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6th European Conference on Computational Mechanics (ECCM 6)

7th European Conference on Computational Fluid Dynamics (ECFD 7)

11 – 15 June 2018, Glasgow, UK

A VIRTUAL CONTROL, MESH-FREE COUPLING

METHOD FOR NON-COINCIDENT INTERFACES

PAUL KUBERRY1, PAVEL BOCHEV1AND KARA PETERSON1

1Computational Mathematics

Sandia National Laboratories, MS-1320,

Albuquerque, NM 87185-1320, USA

{pakuber,pbboche,kjpeter}@sandia.gov

Key words: Mesh tying, interface problems, virtual Neumann controls, moving least-

squares

Abstract.

We present an optimization approach with two controls for coupling elliptic partial

diﬀerential equations posed on subdomains sharing an interface that is discretized inde-

pendently on each subdomain, introducing gaps and overlaps. We use two virtual Neu-

mann controls, one deﬁned on each discrete interface, thereby eliminating the need for

a virtual common reﬁnement interface mesh. Global ﬂux conservation is achieved by in-

cluding the square of the diﬀerence of the total ﬂux on each interface in the objective. We

use Generalized Moving Least Squares (GMLS) reconstruction to evaluate and compare

the subdomain solution and gradients at quadrature points used in the cost functional.

The resulting method recovers globally linear solutions and shows optimal L2-norm and

H1-norm convergence.

1 INTRODUCTION

Spatially non-coincident discrete interfaces having gaps and overlaps arise in multiple

modeling and simulation scenarios. One example is mesh tying where a complex geometric

domain is broken into multiple parts and each part is meshed separately; see, e.g., [4,

11, 1, 10]. Other examples include mortar methods [5, 12] for transmission problems

with curved interfaces. The bulk of the existing approaches are based on appropriate

extensions of traditional Lagrange multiplier formulations to non-coincident interfaces by

designating one of the discrete interfaces as a “master” and enforcing state continuity by

suitable projections onto that interface. Design of such methods is often accompanied by

theoretical and practical diﬃculties to ensure stable and accurate discrete formulations.

These diﬃculties have prompted examination of alternative formulations based on, e.g.,

least-squares principles [1], or nonstandard Lagrange multipliers [11] deﬁned as traces of

Raviart-Thomas elements. A fundamentally diﬀerent and promising approach for prob-

lems with non-coincident interfaces is based on coaching the mesh-tying problem into a

constrained optimization problem. Such formulations switch the roles of the coupling

Paul Kuberry, Pavel Bochev and Kara Peterson

conditions and the subdomain equations and transform the interface problem into a vir-

tual control problem in which the coupling conditions deﬁne the objective, the subdomain

equations deﬁne the constraints, and the interface ﬂux serves as a Neumann control.

Although the intent of the early work on such methods was to obtain non-standard

domain decomposition methods for matching subdomain interfaces [7, 6], the optimization

approach oﬀers some unique advantages in the context of spatially diﬀering interfaces. In

particular, treating the transmission condition as an optimization objective rather than

a constraint is better suited for non-coincident interfaces because it involves minimizing

rather than eliminating the state mismatch. While mathematically the latter requires

a unique interface to compute the exact diﬀerence between the states, the former only

depends on a reasonably good estimate of the mismatch.

To the best of our knowledge, the paper [9] is the ﬁrst application of these ideas for non-

coincident interface problems. The method in this paper uses standard C0piecewise linear

elements, a single Neumann control deﬁned on a virtual common reﬁnement interface and

mapped back to each discrete interface in a conservative manner, and linear extension of

the subdomain states to compute the state mismatch. The method conserves the global

interface ﬂux, is ﬁrst-order accurate in the H1-norm and second-order accurate in the

L2-norm. However, it does not recover globally linear solutions and construction of a

virtual common reﬁnement interface mesh can be complicated in three dimensions.

The paper [2] provides a further development of the optimization mesh tying approach,

which addresses some of this drawbacks. In this paper we use two virtual Neumann

controls, one deﬁned on each discrete interface, thereby eliminating the need for a virtual

common reﬁnement interface mesh. Global ﬂux conservation is achieved by including

the square of the diﬀerence of the total ﬂux on each interface in the objective. The

resulting method provably recovers globally linear solutions and shows optimal H1-norm

convergence. However, the L2-norm rate is suboptimal.

The culprit is the inclusion of terms comprising the mismatch between the ﬁnite element

ﬂux on one of the discrete interfaces and the extension of the ﬂux from the other interface.

These terms were required to obtain a well-posed optimization problem. However, because

the virtual control on each interface provides a Neumann condition for the respective

subdomain problem, optimization forces the control and the ﬁnite element ﬂux to be as

close as possible. This creates an accuracy bottleneck, as the ﬁnite element ﬂux is only

ﬁrst-order accurate.

Removal of this bottleneck requires a more accurate approximation of the discrete ﬂux

on each interface. One option is to use variational ﬂux recovery techniques; see, e.g., [3]

and [8]. However, using these approaches in our formulation would require assembly of a

consistent interface mass matrix and application of its inverse in the objective functional.

Because the inverse mass matrix is dense, the latter would have to be performed iteratively,

potentially rising the computational cost of the scheme. In addition, the recovered ﬂux

would still have to be transferred to the other interface in a way that does not diminish

its accuracy.

In this paper we consider a mesh-free alternative, which uses Generalized Moving Least-

Squares (GMLS) [13] to reconstruct high-order ﬂux approximation and perform the nec-

2

Paul Kuberry, Pavel Bochev and Kara Peterson

essary data transfers between the interfaces. This oﬀers several valuable advantages in

the context of non-coincident interfaces. By treating ﬁnite element degrees as scattered

data sites, the reconstruction process does not require any information about the under-

lying mesh structure and/or ﬁnite element basis functions, and the reconstruction point

is not constrained to be in an element. Because GMLS allows approximation of any lin-

ear functional from this data, we can combine the gradient reconstruction step with the

data transfer step in one single operation requiring the solution of a small, local weighted

least-squares problem. Since these problems are independently deﬁned their solution can

be performed in parallel.

2 Notation and technical preliminaries

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#

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Figure 1:Independent meshing of two subdomains sep-

arated by a curved interface σresults in two spatially non-

coincident interface grids σh

1and σh

2.

We consider a bounded open region Ω ⊂

Rd,d= 2,3 with an interface σ, which

splits the domain into non-overlapping sub-

domains Ω1and Ω2. Each subdomain is

independently partitioned into ﬁnite ele-

ments kn

icomprising conforming ﬁnite ele-

ment meshes Ωh

i,i= 1,2. We assume that

the meshes are constructed by creating ﬁrst

a polygonal approximation of each subdo-

main by placing nodes on their Dirichlet

boundaries Γi=∂Ωi\σ,i= 1,2 and the interface σ.

As a result, each mesh induces a ﬁnite element partition σh

i,i= 1,2 of σ, containing

the element sides sn

ithat have all their vertices in σ. The resulting interface grids σh

1and

σh

2in general are not spatially coincident and may have gaps and/or overlaps; see Fig. 1.

Given a mesh entity µwe denote the sets of all mesh vertices in µby V(µ), e.g.,V(σh

i)

are the vertices in the interface mesh σh

iand V(Ωh

i) is the set of all vertices in the

subdomain mesh Ωh

i.

We denote by H1(Ωi) and H1

Γi(Ωi) the standard Sobolev space of order one on Ωi,

i= 1,2, and its subspace of functions with vanishing trace on Γi, respectively. In this

paper we restrict attention to piecewise linear, bilinear or trilinear nodal C0elements

and denote the corresponding conforming ﬁnite element subspace of H1(Ωh

i) by Hh

i. We

assume that this space is endowed with a Lagrangian basis {Nk

i}. We will also need

the conforming subspace Hh

i,Γof H1

Γi(Ωh

i) and the space Hh

i,σ is spanned by the basis

functions associated with vertices on σh

i. The traces of the functions in this space form

the interface space Th

i, i.e., Th

i=Hh

i,σσi. The coeﬃcient vector of uh

i∈Hh

iis ui∈Rni,

where ni=|Hh

i|, the dimension of Hh

i.

2.1 Generalized Moving Least Squares

In this section we brieﬂy review and specialize the GMLS theory to our needs. This

theory provides a mesh-free approach to reconstruct the action of a linear functional τ

from a scattered set of samples of its argument [13, §4.3].

3

Paul Kuberry, Pavel Bochev and Kara Peterson

Here we restrict attention to functionals representing point values of a scalar ﬁeld or

its ﬁrst derivatives. Speciﬁcally, we consider the functionals τx(u) = u(x) and τx,i(u) =

∂iu(x), where x∈Rdis the evaluation point. GMLS allows to construct approximations

τh

xand τh

x,i of these functionals from point samples u={u(xi)}N

i=1 contained in a suitable

neighborhood of xand such that τh

xand τh

x,i are exact for all p∈Pm, where Pmis the space

of all multivariate polynomials of degree less than or equal to mwith basis p={pi}Q

i=1.

Selection of the data sites is accomplished by a smooth “window” function ω(x,y) whose

support is contained in a ball of radius . The size of this ball depends on the polynomial

degree mand the density of the data sites xi.

One can show that the GMLS approximations of these functionals are given by

τh

x(u) =

Q

X

k=1

ck(u,x)τx(pk) = c|(u,x)p(x),(1)

and

τh

x,i(u) =

Q

X

k=1

ck(u,x)τx,i(pk) = c|(u,x)∂ip(x),(2)

respectively, where the coeﬃcients c|(u,x) solve the following weighted least-squares

problem:

c(u,x) = argmin

b∈RQ

1

2(Rb−u)TW(x) (Rb−u).(3)

In (3) Ris N×Qmatrix with element Rij =pj(xi) and W(x) is N×Ndiagonal matrix:

W(x) = diag(ω(x,x1), . . . , ω(x,xN). It is straightforward to see that

c(u,x) = RTW(x)R−1RW (x)u.(4)

Note that the GMLS coeﬃcients c(u,x) depend on the point xand on the data sample

ubut not on the functional being approximated. As a result, both (1) and (2) use the

same set of coeﬃcients c(u,x), i.e., the linear system (4) must be solved only once for

every point.

In this paper we apply GMLS with m= 2, which is suﬃcient to recover the accuracy

of the linear, bilinear or trilinear ﬁnite element spaces used to discretize the problem, as

well as suﬃciently high order to evaluate partial derivatives with second order accuracy.

Because we consider general unstructured ﬁnite element meshes, we use dynamic window

functions that change the size of their support depending on the node density near the

evaluation point x.

3 Model problem

We consider a scalar elliptic interface problem comprising a pair of subdomain equations

(−∇ · (κi∇ui) = fiin Ωi,i=1,2

ui= 0 on Γi,i=1,2 (5)

4

Paul Kuberry, Pavel Bochev and Kara Peterson

augmented with a standard set of transmission conditions

u1=u2and κ1∇u1·n=κ2∇u2·non σ . (6)

In (5)–(6), nis a unit normal1on σand κiis a positive constant on Ωi.

Following [2] the starting point for our new method is the following optimization prob-

lem:

minimize Jδ(u1, u2, g1, g2) over H1

Γ1(Ω1)×H1

Γ2(Ω2)×L2(σ) subject to (8),(7)

where the constraints are given by the variational equations seek ui∈H1

Γi(Ωi)such that

κi(∇ui,∇vi)Ωi= (fi, vi)Ωi+hgi, viiσ∀vi∈H1

Γi(Ωi), i = 1,2; (8)

and the objective is deﬁned as

Jδ(u1, u2, g1, g2) = 1

2Zσ

(u1−u2)2dS +Zσ

((κ1∇u1−κ2∇u2)·n)2dS

+Zσ

g1dS +Zσ

g2dS2

+δ1Zσ

g2

1dS +δ2Zσ

g2

2dS#(9)

Formally, the objective (9) is deﬁned only for a single interface σ. However, the structure

of Jδis such that it can be easily extended to the case of non-coincident discrete interfaces

σh

1and σh

2. Indeed, the ﬁrst two terms measure the mismatches between the states

and their interface ﬂuxes. Because (7) aims to minimize rather than to eliminate these

mismatches, these terms can be replaced by pairs of terms on σh

1and σh

2, respectively,

measuring the mismatches between the ﬁelds and ﬂuxes on each respective interface and an

appropriate reconstruction of these quantities from the other interface. Thanks to the use

of two independent control variables the last group of terms also admits a straightforward

extension: we simply assign control gito interface σh

i.

To carry out this agenda we need the following operators and spaces:

•A state reconstruction operator Ri, which approximates the values of uh

iat any

point xin the vicinity of σh

i,i= 1,2 and is second order accurate;

•A gradient reconstruction operator Gi, which approximates the values of ∇uh

iat

any point xin the vicinity of σh

i,i= 1,2 and is second order accurate;

•A pair of discrete control spaces L2,h

1,σ and L2,h

2,σ, deﬁned on σ1and σ2, respectively,

for the discretization of the virtual controls.

In [2] we deﬁned Riand Githrough polynomial extensions and chose L2,h

i,σ to be a piecewise

constant space on σh

i. We retain the choice of virtual control spaces, but use instead the

GMLS approximation (1) to deﬁne the Riand Gi.

1The choice of a unit normal on the interface is arbitrary. For example, one can choose the normal

that coincides with the outer unit normal on σconsidered as part of, e.g., ∂Ω1.

5

Paul Kuberry, Pavel Bochev and Kara Peterson

Speciﬁcally, given a point x∈Rdwe set

Riuh

i(x) := τh

x(uh

i),

where τh

xis the GMLS approximation deﬁned in (1). Likewise, we deﬁne

Giuh

i(x) := τh

x,1(uh

i), . . . , τ h

x,d(uh

i),

where τh

x,i is the GMLS approximation of the ith partial derivative deﬁned in (2). These

choices yield the following generalization of the objective (9) to non-coincident interfaces:

Jh

δ(uh

1, uh

2, gh

1, gh

2) = 1

2"Zσh

1

(uh

1−R2uh

2)2dS +Zσh

2

(uh

2−R1uh

1)2dS

+Zσh

1(κ1∇uh

1−κ2G2uh

2)·n12dS +Zσh

2(κ1G1uh

1−κ2∇uh

2)·n22dS

+ Zσh

1

gh

1dS +Zσh

2

gh

2dS!2

+δ1Zσh

1

(gh

1)2dS +δ2Zσh

2

(gh

2)2dS

.

(10)

To summarize, the ﬁrst two pairs of terms in (10) generalize the state misﬁt and the ﬂux

misﬁt terms in (9), and the ﬁfth term controls the total ﬂux misﬁt between the interfaces.

The last two terms generalize the control penalties necessary for the well-posedness of the

optimization problem. As a result, the discretization of (7) on non-coincident interfaces

is given by the following problem:

minimize Jh

δ(uh

1, uh

2, gh

1, gh

2) over Hh

1,Γ×Hh

2,Γ×L2,h

1,σ ×L2,h

2,σ

subject to a discretized form of the weak equations (8) .(11)

4 NUMERICAL EXAMPLES

Patch test This example demonstrates the ability of the method to recover globally

linear solutions. We set u= 3x+ 2yand deﬁne the Dirichlet boundary condition data

and the right hand side by inserting this solution in (5).

For the patch test we use two diﬀerent domain and mesh combinations and an S-curve

interface parameterized by

σ={x= 1 + 0.1 sin(1.5πt); y=t}

In the ﬁrst conﬁguration (Figure 2a), Ω is the rectangle [0,2] ×[0,1] and σis an S-

curve interface through the center of the computational domain. For this example we

use relatively coarse subdomain meshes Ωh

ihaving large and visible gaps and overlaps;

see Fig. 3. The induced interface grids for this “coarse grid” example have element ratio

3 : 4.

In the second conﬁguration (Figure 2b), Ω is the rectangle [0.8,1.2]×[0,1] and σis again

an S-curve interface through the center of the computational domain. The subdomain

grids for this example are relatively ﬁner but still have gaps and overlaps, albeit less

6

Paul Kuberry, Pavel Bochev and Kara Peterson

+1+2

<

(0,0) (2,0)

(2,1)(0,1)

(1,0)

(0.9,1)

(a) Wide domain

+1

+2

<

(0.8,0) (1.2,0)

(1.2,1)(0.8,1)

(1,0)

(0.9,1)

(b) Narrow domain

Figure 2: The ﬁrst conﬁguration (2a) is designed to accentuate the gaps and overlaps, while the second

one (2b) aims to increases the surface area of the interface relative to the subdomain boundaries.

visible than in the ﬁrst example. The induced interface grids for this “ﬁne grid” example

have element ratio 2 : 3. We solve (11) on both conﬁgurations with penalty terms set

to 0. Using GMLS with order m= 2, the optimization formulation recovers the exact

solution to machine precision in both cases; see Figures 3–4. Of note, using GMLS with

order m= 1 in the optimization formulation also recovers the exact solution to machine

precision.

Convergence rates To estimate the convergence rates of (11) we use the same manu-

factured solution and methodology as in [2]. Speciﬁcally, we set

u=x2(y−2)3sin(2πx)−(x−3)3cos(2πx −y).(12)

and deﬁne the right hand sides and Dirichlet boundary conditions by inserting (12) into

(5). We solve (11) using several diﬀerent combinations of subdomain grids to include a

suﬃciently representative range of interface element ratios. For each combination we start

with an initial grid pair Ωh

1and Ωh

2having the desired element ratio on the interface. The

initial pair is then reﬁned successively six times while keeping the interface element ratio

ﬁxed. For all interface ratios in this study we set δ1=δ2= 1e-10 in the objective (10).

Results in (5) conﬁrm that both the H1-norm and L2-norm errors of the new formulation

converge optimally.

5 CONCLUSIONS

We have extended the non-standard domain decomposition method of Gunzburger

and Lee [7, 6] to spatially varying interface discretizations having gaps and overlaps. Our

approach diﬀers in that we have used a separate Neumann control on each subdomains’

interface and included additional terms to the cost functional to target the minimization

of global ﬂux mismatch as well as normal ﬂuxes. We introduced the use of GMLS for

7

Paul Kuberry, Pavel Bochev and Kara Peterson

Figure 3: Patch test for an S-curve interface on coarse grids containing large gaps and overlaps. The

interface grids σh

ihave element ratio 3 : 4.

meshlessly evaluating and comparing function and gradient values between subdomains at

quadrature points for terms in the objective of the minimization problem. This removed an

accuracy bottleneck in the evaluation of the normal ﬂuxes between subdomains while also

avoiding the diﬃculties generally associated with mismatched and noncoincident interfaces

including Taylor series extensions, projections or ray-tracing, or the construction of virtual

interfaces. The method passes a linear patch test and converges optimally in both the

H1norm and L2-norm.

Acknowledgment

Sandia National Laboratories is a multimission laboratory managed and operated by

National Technology and Engineering Solutions of Sandia, LLC., a wholly owned sub-

sidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National

Nuclear Security Administration under contract DE-NA-0003525. SAND2018-6034 C.

The views expressed in the article do not necessarily represent the views of the U.S.

Department of Energy or the United States Government.

This material is based upon work supported by the U.S. Department of Energy, Oﬃce

of Science, Oﬃce of Advanced Scientiﬁc Computing Research and the Laboratory Directed

Research and Development program at Sandia National Laboratories.

REFERENCES

[1] P. Bochev and D. Day. A least-squares method for consistent mesh tying. Int. J.

Num. Anal. Model., 4:342–352, 2007.

8

Paul Kuberry, Pavel Bochev and Kara Peterson

Figure 4: Patch test for an S-curve interface on ﬁne grids containing small gaps and overlaps. The

interface grids σh

ihave element ratio 2 : 3. The ﬁgure is rotated 90 degrees counterclockwise.

[2] Pavel Bochev, Paul Kuberry, and Kara Peterson. A virtual control coupling ap-

proach for problems with non-coincident discrete interfaces. In Ivan Lirkov, Svetozar

Margenov, and Jerzy Wa´sniewski, editors, Proceedings of the 11th International Con-

ference, LSSC 2017, Sozopol, Bulgaria, June 5-9, 2017, volume 10665 of Lecture

Notes in Computer Science, pages 147–155. Springer Berlin Heidelberg, 2018.

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decomposition method for partial diﬀerential equations. Computers & Mathematics

with Applications, 37(10):77 – 93, 1999.

9

Paul Kuberry, Pavel Bochev and Kara Peterson

!"

•Solution:

•,

#$%&'$()&*+,-./0&)1/%

2.34&*5+-6%)+*'$(+7

x2(y2)3sin(2⇡x)(x3)3cos(2⇡xy)

1=2=1=2=⇢=1

1=2=1e-10

10-3 10-2

Mesh Size

10-1

100

H1 Error

S-Curve Interface

1st Order

2 vs.2

2 vs.3

2 vs.4

2 vs.5

2 vs.6

2 vs.7

3 vs.4

3 vs.5

10-3 10-2

Mesh Size

10-4

10-3

10-2

10-1

L2 Error

S-Curve Interface

2nd Order

2 vs.2

2 vs.3

2 vs.4

2 vs.5

2 vs.6

2 vs.7

3 vs.4

3 vs.5

!"

•Solution:

•,

#$%&'$()&*+,-./0&)1/%

2.34&*5+-6%)+*'$(+7

x2(y2)3sin(2⇡x)(x3)3cos(2⇡xy)

1=2=1=2=⇢=1

1=2=1e-10

10-3 10-2

Mesh Size

10-1

100

H1 Error

S-Curve Interface

1st Order

2 vs.2

2 vs.3

2 vs.4

2 vs.5

2 vs.6

2 vs.7

3 vs.4

3 vs.5

10-3 10-2

Mesh Size

10-4

10-3

10-2

10-1

L2 Error

S-Curve Interface

2nd Order

2 vs.2

2 vs.3

2 vs.4

2 vs.5

2 vs.6

2 vs.7

3 vs.4

3 vs.5

Figure 5:Convergence rates of (11) for interface grids having diﬀerent element ratios. In each case the interface element

ratio |σh

1|:|σh

2|is preserved throughout the grid reﬁnement process.

[8] Thomas J.R. Hughes, Gerald Engel, Luca Mazzei, and Mats G. Larson. The con-

tinuous Galerkin method is locally conservative. Journal of Computational Physics,

163(2):467 – 488, 2000.

[9] P. Kuberry, P. Bochev, and K. Peterson. An optimization-based approach for elliptic

problems with interfaces. SIAM Journal on Scientiﬁc Computing, 39(5):S757–S781,

2017.

[10] T. A. Laursen and M. W. Heinstein. Consistent mesh tying methods for topologically

distinct discretized surfaces in non-linear solid mechanics. International Journal for

Numerical Methods in Engineering, 57(9):1197–1242, 2003.

[11] M.L. Parks, L. Romero, and P. Bochev. A novel lagrange-multiplier based method

for consistent mesh tying. Computer Methods in Applied Mechanics and Engineering,

196(35–36):3335 – 3347, 2007.

[12] Michael A. Puso. A 3d mortar method for solid mechanics. International Journal

for Numerical Methods in Engineering, 59(3):315–336, 2004.

[13] Holger Wendland. Scattered data approximation, volume 17. Cambridge university

press, 2004.

10