A multiscale method for inhomogeneous elastic problems with high contrast coecients Zhongqian Wang1 Changqing Ye1 Eric T. Chung1

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A multiscale method for inhomogeneous elastic

problems with high contrast coeﬃcients

Zhongqian Wang1,∗, Changqing Ye1, Eric T. Chung1

1Department of Mathematics, The Chinese University of Hong Kong, Hong Kong Special Administrative Region

Abstract: In this paper, we develop the constrained energy minimizing generalized multiscale ﬁnite element

method (CEM-GMsFEM) with mixed boundary conditions (Dirichlet and Neumann) for the elasticity equations

in high contrast media. By a special treatment of mixed boundary conditions separately, and combining the

construction of the relaxed and constraint version of the CEM-GMsFEM, we discover that the method oﬀers

some advantages such as the independence of the target region’s contrast from precision, while the sizes of

oversampling domains have a signiﬁcant impact on numerical accuracy. Moreover, to our best knowledge, this

is the ﬁrst proof of the convergence of the CEM-GMsFEM with mixed boundary conditions for the elasticity

equations given. Some numerical experiments are provided to demonstrate the method’s performance.

Keywords: CEM-GMsFEM; mixed boundary conditions; high contrast media

1 Introduction

The study of elastic inhomogeneous mixed boundary conditions is a major area of research in the ﬁeld

of inverse problems, which has wide and practical applications in many ﬁelds such as geophysics, oil

exploration, remote sensing, ocean exploration, radar and sonar [3,6,27,28]. In the past decades, many

researchers in diﬀerent ﬁelds have been actively exploring computationally eﬃcient methods for solving

mixed boundary PDEs, for example, avoiding internal nodes or domain discretization to evaluate speciﬁc

solutions [22], multiscale extensions and averaging techniques to approximate models with asymptotically

small chemical patterns [4,26], and inhomogeneous boundary treatments for ﬂuid-related models in

physics [5,13]. A challenging problem that arises in this domain is the solid rheological properties

of materials at the interface, such as the viscoelastic contact interface between transparent plexiglass

components and supports used in aerospace and underwater vehicles, the creep problem at the contact

interface between diﬀerent metallic materials in high-temperature environments, and the mechanical

behavior of materials with high contrast parameters for interfaces with complete bonding [14,15]. These

traditional methods are no longer valid.

A diﬀerent approach to the traditional problem is Multiscale Finite Element Methods (MsFEM)

[2,7,16]. MsFEMs have been developed over the past decades by transforming microscale inhomo-

geneous information into macroscale parameters via homogenization or up-scaling, and then solving at

the macroscale level, which mainly includes Representative Volume Element Methods (RVEMs) [18],

Heterogeneous Multiscale Methods (HMMs) [1,12]. However, these methods are diﬃcult to handle the

grid-dependent elimination of the localization phenomenon. Generalized multiscale ﬁnite element method

(GMsFEM) [11] was then proposed to obtain an eﬃcient eigenvalue approximation using a local-global

model reduction technique, where the constructed global multiscale space can be applied repeatedly to

get an eﬃcient multiscale solution with reduced degrees of freedom at the macroscopic scale. Moreover,

GMsFEM has been widely adopted in the ﬁeld of equivalent parameter prediction, elastic wave propa-

gation, acoustic analysis and gradient theory [8]. Regardless of implementation complexity, the above

∗Corresponding author.

arXiv:2210.11297v1 [math.NA] 20 Oct 2022

methods have a similar issue: the potential high computing cost of handling inhomogeneous issues in

intricate large-scale models.

In this paper, we aim to develop the constraint energy minimizing generalized multiscale Galerkin

method (CEM-GMsFEM) to solve the complex elastic PDEs with mixed inhomogeneous boundary con-

ditions. As far as we know, no previous research has investigated in this ﬁeld. Although the bit-structural

properties of common concrete [17], alloys, and other materials in engineering have obvious eﬀects on

macroscopic properties, these materials also have typical inhomogeneous and multiscale characteristics,

and there are three main diﬃculties in using the FEMs to deal with these complex material boundary

problems:

•In the physical sense, the intrinsic structure equations lack internal length parameters characterizing

the microstructural features of the material, and heterogeneous theory cannot reasonably explain.

•In the numerical calculation, the spatial complexity of the microstructural information of hetero-

geneous materials can cause expensive computation time and the choice of minimum scale and

minimum degrees of freedom is diﬃcult to balance.

•In the validation of the computational method, when using oversampling techniques to construct

multiscale basis functions, it is diﬃcult to ﬁnd a suitable model to improve the accuracy of the

method.

We could use an improved CEM-GMsFEM to address the three limitations presented above [9]. Some

works have demonstrated that the convergence of this method is independent of contrast, and that it

decreases linearly with grid size for an appropriate choice of oversampling size [19]. At the same time,

CEM-GMsFEM is widely used in physical, geographical and environmental engineering [20,21], its online

methods have been introduced to adaptively construct multiscale basis functions in certain regions to

signiﬁcantly reduce errors [10]. In the case of mixed boundary conditions, it is diﬃcult to choose an

adaptive method to handle multiple boundary cases simultaneously and by choosing a suﬃcient number

of oﬄine basis functions which can result in low errors at diﬀerent contrast and grid sizes [23–25]. Using

our recently proposed method and a special online basis construction for the oversampling regions, we

show that the errors can be reduced suﬃciently by an appropriate choice of the oversampling regions.

The paper is organized as follows. In Section 2, we introduce PDEs for mixed boundary conditions

and the notations of grids. In Section 3, the CEM-GMsFEM used in this paper and the procedure for

handling the mixed boundary conditions are presented. In Section 4, numerical analysis are given in

Section 4. In Section 5, we conduct the conclusions.

2 Problem formulation and ﬁne grid approximation

In this section, we give mathematical models of anisotropic elastic materials with inhomogeneous Dirichlet

and Neumann boundary conditions. In the following equations, the domain Ω ∈Rd(d= 2,3) denotes an

elastic body, and ∂Ω denotes its boundary with ∂Ω=Γa∪Γb. The linear elasticity problem consists of

ﬁnding the displacement u, such that:











-div (σ(u)) = f, in Ω,

u=h, on Γa,

σ(u)·n=g, on Γb,

σ(u)=2µε (u) + λ∇ · uI,in Ω,

ε(u) = 1

2h∇u+ (∇u)Ti,in Ω,

(1)

where a body force fis considered in the domain Ω,

the surface force gis on the Neumann boundary part,

his on the Dirichlet boundary part,

σand εare stress and strain tensor,

nis the outward normal vector along Γb,

λand µare the Lam´e coeﬃcients.

Then this inhomogeneous problem with Dirichlet conditions are incorporate through the decomposi-

tion ˜u=u−hsuch that ˜u= 0 on Γa,i.e.,

˜u∈H1

a(Ω) := v∈H1(Ω) |v= 0 on Γa.

We rewrite the problem (1) in a variation formulation: ﬁnd ˜u∈H1

a(Ω), such that

ZΩ

σ(˜u) : grad (v) dx=ZΩ

f·vdx+ZΓb

g·vds−ZΩ

σ(h) : grad (v) dx, ∀v∈H1

a(Ω) .(2)

Let φ1, ..., φnbe the basis set for H1

a(Ω) ,then ˜uhsatisﬁes

Ah˜uh=Lh,

where Ahis symmetric, positive deﬁnite matrix with

Ah,ij =a(φi, φj) = ZΩ

σ(φi) : (φj) dx,

and Lhis a vector with i-th component

l(φi) = ZΩ

f·φidx+ZΓb

g·φids−ZΩ

σ(h) : grad (φi) dx.

Now we present GMsFEM. In this paper, we will develop and analyze the continuous Galerkin(CG) cou-

pling after the construction of local basis functions. In essence, the CG coupling will need vertextrased

local basis functions, then we give some mesh notations as follows. Let THbe a standard quadrilateral-

ization of the domain Ω, where we call the THcoarse grid, H > 0 being the coarse mesh size. Elements

of THare called coarse lattice blocks. The set of all coarse gird edges is denoted by EH, and the set

of all coarse gird nodes is denoted by SH.We let Tha ﬁne mesh be the conformal reﬁnement of the

quadrilateral and h > 0 the size of the ﬁne mesh. We use Ehto denote the set of facets in Thwith

Eh=Eh

a∪ Eh

b. In addition, we denote the number of ﬁne grid nodes by NHand the number of ﬁne grid

blocks by N. If we consider a two-dimensional region of space [0,1] ×[0,1] with Nx and Ny partitions

in the xand ydirections respectively, and nx and ny partitions on each ﬁne grid, we have the following

equation: NH=Nx ∗N y ∗nx ∗ny, N = (N x ∗nx + 1) ∗(Ny ∗ny + 1) .We note that the reﬁnement for

the use of conformity is only intended to simplify the discussion of the method and is not a restriction

on it. As is shown in Figure 1, we deﬁne Ki,m as an oversampled domain on each Ki∈ T H:

Ki,m =[Kj∈ T H|Kj∩Ki6= 0∪Ki,m−1,

where Kiis the closure of Ki,and the initial value Ki,0=Kifor each element.

Figure 1: Illustration of the oversampling domain

3 The construction of the CEM-GMsFEM basis function

In this section, we describe the construction process of the CEM-GMsFEM, ﬁrst constructing auxiliary

basis functions, and then moving on to multiscale basis functions throughout the oversampling area

utilizing constrained energy minimization. We also provide an innovative approach to inhomogeneous

elastic equations with natural and Dirichlet boundary conditions.

3.1 Auxiliary basis function

Let V(Kj) be the snapshot space on each coarse grid block Kj, and we use the method of the spectral

problem to solve the basis functions on Kj: ﬁnd θi

j, φi

j∈R×V(Kj) such that for all v∈V(Kj),

aiφi

j, v=θi

jsiφi

j, v,∀v∈V(Kj),(3)

where

aiφi

j, v=ZKj

σφi

j:(v) dx, (4)

and

siφi

j, v=ZKj

˜κφi

j·vdx(5)

with ˜κ=PN1

i=1 (λ+ 2µ)∇χi· ∇χi, N1is the number of all coarse grids, {∇χi}N1

i=1 is a set of partition of

unity functions on the coarse grid.

Let the eigenvalues be in the following order:

θ1

j≤θ2

j≤θ3

j≤... ≤θi

j≤...,

then the local auxiliary space Vaux (Kj) is deﬁned by gjeigenvalue functions as following

Vaux (Kj) = span φi

j|1≤j≤gj.(6)

We remark that

j=1

V(Kj), Vaux =

j=1

Vaux (Kj).

Let sφi

j, v=PN1

j=1 siφi

j, v,we deﬁne that ψ∈Vis φi

j-orthogonal if

sψ, φi

j=(1, j06=j,

0, j0=j.

In addition, we deﬁne a projection operator π=PN1

i=1 πifrom space Vto Vaux by

πi(v) =

j=1

siv, φi

j

siφi

j, φi

jφi

3.2 Oﬄine multiscale basis functions

In practice, many functions are discontinuous in the domain, so after constructing the auxiliary basis func-

tions, we consider an oversampling method to extend these functions that are discontinuous in the domain

onto the oversampling region Kj,m. For φi

j∈H1

a(Kj,m) := v∈H1(Kj,m)|v= 0 on Γa∩∂Kj,m,we

give the relaxed CEM-GMsFEM: ﬁnd ξI,i

cem,j,m,such that

ξI,i

cem,j,m = argmin a(ξ, ξ) + sπξ −φi

j, πξ −φi

j|ξ∈H1

a(Kj,m).(7)

We could also constrain the above problem:

ξII,i

cem,j,m = argmin a(ξ, ξ)|ξ∈H1

a(Kj,m), ξ is φi

j-orthogonal.(8)

In the following, we describe the constructed form of CEM-GMsFEM in two parts.

Part.1. Relaxed CEM-GMsFEM

Note the problem Eq.(7) is equivalent to the local problem: ﬁnd ξI,i

cem,j,m ∈H1

a(Kj,m),such that

aξI,i

cem,j,m, z1+sπξI,i

cem,j,m, π (z1)=sφi

j, π (z1),∀z1∈H1

a(Kj,m).(9)

Then, we obtain the following matrix representation from the above formulation

Az1+Mz2MT

z2z1=Mz2,(10)

where

Az1:= (a(ξi, ξj)) ,Mz2:= s((qi, qi)) ,

θ, z1,z2are vectors of coeﬃcients for the approximations, and Mz2is the internally reordered matrix

of Mz2.

Finally we note that the global multiscale space is deﬁned by

Vcem := span nξI,i

cem,j,m|1≤i≤gj,1≤j≤N1o.

The relaxed CEM-GMsFEM bases are deﬁned by:

ξI,i

cem,j = argmin na(ξ, ξ) + sπξ −φI,i

cem,j , πξ −φi

j|ξ∈H1

a(Ω)o,(11)

which is equivalent to the following problem: ﬁnd ξI,i

cem,j ∈V, such that

aξI,i

cem,j , z1+sπξI,i

cem,j , π (z1)=sφi

j, π (z1),∀z1∈H1

a(Ω) .(12)

Furthermore, we deﬁne the global space of multiscale basis functions as

Vglo = span nξI,i

cem,j |1≤i≤gj,1≤j≤N1o.

Part.2. Constrained CEM-GMsFEM

Note the problem Eq.(8) is equivalent to the local problem: ﬁnd ξII,i

cem,j,m ∈H1

a(Kj,m),

θ∈Vaux (Kj,m) such that

aξII,i

cem,j,m, z1+s(z1, θ) = 0,∀z1∈H1

a(Kj,m),

sξII,i

cem,j,m −φi

j, z2= 0,∀z2∈Vaux (Kj,m).

(13)

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AmultiscalemethodforinhomogeneouselasticproblemswithhighcontrastcoecientsZhongqianWang1;,ChangqingYe1,EricT.Chung11DepartmentofMathematics,TheChineseUniversityofHongKong,HongKongSpecialAdministrativeRegionAbstract:Inthispaper,wedeveloptheconstrainedenergyminimizinggeneralizedmultiscaleniteelement...

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A multiscale method for inhomogeneous elastic problems with high contrast coecients Zhongqian Wang1 Changqing Ye1 Eric T. Chung1

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