Abstract and Figures

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 alternative 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.
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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 ΦUwith basis φ={φ1, . . . , φq}; (iii) a Φ-unisolvent1set of sam-
1We recall that Φ-unisolvency implies {φΦ|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×U7→ R+∪ {0}.
GMLS seeks an approximation eτ(u) of the target τ(u)Uin terms of
the sample vector u:= (λ1(u), . . . , λn(u)) Rn, such that eτ(φ) = τ(φ) for all
φΦ, 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 BRn×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)bbRn.
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
bRq
1
2|Bbu|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 eiRnbe the ith Cartesian unit vector and let uτ
i:= c(ei;τ)·φΦ. 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 Uand V, respectively. We
consider the following abstract variational equation: given fVfind uU
such that
a(u, v) = f(v)vV , (3)
where a(·,·) : U×VRis 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 uU
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)vV ,
4 P. Bochev et al.
respectively. Combining these representations yields the following approximation
of (3): find uUsuch that ea(u, v) = e
f(v) for all vV, or equivalently,
cV(v;τ)·a(u, φV) = cV(v;τ)·f(φV)vV. (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 uUsuch that ea(u, vτ
i) = e
f(vτ
i)
for all vτ
iSτ
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 uτSτ
Usuch that
ea(uτ, vτ
i) = e
f(vτ
i)vτ
iSτ
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
KRnV×nUand FRnVhave 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 . 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 and u= 0 on Γ , (9)
where 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(),
a(u, v) = Z
εu· ∇v+ (b· ∇u)vdx and f(v) = Z
fvdx.
Let hand Xηdenote 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 hand 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(|bkxj|),
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
iRnk. 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] = KkhSkand 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
Kkheak(uk, vk) and e
f([v]) := X
Kkhe
fk(vk),
where uk, vkSk. 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
Kkh
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
KkhZKk
u· ∇vdx ZKk
ub· ∇vdx +ZKk
~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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This paper describes the new diffuse approximation method, which may be presented as a generalization of the widely used finite element approximation method. It removes some of the limitations of the finite element approximation related to the regularity of approximated functions, and to mesh generation requirements. The diffuse approximation method may be used for generating smooth approximations of functions known at given sets of points and for accurately estimating their derivatives. It is useful as well for solving partial differential equations, leading to the so called diffuse element method (DEM), which presents several advantages compared to the finite element method (FEM), specially for evaluating the derivatives of the unknown functions.