mapping for two neighboring surfaces, A

c

,1

and A

c

,2

, respects the continuity within T

h

ð

C

1

Þ (i.e. two points

g

1

and

g

2

near the common edge remain along

that edge in h

3

– where n

1

¼ A

c

;1

g

H

1

; n

2

¼ A

c

;2

g

H

2

, and g

H

1

; g

H

2

are the corresponding points of

g

1

,

g

2

in the master element surface).

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7579

M

h

¼ z : T

h

ðNÞ!R

d

z 2 C

ðT

h

ðNÞÞ; zj

#

2 P

p

ð#Þ½

d

;

8

# 2T

h

ðNÞ

no

;

where

C

ðT

h

ðNÞÞ :¼ z : T

h

ðNÞ!R

d

z 2CP

C

C

i

N

ðtÞT

h

ð

NÞ

hi

d

;

8

i 2½1; 2;

8

t 2 I

:

The selection of M

h

ð

N

Þ based on a visible embedding of the ﬂuid or solid trace space (or the trace itself) allows easy construc-

tion of the discrete space. Implicit to this construction is that,

j

ðkÞ 6 max

j

ð

v

Þ;

j

ðuÞ½: ð37Þ

In general, we choose M

h

ð

N

Þ nested inside the richest trace space (though this choice may not always be obvious as it

depends on polynomial degree and discretization).

4.1.4. Lagrangian/ALE mapping discretizations

The ﬂuid and solid problems are both set on moving domains (ALE or Lagrangian), which vary with time. In Sections 3.1

and 3.2, domain motion was integrated into the ﬂuid and solid mechanical formulations by introducing a reference domain,

K

i

, that may be mapped to the physical domain,

X

i

, i = 1, 2. The reference to physical mappings are also given by polynomials

so that the mapping, at any time t

n

, n 2 [0,N], satisﬁes,

P

X

i

K

i

ðt

n

Þ2K

h;

j

i

ðPÞ

ð

K

i

; T

h

Þ; K

h;

j

i

ðPÞ

ð

K

i

; T

h

Þ :¼ S

h;

j

i

ðP Þ

ð

K

i

; T

h

Þ

hi

d

: ð38Þ

We note that, for the ﬂuid

j

1

ð

K

Þ 6

j

1

ðPÞ and for the solid,

j

2

ðP

K

Þ¼max

j

2

ðPÞ;

j

ðuÞðÞ.

5

A piecewise continuous interpolation

of the mapping is used through time,

P

X

i

K

i

ð; tÞ :¼

t

n

t

D

n

t

P

X

i

K

i

ð; t

n1

Þþ

t t

n1

D

n

t

P

X

i

K

i

ð; t

n

Þ; t 2ðt

n1

; t

n

;

D

n

t

¼ t

n

t

n1

; ð39Þ

resulting in continuous piecewise linear displacements (for the solid) and piecewise constant domain velocities (for the

ﬂuid) through the interval I.

As the ﬂuid and solid systems are coupled about their respective discretizations of

C

C

, the domain movements are inex-

tricably linked. While the solid map is given by the displacement ﬁeld u

h

, the ﬂuid map is arbitrary so long as it adheres to

the movement on the coupled boundary. The selection of this map satisﬁes the weak Laplacian problem [58,54],

a

t

n1

ðd

h

; y

h

Þ¼0;

8

y

h

2 V

h

0

; ð40Þ

for d

h

2 V

h

and d

h

¼

D

n

t

v

h;n

on

C

1

. The ALE mapping is then updated,

P

X

1

K

1

ðt

n

Þ¼d

h

þP

X

1

K

1

ðt

n1

Þ;

while the solid mapping is updated by the computed displacement u

h,n

,

P

X

2

K

2

ðt

n

Þ¼u

h;n

þP

X

2

K

2

ðt

n1

Þ:

4.2. Discrete weak form

Following the continuous weak form in Deﬁnition 4, and using the discrete conservative ALE Navier–Stokes system (Eqs.

(14a) and (14b)) along with the discrete quasi-static ﬁnite elasticity system ((26a) and (26b)), the discrete coupled system

follows directly from Section 3.4.

Find ð

v

h

; u

h

Þ2Y

h

I;D

; ðp

h

;

u

h

; k

h

Þ2Z

h

such that, for all n =1,...,N

I

, X

h,n

=(v

h,n

, u

h,n

) and Z

h,n

=(p

h,n

; u

h,n

, k

h,n

) satisfy Eq. (41)

on [t

n1

,t

n

]=I

n

I, for any ðy

h

1

; y

h

2

Þ¼Y

h

2 Y

h

0

and ðq

h

1

; q

h

2

; q

h

Þ¼Q

h

2 Z

h

.

A

I

n

ðX

h;n

; Y

h

ÞþB

I

n

ðZ

h;n

; Y

h

Þ¼F

I

n

ðY

h

Þ; ð41aÞ

B

I

n

ðQ

h

; LX

h;n

Þ¼0; ð41bÞ

where the operators – A

I

n

; B

I

n

and F

I

n

– are deﬁned in Eqs. (30)–(32), and the discrete spaces – Y

h

I;D

; Y

h

0

, and Z

h

– follow as

the discrete variants of those deﬁned in Section 3.4.

4.3. Solving the global system

Let f be a functional, where f is the subtraction of the equations in the coupled system (41), i.e.

5

For consistency, if the ﬂuid j

1

ðP

K

Þ < jðvÞ, we must reduce the degree of the velocity space near Dirichlet boundaries where no ﬂux of ﬂuid is required.

7580 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

f X

n

; YðÞ:¼ A

I

n

ðX

h;n

; Y

h

ÞþB

I

n

ðZ

h;n

; Y

h

ÞF

I

n

ðY

h

ÞB

I

n

ðQ

h

; LX

h;n

Þ; ð42Þ

where X

n

=(X

h,n

, Z

h,n

) are the state variables and Y =(Y,Q) the test functions. Consequently, the coupled system must satisfy,

for any choice Y 2 Y

h

0

Z

h

,

f ðX

n

; YÞ¼0:

Due to the ﬁnite dimension of the test space, Y

h

0

Z

h

, and linear dependence of f on Y, the constraint can be satisﬁed by

ensuring f = 0 for each basis function in the velocity, displacement, pressure and Lagrange multiplier spaces. The result is

a vector of constraints (one for each basis function), which may be assembled into a vector function, F , that satisﬁes,

FðX

n

Þ¼0: ð43Þ

Eq. (43) may be solved using an iterative approach, such as the global Newton–Raphson method [22,19]. For convenience, let

X

n

be deﬁned by the weighted sum of all basis functions for all state variables, i.e.

ðX

n

Þ

T

¼

v

n

W

X

¼

v

h;n

u

h;n

p

h;n

u

h;n

k

h;n

0

B

B

B

B

B

B

@

1

C

C

C

C

C

C

A

;

v

n

¼

V

n

U

n

P

n

u

n

k

n

0

B

B

B

B

B

B

@

1

C

C

C

C

C

C

A

; W

X

¼

W

v

W

u

W

p

W

u

W

k

0

B

B

B

B

B

B

@

1

C

C

C

C

C

C

A

; ð44Þ

As a result, the iterative update to the solution X

n

is accomplished by updating the coefﬁcient vector,

v

n

. Given an initial

guess,

v

n,0

, the approximate solution X

n

at the nth time step is,

ðX

n

Þ

T

¼ lim

k!1;k2N

þ

v

n;k

W

X

;

v

n;k

:¼

v

n;k1

þ

a

k

d

v

n;k

;

where the scalar vector d

v

n,k

is the Newton update and

a

k

a scalar parameter. The Newton update, d

v

n,k

, is selected as the

solution to Eq. (45), where

r

v

n;k1

is the gradient with respect to each scalar coefﬁcient of

v

n;k1

; r

v

n;k1

FðX

n;k1

Þ the Jacobian,

and X

n,k1

=(

v

n,k1

W

X

)

T

.

r

v

n;k1

FðX

n;k1

Þd

v

n;k

¼FðX

n;k1

Þ: ð45Þ

Having solved Eq. (45) for d

v

n,k

, the scalar parameter,

a

k

, is selected to ensure a monotonic decrease in the residual, i.e.

max

a

k

2ð0;1

jF ðy

n;k

Þj < jF ðy

n;k1

Þj; ð46Þ

measured in the l

2

-vector norm, jj.

4.3.1. Matrix system

Each iteration of the Newton–Raphson procedure requires that the linear system seen in Eq. (45) be solved for the update

vector. This is solved by approximating the Jacobian by the block matrix A

n;k

,

r

v

n;k1

FðX

n;k1

ÞA

n;k

:¼

A

n;k

ð

e

B

n;k

Þ

T

B

n;k

0

!

; ð47Þ

resulting in the update vector,

d

v

n;k

¼ðA

n;k

Þ

1

FðX

n;k1

Þ: ð48Þ

The block matrix is composed of components A

n,k

, B

n,k

, and

e

B

n;k

as shown in Eq. (47). These block components represent the

contributions of the operators A

I

and B

I

introduced in Problem 4. As the operators, A

I

and B

I

, are composed of ﬂuid and

solid terms, so to are the block components, i.e.

A

n;k

¼

A

n;k

v

0

0 A

n;k

u

!

; B

n;k

¼

B

n;k

v

0

0 B

n;k

u

C

v

C

u

0

B

@

1

C

A

; ð

e

B

n;k

Þ

T

¼

ðB

n;k

v

Þ

T

0 ðC

v

Þ

T

0 ð

e

B

n;k

u

Þ

T

ðC

u

Þ

T

!

:

The ﬂuid matrix A

n;k

v

can be written in d d distinct blocks such that,

A

n;k

v

¼

A

n;k

v

;11

A

n;k

v

;1d

.

.

.

.

.

.

.

.

.

A

n;k

v

;d1

A

n;k

v

;dd

0

B

B

B

@

1

C

C

C

A

; ð49Þ

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7581

and A

n;k

v

;lm

is an N

v

N

v

matrix deﬁned as (where integrals on

X

(t

n

) are approximated using the domain update determined by

the previous Newton iterate),

A

n;k

v

;lm

ij

¼

q

m

t

n

ðw

j

v

e

m

; w

i

v

e

l

Þþ

q

Z

t

n

t

n1

c

s

ð

v

h;n;k1

w

h;n;k1

; w

j

v

e

m

; w

i

v

e

l

Þd

s

þ

q

Z

t

n

t

n1

c

s

ðw

j

v

e

m

0;

v

h;n;k1

; w

i

v

e

l

Þd

s

þ

l

Z

t

n

t

n1

a

s

ðw

j

v

e

m

; w

i

v

e

l

Þd

s

: ð50Þ

Further, the ﬂuid and solid pressure blocks, B

n;k

v

and

e

B

n;k

u

, are 1 d block matrices where each component, B

n;k

v

;l

, is given by,

B

n;k

v

;l

ij

¼

Z

t

n

t

n1

b

s

ðw

i

P

; w

j

v

e

l

Þd

s

; ð51Þ

and similarly for the solid component

e

B

n;k

u;l

.

The solid system produces sub-components, A

n;k

u;lm

, for which analytic computation of the true Jacobian contributions is

complicated by the inherent non-linearity of many hyperelastic laws. Further, due to the generality required of the hyper-

elastic law to aptly model varying materials, a more generally applicable method is preferred.

Consequently, the d

2

solid sub-components to A

n;k

u

are computed using a ﬁnite difference approximation. Each N

u

N

u

subcomponent, A

n;k

u;lm

, may be expressed as,

A

n;k

u;lm

ij

¼

1

2

Z

I

n

s

s

ðu

h;þ

; w

i

u

e

l

Þs

s

ðu

h;

; w

i

u

e

l

Þd

s

þ

1

2

Z

I

n

b

þ

s

ð

u

h;n;k1

; w

i

u

e

l

Þb

s

ð

u

h;n;k1

; w

i

u

e

l

Þd

s

; ð52Þ

where 0 <

1 is a differencing parameter (typically

=10

4

h), b

+

and b

denote perturbations of the current domain (see

Remark 4), and for any

s

2 [t

n1

,t

n

]),

u

h;

ð

s

Þ :¼ u

h;n;k1

ð

s

Þ

s

t

n1

D

n

s

w

j

u

e

m

: ð53Þ

Remark 4. As the solid follows the Lagrangian formulation, integrals and gradients on

X

are functions of u

h

. Hence, to

estimate the Jacobian, perturbations of u

h

must be accounted for. Denoting the perturbed gradient and mapping Jacobian as,

G

s

:¼

r

g

u

h;

þ I

T

r

g

; J

s

:¼ det

r

g

u

h;

þ I

;

the computation of s

s

ðu

h;þ

; w

i

u

e

l

Þ, for example, utilizes G

þ

s

and J

þ

s

in place of G

s

and J

s

(and similarly for b

+

).

Considering the block contributions to the Jacobian based on the solid pressure variables,

B

n;k

u;l

ij

¼

1

2

m

þ

t

n

ðw

i

u

; 1Þm

t

n

ðw

i

u

; 1Þ

hi

;

e

B

n;k

u;l

ij

¼

Z

I

n

b

s

ðw

i

u

; w

j

u

e

l

Þd

s

; ð54Þ

where, following Remark 4,

m

t

n

ðw

i

u

; 1Þ¼

Z

K

2

w

i

u

J

t

n

d

g

:

Lastly, the Lagrange multiplier is introduced by matrices C

v

and C

u

. These matrices are composed of d d block matrices C

v

,lm

and C

u,lm

, where,

C

n

v

;lm

ij

¼

Z

t

n

t

n1

l

s

ðw

i

k

e

l

;

^

w

j

v

e

m

Þd

s

; C

u;lm

ij

¼

Z

t

n

t

n1

l

s

ðw

i

k

e

l

;

^

w

j

u

e

m

Þd

s

: ð55Þ

Notice that this deﬁnition of A

n;k

, and its sub-components, is merely an approximation of the Jacobian. To accommodate a

general solid stress, a differencing scheme is adopted to give an accurate estimate of the solid contribution. Further, a ﬁrst

order approach is taken to incorporate the inﬂuence of ﬂuid domain movement, as all integral terms for the ﬂuid contribu-

tion are dependent on the ALE mapping at the k 1 Newton iterate, ignoring any dependence on v

h,n,k

. This is compensated

for in the global Newton Scheme by updating the ALE motion in the selection of

a

k

(as we now have v

h,n,k

to evaluate Eq. (46)

at the kth iterate) and does not seem to affect convergence.

5. Stability estimates

Assuming solutions to the coupled system exist and are unique, a priori energy estimates demonstrate energy conserva-

tion of the scheme and ensure well-posedness by bounding the solutions entirely by given data. Moreover, as energy can

only be transferred at the coupled interface (not created), the added Lagrange multiplier must not contribute to the energy

bound. Either creation or destruction of energy by the Lagrange multiplier yields a scheme incapable of energy conservation.

7582 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

With the method outlined, the discrete weak form coupled problem can be shown to satisfy Lemma 1, ensuring stability and

mechanical energy conservation of the weak system (note, kk

n,

X

and kk

n,

X

are the norms and semi-norms on the nth Hilbert

space [60], and otherwise kk is a norm on the space given in the subscript).

Lemma 1. Consider the discrete coupled system (41), with homogeneous Dirichlet boundary conditions. Suppose the operator s

satisﬁes the Gårding-type inequality (Remark 5 of Appendix A), then the discrete solutions

v

n

h

and u

n

h

satisfy,

a

1

4

ku

n

h

k

2

1;

K

2

þ

.

k

v

n

h

k

2

0;

X

1

ðt

n

Þ

þ

a

a

Z

t

n

0

k

v

h

k

2

1;

X

1

ð

s

Þ

d

s

6 C

1

ku

0

h

k

2

1;

K

2

þk

v

0

h

k

2

0;

X

1

ð0Þ

þkf

1

k

2

L

2

ðI;H

1

ð

X

1

ÞÞ

þkf

2

k

2

L

1

ðI;H

1

ðK

2

ÞÞ

þkf

2

k

2

H

1

ðI;H

1

ðK

2

ÞÞ

þ C

s

;

for any n 2 [1,N], where C

1

2 R

þ

is a constant independent of u and v. Stability is not guaranteed (as

X

1

depends on the solution).

However, if f

1

is spatially constant on

K

1

, and the data f

1

2 L

2

(I; L

2

(

K

1

)) and f

2

2 L

1

(I; H

1

(

K

2

)) \ H

1

(I;H

1

(

K

2

)), the energy

estimate yields unconditional stability.

Proof. See Appendix A. h

6. Results and discussion

Presented in this section are four tests aimed at demonstrating stability, robustness, and convergence of the coupling

scheme for a variety of mesh types (tetrahedral/hexahedral, linear/curvilinear) and function spaces (general Taylor–Hood

elements [28,15]). Using a simple artiﬁcially coupled ﬂuid–ﬂuid system, the importance of the third Lagrange multiplier

space is highlighted. Further, the inﬂuence of non-linearities on the problem, and its impact on error is addressed. Subse-

quently, a convergence analysis was conducted in two ﬂuid–solid mechanical systems. Based on this analysis, the potential

beneﬁt of using non-conforming grids on error reduction is elucidated. As many practical applications have a much stronger

need for non-conformity, the method is assessed in a simpliﬁed elliptical model of the left-heart.

The numerical scheme outlined in Section 4 was coded in a parallel MPI-based Fortran90 code. The linear systems were

solved using MUMPS

6

, a Multifrontal Massively Parallel Solver [1]. Extensive veriﬁcation – progressively testing the ﬂuid (lin-

ear, non-linear, transient, and ALE) and solid mechanical frameworks – and convergence analyses on both ﬂuid and solid

mechanical systems can be found in [54].

6.1. Numerical behavior of the third saddle point

For linear systems (or each linearized Newton Step), the solvability of Eq. (48) depends on the invertibility of the block {A}

and X ¼ BA

1

e

B

T

– the so-called Schur complement [41,70,23]. In the coupled mechanical system, the introduction of the

Lagrange multiplier, k, alters the structure of the Schur complement, which must maintain full rank for the solutions to

the system to be unique – a direct result of the inf–sup condition (see Sections 3.3 and 4.1.3). Moreover, beyond uniqueness,

understanding the coupling effect that the Lagrange multiplier introduces on problems with non-conforming grids, different

interpolations, etc. is crucial. Analyzing the inﬂuence of these factors beyond simple linear theory within non-linear systems

is vital for practical implementation and use of the method.

To assess the effects of the added Lagrange multiplier, a ﬂuid problem was solved and subsequently partitioned in two

coupled domains as seen in Fig. 3. For this problem, the system produces an internal vortex, whose symmetry is lost with

increased Reynolds number ðRÞ. Using identical grids with equal interpolations on either side (in this case P

2

P

1

), a num-

ber of Lagrange multiplier spaces may be tested, including P

0

; P

1

, discontinuous P

1

; P

2

; P

2

T

(which, relating to the trace on

the boundary, omits those nodal functions along Dirichlet boundaries), and P

3

.

As expected, P

0

; P

1

and P

2

T

all yielded solutions, while discontinuous P

1

; P

2

and P

3

over constrained the system, leading

to spurious oscillations in the Lagrange multiplier. However, contrary to intuition, P

0

; P

1

and P

2

T

– despite widely varying

richness – produce nearly equivalent degrees of coupling. This effect may be attributed to the apparent symmetry of the

spaces, each of which have identical spurious modes.

For a more thorough examination, consider the same problem with the ﬂuid on the left far richer (containing 512 hexa-

hedral elements and P

3

P

2

interpolation of velocity/pressure) than that on the right (containing 64 hexahedral elements

and P

2

P

1

interpolation of velocity/pressure). In this case, many more embeddings of spaces are possible, including the

Lagrange multiplier spaces P

k

embedded on the right grid, and P

k

embedded on the left grid (where 0 6 k 6 4 were at-

tempted). The results of this analysis are quantiﬁed in Table 1 and displayed in Fig. 1.

As anticipated, embeddings of the Lagrange multiplier in identiﬁable subspaces of the coarser right grid (selections of

k = 0,1,2) all produced solvable systems. Further, elimination of Dirichlet nodes using P

2

T

versus P

2

did not inﬂuence solv-

ability due to the richness of the left space. While difﬁcult to prove conclusively, choices of k = 3,4 also produced solutions,

6

Available through http://graal.ens-lyon.fr/MUMPS/avail.html

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7583

which is, again, most likely due to the richness of the right space. However, from Table 1, it is seen that neither of these

spaces perform particularly well at coupling the two problem domains. Further, the increase in polynomial order beyond

that of either trace space provides no beneﬁt.

In contrast, choosing the Lagrange multiplier as an embedding in the richer left grid, selections of k = 0,1,2 all produce

solvable systems. In this case, the elimination of Dirichlet nodes using P

3

T

versus P

3

is critical, as the later yields an insolvable

system (as does P

4

). Moreover, embedding in the richer left grid, in all cases, produces vastly superior coupling results (see

Table 1 and Fig. 4). From the linear study [57], it was suggested that the optimal space is in fact embedding the Lagrange

multiplier space in the left grid using P

3

T

functions (removing all Dirichlet nodes). Indeed, from Table 1 and Fig. 4, this choice

results in vastly superior coupling results as P

3

T

includes the trace of all functions along the boundary in each domain.

The performance of P

3

T

is further distinguished by increased R, seen in Table 2 and Fig. 5, where error remains below

machine precision under a 100-fold increase in the Reynolds number. Perhaps even more signiﬁcantly, the non-linear con-

vergence of the method is lost when the Lagrange multiplier space is too weak relative to the kinematic trace spaces.

Many other methods for non-conforming domains, such as those proposed in [9,8], rely on prescribing functions at nodes

to induce coupling. Analysis of these types of approaches have suggested that such nodal-based coupling is not optimal [11].

If the kinematic spaces differ signiﬁcantly either due to mesh size or polynomial order, suboptimal coupling (particularly for

non-linear systems) can negatively impact both stability and accuracy – effectively limiting the degree of non-conformity. In

contrast, the method presented here is seen to be stable even when the ﬂuid-to-solid element ratio (the average number of

ﬂuid elements paired to a solid element) is high (as seen in the ﬁnal example).

Fig. 3. Domain diagrams and boundary conditions (BC). Here the triple arrows denote ﬂow/inﬂow directions, circles denote zero traction, hashes denote no

slip Dirichlet conditions, and jagged lines denote coupling along marked boundaries. (Top Left) Stress driven cavity model. Tangential tractions are applied to

the top/bottom. All side walls are no slip Dirichlet boundaries. The domain is artiﬁcially broken in half along the marked coupling boundary. (Top Right) Fluid

ﬁlled hyperelastic box model. A displacement is prescribed (Eq. (56)) along to bottom of the box. The ﬂuid (

X

1

) – entirely contained in the solid (

X

2

) – begins

with an initial downward velocity. The zero traction is applied on all outer walls of the box. (Bottom Left) Hyperelastic channel model. The solid (

X

2

) has

coupled/zero traction BCs on the inner/outer channel walls. The back plane of the solid is held ﬁxed, while the front is allowed to move in plane (zero

penetration and tangential traction). The ﬂuid (

X

1

) is driven into the system by an axial traction. (Bottom Right) Elliptical Left-Heart Model. Held ﬁxed on the

upper planar surface, the solid (

X

2

) has coupled/zero traction BCs on the inner/outer walls. The ﬂuid (

X

1

) is driven into the elliptical model by a prescribed

Dirichlet function (red boundary). The remaining head of

X

1

is held ﬁxed by no slip Dirichlet conditions. (For interpretation of the references in colour in

this ﬁgure legend, the reader is referred to the web version of this article.)

Table 1

L

2

norm error on the coupling interface for varied Reynold’s Number, R (taken as the ratio of density to

viscosity), and orders of interpolation for the Lagrange Multiplier space embedded in the right domain

(denoted R) and left domain (denoted L).

M

h

RðR ¼ 1Þ

RðR ¼ 10

2

Þ LðR ¼ 10

2

Þ

Error with M

h

of kv

L

v

R

k

0,

X

1 1.62 10

4

1.27 10

1

9.52 10

3

2 2.03 10

4

1.41 10

1

4.09 10

3

3 1.30 10

3

2.12 10

1

9.99 10

16a

4 1.28 10

3

2.16 10

1

Singular

a

P

3

T

results reported.

7584 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

6.2. Convergence of the ﬂuid–solid mechanical system

In our previous paper underpinning this work [57], optimal error estimates were shown – both theoretically and numer-

ically – for the coupled mechanical system in the linear setting. To verify convergence properties of the coupled system in a

non-linear context, two problems are considered (see Fig. 3). In the ﬁrst problem, a hyperelastic box with initial momentum,

is quickly decelerated leading to complex internal ﬂows and oscillatory behavior (see Figs. 6 and 8). The second problem con-

siders ﬂow through a hyperelastic channel. Though this problem also exhibits oscillatory behavior, results are examined once

the model reaches steady-state (gauged as maximal movement of the coupling interface of 10

8

). In both cases, the passive

solid body is modelled as a hyperelastic Neo-Hookean material (with C

m

= 5), and the ﬂuid as a Newtonian (with

q

=1,

l

= 0.005 for the hyperelastic box and

l

= 0.0075 for the hyperelastic channel).

The ﬁrst model is that of impact in a ﬂuid ﬁlled hyperelastic box, see (see Fig. 3). Starting with an initial downward veloc-

ity of v = (0,0,1)

T

, the momentum of the system is then curtailed by the exponential deceleration of the bottom surface of

the solid (seen in Eq. (56)). As the ﬂuid is entirely contained within the solid, this deceleration causes complex internal ﬂows.

Fig. 4. Midplane view of coupled solutions using coupling Lagrange multiplier spaces using (top) P

1

embedded in right domain, (bottom) P

3

T

embedded in

the left domain. White lines denote element boundaries. Displayed are magnitudes of velocity ranging from zero (red) to 3/10 (blue). Nine contour bands

are evenly distributed through this range, labelling all velocity magnitudes in

3

10

ðn=N 0:02Þ;

3

10

ðn=N þ 0:02Þ

black (for n =1,..., 9 and N = 10). (For

interpretation of the references in colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 2

L

2

norm error on the coupling interface for varied Reynold’s Number, R (taken as the ratio of density to viscosity),

and orders of interpolation for the Lagrange Multiplier space embedded in the left domain.

R=M

h

P

1

P

2

P

3

T

Error with R of kv

L

v

R

k

0,

X

1 9.39E05 4.12E05 3.90E18

10 9.39E04 4.12E04 4.86E17

50 4.71E03 2.06E03 9.99E16

100 9.52E03 4.09E03 9.99E16

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7585

u ¼ð0; 0; uÞ

T

; uðx; tÞ¼

Z

t

0

j

v

0

je

b

s

d

s

; x 2

D

2

: ð56Þ

The rate at which the solid face slows is given by the parameter b.Asb ? 0, the velocity of the solid face approaches

(0,0, 1)

T

for all time, while as b ? 1, the velocity of the solid decays instantly (simulating true impact). As simulating true

impact causes a near singularity in the temporal boundary conditions – in turn, complicating convergence analysis – a milder

selection of b was used (see Fig. 8).

As the solid face slows, momentum in the ﬂuid is, in part, dissipated by viscous effects. The remaining momentum is

transformed into potential energy stored within the solid. Potential energy is seen by the displacement of the body, which

distends symmetrically on all sides and pulls the upper surface downward. As momentum decays, the potential energy of the

solid is returned to the ﬂuid, causing inward ﬂow along the sides and upward ﬂow at the top. Once again, the momentum of

the ﬂuid causes the solid to distend (this time inward on the sides and outward on the top), and the system continues to

oscillate until all energy has been dissipated by viscous effects (see Fig. 8). In the limit, as t ? 1, the solution to the system

is a stagnant ﬂuid with the hyperelastic box with u = (0,0,—v

0

—/b)

T

.

To assess convergence of the coupled model, ﬁve (ﬂuid/solid) tessellations with (64/96 ), (216/216 ), (512/768 ), (4095/

6144 ), and (20,736/13,824 ) hexahedral elements were used. The velocity/pressure in the ﬂuid and displacement/pressure in

the solid were both interpolated using P

2

P

1

Taylor–Hood functions. The spatial map of the ﬂuid–solid system was taken

as trilinear and triquadratic, respectively. The third Lagrange multiplier space was selected as an embedding of the ﬂuid trace

space using P

2

functions (note that, as no Dirichlet conditions are present P

2

yields the same space as P

2

T

). Solutions for the

ﬁrst four discretizations were compared at t = 0.1 with the ﬁnest grid solution (ran with

D

s

1 10

4

). Solutions on the

various grids are seen in Fig. 6, while the measures of error are supplied in Table 3.

Fig. 5. Growth of error with increasing Reynolds Number, R. Colors indicate the error between ﬂuid velocity computed in each domain for (from left to

right) R¼f1; 10; 50; 100g choosing the Lagrange multiplier space to be embedded in the left domain and interpolated using P

1

functions.

Fig. 6. Convergence of solutions to the ﬂuid ﬁlled hyperelastic box model on various computational domains, ranging from coarsest (left) to ﬁnest (right).

Magnitude of the solid is colour blue to seafoam green with contour bars to illustrate changes in the solution. Velocity magnitude of the ﬂuid is also shown,

ranging from blue to red. (For interpretation of the references in colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 3

Maximum L

2

norm and H

1

semi-norm between ﬂuid and solid domains at time t = 0.1. Columns indicate the norm values on various grid reﬁnements, while

rows scale temporally as powers of h. The ﬁnal column gives the optimal scaling based on interpolation theory.

D

t

/h 1/2 1/3 1/4 1/8 O

kk

0

kk

0

kk

0

kk

0

Convergence of max(kv v

h

k

0,

X

, ku u

h

k

0,

X

)

h 1.09 10

3

7.87 10

4

5.98 10

4

3.13 10

4

h

h

2

1.09 10

3

5.46 10

4

2.99 10

4

6.45 10

5

h

2

h

3

1.09 10

3

3.81 10

4

1.41 10

4

1.00 10

5

h

3

D

t

/h 1/2 1/3 1/4 1/8 O

jj

1

jj

1

jj

1

jj

1

Convergence of max(jv v

h

j

1,

X

, ju u

h

j

1,

X

)

h 1.34 10

2

8.32 10

3

6.31 10

3

3.42 10

3

h

h

2

1.34 10

2

6.79 10

3

4.19 10

3

1.75 10

3

h

2

h

3

1.34 10

2

5.97 10

3

3.41 10

3

1.72 10

3

h

2

7586 D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

The hyperelastic channel model (see Fig. 3) begins with a quiescent ﬂuid and solid, which is subjected to an axial pressure

gradient. With time, the channel dilates; however this process is heavily inﬂuenced by the momentum of the ﬂuid, causing

over and under dilation relative to the eventual steady-state (reached after 20 s). A unit inward traction was applied to one

end of the channel, causing a peak velocity of 2 and R130. The approximate percentage strain in the solid mechanical

model (taken as the maximal displacement over inner channel dimension) was 50%.

To assess the convergence of the model for varying levels of non-conformity, four ﬂuid and four solid tessellations with

{4,32,256,2048} and {12,40,320,2560} hexahedral elements were coupled in different combinations to approximate the

solution. Both ﬂuid and solid bodies were interpolated using P

2

P

1

Taylor–Hood functions. In each case, the third Lagrange

multiplier space was selected as an embedding of richest trace space using P

2

T

functions. Further, both domains used triqua-

dratic spatial maps. Solutions on the various grids are shown in Fig. 7. The error, measured relative to the ﬁnest grid, is

reported in Table 4.

Convergence of the method is demonstrated in both the hyperelastic box and channel examples (see Tables 3 and 4). In

these examples, the interpolation spaces are P

2

P

1

; which, under optimal convergence conditions [6,60,57], should lead to

error convergence scaling like h

2

and h

1

in the L

2

and H

1

-norms, respectively. However, these results are limited by the La-

grange multiplier spaces. Indeed, within the displacement/velocity spaces there are projections (such as the H

1

-projection)

which converge like h

3

and h

2

in the L

2

and H

1

-norms. As the method outlined takes an implicit Euler approach to temporal

discretization, error in the hyperelastic box (see Table 3) shows that, for appropriate scaling of the temporal discretization

parameter, optimal error convergence rates are observed. Indeed, these results show convergence rates closer to that of the

optimal projection, suggesting it is nearly weakly divergence free.

While convergence in the hyperelastic box model was assessed by further reﬁning an initial model, in the hyperelastic

channel model, convergence is considered for different mesh discretizations. Certainly, optimal convergence is again seen

in Table 4 along the diagonal (corresponding to equivalent reﬁnement of the entire model). However, Table 4 also demon-

strates the utility of the non-conforming domains, as the error stems predominantly from error in the ﬂuid model. Examining

the last row of Table 4 the model coupling the coarsest solid and ﬁnest (excluding the benchmark grid) ﬂuid produces an H

1

error of 1.14 10

1

, while coupling the ﬁnest ﬂuid and solid grids (excluding the benchmark grid) – effectively increasing

the system size 215% – gives nearly equivalent error. As a result, using varying reﬁnement provides a much more efﬁcient

means to solve the system, while maintaining coupling accuracy to machine precision.

6.3. Elliptical left-heart model

To demonstrate the functionality of the method for more complex problems, both in terms of non-conformity of the inter-

face as well as the non-linearity of the system, a simpliﬁed elliptical model of the left ventricle of the heart is considered (see

Fig. 3). In this model, ﬂow is driven into a hyperelastic chamber, simulating the conditions common under diastolic ﬁlling of

the left ventricle. The ﬂuid is modelled as Newtonian (with a R1; 000) on an anisotropic linear tetrahedral grid (using

P

2

P

1

interpolation of velocity and pressure) consisting of 21,000 elements. The solid, an anisotropic Neo-Hookean mate-

rial is modelled on a curvilinear hexahedral grid (using P

3

P

1

interpolation for displacement and pressure) consisting of

180 elements. The quasi-static formulation is used in this example. This may be justiﬁed by a simple dimensional analysis

which shows that the rate of momentum in the ﬂuid dominates that observed in the solid.

The third Lagrange multiplier is selected as an embedding in the richer ﬂuid space using P

2

T

. While the coupling interface

from the ﬂuid side is composed of 1920 triangular elements, the solid side consists of only 180. Due to the anisotropic nature

of the tetrahedral grid, the ratio of ﬂuid-to-solid elements ranges from 4 to 64. Results of this simulation at various time

points in the diastolic period are shown in Fig. 9.

While the previous examples demonstrate the methods functionality, the elliptical left-heart model shows that this

behavior holds under even more complex conditions. Not only are the inherent interpolation spaces different but the discret-

ization sizes widely differ. The model problem also exhibits much more complex dynamics due to the increased non-

linearity of the ﬂuid model. Under these conditions, the method is still seen to perform well, yielding precise (to machine

precision) coupling.

Fig. 7. Convergence of upper quadrant solutions to the steady-state hyperelastic channel model at different grid resolutions: (Left) coarsest ﬂuid and solid

models, (Middle) ﬁnest ﬂuid and coarsest solid models, (Right) benchmark ﬁne grid solution. In the ﬂuid, isoplanes of velocity magnitude are shown ranging

from 0.2 (translucent green-yellow) to 1.6 (red) in increments of 0.2. The solid displacement magnitude is plotted ranging from 0 (blue) to 0.1 (seafoam

green). Nine contour lines, marked in black, are evenly distributed in the range [0,0.14]. (For interpretation of the references in colour in this ﬁgure legend,

the reader is referred to the web version of this article.)

D. Nordsletten et al. / Journal of Computational Physics 229 (2010) 7571–7593

7587

7. Conclusion

In this paper, a monolithic technique was introduced for problems of coupled mechanics. This technique, through the use

of a third Lagrange multiplier, allows for accurate coupling (in many cases to machine precision) of highly non-conforming

domains, while preserving stability properties typically enjoyed by other monolithic schemes. The method is also shown to

converge optimally based on error estimates derived for simpler linear systems. The freedom of this non-conformity allows

for the coupling of function spaces and tessellations independently tailored for the physics of the model. For example, in bio-

mechanical systems, such as the sample system of Section 6.3, where the characteristic behavior of ﬂuids and solids demand

disparate discretization schemes, non-conformity of the coupling bodies provides an accurate and efﬁcient numerical ap-