A virtual control, mesh-free coupling method for non-coincident interfaces

Abstract

We present an optimization approach with two controls for coupling elliptic partial differential equations posed on subdomains sharing an interface that is discretized independently 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 including 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.

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Full text

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
!&
$
!
!
!
#
$
'#'&
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,σσi. 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
2Zσ
(u1−u2)2dS +Zσ
((κ1∇u1−κ2∇u2)·n)2dS
+Zσ
g1dS +Zσ
g2dS2
+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)·n12dS +Zσh
2(κ1G1uh
1−κ2∇uh
2)·n22dS
+ 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.
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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.
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9

Paul Kuberry, Pavel Bochev and Kara Peterson
!"
•Solution:
•,
#$%&'$()&*+,-./0&)1/%
2.34&*5+-6%)+*'$(+7
x2(y2)3sin(2⇡x)(x3)3cos(2⇡xy)
1=2=1=2=⇢=1
1=2=1e-10
10-3 10-2
Mesh Size
10-1
100
H1 Error
S-Curve Interface
1st Order
2 vs.2
2 vs.3
2 vs.4
2 vs.5
2 vs.6
2 vs.7
3 vs.4
3 vs.5
10-3 10-2
Mesh Size
10-4
10-3
10-2
10-1
L2 Error
S-Curve Interface
2nd Order
2 vs.2
2 vs.3
2 vs.4
2 vs.5
2 vs.6
2 vs.7
3 vs.4
3 vs.5
!"
•Solution:
•,
#$%&'$()&*+,-./0&)1/%
2.34&*5+-6%)+*'$(+7
x2(y2)3sin(2⇡x)(x3)3cos(2⇡xy)
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.
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press, 2004.
10