Stabilization-free serendipity virtual element method for plane elasticity

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Stabilization-free serendipity virtual element method for plane

elasticity

Alvin Chena,∗, N. Sukumarb,∗

aDepartment of Mathematics, University of California, Davis, 95616, CA, USA

bDepartment of Civil and Environmental Engineering, University of California, Davis, 95616, CA, USA

Abstract

We present a higher order stabilization-free virtual element method applied to plane elasticity

problems. We utilize a serendipity approach to reduce the total number of degrees of freedom

from the corresponding high-order approximations. The well-posedness of the problem is

numerically studied via an eigenanalysis. The method is then applied to several benchmark

problems from linear elasticity and we show that the method delivers optimal convergence

rates in L2norm and energy seminorm that match theoretical estimates as well as the

convergence rates from higher order virtual element methods.

Keywords: virtual element method, polygonal meshes, stabilization-free hourglass control,

strain projection, spurious modes, serendipity elements

1. Introduction

The Virtual Element Method (VEM) is an extension of the classical ﬁnite element method

(FEM) to arbitrary polygonal meshes. In early works [1,3,5,6,7,9,17], the method was

developed for scalar elliptic boundary-value problems as well as extended to linear elastic-

ity. These methods when applied to high-order VEM required additional internal degrees

of freedom; however, from the study of serendipity ﬁnite elements [2], it was shown that

the number of internal degrees of freedom can be greatly reduced. This property is also

important for the extension to three-dimensional VEM. For polyhedral elements, the face

degrees of freedoms cannot be reduced using standard static condensation techniques, but

the serendipity approach can still be applied to reduce these face degrees of freedom [10].

In [8,10] and [15], the serendipity virtual element method was developed for scalar problems

and nonlinear elasticity, respectively. Unlike serendipity FEM, the serendipity VEM is found

to be robust for general polygonal meshes, including on meshes with distorted elements. In

all prior studies with the VEM, a stabilization term is required in order to ensure that the

element stiﬀness matrix has the correct rank. Recently in [12], a stabilization-free VEM was

introduced for the Poisson equation, and extended to linear plane elasticity in [13]. The

main idea in this approach is to modify the approximation space to be able to compute a

higher order polynomial L2projection of the strain. A secondary projection operator is used

∗Corresponding authors

Email addresses: avnchen@ucdavis.edu (Alvin Chen), nsukumar@ucdavis.edu (N. Sukumar)

Preprint submitted to Elsevier November 21, 2022

arXiv:2210.02653v2 [math.NA] 17 Nov 2022

to ﬁll in the additional degrees of freedom introduced by the higher order polynomials. A

similar approach was explored in [16]; however, instead of using additional projections, they

used static condensation to eliminate the extra degrees of freedom. In the standard VEM,

the stabilization term is problem dependent and there is no general method of constructing

it; therefore, devising a stabilization-free virtual element method is desirable.

In this paper, we combine the serendipity elements [8] to extend the stabilization-free

techniques from [12,13] to higher order methods for two-dimensional linear elasticity. The

resulting method will not have a stabilization term and in many cases will not require any

additional internal degrees of freedom. In Section 2, we introduce the model problem of

plane elasticity. In Sections 3and 4we set up the necessary polynomial spaces and recall

some properties of the serendipity VEM. In Section 5, we deﬁne the higher order VEM

spaces from [12,13] and in Section 6we discuss the construction and implementation of

the projections, element stiﬀness and forcing terms. Section 7contains a numerical study

of an upper bound for the order of polynomial enhancement for second- and third-order

methods. In Section 8, we apply the second- and third-order methods to the patch test, two

manufactured problems, a beam under a sinusoidal load and an inﬁnite plate with circular

hole under uniaxial tension. The rate of convergence for the manufactured problems and the

beam problem agree with theoretical estimates; however, consistent with expectations [4,11],

the presence of the curved boundary in the inﬁnite plate with circular hole problem results

in suboptimal convergence rates. We close with some concluding remarks in Section 9.

2. Elastostatic Model Problem and Weak Form

We consider an elastic body that occupies the region Ω ⊂R2with boundary ∂Ω. Assume

that the boundary ∂Ω can be written as the disjoint union of two parts ΓDand ΓNwith

prescribed Dirichlet and Neumann conditions on ΓDand ΓN, respectively. The strong form

for the elastostatic problem is:

∇ · σ+f=0in Ω,σ=σTin Ω,(1a)

ε(u) = ∇su=1

2(∇u+∇uT),(1b)

σ(u) = C:ε(u),(1c)

u=u0on ΓD,(1d)

σ·n=t0on ΓN,(1e)

where f∈[L2(Ω)]2is the body force per unit volume, σis the Cauchy stress tensor, εis the

small-strain tensor with ∇s(·) being the symmetric gradient operator, uis the displacement

ﬁeld, u0and t0are the imposed essential boundary and traction boundary data, and nis

the unit outward normal on the boundary. Linear elastic constitutive material relation (C

is the material moduli tensor) and small-strain kinematics are assumed.

The associated weak form of the boundary-value problem posed in (1) is to ﬁnd the

displacement ﬁeld u∈U, where U:= {u:u∈[H1(Ω)]2,u=u0on ΓD}, such that

a(u,v) = b(v)∀v∈U0,(2a)

where U0= [H1

0(Ω)]2and

a(u,v) = ZΩ

σ(u) : ε(v)dx,(2b)

b(v) = ZΩ

f·vdx+ZΓN

t0·vds. (2c)

In (2), H1(Ω) is the Hilbert space that consists of square-integrable functions up to order 1

and H1

0(Ω) is the subspace of H1(Ω) that contains functions that vanish on ΓD.

3. Mathematical preliminaries

Let Thbe the decomposition of the region Ω into nonoverlapping polygons with standard

mesh assumptions [5]. For each polygon E∈ T h, we denote its diameter by hEand its

centroid by xE. Each polygon Econsists of NEvertices (nodes) with NEedges. Let the

coordinate of each vertex be xi:= (xi, yi). We denote the i-th edge by eiand let the set EE

represent the collection of all edges of E.

3.1. Polynomial basis

Over each element E, we deﬁne [Pk(E)]2as the space of of two-dimensional vector-valued

polynomials of degree less than or equal to k. On each E, we choose the scaled monomial

vectorial basis set as:

M(E) = 1

0,0

1,−η

ξ,η

ξ,ξ

0,0

ξ,...,ηk

0,0

ηk,(3a)

where

ξ=x−xE

, η =y−yE

.(3b)

The α-th element of the set c

M(E) is denoted by mα, and we deﬁne the matrix ˜

Npthat

contains the basis elements as

Np:= 1 0 −η η ξ 0. . . ηk0

0 1 ξ−ξ0ξ . . . 0ηk.(3c)

We also deﬁne the space P`(E)2×2

sym that represents 2 ×2 symmetric matrix polynomials

of degree less than or equal to `. We adopt Voigt notation to represent symmetric 2 ×2

matrices as an equivalent 3 ×1 vector. Let Abe a symmetric 2 ×2 matrix whose Voigt

representation is A:

A=a11 a12

a12 a22,A=





a11

a22

a12



.

On using Voigt notation for P`(E)2×2

sym, the basis set over element Eis written as:

M2×2(E) = 







0



,





0



,





1



,





0



,





0



,





ξ



,...,





η`

0



,





η`

0



,





η`





.(4a)

We denote the α-th vector in this set as c

mαand deﬁne the matrix Npthat contains these

basis elements as

Np:= 

1 0 0 ξ0 0 . . . η`0 0

0 1 0 0 ξ0. . . 0η`0

00100ξ . . . 0 0 η`

.(4b)

On using Voigt notation, the stress and strain tensor are represented as:

σ=





σ11

σ22

σ12.



,ε=





ε11

ε22

2ε12



.

Now with the vector representation of stress and strain, we can write the strain-displacement

relation and the constitutive law in matrix form as:

σ=Cε,ε=Su,(5a)

where Sis a matrix diﬀerential operator that is given by

S=



∂

∂x 0

0∂

∂y

∂

∂y

∂

∂x



,(5b)

and Cis the associated matrix representation of the material tensor that is given by

C=EY

(1 −ν2)

1ν0

ν1 0

0 0 1−ν



(plane stress),

C=EY

(1 + ν)(1 −2ν)

1−ν ν 0

ν1−ν0

0 0 1−2ν



(plane strain),

where EYis the Young’s modulus and νis the Poisson’s ratio of the material.

3.2. Properties of serendipity virtual elements

We recall some results on serendipity virtual element methods for scalar problems from [8].

Let Ebe a polygon with NEedges and let ηEbe the minimum number of unique lines to

cover ∂E. For a k-th order method there are a total of kNEboundary degrees of freedom

and k(k−1)

2internal degrees of freedom. The idea of serendipity VEM is that we are able to

fully deﬁne a computable projection operator by only retaining a subset of all the degrees of

freedom. To do this, we introduce two propositions as proven in [8].

Proposition 1. For k < ηE, if the set of Sdegrees of freedom {δ1, δ2,...δS}contains all of

the kNEboundary degrees of freedom, then the following property holds true:

δ1(pk) = δ2(pk) = · · · =δS(pk) = 0 =⇒pk≡0∀pk∈Pk(E),(6)

where δi(·)is the i-th degree of freedom of its argument.

Proposition 2. For k≥ηE, if the set of Sdegrees of freedom {δ1, δ2, . . . , δS}contains all

kNEboundary degrees of freedom and contain all internal moments of order ≤k−ηE, then

the set satisﬁes

δ1(pk) = δ2(pk) = · · · =δS(pk) = 0 =⇒pk≡0∀pk∈Pk(E).(7)

Once these degrees of freedom are chosen, we can construct a serendipity projection

operator ΠS

k,E such that it satisﬁes the properties:

ΠS

k,E can be fully computed using δ1, δ2, . . . , δS,(8a)

and

ΠS

k,E pk=pk∀pk∈Pk(E).(8b)

This operator is used to deﬁne a serendipity virtual element space for a vector ﬁeld and to

construct two L2operators.

Remark 1. For k= 2, on any polygonal element Eit is suﬃcient to take {δ1, δ2, . . . , δS}

to be the vertex and edge degrees of freedom. For k= 3, if Eis at least a quadrilateral

with four distinct sides then it is also suﬃcient to take {δ1, δ2, . . . , δS}as the vertex and edge

degrees of freedom.

4. Projection operators

We ﬁrst deﬁne the serendipity projection of the displacement ﬁeld as proposed in [8].

We then present the derivation of two L2projection operators, the L2projection of the

displacement and the L2projection of the strain.

4.1. Serendipity projection

For any element E, denote H1(E) := [H1(E)]2and C0(¯

E) := [C0(¯

E)]2. Let Sbe the

number of suﬃcient degrees of freedom for a scalar function as deﬁned in Proposition 1

and 2, and then deﬁne the operator D:H1(E)∩C0(¯

E)→R2Sby

D(v) = δ1(v), δ2(v), . . . , δ2S(v),(9)

where δi(v) is the i-th degree of freedom of the vector ﬁeld v. We deﬁne the serendipity

projection operator ΠS

k,E :H1(E)∩C0(¯

E)→[Pk(E)]2as the unique function that satisﬁes

the orthogonality condition:

D(ΠS

k,E v−v), D(mα)R2S= 0 ∀mα∈[Pk(E)]2.(10a)

On writing out the expressions, we get the equivalent system:

j=1

δj(ΠS

k,E v)δj(mα) =

j=1

δj(v)δj(mα)∀mα∈[Pk(E)]2.(10b)

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Stabilization-freeserendipityvirtualelementmethodforplaneelasticityAlvinChena,,N.Sukumarb,aDepartmentofMathematics,UniversityofCalifornia,Davis,95616,CA,USAbDepartmentofCivilandEnvironmentalEngineering,UniversityofCalifornia,Davis,95616,CA,USAAbstractWepresentahigherorderstabilization-freevirtuale...

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Stabilization-free serendipity virtual element method for plane elasticity

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