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Meshfree discretization of surface partial differential equations is 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. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. 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.
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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
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
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,
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:
uκsφ= 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=
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 pVh,
where VhVis 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
(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(|˜xxi|), 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
˜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
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
˜x(φ) = X
4 Nathaniel Trask, Paul Kuberry
where Bx) 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 xiXh, 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=hxiξ1, 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
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
LB φ(xi) = X
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
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.
ij (φ) = κxi+xj
2(φjφi) (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
As reconstruction space, select the gradient of mth
2-order scalar polynomi-
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
αij λj(u)
To discretize the full div-grad problem, we chain the surface divergence
and gradient together as
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 prefinement 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
i=1 φ2
Fig. 1 Solution to manufactured smooth solution on spherical manifold for scalar variable
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 h0.035. 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 ||uuexact||2hm.
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 ([κφ·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 ({i}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=κsφof five-strip problem on the cylinder, corresponding
to a resolution of 8 points per strip.
the analytic solution φ=, 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= 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}, 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) = Rκu·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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... Localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have become increasingly popular over the last ten years for approximating SDOs and solving surface partial differential equations (PDEs); see, for example, [10][11][12][13][14] for GMLS and [15][16][17][18][19][20][21][22][23][24][25][26] for RBF-FD. These methods can be applied to surfaces defined by point clouds, without having to form a triangulation of the surface like surface finite element methods [27] or a level-set representation of the surface like embedded finite element methods [28]. ...
... There has been little attention given in the literature on RBF-FD methods for how to do these approximations; commonly it is assumed that they are computed by some separate techniques (e.g., [15,18,24]). However, for GMLS, these approximations are incorporated directly in the methods (e.g., [11,12,14]). The third purpose of this work is use the ideas from GMLS to develop a new RBF-FD technique for approximating the tangent space directly using PHS+Poly. ...
... The formulation of GMLS on a manifold was introduced by Liang & Zhao [11] and further refined by Trask, Kuberry, and collaborators [12,14]. It uses local coordinates to approximate SDOs as defined in (5)-(7) and requires a method to also approximate the metric terms. ...
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Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational cost, and they can be applied to surfaces defined only by point clouds. However, there have yet to be any studies that perform a direct comparison of these methods for approximating surface differential operators (SDOs). The first purpose of this work is to fill that gap. For this comparison, we focus on an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly) since they are most closely related to the GMLS method. Additionally, we use a relatively new technique for approximating SDOs with RBF-FD called the tangent plane method since it is simpler than previous techniques and natural to use with PHS+Poly RBF-FD. The second purpose of this work is to relate the tangent plane formulation of SDOs to the local coordinate formulation used in GMLS and to show that they are equivalent when the tangent space to the surface is known exactly. The final purpose is to use ideas from the GMLS SDO formulation to derive a new RBF-FD method for approximating the tangent space for a point cloud surface when it is unknown. For the numerical comparisons of the methods, we examine their convergence rates for approximating the surface gradient, divergence, and Laplacian as the point clouds are refined for various parameter choices. We also compare their efficiency in terms of accuracy per computational cost, both when including and excluding setup costs.
... Equation (33) is the cornerstone of this work; it reveals an intimate connection between the graph exterior calculus we have used in [67] as a basis for learning physics and the finite element exterior calculus. Whereas in [68] data was used to associate a metric with a given graph, (33) shows that the graph itself and the associated metric information may be inferred directly from the Whitney forms. The entries of the matrices M k encode graph sparsity by weighting boundary interactions between partitions, and therefore the geometric problem of identifying control volumes is equivalent to a graph discovery problem. ...
... The manufactured problem serves as a comparison to standard methods for solving finite element problems; by manufacturing a solution we can quantify convergence with respect to fine and coarse resolution. The five-strip problem is a classical patch test for mixed FEM spaces to verify H(div) conformity for problems with discontinuous material properties [33,68]; discretizations assuming regularity beyond H(div) exhibit oscillatory solutions. The battery problem models the flow of current through a lithium-ion matrix under a unit voltage drop. ...
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Control volume analysis models physics via the exchange of generalized fluxes between subdomains. We introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between the approach and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in R 2 establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation. Keywords: scientific machine learning, Whitney forms, finite element exterior calculus, partition of unity, data-driven modeling 1. Overview and background In recent years a number of works aim to use machine learning (ML) to develop data-driven models and surrogates for physical systems. Broadly, these use data to either: to develop predictive models where first-principles derivation is prohibitively expensive or complex; or to learn a surrogate model for expensive/complex simulations to facilitate tasks requiring repeated evaluation of a forward model (e.g. uncertainty quantification, optimization, inverse modeling). In multiscale modeling, many seek data-driven homogenization techniques to represent mesoscale geometry at continuum scales while avoiding oversimplifications of geometry [45, 50, 76, 55, 14, 35, 24, 27]. While ML has provided a number of remarkable new capabilities in these fields, the rigorous numerical analysis typical of traditional finite element (FEM) simulation is generally lacking. ML-based data-driven models are typically unable to provide the guaranteed convergence, stability, and preservation of mathematical/physical * Corresponding author (
... The demonstrated techniques used in high-order interpolators use spline quasiinterpolants (de Boor, 1990), bi-cubic splines (Hanke et al., 2016;Craig et al., 2017), the standard radial-basis function spaces (Flyer and Wright, 2007;Bungartz et al., 2016), the moving least squares (MLS) method (Lancaster and Salkauskas, 1981), and MLS variants such as the modified MLS (MMLS) (Joldes et al., 2015;Slattery, 2016), which originate from high-order reconstruction methods. Recent extensions to MLS for producing efficient high-order remap involve locally reconstructing the manifold geometry from a point set representation and then generating a compact stencil in the local coordinate chart (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). ...
... Traditionally, GMLS uses a basis that is defined as a function of the spatial dimension from which a point cloud is sampled. However, in this work, reconstruction of functions sampled on a manifold permits generating a compact stencil in a local coordinate chart, which is one dimension smaller (Liang and Zhao, 2013;Suchde and Kuhnert, 2019;Trask and Kuberry, 2020;Gross et al., 2020). The savings in net computational floating-point operations (flops) is a factor of p 3 in R 2 compared to a traditional basis in R 3 , where p is the order of the basis. ...
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Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid (Hill et al., 2004), TempestRemap (Ullrich and Taylor, 2015), generalized moving least squares (GMLS) (Trask and Kuberry, 2020) with post-processing filters, and WLS-ENOR (Li et al., 2020). By repeated iterative application of the high-order remapping methods to the test fields, we verify the accuracy of each scheme in terms of their observed convergence order for smooth data and determine the bounded error propagation using challenging, realistic field data on both uniform and regionally refined mesh cases. In addition to retaining high-order accuracy under idealized conditions, the methods also demonstrate robust remapping performance when dealing with non-smooth data. There is a failure to maintain monotonicity in the traditional L2-minimization approaches used in ESMF and TempestRemap, in contrast to stable recovery through nonlinear filters used in both meshless GMLS and hybrid mesh-based WLS-ENOR schemes. Local feature preservation analysis indicates that high-order methods perform better than low-order dissipative schemes for all test cases. The behavior of these remappers remains consistent when applied on regionally refined meshes, indicating mesh-invariant implementations. The MIRA intercomparison protocol proposed in this paper and the detailed comparison of the four algorithms demonstrate that the new schemes, namely GMLS and WLS-ENOR, are competitive compared to standard conservative minimization methods requiring computation of mesh intersections. The work presented in this paper provides a foundation that can be extended to include complex field definitions, realistic mesh topologies, and spectral element discretizations, thereby allowing for a more complete analysis of production-ready remapping packages.
... We apply the meshless GFDM extrinsically to approximate the spatial surface differential operators and KDC method for the temporal discretization. We briefly review several existing approximation schemes for the surface differential operators in the literature, including the embedding treatment [32,33], intrinsic treatment [34][35][36] and extrinsic treatment [37,38]. The embedding approaches are usually computationally expensive due to the discretization of the embedding space; and designing efficient embedding strategy becomes even more challenging for moving surfaces. ...
... Instead of the direct approximation, we use the definition of Laplace-Beltrami operator S := ∇ S · ∇ S and define the Laplace-Beltrami operator acting on the variable u in R 3 as S u = ∇ S · (∇ S u) . (34) Here, the approximation of the surface Laplacian is realized from the composition of two independent approximations of the surface divergence and surface gradient. In other words, the Laplace-Beltrami problem is considered as a div-grad problem [40,42] and the GFDM is used to approximate both the surface divergence and surface gradient. ...
This paper presents an efficient collocation method which combines the generalized finite difference method (GFDM) with the Krylov deferred correction (KDC) method for the long-time simulation of heat and mass transport on evolving surfaces. The KDC method utilizes a pseudo-spectral-type temporal collocation formulation to discretize the time-dependent surface heat and mass transport equation in each time marching step, where the time derivatives at the collocation points are introduced as the new unknown variables. A low-order time marching scheme is then applied as an effective preconditioner in the Jacobian-Free Newton-Krylov framework to decouple the spatial surface PDEs at different collocation nodes. Each decoupled surface PDE is then solved by the meshless GFDM, where both the continuous-form evolving surfaces defined by parametric equations and discretized-form evolving surfaces composed of point clouds are considered in the GFDM spatial discretization. Numerical experiments show that the combined GFDM-KDC solver is a promising numerical scheme for long-time evolution simulation of heat and mass transport on intractable evolving surfaces.
... These techniques include surface finite element (SFE) [16], embedded finite element (EFE) [8,33], and closest point (CP) [29] methods. More recently, various meshfree (or meshless) methods have also been developed for PDEs on general surfaces that use a local stencil approach, including radial basis function-finite differences (RBF-FD) [2,26,37,41,42,50], generalized finite differences (GFD) [45], and generalized moving least squares (GMLS) [20,27,46]. These methods can be applied for surfaces represented only by point clouds and do not require a surface triangulation like SFE methods or a level-set representation of the surface like EFE methods. ...
... 2. Localized meshfree discretizations. Several localized meshfree methods have been developed for approximating the solution of (1.1), e.g., [2,26,27,37,41,42,46]. For the sake of brevity, we limit the focus of this study to two localized meshfree methods: polyharmonic spline (PHS)based RBF-FD with polynomials and GFD. ...
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We develop a new meshfree geometric multilevel (MGM) method for solving linear systems that arise from discretizing elliptic PDEs on surfaces represented by point clouds. The method uses a Poisson disk sampling-type technique for coarsening the point clouds and new meshfree restriction/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. MGM is applicable to discretizations of elliptic PDEs based on various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We test MGM both as a standalone solver and preconditioner for Krylov subspace methods on several test problems using RBF-FD and GFD, and numerically analyze convergence rates, efficiency, and scaling with increasing point cloud sizes. We also perform a side-by-side comparison to algebraic multigrid (AMG) methods for solving the same systems. Finally, we further demonstrate the effectiveness of MGM by applying it to three challenging applications on complicated surfaces: pattern formation, surface harmonics, and geodesic distance.
... Along another line, mesh-free methods are proposed to avoid mesh generation. Radial basis function are commonly used for interface problems [33,38], such as the generalized moving least squares approach [14,22,[35][36][37]45]. These methods incorporate techniques such as discontinuous derivative basis functions [45], gradient information [37], adaptive refinement strategies [22], and other techniques to improve the effectiveness. ...
Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method. This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions. It evaluates partial differential equations, boundary conditions, and interface conditions on collocation points in equal footing, and solves a linear least-squares system to obtain the approximate solution. To address the issue of low regularity, two sets of random feature functions are used to approximate the solution on each side of the interface, which are then coupled together via interface conditions. We validate our method through a series of increasingly complex numerical examples. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries.
... Appropriate reformulations and techniques are needed to deal with surface PDEs. They can be classified into three categories: intrinsic [10][11][12][13], embedding [14,15] and extrinsic [16][17][18]. Intrinsic methods are suitable for solving surface partial differential equations (PDEs) because they define and compute differential operators purely in terms of quantities intrinsic to the surface itself. ...
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This paper introduces the first attempt to employ a localized meshless method to analyze time-harmonic acoustic wave propagation on curved surfaces with periodic holes/inclusions. In particular, the generalized finite difference method is used as a localized meshless technique to discretize the surface gradient and Laplace-Beltrami operators defined extrinsically in the governing equations. An absorbing boundary condition is introduced to reduce reflections from boundaries and accurately simulate wave propagation on unclosed surfaces with periodic inclusions. Several benchmark examples demonstrate the efficiency and accuracy of the proposed method in simulating acoustic wave propagation on surfaces with diverse geometries, including complex shapes and periodic holes or inclusions.
... Methods with rigorous convergence results include the wide stencil schemes for Hamilton-Jacobi equations and elliptic PDEs [62], which were originally defined on regular grids and have subsequently been extended to unstructured point clouds [43,47], and the point integral method [55]. Other works without convergence guarantees include upwind schemes for Hamilton-Jacobi equations on unstructured meshes [68], mesh-free generalized finite difference methods [71,72], least squares manifold approximation methods [56,75,78], the local mesh method [53], radial basis function methods [45,48,63,64], and a recent approach using graph Laplacians and deep learning [57]. A general survey of meshfree methods in PDEs is given in [28]. ...
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We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.
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Experimental and field investigations for solution mining processes have improved intensely in recent years. Due to today's computing capacities, three-dimensional simulations of potential salt solution caverns can further enhance the understanding of these processes. They serve as a "virtual prototype" of a projected site and support planning in reasonable time. In this contribution, we present a meshfree Generalized Finite Difference Method (GFDM) based on a cloud of numerical points that is able to simulate solution mining processes on microscopic as well as macroscopic scales, which differ significantly in both the spatial and temporal scale. Focusing on anticipated industrial requirements, Lagrangian and Eulerian formulations including an Arbitrary Lagrangian-Eulerian (ALE) approach are considered.
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We apply a local maximum entropy (LME) approximation scheme to the fourth order phase-field model for the traditional Cahn-Hilliard theory. The discretization of the Cahn-Hilliard equation by Galerkin method requires at least $C^1$-continuous basis functions. This requirement can be fulfilled by using LME shape functions which are $C^\infty$-continuous. In this case, the primal variational formulations of the fourth-order partial differential equation is well defined and integrable. Hence, there is no need to split the fourth-order partial differential equation into two second-order partial differential equations; this splitting scheme is a common practice in mixed finite element formulations with $C^0$-continuous Lagrange shape functions. Furthermore, we use a general and simple numerical method such as statistical manifold learning techniques that allows dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology, and to solve Cahn-Hilliard equation on general surfaces. Finally, the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples.
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Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF- FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231(14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
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We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.
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We present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points $\bm{x}_i$. This graph defines a local primal-dual grid complex with a \emph{virtual} dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a Generalized Moving Least Squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$ and $H^1$ norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.
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We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and inter-monolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.
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The lateral mobility of membrane inclusions is essential in biological processes involving membrane-bound macromolecules, which often take place in highly curved geometries such as membrane tubes or small organelles. Probe mobility is assisted by the lateral fluidity, which is thought to be purely viscous for lipid bilayers and synthetic systems such as polymersomes. In previous theoretical studies, the hydrodynamical mobility is estimated assuming a fixed membrane geometry. However, fluid membranes are very flexible out-of-plane. By accounting for the deformability of the membrane and in the presence of curvature, we show that the lateral motion of an inclusion produces a normal force, which results in a nonuniform membrane deformation. Such a deformation mobilizes the bending elasticity, produces extra lateral viscous and elastic forces, and results in an effective lateral viscoelastic behavior. The coupling between lateral and out-of-plane mechanics is mediated by the interfacial hydrodynamics and curvature. We analyze the frequency and curvature dependent rheology of flexible fluid membranes, and interpret it with a simple four-element model, which provides a background for microrheological experiments. Two key technical aspects of the present work are a new formulation for the interfacial hydrodynamics, and the linearization of the governing equations around a cylindrical geometry.
The Compadre Toolkit provides a performance portable solution for the parallel evaluation of computationally dense kernels. The toolkit specifically targets the Generalized Moving Least Squares (GMLS) approach, which requires the inversion of small dense matrices. The result is a set of weights that provide the information needed for remap or entries that constitute the rows of some globally sparse matrix. This toolkit focuses on the 'on-node' aspects of meshless PDE solution and remap, namely the parallel construction of small dense matrices and their inversion. What it does not provide is the tools for managing fields, inverting globally sparse matrices, or neighbor search that requires orchestration over many MPI processes. This toolkit is designed to be easily dropped-in to an existing MPI (or serial) based framework for PDE solution or remap, with minimal dependencies (Kokkos and either Cuda Toolkit or LAPACK).
A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.