Mixed Isogeometric Methods for Linear Elasticity with Weakly Imposed Symmetry Jeremias Arf

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Mixed Isogeometric Methods for Linear Elasticity with Weakly

Imposed Symmetry

Jeremias Arf †

October 20, 2022

Keywords— elasticity, de Rham complex, mixed method, isogeometric analysis

Abstract

We consider and discretize a mixed formulation for linear elasticity with weakly imposed

symmetry in two and three dimensions. Whereas existing methods mainly deal with simplicial

or polygonal meshes, we take advantage of isogeometric analysis (IGA) and consequently allow

for shapes with curved boundaries. To introduce the discrete spaces we use isogeometric discrete

diﬀerential forms deﬁned by proper B-spline spaces. For the proposed schemes a proof of well-

posedness and an error estimate are given. Further we discuss our ansatz by means of diﬀerent

numerical examples.

1 Introduction

In general it is hard to deﬁne numerical methods for partial diﬀerential equations (PDEs) which

guarantee convergence and stability such that the approximate solution depends continuously on

(initial-)boundary conditions as well as on source functions. Thus, it is remarkable that Arnold,

Falk and Winther established with the Finite Element Exterior Calculus (FEEC) [2, 3] an abstract

guideline for the design of several mixed formulations in order to have both, stability and conver-

gence. The underlying structure is related to the so-called Hodge-Laplacians associated to some

Hilbert complex and the structure-preserving discretization of such complexes. In [3] the needed key

properties for ﬁnite-dimensional ansatz and test function spaces are derived and moreover the poly-

nomial diﬀerential forms on simplicial meshes are introduced. Latter spaces are fundamental objects

in the development of mixed methods. However, as already indicated the underlying approach of

FEEC is quite abstract and the results allow for other discrete spaces. For instance, Buﬀa et al. [16]

utilized isogeometric analysis (IGA) to deﬁne structure-preserving discrete de Rham chains. With

such discrete isogeometric diﬀerential forms one obtains stable approximations e.g. within the ﬁeld

of Maxwell’s equations, where curved geometries can be handled due to the IGA ansatz. Although

we ﬁnd in literature diﬀerent Hilbert complexes corresponding to interesting PDEs, like the Hessian

complex in linearized General Relativity (see [26]), it is not always easy to apply directly the steps

from [3]. Especially the deﬁnition of discrete subcomplexes equipped with bounded cochain projec-

tions might be complicated. This point becomes evident if complexes with higher derivatives are

studied. One well-known example is the elasticity complex in which second derivatives appear; cf.

[23]. Admittedly, suitable Finite Elements for this chain in 2Dwere introduced, e.g. in [4, 7], but one

needs a lot of degrees of freedom per element and the basis deﬁnition is not straightforward. Another

†TU Kaiserslautern, Dept. of Mathematics, Gottlieb-Daimler-Str. 48, 67663 Kaiserslautern,

Germany ( E-mail: arf@mathematik.uni-kl.de).

arXiv:2210.10192v1 [math.NA] 18 Oct 2022

way to use FEEC is the ansatz complexes from complexes, where two or more Hilbert complexes are

used to discretize PDEs. This approach can be seen in [26], but also in [6] in the context of linear

elasticity. Due to the increased number of complexes the authors in the last two publications are

able to reduce the discretization problem to classical de Rham sequences, which are well-understood

and easy to approximate.

We want to follow this idea of coupled complexes in this article. To be more precise, we use the

connection between de Rham chains and linear elasticity as described in [6, 19] and combine the

underlying formulation with weakly imposed symmetry with the concept of isogeometric diﬀerential

forms from [16]. Whereas already existing methods like in [6, 27, 19] mainly use Finite Elements

on simplicial or polygonal meshes, we want to exploit IGA to incorporate computational domains

with curved boundaries in 2Dand 3D. Even if we are mainly interested in linear elasticity, we start

the article with a rather abstract setting, since it improves clarity. The document has the following

structure.

Before we specify the abstract problem in Sec. 2, we brieﬂy introduce some notation. In Sec. 3

we apply the ideas of Sec. 2 to the example of linear elasticity. For this purpose we deﬁne suitable

B-spline spaces and prove the well-posedness of the scheme with weak symmetry. Key part of the

well-posedness proof is an inf-sup stability result which is veriﬁed utilizing the macroelement tech-

nique. In Sec. 3.4 we consider several numerical examples for the linear elasticity application.

We ﬁnish with a short conclusion at the end of the document in Sec. 4 .

Our main references are [6, 19, 11, 12, 16] and we use results from them at diﬀerent places in the

article.

Notation

In the section we introduce some notation and deﬁne several spaces.

For some bounded Lipschitz domain D⊂Rd, d ∈Nwe write for the standard Sobolev spaces

H0(D) = L2(D), Hk(D), k ∈N, where L2(D)stands for the Hilbert space of square-integrable

functions endowed with the inner product h·,·i =h·,·iL2. The norms k·kHkdenote the classical

Sobolev norms in Hk(D), where k·kH0=k·kL2. In case of vector- or matrix-valued mappings we

can deﬁne Sobolev spaces, too, by requiring the component functions to be in suitable Sobolev

spaces. We use the following notation: for v:= (v1, . . . , vd), v ∈Hk(D, Rd): ⇔vi∈Hk(D),∀i

and for M:=Mij d,d

i,j=1, M ∈Hk(D, M) : ⇔Mij ∈Hk(D),∀i, ∀j. In particular, depending on

the situation we have M=R2×2or M=R3×3in the document below. The inner product h·,·iL2

introduces straightforwardly an inner product on L2(D, Rd). For the deﬁnitions of the next spaces

and norms we follow partly [25] to introduce further notation. First, let us consider vector-valued

mappings. Then if d= 3 we set

H(D, curl):={v∈L2(D, R3)| ∇ × v∈L2(D, R3)},kvk2

curl :=kvk2

L2+k∇ × vk2

L2,

and H(D, div):={v∈L2(D, Rd)| ∇ · v∈L2(D)},kvk2

div :=kvk2

L2+k∇ · vk2

L2if d∈N.

Above we wrote ∇for the classical nabla operator. The deﬁnitions for H(D, curl), H(D, div),and

the corresponding norms can be generalized to the matrix setting by requiring that all the rows

(as vector-valued mappings) are in the respective spaces. Here the curl ∇× and divergence ∇· act

row-wise, too. We write H(D, curl,M), H(D, div,M)and

H(D, div,S):=L2(D, S)∩H(D, div,M),

where we denote with L2(D, S)the space of square-integrable symmetric matrix ﬁelds. If we write

0(D, Rd)we mean the space of H1functions with component-wise zero boundary values in the

sense of the trace theorem; see [29] for more information. But if we have a subscript ’0’ of the form

0(D, Rd)it stands for the L2functions with zero mean.

Later the next operators become useful

Skew("m11 m12

m21 m22#):=m21 −m12,Skew(





m11 m12 m13

m21 m22 m23

m31 m32 m33





):=





m32 −m23

m13 −m31

m21 −m12





,

which can be applied to matrix ﬁelds.

Further, for a matrix M= (Mij )we use an upper index Mjto denote the j-th column and a lower

index Mifor the i-th row. Finally, we introduce the 2Dcurl operator

curl(v):= (∂2v, −∂1v)T,curl((v1, v2)T):="∂2v1−∂1v1

∂2v2−∂1v2#,

i.e. ∂iis the partial derivative w.r.t. the i-th coordinate.

After stating some basic notation we proceed with the explanation of the abstract problem.

2 Abstract framework

In this section we study a special class of variational formulations derived from coupled Hilbert

complexes. Although we are interested in the case of linear elasticity later, we choose here an

abstract notion for reasons of clearness.

For the theoretical background we use and recommend the publications [3, 19].

Deﬁnition 1 (Closed Hilbert complexes).A closed Hilbert complex is a sequence of Hilbert spaces

(Wk)ktogether with closed densely deﬁned linear operators dk:Vk⊂Wk→Wk+1, where Vkis

the domain of dkand further we have closed ranges R(dk)⊂Ker(dk+1)⊂Wk+1. This means

dk+1 ◦dk= 0,∀k.

The domain spaces (Vk, dk)deﬁne the corresponding domain complex which determines a bounded

Hilbert complex, where the Vkare equipped with the graph inner product hv, wiV:=hv, wi+

hdkv, dkwi. We write for the associated norms kvk2

V:=kvk2+



dkv



2. Here and in the following k·k

stands for the norm induced by the underyling Hilbert spaces Wkand we write just h·,·i =h·,·iWk.

Such complexes are key objects within the theory of FEEC, where mixed formulations of Hodge

Laplacians are considered. Especially the stable discretization of various mixed problems are closely

related to Hilbert complexes. That is why we will exploit this notion, too, in order to deﬁne an

abstract mixed weak formulation below.

2.1 Continuous problem

Now we restrict ourselves to a special case of Hilbert complexes as explained in the assumption below.

Assumption 1 (Coupled Hilbert complexes ).Let (Vk, dk)and (¯

Vk,¯

dk)be the domain complexes

associated to the closed Hilbert complexes (Wk, dk)and (¯

Wk,¯

dk)s.t.:

I. ¯

dn−1:¯

Vn−1→¯

Vnand dn−1:Vn−1→Vnare surjective.

II. There are bounded mappings Sk:Vk→¯

Vk+1, k ∈ {n−2, n −1}with Sn−1dn−2=¯

dn−1Sn−2.

III. The mapping ¯

dn−1Sn−2:Vn−2→¯

Vnis surjective.

The next lemma adapted from [5] connects an inf-sup condition with the existence of a bounded

right inverse. This will show us that assumption point III is fulﬁlled if a special inf-sup condition is

valid.

Lemma 1. Let R, Q be two Hilbert spaces. Let T:R→Qbe a bounded linear mapping w.r.t the

norms k·kR,k·kQinduced by the inner products. Further assume

inf

q∈Qsup

w∈R

hT w, qiQ

kwkRkqkQ≥C†>0.

Then there is for all q∈Qan element w∈Rs.t. T w =qand kwkR≤1

C†kqkQ.

Proof. We follow the proof steps of [5, Lemma 2].

For reasons of clariﬁcation we write for the Qand Rinner product h·,·iQ,h·,·iRrespectively. The

Hilbert space adjoint T∗exists and satisﬁes

C†kqkQ≤sup

w∈R

hT w, qiQ

kwkR

= sup

w∈R

hw, T ∗qiR

kwkR

=kT∗qkR, q ∈Q.

Thus we have injectivity of T∗and the left-inverse of T∗is bounded with operator norm ≤1/C†.

This implies the surjectivity of Tand the right-inverse T†is bounded by 1/C†.

Corollary 1 (Auxiliary inf-sup condition).If there is a constant CIS >0s.t.

inf

q∈¯

Vnsup

w∈Vn−2

h¯

dn−1Sn−2w, qi

kwkVkqk≥CIS >0,(1)

then the property III is valid.

Proof. This follows directly from Lemma 1 with T=¯

dn−1Sn−2:Vn−2→¯

Vn.

In Fig. 1 the relations between the diﬀerent spaces are illustrated. For the spaces and operators

Vn−2

. . .

{0}

dn−3

Vn−1

dn−2

dn−1

Vn−1¯

dn−1

Sn−2Sn−1

Figure 1: The diagram commutes and relates the diﬀerent spaces.

in Assumption 1 we can introduce an abstract mixed weak form.

Deﬁnition 2 (Abstract mixed weak form).Find (σ, u, p)∈Vn−1×Vn×¯

Vns.t.

A(σ, τ) + hu, dn−1τi+hp, Sn−1τi=ln−1(τ),∀τ∈Vn−1,

hdn−1σ, vi=ln(v),∀v∈Vn,(2)

hSn−1σ, qi= 0,∀q∈¯

Vn,

where lk:Wk→Rare continuous linear forms and A:Wn−1×Wn−1→Ris a continuous and

coercive bilinear form.

To have a well-deﬁned problem above the obvious question arises: When is the weak form from

above well-posed, i.e. which conditions lead to a unique solution? Since we observe a saddle-point

problem structure, we just have to apply the Brezzi conditions from [13] which can be summarized

in our context as follows.

Deﬁnition 3 (Brezzi stability conditions).There are positive constants CS1, CS2s.t.

(S1) A(τ, τ )≥CS1kτk2

V,∀τ∈kB,

kB :={τ∈Vn−1| hdn−1τ, vi+hSn−1τ, qi= 0,∀(v, q)∈Vn×¯

Vn},

(S2) inf

(v,q)∈Vn×¯

Vnsup

τ∈VnhSn−1τ, qi+hdn−1τ, vi

kτkVkvk+kqk≥CS2>0.

The ﬁrst condition (S1) is obviously fulﬁlled by Assumption 1, i.e. by the surjectivity of dn−1:Vn−1→

Vn, i.e.

A(τ, τ )≥Ckτk2=Ckτk2

V,∀τ∈kB.

The inf-sup condition (S2) seems more problematic and deserves a closer look.

Lemma 2 (Well-posedness).In view of Assump. 1 and Def. 2 we obtain the well-posedness of the

saddle-point problem (2) in the mentioned deﬁnition.

Proof. With the remarks above, it is enough to check the validity of (S2). We orient ourselves towards

the proof of Theorem 7.2 in [19].

Let (v, q)∈Vn×¯

Vn\{0×0}arbitrary, but ﬁxed. By the surjectivity assumption regarding dn−1

we ﬁnd τ∈Vn−1with v=dn−1τ. Then, in view of the surjectivity of ¯

dn−1Sn−2one can choose

ρ∈Vn−2s.t. ¯

dn−1Sn−2ρ=q−Sn−1τ∈¯

Vn.

Finally, we set

σ:=dn−2ρ+τ∈Vn−1.

In particular, we see dn−1σ= 0 + dn−1τ=vand

Sn−1σ=Sn−1dn−2ρ+Sn−1τ=¯

dn−1Sn−2ρ+Sn−1τ

=q−Sn−1τ+Sn−1τ

=q.

By the arbitrariness of (v, q)we get that

T:Vn−1→Vn×¯

Vn, κ 7→ dn−1κ×Sn−1κ

is a continuous and surjective mapping between Banach spaces. By the open mapping theorem we

have the existence of a 0< δ s.t.

Bδ(0 ×0) ⊂T(B1(0)),

where Br(·)denote the respective open r-neighborhoods.

Let kvk+kqk=C > 0. Then we know from above the existence of σ∈Vn−1with T σ = (v, q).

Hence, δ

2C(v, q)∈Bδ(0 ×0) ⊂T(B1(0)).

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MixedIsogeometricMethodsforLinearElasticitywithWeaklyImposedSymmetryJeremiasArfOctober20,2022Keywordselasticity,deRhamcomplex,mixedmethod,isogeometricanalysisAbstractWeconsideranddiscretizeamixedformulationforlinearelasticitywithweaklyimposedsymmetryintwoandthreedimensions.Whereasexistingmethodsma...

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Mixed Isogeometric Methods for Linear Elasticity with Weakly Imposed Symmetry Jeremias Arf

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