PreprintPDF Available

# Compatible meshfree discretization of surface PDEs

Authors:

## Abstract and Figures

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 R^d. 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.
Content may be subject to copyright.
Noname manuscript No.
(will be inserted by the editor)
Compatible meshfree discretization of surface PDEs
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
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
Government.
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)
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=
{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 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
qVh
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(|˜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
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
xiBx)
αiλi(φ),(5)
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
p|gh|ip|gh|gij
hjφ
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
jB(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
κsφ·dl(8)
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)
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
s·u=X
j
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
||φ||2=sPN
i=1 φ2
i
N.(11)
Fig. 1 Solution to manufactured smooth solution on spherical manifold for scalar variable
φ.
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
h2-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,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.
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.
References
1. Amiri, F., Mill´an, D., Shen, Y., Rabczuk, T., Arroyo, M.: Phase-ﬁeld modeling of frac-
ture in linear thin shells. Theoretical and Applied Fracture Mechanics 69, 102–109
(2014)
2. Amiri, F., Ziaei-Rad, S., Valizadeh, N., Rabczuk, T.: On the use of local maximum en-
tropy approximants for cahn–hilliard phase-ﬁeld models in 2d domains and on surfaces.
Computer Methods in Applied Mechanics and Engineering 346, 1–24 (2019)
3. Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M.: Compatible
spatial discretizations, vol. 142. Springer Science & Business Media (2007)
4. Bochev, P.B., Gunzburger, M.D.: Least-squares ﬁnite element methods, vol. 166.
Springer Science & Business Media (2009)
5. Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut ﬁnite element method for partial
diﬀerential equations on surfaces: The laplace–beltrami operator. Computer Methods
in Applied Mechanics and Engineering 285, 188–207 (2015)
6. Edwards, H.C., Trott, C.R., Sunderland, D.: Kokkos: Enabling many-
core performance portability through polymorphic memory access pat-
terns. Journal of Parallel and Distributed Computing 74(12), 3202
– 3216 (2014). DOI https://doi.org/10.1016/j.jpdc.2014.07.003. URL
http://www.sciencedirect.com/science/article/pii/S0743731514001257. Domain-
Speciﬁc Languages and High-Level Frameworks for High-Performance Computing
7. Flyer, N., Lehto, E., Blaise, S., Wright, G.B., St-Cyr, A.: A guide to rbf-generated ﬁnite
diﬀerences for nonlinear transport: Shallow water simulations on a sphere. Journal of
Computational Physics 231(11), 4078–4095 (2012)
8. Ganesan, S., Tobiska, L.: Arbitrary lagrangian–eulerian ﬁnite-element method for com-
putation of two-phase ﬂows with soluble surfactants. Journal of Computational Physics
231(9), 3685–3702 (2012)
9. Hansbo, P., Larson, M.G., Zahedi, S.: A cut ﬁnite element method for a stokes interface
problem. Applied Numerical Mathematics 85, 90–114 (2014)
10. Heroux, M., Bartlett, R., Hoekstra, V.H.R., Hu, J., Kolda, T., Lehoucq, R., Long, K.,
Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J.,
Williams, A.: An Overview of Trilinos. Tech. Rep. SAND2003-2927, Sandia National
Laboratories (2003)
11. Hu, J.J., Prokopenko, A., Siefert, C.M., Tuminaro, R.S., Wiesner, T.A.: MueLu multi-
grid framework. http://trilinos.org/packages/muelu (2014)
12. Krysl, P., Belytschko, T.: Analysis of thin shells by the element-free galerkin method.
International Journal of Solids and Structures 33(20-22), 3057–3080 (1996)
13. Kuberry, P., Bosler, P., Trask, N.: Compadre toolkit version 1.0.1 (2019). DOI
10.5281/zenodo.2560287. URL https://doi.org/10.5281/zenodo.2560287
14. Lai, R., Liang, J., Zhao, H.: A local mesh method for solving pdes on point clouds.
Inverse Problems & Imaging 7(3) (2013)
15. Lauritzen, P.H., Jablonowski, C., Taylor, M.A., Nair, R.D.: Numerical techniques for
global atmospheric models, vol. 80. Springer Science & Business Media (2011)
16. Liang, J., Zhao, H.: Solving partial diﬀerential equations on point clouds. SIAM Journal
on Scientiﬁc Computing 35(3), A1461–A1486 (2013)
17. Masud, A., Hughes, T.J.: A stabilized mixed ﬁnite element method for darcy ﬂow.
Computer methods in applied mechanics and engineering 191(39-40), 4341–4370 (2002)
18. Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diﬀuse
derivatives. IMA Journal of Numerical Analysis 32(3), 983–1000 (2012)
19. Petras, A., Ling, L., Ruuth, S.J.: An rbf-fd closest point method for solving pdes on
surfaces. Journal of Computational Physics 370, 43–57 (2018)
20. Rahimi, M., DeSimone, A., Arroyo, M.: Curved ﬂuid membranes behave laterally as
eﬀective viscoelastic media. Soft Matter 9(46), 11033–11045 (2013)
21. Ringler, T., Petersen, M., Higdon, R.L., Jacobsen, D., Jones, P.W., Maltrud, M.: A
multi-resolution approach to global ocean modeling. Ocean Modelling 69, 211–232
(2013)
22. Salmon, R.: Lectures on geophysical ﬂuid dynamics. Oxford University Press (1998)
23. Shankar, V., Wright, G.B.: Mesh-free semi-lagrangian methods for transport on a sphere
using radial basis functions. Journal of Computational Physics 366, 170–190 (2018)
24. Shankar, V., Wright, G.B., Kirby, R.M., Fogelson, A.L.: A radial basis function (rbf)-
ﬁnite diﬀerence (fd) method for diﬀusion and reaction–diﬀusion equations on surfaces.
Journal of scientiﬁc computing 63(3), 745–768 (2015)
25. Sigurdsson, J.K., Atzberger, P.J.: Hydrodynamic coupling of particle inclusions embed-
ded in curved lipid bilayer membranes. Soft matter 12(32), 6685–6707 (2016)
26. Stolarski, H., Belytschko, T., Lee, S.H.: A review of shell ﬁnite elements and corotational
theories. Computational mechanics advances 2(2), 125–212 (1995)
27. Stone, H.: A simple derivation of the time-dependent convective-diﬀusion equation for
surfactant transport along a deforming interface. Physics of Fluids A: Fluid Dynamics
2(1), 111–112 (1990)
28. Suchde, P., Kuhnert, J.: A meshfree generalized ﬁnite diﬀerence method for surface
pdes. arXiv preprint arXiv:1806.07193 (2018)
29. Trask, N., Maxey, M., Hu, X.: A compatible high-order meshless method for the stokes
equations with applications to suspension ﬂows. Journal of Computational Physics 355,
310–326 (2018)
30. Trask, N., Perego, M., Bochev, P.: A high-order staggered meshless method for elliptic
problems. SIAM Journal on Scientiﬁc Computing 39(2), A479–A502 (2017)
31. Walker, S.W.: The shape of things: a practical guide to diﬀerential geometry and the
shape derivative, vol. 28. SIAM (2015)
32. Wendland, H.: Scattered data approximation, vol. 17. Cambridge university press (2004)
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
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.
Article
Full-text available
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.
Article
Full-text available
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.
Article
Full-text available
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.
Article
Full-text available
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.
Article
Full-text available
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.
Article
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.
Article
The development of shell finite elements is reviewed. As a background for these discussions, some of the controversies and difficulties in classical shell theories are reviewed and the features of shell bending behavior which were elucidated analytically are recalled. The description of shell elements focuses on three types of methods: mixed methods, assumed strain methods and discrete Kirchhoff methods. In Section 4, a corotational theory and its application to shells are presented; as part of this development, the kinematics of degenerated continuum formulations are compared to the Koiter-Sanders theory.
Article
We present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e. point clouds, without parameterization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis using K nearest neighbors. Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.