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Mesh-hardened ﬁnite element analysis through a

Generalized Moving Least-Squares

approximation of variational problems

P. Bochev, N. Trask, P. Kuberry, and M. Perego

Center for Computing Research,

Sandia National Laboratories?,

Albuquerque, NM 87125, USA,

{pbboche,natrask,pakuber,mperego}@sandia.gov,

Abstract. In most ﬁnite element methods the mesh is used to both

represent the domain and to deﬁne the ﬁnite element basis. As a result

the quality of such methods is tied to the quality of the mesh and may

suﬀer when the latter deteriorates. This paper formulates an alterna-

tive approach, which separates the discretization of the domain, i.e., the

meshing, from the discretization of the PDE. The latter is accomplished

by extending the Generalized Moving Least-Squares (GMLS) regression

technique to approximation of bilinear forms and using the mesh only

for the integration of the GMLS polynomial basis. Our approach yields a

non-conforming discretization of the weak equations that can be handled

by standard discontinuous Galerkin or interior penalty terms.

Keywords: Galerkin methods, Generalized Moving Least Squares, Non-

conforming Finite Elements.

1 Introduction

The vast majority of ﬁnite element methods uses the mesh to both approximate

the computational domain and to deﬁne the shape functions necessary to dis-

cretize the weak forms of the governing PDEs. These dual roles of the mesh are

often in conﬂict. On the one hand, the properties of the discrete equations de-

pend strongly on the quality of the underlying mesh and may deteriorate to the

point of insolvability on poor quality grids. For example, high-aspect or “sliver”

elements lead to nearly singular shape functions, which result in ill-conditioned

or even singular discrete equations [4, 13]. On the other hand, automatic gener-

ation of high-quality grids remains a challenge. Currently, hexahedral grids can

deliver robust results but require prohibitive manual eﬀorts. Conversely, tetra-

hedral grids can be constructed more eﬃciently but their quality may be insuf-

ﬁcient for traditional Finite Element Analysis (FEA) due to poor aspect ratios.

?Sandia National Laboratories is a multimission laboratory managed and operated by National

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

International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration

under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any

subjective views or opinions that might be expressed in the paper do not necessarily represent

the views of the U.S. Department of Energy or the United States Government.

2 P. Bochev et al.

Summarily, meshing can consume signiﬁcant resources, creating a computational

bottleneck in the ﬁnite element workﬂow [9]. Moreover, in some circumstances

such as Lagrangian simulations of large-deformation mechanics [12], distorted

grids are unavoidable. As a result, hardening ﬁnite element methods against sub-

standard grids can have signiﬁcant impacts towards enabling automated CAD-

to-solution capabilities by reducing or even removing the performance barriers

created by the mesh-quality requirements of conventional FEA.

Attaining these goals requires either reducing or altogether eliminating the

dependency of the ﬁnite element shape functions on the underlying mesh. In

this paper we aim for the latter by extending Generalized Moving Least Squares

(GMLS) [15] regression techniques to approximate the weak variational forms of

the PDE problems that are at the core of FEA.

In so doing our approach limits the role of the underlying mesh to perform-

ing numerical integration and enables generation of well-conditioned discrete

problems that are independent of its quality. These problems are obtained by

(i) solving a small local quadratic program on each element; (ii) substituting

the test and trial functions by the polynomial basis of the GMLS reproduction

space, and (iii) integrating the resulting products of polynomials, which can

be accomplished with relatively few quadrature points. The approximate weak

forms generated by this process fall into the category of non-conforming FEM,

which are supported by a mature and rigorous stability theory and error anal-

ysis. This allows us to borrow classical “oﬀ-the-shelf” stabilization techniques

from, e.g., Discontinuous Galerkin [2,8] or Interior Penalty [16,1] methods.

These traits, i.e., formulations that require only integration of polynomials

and can be stabilized by standard non-conforming FEM terms, set our approach

apart from other techniques, such as meshfree Galerkin methods [5, 11, 3, 10]

that also aim to alleviate mesh quality issues. These methods use GMLS, or

similar regression techniques, to deﬁne meshfree shape functions which replace

the standard mesh-based ﬁnite element bases in the weak forms. However, the

meshfree shape functions are not known in closed form and are non-polynomial.

As a result, their integration requires a relatively large number of quadrature

points, which increases the computational cost of such schemes, as every shape

function evaluation involves a solution of a small linear algebra problem. This

has prompted consideration of reduced order integration [7, 6]; however, such

integration leads to numerical instabilities due to underintegration and requires

application-speciﬁc stabilizations.

2 Generalized Moving Least Squares (GMLS) regression

GMLS is a non-parametric regression technique for the approximation of bounded

linear functionals from scattered data [15, Section 4.3]. A typical GMLS set-

ting includes (i) a function space Uwith a dual U∗; (ii) a ﬁnite dimensional

space Φ⊂Uwith basis φ={φ1, . . . , φq}; (iii) a Φ-unisolvent1set of sam-

1We recall that Φ-unisolvency implies {φ∈Φ|λi(φ) = 0, i = 1,...,n}={0}.

GMLS for variational problems 3

pling functionals S={λ1, . . . , λn} ⊂ U∗; and (iv) a locally supported kernel

w:U∗×U∗7→ R+∪ {0}.

GMLS seeks an approximation eτ(u) of the target τ(u)∈U∗in terms of

the sample vector u:= (λ1(u), . . . , λn(u)) ∈Rn, such that eτ(φ) = τ(φ) for all

φ∈Φ, i.e., the approximation is Φ-reproducing. To deﬁne eτ(u) we need the

vector τ(φ)∈Rqwith elements (τ(φ))i=τ(φi), i = 1, . . . , q, the diagonal

weight matrix W(τ)∈Rn×nwith element Wii (τ) = w(τ;λi), and the basis

sample matrix B∈Rn×qwith element Bij =λi(φj); i= 1, . . . , n;j= 1, . . . , q.

Let |·|W(τ)denote the Euclidean norm on Rnweighted by W(τ), i.e.,

|b|2

W(τ)=b|W(τ)b∀b∈Rn.

The GMLS approximant of the target is then given by

eτ(u) := c(u;τ)·τ(φ),(1)

where the GMLS coeﬃcients c(u;τ)∈Rqsolve

c(u;τ) = argmin

b∈Rq

1

2|Bb−u|2

W(τ).(2)

It is straightforward to check that c(u;τ) = (B|W(τ)B)−1(B|W(τ))u.We

refer to [14] for information about the eﬃcient and stable solution of (2). Lastly,

let ei∈Rnbe the ith Cartesian unit vector and let uτ

i:= c(ei;τ)·φ∈Φ. We

call the set Sτ={uτ

1, . . . , uτ

n} ⊂ Φa GMLS reciprocal of Srelative to τ.

3 GMLS approximation of variational equations

Let Uand Vdenote Hilbert spaces with duals U∗and V∗, respectively. We

consider the following abstract variational equation: given f∈V∗ﬁnd u∈U

such that

a(u, v) = f(v)∀v∈V , (3)

where a(·,·) : U×V→Ris a given bilinear form. We refer to Uand Vas the

trial and the test space, respectively. To approximate (3) we will use two separate

instances of the GMLS regression for the test and trial spaces, respectively. To

diﬀerentiate between these instances we tag their entities with a sub/superscript

indicating the underlying space, e.g., SUand cU(u) denote a sampling set and

a coeﬃcient vector, respectively, for the trial space. One exception to this rule

will be the reciprocal GMLS functions uτ

iand vτ

i.

We obtain the GMLS approximation of (3) in two steps. For any ﬁxed u∈U

the form a(u, ·) deﬁnes a bounded linear functional on V, i.e., a(u, ·)∈V∗. We

shall assume that the kernel wis such that W(a(u, ·)) = W(f). For this reason

we retain the generic label τto indicate dependence of various GMLS entities

on their respective target functionals. Then, the GMLS approximants of a(u, ·)

and fcan be written in terms of the same GMLS coeﬃcient vector as

ea(u, v) := cV(v;τ)·a(u, φV) and e

f(v) = cV(v;τ)·f(φV)∀v∈V ,

4 P. Bochev et al.

respectively. Combining these representations yields the following approximation

of (3): ﬁnd u∈Usuch that ea(u, v) = e

f(v) for all v∈V, or equivalently,

cV(v;τ)·a(u, φV) = cV(v;τ)·f(φV)∀v∈V. (4)

The weak problem (4) has inﬁnitely many “equations” and “variables”. To reduce

the number of equations we restrict the test space in (4) to the GMLS reciprocal

set Sτ

Vto obtain the following problem: ﬁnd u∈Usuch that ea(u, vτ

i) = e

f(vτ

i)

for all vτ

i∈Sτ

Vor, which is the same,

cV(ei;τ)·a(u, φV) = cV(ei;τ)·f(φV)i= 1, . . . , nV.(5)

This completes the ﬁrst step. The second step discretizes the trial space by

restricting the search for a solution to the reciprocal GMLS space Sτ

U, i.e., we

consider the problem: ﬁnd uτ∈Sτ

Usuch that

ea(uτ, vτ

i) = e

f(vτ

i)∀vτ

i∈Sτ

V.(6)

where uτ:= PnU

j=1 ajuτ

j.Using (5) and uτ

j:= c(ej;τ)·φone can write (6) as

nU

X

j=1

(cV(ei;τ)·a(φU,φV)·cU(ej;τ)) aj=cV(ei;τ)·f(φV)i= 1, . . . , nV.(7)

It is easy to see that this problem is equivalent to the following nV×nUsystem

of linear algebraic equations

Ka=F(8)

where a={a1, . . . , aU

n} ∈ RnUare the GMLS degrees-of-freedom (DOF), while

K∈RnV×nUand F∈RnVhave elements

Kij =cV(ei;τ)·a(φU,φV)·cU(ej;τ) and Fi=cV(ei;τ)·f(φV),

respectively. Problems (6) and (8) can be viewed as GMLS analogues of a con-

forming Petrov-Galerkin discretization of (3) and its equivalent linear algebraic

form. In this context, the reciprocal ﬁelds uτ

jand vτ

iare analogues of modal

bases for the trial and test spaces. As a result, in what follows we shall generate

the necessary approximants by simply restricting bilinear forms and right hand

side functionals to the reciprocal spaces. Just as in the ﬁnite element case “as-

sembling” (8) amounts to computing the action of the bilinear form a(·,·) and

the right hand side functional fon the polynomial basis functions φUand φV.

However, application of (8) for the numerical solution of PDEs is subject

to additional considerations, if one wishes to obtain a computationally eﬀective

scheme. This has to do with the fact that in the PDE context a(·,·) and f

usually involve integration over a domain Ω. In such a case one would have to

consider a GMLS regression with a kernel wwhose support contains the entire

problem domain. Unfortunately, this renders (8) dense, making the discretization

impractical for all but small academic problems.

The key to obtaining computationally eﬃcient discretizations from (6), resp.

(8) is to apply the GMLS formulation locally. In the following section we spe-

cialize the approach to generate a non-conforming scheme for a model PDE.

GMLS for variational problems 5

3.1 Application to a model PDE

Consider the advection-diﬀusion equation with homogeneous Dirichlet boundary

conditions

−ε∆u +b· ∇u=fin Ωand u= 0 on Γ , (9)

where Ω⊂Rd,d= 1,2,3 is a bounded region with Lipschitz continuous bound-

ary Γ,bis a solenoidal vector ﬁeld, and fis a given function. The weak form

of (9) is given by the abstract problem (3) with U=V=H1

0(Ω),

a(u, v) = ZΩ

ε∇u· ∇v+ (b· ∇u)vdx and f(v) = ZΩ

fvdx.

Let Ωhand Xη⊂Ωdenote a conforming partition of the computational do-

main into ﬁnite elements {Kk}Ne

k=1 and a point cloud comprising points {xi}Np

i=1,

respectively. We seek an approximation of uon the point cloud, i.e., the DoFs

are associated with Xηrather than the underlying mesh. Furthermore, no rela-

tionship is assumed between Ωhand Xη, in practice though one may deﬁne Xη

using mesh entities such as element vertices, element centroids, etc..

Using the additive property of the integral a(u, v) = PNe

k=1 ak(u, v) and

f(v) = PNe

k=1 fk(v), where ak(·,·) and fk(·) are restrictions of a(·,·) and f(·)

to element Kk. To discretize (9) we will apply GMLS locally to approximate

ak(·,·) and fk(·). Since U=Vwe can use the same regression process for the

trial and test spaces and drop the sub/superscripts used earlier to distinguish

between them. We deﬁne the local GMLS kernel as w(Kk,xj) := ρ(|bk−xj|),

where bkis the centroid of Kkand ρ(·) is a radially symmetric function with

supp ρ=O(h). This kernel satisﬁes the assumption W(ak(u, ·)) = W(fk). The

GMLS approximants of ak(·,·) and fk(·) will be constructed from point samples

close to bkusing the local sampling set Sk={δxjw(Kk,xj)>0}with cardi-

nality nk. We assume that the support of ρis large enough to ensure that Skis

Pm-unisolvent. We also have the GMLS recirpocal set Sk={uk

1, . . . , uk

nk}with

uk

i:= c(ek

i;bk)·φand ek

i∈Rnk. We obtain the local GMLS approximants of

the elemental forms by restricting each ak(·,·) to Sk×Sk, i.e.,

eak(uk

j, uk

i) := c(ek

i;bk)·ak(φ,φ)·c(ek

j;bk)

Tha matrix ak(φ,φ)∈Rnq×nqhas element

(ak(φ,φ))st =ZKk

∇φs· ∇φtdx

Likewise, we have that e

fk(uk

i) = c(ek

i;bk)·fk(φ) where fk(φ)∈Rnqwith

(fk(φ))s=ZKk

f φsdx .

The local approximants eak(·,·) and e

fk(·) give rise to a local matrix Kk

ij =

c(ek

i;bk)·ak(φ,φ)·c(ek

j;bk) and a local vector Fk

i=cV(ei;τ)·f(φ),respec-

tively, which are analogues of the element stiﬀness matrix and load vector in

FEA.

6 P. Bochev et al.

Fig. 1. Comparison of a Moving Least Squares basis function (black) and a composite

reciprocal basis function [u]i(red) in one-dimensions for Φ=P2and two diﬀerent

kernels.

To deﬁne the global approximants of a(·,·) and f(·) from the local ones we

ﬁrst need to deﬁne a global discrete space to supply the global test and trial

functions. We construct this space as [S] = ∪Kk∈ΩhSkand denote its elements

by [u]. Stacking all local DoF in a single vector [a] := {a1,...,aNe}produces

the global DoF set for [u]. We now deﬁne the global approximants by summing

over all elements, i.e.,

ea([u],[v]) := X

Kk∈Ωheak(uk, vk) and e

f([v]) := X

Kk∈Ωhe

fk(vk),

where uk, vk∈Sk. In general, a sampling functional δxjcan belong to multiple

local sampling sets Sk, which means that [u] will be multivalued at xj. In fact,

one can show that the global approximants ea(·,·) and e

f(·) can be generated

by using a composite “basis” of the global space [S] assembled from the local

reciprocal bases as

[u]i:= X

Kk∈Ωh

χkuk

i,

where χkis the characteristic function of element Kk. Figure 1 shows an example

of a composite global basis function in one dimension and compares it to a

Moving Least Squares basis function used in many meshfree Galerkin methods;

see, e.g., [5, 11].

The multivalued character of the global approximation space [S] means that

ea(·,·) and e

f(·) are non-conforming approximations of a(·,·) and f(·), resembling

the type of “broken” forms one sees in Discontinuous Galerkin (DG) and interior

penalty methods. The similarity between ea(·,·) and a broken DG form indicates

that the former may not be stable without any additional modiﬁcations. At the

same time, this similarity also suggests that standard DG terms could be used

to stabilize ea(·,·). Below we describe one possible scheme that results from this

approach, focusing on the handling of the local bilinear forms and skipping for

brevity the modiﬁcations to fk(·)

Following [8] we integrate the advective term in the element forms ak(·,·) and

use the uwpind trace ~u on each boundary facet to obtain the upwind element

GMLS for variational problems 7

Fig. 2. Left: convergence of the nonconforming “DG” scheme for diﬀerent polynomial

orders. Right: nonconforming “DG” solution of one-dimensional advection-diﬀusion

problem for increasing P´eclet numbers.

form

~ak(u, v ) = X

Kk∈ΩhZKk

∇u· ∇vdx −ZKk

ub· ∇vdx +Z∂Kk

~u vb·nkdS

To stabilize the diﬀusive term we use the interior penalty method [1]. These

steps transform the element forms into the following stabilized, “DG” versions

aDG

k(u, v) = ~ak(u, v)−X

FZF

{{∇u}} · [[v]]dS +ZF

v·[[u]]dS −δ

hZF

[[u]] ·[[v]]dS,

where the sum is over all element facets in the mesh, {{·}} is the average operator,

[[·]] is the jump operator, and δis stabilization parameter; see [8, p.1261].

We then restrict the elemental DG forms aDG

k(u, v) to the composite recipro-

cal space, i.e., [S]×[S] to obtain their local GMLS approximants eaDG

k(·,·). Sum-

mation of the latter over all elements then yields the global DG form eaDG (·,·).

4 Numerical examples

To demonstrate the approach we’ve implemented the “DG” scheme from §3.1

in one-dimension using the element centroids to deﬁne the point cloud Xη. The

left plot in Fig. 2 highlights the optimal convergence of the scheme for several

diﬀerent polynomial reproduction spaces. We see that in each case the numerical

solution attains the best approximation-theoretic rate for the respective polyno-

mial order.

The right plot in Fig. 2 demonstrates the scheme for increasing P´eclet num-

bers. Solution plots in this ﬁgure reveal that the simple upwind strategy adopted

in our implementation is adequate for low to moderate P´eclet numbers. Future

work will consider improved upwinding for strong advection-dominated prob-

lems, alternatives to the interior penalty stabilization, and extension to higher

dimensions.

8 P. Bochev et al.

Acknowledgments

This material is based upon work supported by the U.S. Department of En-

ergy, Oﬃce of Science, Oﬃce of Advanced Scientiﬁc Computing Research under

Award Number DE-SC-0000230927, and the Laboratory Directed Research and

Development program at Sandia National Laboratories.

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