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Compatible meshfree discretization of surface PDEs

Nathaniel Trask ·Paul Kuberry

Received: date / Accepted: date

Abstract Meshfree discretization of surface partial diﬀerential equations are

appealing, due to their ability to naturally adapt to deforming motion of the

underlying manifold. In this work, we consider an existing scheme proposed

by Liang et al. reinterpreted in the context of generalized moving least squares

(GMLS), showing that existing numerical analysis from the GMLS literature

applies to their scheme. With this interpretation, their approach may then be

uniﬁed with recent work developing compatible meshfree discretizations for

the div-grad problem in Rd. In this way, we obtain a compatible meshfree

discretization of elliptic problems on manifolds which is naturally stable for

problems with material interfaces, without the need to introduce numerical

dissipation or local enrichment near the interface. We provide convergence

studies illustrating the high-order convergence and stability of the approach

for manufactured solutions and for an adaptation of the classical ﬁve-strip

benchmark to a cylindrical manifold.

Keywords Generalized moving least squares ·Compatible discretization ·

Surface PDE ·Meshfree

N. Trask

Center for Computing Research, Sandia National Laboratories

Mailstop 1320, P.O. Box 5800, Albuquerque, NM 87125-1320

E-mail: natrask@sandia.gov

P. Kuberry

Center for Computing Research, Sandia National Laboratories

Mailstop 1320, P.O. Box 5800, Albuquerque, NM 87125-1320

Sandia National Laboratories is a multimission laboratory managed and operated by Na-

tional Technology and Engineering Solutions of Sandia, LLC.,a wholly owned subsidiary of

Honeywell International, Inc., for the U.S. Department of Energys National Nuclear Security

Administration under contract DE-NA-0003525. This paper describes objective technical re-

sults and analysis. Any subjective views or opinions that might be expressed in the paper do

not necessarily represent the views of the U.S. Department of Energy or the United States

Government.

2 Nathaniel Trask, Paul Kuberry

1 Background and model problem

A wide class of mechanics problems are modelled by surface partial diﬀerential

equations (PDEs), whereby transport and material response are restricted to

a manifold M ⊂ Rd. For example, mechanical response of thin structures are

often modeled eﬃciently by plate or shell formulations of elasticity[26,12,1].

Diﬀusive transport is often modelled via the Laplace-Beltrami problem[31];

for example in the context of surfactant transport on interfaces in multiphase

ﬂows[27,8]. In biomechanics and material science, the mechanics of lipid bilay-

ers are governed by the Stokes equations, wherein the ﬂow is restricted to lie

within a manifold whose conﬁguration may evolve in response to ﬂuid ﬂow[25,

20]. In ocean/atmosphere dynamics, meshfree techniques are used to solve the

shallow-water equations in the context of semi-Lagrangian schemes[22,15,7,

23].

To solve surface PDEs, one typically requires a means to compute metric

information characterizing the local curvature and conﬁguration of the man-

ifold. For many of these applications, the manifold evolves in a Lagrangian

fashion, and it is desirable to obtain a scheme wherein this metric information

is easily computed as the conﬁguration evolves. Recent interest has formed

around the cut-FEM method, in which the manifold is evolved as a level set

which may then be treated as an immersed problem in an ambient Cartesian

ﬁnite element simulation[9,5].

Meshfree methods on the other hand oﬀer an attractive alternative in which

a point cloud sampling of the manifold is used to characterize both the mani-

fold itself and also the solution of the PDE. The points may thus evolve under

Lagrangian motion, and meshfree techniques may be used to recalibrate metric

information “on-the-ﬂy”. Several meshfree techniques have emerged relatively

recently for this class of problems, see for example [23,24, 2, 20, 19, 28,16,14].

Oftentimes however, these techniques rely on approaches such as hyperviscos-

ity and under-integration to handle the lack of stability theory that trouble

some meshfree discretizations. These methods may be characterized as treat-

ing the manifold intrinsically, working in coordinates internal to the manifold,

or extrinsically, working in an ambient space and projecting back to the man-

ifold. The approach considered in this work is an intrinsic method.

In recent work we have shown that generalized moving least squares (GMLS)

may be used together with a primal/dual strategy to obtain accurate dis-

cretization of H(div)-type problems[30, 29]. We informally categorize this a

spatially compatible meshfree discretization because its computational proper-

ties parallel those of div-compatible mesh-based schemes [3]. In the current

work, we re-interpret a meshfree scheme for surface PDE used by Liang et al.

[16] as a GMLS discretization. We then introduce an extension of our compat-

ible meshfree approach into this surface PDE setting, and provide numerical

results highlighting the high-order accuracy and stability of the approach.

For the sake of brevity, we will consider as a simpliﬁed model H(div)-

problem the Laplace-Beltrami problem without boundary conditions, written

Compatible meshfree discretization of surface PDEs 3

in mixed form:

∇s·u=f(1)

u−κ∇sφ= 0,(2)

where ∇sdenotes the surface gradient, and κdenotes a diﬀusion parameter

that is potentially discontinuous.

2 Generalized moving least squares

We ﬁrst brieﬂy summarize the GMLS framework, referring to previous works

for technical details [16,30, 18, 32].

Consider φof function class V. Consider a collection of samples Λ=

{λi(φ)}N

i=1 corresponding to a quasi-uniform[32] collection of data sites Xh=

{xi} ⊂ Rdcharacterized by ﬁll distance h. To approximate a given linear target

functional τ˜xassociated with a target site ˜x, we seek a reconstruction p∈Vh,

where Vh⊂Vis a ﬁnite dimensional space chosen to provide good approxi-

mation properties, with basis P={P}dim(Vh)

i=1 . We perform this reconstruction

in the following weighted `2sense:

p= argmin

q∈Vh

N

X

i=1

(λi(φ)−λi(q))2ω(λi, τ˜x) (3)

where ωis a locally supported positive function. In this paper we restrict

attention to ω=Φ(|˜x−xi|), where |·| denotes the Euclidean norm and

Φ(r) = (1 −r/)4

+, where is a parameter controlling the support of ω, and

f+denotes the positive ﬂoor function.

With the optimal reconstruction pin hand, the target functional is ap-

proximated via τ˜x(φ)≈τh

˜x(φ) := τ˜x(p). As an unconstrained `2-optimization

problem, this process admits the explicit form

τh

˜x(φ) = τ˜x(P)|(Λ(P)|WΛ(P))−1Λ(P)|WΛ(φ),(4)

where we denote:

–τ˜x(P)∈Rdim(Vh)is a vector with components consisting of the target

functional applied to each basis function.

– W ∈RN×Nis a diagonal matrix with diagonal entries consisting of {ω(λi, τ˜x)}i=1,...,N .

–Λ(P)∈RN×dim(Vh)is a rectangular matrix whose (i, j) entry corresponds

to the application of the ith sampling functional applied to the jth basis

function.

–Λ(φ)∈RNis a vector consisting of the Nsamples of the function φ.

We note that by taking the contraction of the tensors appearing in Equation 4

and exploiting the compact support of ω, we may interpret the output of the

GMLS process as a ﬁnite diﬀerence-like stencil of the form

τh

˜x(φ) = X

xi∈B(˜x)

αiλi(φ),(5)

4 Nathaniel Trask, Paul Kuberry

where B(˜x) denotes the -ball neighborhood of the target site ˜x. Therefore,

GMLS admits an interpretation as an automated process for generating gen-

eralized ﬁnite diﬀerence methods on unstructured point clouds. We note that

the computational cost of solving the GMLS problem amounts to inverting

a small linear system which may be assembled using only information from

neighbors within the support of ω, and construction of such stencils across the

entire domain is embarrassingly parallel.

3 Liang’s scheme recast as GMLS problem

We next show that the scheme used by Liang et al. may be recast as a GMLS

problem. This means that accuracy results from GMLS literature may imme-

diately be applied to their scheme. For the sake of brevity, we only draw a

high-level comparison and refer the interested reader to Liang’s original work

for details[16]. In their work they consider the setting κ= 1.

We assume for the remainder of this work that Xhsamples a continuous

two-dimensional manifold M ⊂ R3. For a given point xi∈Xh, Liang seeks

an approximation to the Laplace-Beltrami operator, deﬁned in contravariant

form as

∆LB φ=1

p|g|∂ip|g|gij ∂jφ,

where gij denotes the inverse metric tensor and |g|denotes its determinant.

First, they solve an eigenvalue problem to obtain an approximation to

the tangent plane TxiM. The speciﬁc calculation of this approximate tangent

plane is unimportant for the current work; we assume we are given two linearly

independent vectors ξ1, ξ2which are suﬃciently “close” to the actual tangent

plane, whose cross product deﬁnes a vector normal to the manifold, ˆn. Their

scheme then consists of solving two GMLS problems. First, an approximation

ghto the metric tensor is obtained to approximate Mlocally in a neighborhood

of xi. This approximate metric tensor is then used to deﬁne a second GMLS

problem calculating the action of the Laplace-Beltrami operator.

First, we adapt as coordinate system the triple (ξ1, ξ2,ˆn). We seek a pa-

rameterization of the manifold from local coordinates Γ(ξ1, ξ2) : TxiM→M,

and assume that this mapping takes the form

Γ(ξ1, ξ2) = hξ1, ξ2, q(ξ1, ξ2)i(6)

where qis an unknown smooth function which we would like to infer. If qis

known, then the metric tensor gij is deﬁned componentwise as

gij =Γξi·Γξj,

where Γξi=h∂xiξ1, ∂xiξ2, ∂xiq(ξ1, ξ2)idenotes the gradient of the parameter-

ization with respect to the local coordinate ξi.

We deﬁne the ﬁrst GMLS problem as follows:

Compatible meshfree discretization of surface PDEs 5

–As samples Λ, select the point evaluation of Γat all points in the -ball

neighborhood of xi.

–As target functional, select the point evaluation of the metric tensor at xi.

–As reconstruction space, select the collection of mth

1-order polynomials,

where m1is an integer parameter.

Upon recovering the approximate metric tensor gh, we may compute and store

its inverse.

The second GMLS problem may then be deﬁned:

–As samples Λ, select the point evaluation of φin the neighborhood of of xi

–As target functional, select

τxi(φ) = 1

p|gh|∂ip|gh|gij

h∂jφ

–As reconstruction space, select the collection of mth

2-order polynomials,

where m2is another integer parameter.

In this manner, the GMLS process provides a local, high-order stencil for

the Laplace-Beltrami problem of the form

∆h

LB φ(xi) = X

j∈B(xi)

αij φj,(7)

which may be assembled into a global stiﬀness matrix as in standard mesh-

based schemes. The key feature of this approach is that the coeﬃcients in

the stencil encode a high-order reconstruction of the manifold, providing a

high-order scheme that may be eﬃciently generated “on the ﬂy” from a qua-

siuniform point cloud sampling of the manifold.

4 Compatible generalization of Liang’s scheme

In [30] we presented a compatible meshfree discretization of the div-grad prob-

lem in Rd. The key premise of our approach is the treatment of the mixed

problem (Equation 1) on a virtual primal/dual grid. In the manifold setting,

we solve the ﬁrst GMLS problem in Liang’s scheme without modiﬁcation, so

that we again have access to ghat the point xi. We consider the -ball graph of

neighbors in the vicinity of xias a surrogate for a mesh, informally identifying

nodes as virtual dual cells and virtual dual faces. We interpret the nodes and

edges of this graph as a physical mesh constituted of only zero- and one-chains.

To discretize the surface gradient over the edge eij connecting xiand xjon

the primal grid, we deﬁne the following mimetic gradient operator

GRADij (φ) = Zeij

κ∇sφ·dl(8)

6 Nathaniel Trask, Paul Kuberry

We make the added assumption that κmay be approximated as piecewise

constant over the edge, yielding the following discrete gradient from the fun-

damental theorem of calculus.

GRADh

ij (φ) = κxi+xj

2(φj−φi) (9)

In the scenario where κis a constant, GRADij(φ) = GRADh

ij (φ) exactly for

any integrable φ.

To obtain a surface divergence operator on the virtual dual grid, we deﬁne

the following GMLS problem:

–As target functional, select the surface divergence,

τ(u) = ∇s·u(xi) = 1

p|g|∂ip|g|ui

–As reconstruction space, select the gradient of mth

2-order scalar polynomi-

als,

Vh=nu=∇q, where q∈(Pm2)do

–As sampling functionals, select λj(u) = Reij u·dl.

Solving the GMLS problem, we obtain again a stencil for the surface di-

vergence of the form

∇s·u=X

j

αij λj(u)

To discretize the full div-grad problem, we chain the surface divergence

and gradient together as

∇s·u=X

j

αij GRADh

ij (φ).(10)

5 Numerical approach

In the results that follow, we will consider both spherical and cylindrical man-

ifolds. To document convergence, we seek a hierarchy of quasi-uniform point

clouds parameterized by a ﬁll distance h. To do this in the spherical case we

take a quasi-uniform MPAS[21] grid and generate points at cell centers. In the

cylindrical case we take a Cartesian lattice with grid spacing hand apply the

mapping (x, y)→(rcos θ, z ), where ris the cylinder radius.

We select for parameters m1=m2=m, so that the same polynomial

order is used for both the manifold and function reconstructions, and mmay

be varied to benchmark the high-order convergence in a p−reﬁnement setting.

It is well-understood in GMLS that the existence of a solution to Equation

3 stems from unisolvency of the samples over the reconstruction space[32];

practically this means that the support must be chosen suﬃciently large,

Compatible meshfree discretization of surface PDEs 7

to obtain suﬃcient samples, to uniquely obtain a polynomial ﬁt. We select

the support adaptively following a process detailed in previous work[29]. In

short, for each particle we ﬁnd the distance to the dim(Vh)-st nearest neighbor

rneigh, and then set the support as 1.5rneigh . In this manner, we are guaranteed

to obtain suﬃcient neighbors for unisolvency, provided the underlying point

cloud is quasi-uniform.

At each point of the point cloud, the GMLS approach is used to generate

a local stencil, which is summed into a row of a sparse global stiﬀness matrix.

We use the Compadre toolkit [13] and multiple packages from Trilinos[10]

(MueLu[11], Belos, Tpetra, Teuchos, and Kokkos[6]) to exploit latent paral-

lelism in this construction: each small optimization problem may be solved on

GPUs, and the global matrix is distributed across multiple processors. At this

point, the matrix may be solved using standard algebraic multigrid precondi-

tioning techniques.

To document convergence, we deﬁne the following `2norm

||φ||2=sPN

i=1 φ2

i

N.(11)

Fig. 1 Solution to manufactured smooth solution on spherical manifold for scalar variable

φ.

8 Nathaniel Trask, Paul Kuberry

6 Results: Manufactured solution on a sphere

We consider ﬁrst the case on the unit sphere in which κ= 1. Using the

method of manufactured solutions, we denote the true solution φ=YM

l(θ, r),

f=λl,M φ, where YM

ldenotes a spherical harmonic with associated eigenvalue

λl,M . We select as parameters l= 5 and M= 4. A representative point cloud

is given in Figure 1, corresponding to a resolution of h≈0.035. In Table 6,

we summarize the convergence of the solution in the `2-norm with respect to

h, and observe algebraic convergence of the form ||u−uexact||2∝hm.

m = 2 m = 4 m = 6

h`2-error Rate `2-error Rate `2-error Rate

0.07 1.3199e-03 - 4.0960e-04 - 6.9184e-07 -

0.035 3.2997e-04 2.00 4.5474e-05 3.17 1.0538e-08 6.04

0.0175 8.2581e-05 2.00 4.9164e-06 3.21 1.6135e-10 6.03

0.00875 2.1249e-05 1.96 2.9869e-07 4.04 3.6536e-12 5.46

Table 1 Convergence in `2norm with respect to particle spacing h, illustrating (m)th-order

convergence with respect to the order of reconstruction min the GMLS problem.

7 Results: Five-strip problem on a cylinder

We consider now a case in which κis piecewise constant. For such problems

involving material interfaces, there is an implicit interface condition that the

jump in ﬂux is zero ([κ∇φ·ˆ

n] = 0). In Rd, the ﬁve-strip test is a standard

benchmark to test H(div)-conformity, and has been used in the past to high-

light areas where e.g. Lagrangian ﬁnite elements will qualitatively fail but

compatible spatial discretizations will reproduce the solution exactly[17, 4]. In

the Euclidean setting, the problem is set up by partitioning the unit square

into ﬁve strips ({Ωi}i∈{1,...,5}) of equal height ∆y = 0.2, each with corre-

sponding diﬀusion coeﬃcient µi∈ {16,6,1,10,2}. A ﬂux boundary condition

is imposed consistent with an analytic solution φ= 1 −x. It is easily seen

that ∇φis orthogonal to the normal associated with the material interfaces,

and thus the interface condition is trivially satisﬁed. Thus, the problem stands

as a form of patch test for interfacial problems. We have shown in previous

work that in Rd, the staggered compatible meshfree approach pursued in this

work provides predictive results for this problem reminiscent of mesh-based

spatially compatible approaches[30].

We generalize the classical benchmark by constructing a similar solution

on the cylinder with the property that the interface condition is similarly

upheld. We consider a cylinder with height 2 and unit diameter, and partition

the cylinder into a stack of ﬁve cylinders, each with height ∆z = 0.4 and

diﬀusion coeﬃcient µi∈ {16,6,1,10,2}. We construct the problem to provide

Compatible meshfree discretization of surface PDEs 9

Fig. 2 Plot of analytic ﬂux u=κ∇sφof ﬁve-strip problem on the cylinder, corresponding

to a resolution of 8 points per strip.

the analytic solution φ=rθ, so that ∇sφ= ˆeθ, and ﬂux is again parallel to

the interface. For all results we select m= 2.

The cylinder diﬀers from the Euclidean case in that there are no boundaries

from which the ﬂow may be driven by a Neumann condition, and additionally

the exact solution φ=rθ is discontinuous at θ= 0. To address this, we impose

that the analytic solution hold exactly as a Dirichlet condition over the sector

θ∈ {−π

2,π

2}, so that there is no ambiguity in handling the discontinuity at

θ= 0. To enforce this Dirichlet condition, particles lying in the sector are

identiﬁed, and their rows in the global stiﬀness matrix replaced with zeros

and a one on the diagonal, and the corresponding entry on the right-hand-side

replaced with the analytic solution.

A characteristic point cloud, with corresponding ﬂuxes, is plotted in Figure

2. We plot the proﬁle of ﬂuxes along the θ=πline, using nearest point

interpolation to post-process probe data from the point cloud. We include

both coarse resolution results corresponding to two points per strip, and ﬁne

resolution results corresponding to 32 points per strip. From this proﬁle, we

see that the gradient is non-oscillatory in the vicinity of the jump, similar to

the results one would obtain with a spatially compatible mesh-based approach.

To postprocess the ﬂuxes, we must ﬁrst recover the surface gradient of the

solution. The scheme only provides as a solution point-values of φ; the ﬂuxes

uare only available in terms of their integrals over edges λij(u) = Rκ∇u·dl.

10 Nathaniel Trask, Paul Kuberry

Fig. 3 Convergence of post-processed ﬂux to analytic solution along probe at θ=π. Coarse

and ﬁne grids correspond to two and thirty points per strip, respectively. Numerical ﬂux is

non-oscillatory near material interfaces for both coarse and ﬁne resolution.

Fill distance hFlux `2-error Rate

0.2 0.05621 -

0.1 0.01472 1.93

0.05 0.00369 2.00

0.025 0.00092 2.01

0.0125 0.00023 2.00

Table 2 Convergence with respect to hof `2-norm of ﬂux magnitude normalized by maxi-

mum ﬂux, illustrating second-order convergence for the choice m= 2.

To obtain instead the point evaluation of the ﬂux at each particle, we compute

the following GMLS problem in post-processing:

–As target functional, select the point evaluation,

τ(u) = κ(xi)∇sφ(xi)

–As reconstruction space, select the space of mth

2-order scalar polynomials.

–As sampling functionals, select the point evaluation λj(u) = φ(xj).

In this manner, we obtain a point evaluation of the ﬂux at each point.

We quantify convergence of ﬂuxes in Table 7 for the choice m= 2, where

we observe algebraic convergence of the ﬂux magnitude in the `2norm with

respect to h.

Compatible meshfree discretization of surface PDEs 11

8 Conclusion

In this work, we have used the GMLS framework to unify both Liang’s ap-

proach for solving surface PDEs and our previous compatible meshfree scheme

for Euclidean space, to obtain a new compatible meshfree scheme applicable

to general manifolds. The scheme has been shown to directly generalize re-

sults from the Euclidean setting. In other works, we have used the Euclidean

compatible meshfree scheme as a foundation for developing schemes for the

stationary Stokes problem[29]. We intend to similarly extend the current work

to obtain a meshfree capability for stationary Stokes on manifolds. Such a

framework is foundational to developing mechanics solvers capable of operat-

ing in near incompressible regimes.

Acknowledgements The authors acknowledge support under the Sandia National Lab-

oratories LDRD program, and thank Dr. Pavel Bochev for reviewing an early draft of the

work. On behalf of all authors, the corresponding author states that there is no conﬂict of

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

Department of Energy or the United States Government.

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