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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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