THE ISOGEOMETRIC MULTISCALE FINITE ELEMENT METHOD FOR HOMOGENIZATION PROBLEMS

VIETNAM NATIONAL UNIVERSITY, HO CHI MINH CITYHO CHI MINH CITY UNIVERSITY OF SCIENCE HOANG TUONG THE ISOGEOMETRIC MULTISCALE FINITE ELEMENT METHOD FOR HOMOGENIZATION PROBLEMS MSc THESIS I

VIETNAM NATIONAL UNIVERSITY, HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF SCIENCE

HOANG TUONG

THE ISOGEOMETRIC MULTISCALE

FINITE ELEMENT METHOD

FOR HOMOGENIZATION PROBLEMS

MSc THESIS IN MATHEMATICS

Ho Chi Minh city - 2012

First and foremost, I would like to express my sincere gratitude to my advisor

Dr Nguyen Xuan Hung for supporting my research, for his patience, ment, and enthusiasm Without his guidance, this thesis would not have beencompleted

encourage-I would like to thank Prof Dang Duc Trong - Dean of the Faculty of ematics and Computer Science, and Dr Nguyen Thanh Long for introducing

Math-me to Partial Differential Equations (PDEs) I will never forget the wonderfullectures I have learned during the course as a Master student

I would like to thank all of my classmates, who were side by side with me inthe Master program The time being with you is one of the most memorable time

in my life To Mr Do Huy Hoang, the oldest member, his passion of learningwill always inspires me

I would like to thank Mr Thai Hoang Chien, Mr Tran Vinh Loc and all themembers in Division of Computational Mechanics, Ton Duc Thang Universityfor the helpful discussion of Isogeometric Analysis (IGA) when I first begin myjourney doing the research

Last but not the least, I would like to thank my parents for giving birth to meand continuously supporting me spiritually throughout my life

Ho Chi Minh city, September, 2012

Hoang Tuong

1.1 Heterogeneous material 1

1.2 Multiscale modeling 2

1.3 Homogenization theory 3

1.3.1 Setting of the problem 4

1.4 Finite Element Analysis (FEA) 7

1.5 Isogeometric Analysis (IGA) 8

v

1.7 The Isogeometric Analysis Heterogeneous Multiscale Method

(IGA-HMM) 11

2 Preliminary results on homogenization theory 13 2.1 Main convergence results 15

2.2 Proof of the main convergence results 16

2.3 Convergence of the energy 20

3 The Finite Element Heterogeneous Multiscale Method (FE-HMM) 22 3.0.1 Model problems 22

3.1 The finite element heterogeneous multiscale method (FE-HMM) 23 3.1.1 Macro finite element space 24

3.1.2 Micro finite element space 25

3.1.3 The FE-HMM method 26

3.2 The motivation behind the FE-HMM 26

3.3 Convergence of the FE-HMM method 28

3.3.1 Priori estimates 28

3.3.2 Optimal micro refinement strategies 29

3.4 Numerical experiments 29

3.4.1 2D-elliptic problem with non-uniformly periodic tensor 29

3.4.2 2D-elliptic problem with uniform periodic tensor 31

4.1 NURBS-based isogeometric analysis fundamentals 42

4.1.1 Knot vectors and basis functions 42

4.1.2 NURBS curves and surfaces 43

4.1.3 Refinement 45

4.2 An isogeometric analysis heterogeneous multiscale method (IGA-HMM) 46

4.2.1 Model problems 46

4.2.2 Drawbacks of the FE-HMM method 47

4.2.3 The isogeometric analysis heterogeneous multiscale method (IGA-HMM) 47

4.2.4 Priori Error Estimates 51

4.3 Numerical validation 53

4.3.1 Problem 1 53

4.3.2 Problem 2: IGA-HMM applied for curved boundary domains 60 4.3.3 Problem 3: An efficiency of IGA-HMM with a flexible de-gree elevation 63

4.3.4 An higher order of IGA-HMM in both macro and micro patch space 67

Conclusions and future work 73 Appendix 75 Control data for NURBS objects 75

vii

List of Figures

1.1 Heterogeneous material (www.advancedproductslab.com) 1

1.2 Multiscale systems (www.scorec.rpi.edu) 2

1.3 Multiscale modeling (http://www.efrc.udel.edu) 3

1.4 Periodic domain modelling 4

1.5 An illustration of CAD objects using NURBS (www.tsplines.com) 9 1.6 An illustration of geometry description in IGA 9

1.7 Communication with CAD: a comparision between FEA and IGA (Hughes-Cottrell-Bazilevs, CMAME, 2005) 10

3.1 [4] In every element of the FE-HMM, the contribution to the stiff-ness matrix of the macroelements (a) is given by the solutions of the microproblems (b), which are computed using numerical quadra-ture on every microelement (c) 24

3.2 Domain of the problem (3.13) 30

3.3 Conductivity tensor and its oscillation of problem (3.13) withε= 0.1 30 3.4 H1 (energy) error between Dirichlet coupling conditions for δ = 1.1ε, δ = 53ε and Periodic coupling condition forδ =ε 33

3.5 L2 error between Dirichlet coupling conditions for δ = 1.1ε, δ = 53ε

1.1ε, δ = 53ε and Periodic coupling condition forδ =ε 34

3.7 L2 norm between Dirichlet coupling conditions for δ= 1.1ε, δ = 53ε

3.8 H1 error when using periodic coupling conditions for δ = 1.1ε, δ =

ix

4.4 Solution of the problem 4.3.1 55

4.17 Convergence of the energy norm and max norm, purely using gree elevation 66

4.18 H1 error of the thermal quarter annulus problem, using NURBS ofdegree 5 in micro space 70

degree 5 in micro space 71

4.20 CPU time solving thermal quarter annulus problem, using NURBS

of degree 5 in micro space 72

xi

List of Tables

constraint, N mac fixed.) 31

Dirich-let one ( δ = 53ε, ε= 0.005, Nmac fixed) 32

refinement strategy 63

4.7 L2 error of the thermal quarter annulus problem, using H1 microrefinement strategy 64

CPU-time 67

degree 5 in micro space 70

degree 5 in micro space 71

4.11 CPU time solving the thermal quarter annulus problem, using

xiii

: α - th derivative of u, for multi - index α

Du ≡ ∇u= (ux1, , uxn) : gradient vector of u,

u ∈ C k(Ω)| uhas compact support

Cc∞(Ω) = {u ∈ C∞(Ω)| uhas compact support}(sometimes also denoted by D(Ω))

Wk,p(Ω), Hk(Ω), etc.(k = 0,1,2, ,1 6p < ∞) denote Sobolev spaces

Figure 1.1: Heterogeneous material (www.advancedproductslab.com)

Heterogeneity can be the variation in compositions, such as in compositesmaterial or porous media, see Fig.1.1 These heterogeneities lead to a nonuniformdistribution of local materials properties Heterogeneity is also present at variouslength scales and in different forms For example, polycrystalline materials arecomposed of many crystalities of varying size and orientation.

When modeling phenomena, traditional models usually focus on one scale If weinterested in the macro behavior of a system, we model the effect of the smallerscales by some (simple) constitutive relations On the other hand, if the microeffect is our interest, we usually presume that there is no change in the macroscales However, for a more complex problem which has important features atmultiple (spatial or temporal) scales, we need to include into the models theinformation not only from a single one Multiscale modelling is used in this case,see Fig 1.2

Figure 1.2: Multiscale systems (www.scorec.rpi.edu)

2

The aim of multiscale modeling is to give the properties or behaviors of a tem on one level using the information from different levels Each level addresses

sys-a phenomenon over sys-a specific window of length sys-and time On esys-ach level psys-articulsys-arapproaches are used for description of the system, see Fig 1.3 Multiscale mod-eling is especially important in material engineering because we usually need topredict material properties or system behavior based on knowledge of the microstructure and properties of elementary processes We refer to [15, 30] for moredetails

Figure 1.3: Multiscale modeling (http://www.efrc.udel.edu)

In various fields of science and technology such as Mechanics, Physics, Chemistryand Engineering, when studying composite materials, macroscopic properties ofcrystalline or polymer structures, nuclear reactor design, etc ones have to dealwith boundary value problems with heterogeneous media If the period of the

structure is small in comparison with the size of the region in which the system

is studying, it can be characterized by a small parameter which is the ratio of theperiod of the structure to a typical length in the region Starting from the mi-croscopic description of the problem, we want to find the macroscopic or effectivebehavior of the system This process of seeking an average formulation is calledhomogenization

The theory of homogenization has been studied for many years Some ematicians in this field include: A Benssoussan, J.L.Lions, G Papanicolaou [9],

math-G Dal Maso [12], De Giorgi [13], F.Murat, L.Tarta [24], G Nguetseng [25, 22],

G Allaire [5] , V.V.Zhikov, S.M Kozlov, O.A Oleinik [29],

1.3.1 Setting of the problem

We consider a model problem of diffusions or conductivity in a periodic medium(for example, an heterogeneous domain obtain by mixing periodically two differentphases, one being the matrix and the other is the inclusions, see figure Fig 1.4)

ε

0000 0

10 1 00 0 00 0 00

00 00 0 00 0

0 00 0 00 00 0

00000

0

00 0 0 00 0

00000 0

0 00 00

0 000 000 111 111

00 00 00 00 000

00 0 00 00 11 11

00 000

Ω

Figure 1.1: A perio domain.

where u

 (x) is the unknown modeling the potential or the temperature.

Remark 1.1.1 From a al point of view, problem (1.2) is well posed in the

sense that, if the sour e term f(x) belongs to the sp e L

2

of square integrable tions on then the Lax-Milgram lemma implies e and uniqueness of the solution

where the onstant C does not depend on .

The domain with its y A

x



, is highly heterogeneous with perio heterogeneities of  Usually one does not need the full details of the variations

of the potential or temperature u

 , but rather some global of a eraged behavior of the domain as an homogeneous domain In other words, an e or equivalent

y of is sought From a n point of view, solving equation (1.2) by any method will require too m h eort if  is small the number of elements

(or degrees of freedom) for a xed level of grows like 1=

N It is thus preferable

to a erage or homogenize the properties of and an approximation of u



on a mesh Averaging the solution of (1.2) and nding the e properties of the domain is what we homogenization.

There is a of methodology between the traditional ph h of homogenization and the theory of homogenization In the lit-

erature, the representative volume element (RVE) method is often used (see [5 ℄,

or Chapter 1 in [12 ℄) Roughly speaking, it in taking a sample of the

heteroge-neous medium of size m h larger than the heterogeneities, but still m h smaller than the

Figure 1.4: Periodic domain modelling

The periodic domain is called Ω (a bounded open set in Rd , where d > 1 isthe space dimension), its periodic period ε (a positive number which is assumed

to be very small in comparison with the size of the domain), and the rescaled

4

unit periodic cell Y = (0,1)d The conductivity in Ω is not constant but variesperiodically with period in each direction.

The matrix is characterized by a second-order conductivity tensor a(y) where

y=x/ε ∈ Y is called the fast variable, while x ∈Ω is called the slow variable.Since the component conductors do not need to be isotropic, the matrix a(y)can be any second-order tensor that is bounded and positive define, i.e thereexist two positive constant β >α >0 such that for any vector ξ ∈Rd, and at anypoint y ∈ Y

Denoting byf(x) the source term (a scalar function defined in Ω), and enforced

a Dirichlet doundary condition (for simplicity), our model problem of conductivity

mechanics, electrostatics, where u is displacement field or electric field, spectively, see Table 1.1

re-Table 1.1: Significant of u and f in application

In this thesis, we restrict our research on static heat problems, where u is thetemperature field We emphasize that the applications for other problems aresimilar

To solve the problem (1.1), due to the requirement on regularities of the tion and in the numerical approaches, we consider the corresponding weak form,where (1.1) is replaced by a variational formulation, namely

The domain Ω with its conductivity a(x/ε) is highly-oscillating with periodicheterogeneities of length scale ε Usually one does not need the full details ofthe variations of the potential or temperature, but rather some global averaged behaviorof the domain Ω considered as an homogeneous domain In other words,

6

an effective or equivalent macroscopic conductivity of Ω is sought.

Observe also that making the heterogeneities smaller and smaller means that

we ’homogenize’ the mixture and from the mathematical point of view this meansthat ε tends to zero Taking ε is the mathematical ’homogenization’ of problem(1.1)

Many natural questions arise:

1 Does the temperature uε converge to some limit function u0?

2 If that is true, does u0 solve some limit boundary value problem?

3 Are then the coefficients of the limit problem constant?

4 Finally, is u0 a good approximation of uε?

Answering these questions is the aim of the mathematical theory of tion These questions are very important in the applications since, if one can givepositive answers, then the limit coefficients, as it is well known from engineersand physicists, are good approximations of the global characteristics of the com-posite material, when regarded as an homogeneous one Moreover, replacing theproblem by the limit one allows us to make easy numerical computations

Partial differential equations model a wide range of problems in biological andphysical sciences Many interesting (and realistic) PDE models are complicatedand cannot be solved analytically Finite Element Method (FEM) is one of themost popular and powerful tools to deal with such PDEs Finite Element Analysis(FEA) was first developed in 1943 by R Courant, who utilized the Ritz method of

numerical analysis and minimization of variational calculus to obtain approximatesolutions to vibration systems Since then, FEA has been used and developed bymany people all over the world from many different fields, from mathematics toengineering: I Babuska, F.Brezzi, T.Belytschko, L.Demkowicz, J.Douglas, T.J.R.Hughes, J.T.Oden, R.L.Taylor, O.C Zienkiewiczs, Some references for FEA are[10, 20].

The isogeometric analysis was first proposed by Hughes and co-workers [21] andnow has attracted the attention of academic as well as industrial engineeringcommunity all over the world The IGA allows to closely link the gap betweenComputer Aided Design (CAD) and Finite Element Analysis (FEA) It meansthat the IGA uses the same basis functions to describe both the geometry of do-main (CAD) and the approximate solution Being different from basis functions

of the standard FEM based on Lagrange polynomial, isogeometric approach lizes more general basis functions such as B-splines and Non-Uniform RationalB-splines (NURBS) that are common in CAD geometry The exact geometry istherefore maintained at the coarsest level of discretization and the re-meshing isperformed on this coarsest level without any communication with CAD geometry,see Fig 1.7 Furthermore, B-splines (or NURBS) provide a flexible way to makerefinement, de-refinement, and degree elevation [21] They allow us to achieveeasily the smoothness of arbitrary continuity in comparison with the traditionalFEM For a reference on IGA, we recommend the excellent book [11] and we refer

uti-to the NURBS book [27] for geometric description

8

Sederberg et al., 2003&2004

T-splines

Breaking the tensor-product structure: T-splines

T-spline goals:

• Efficient local refinement

• Trimmed NURBS to T-spline conversion

• Easier volume meshing (hopefully )

Figure 1.5: An illustration of CAD objects using NURBS (www.tsplines.com)

IGA: geometry description and mesh refinement

R V´ azquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, 2010 7 / 33

Figure 1.6: An illustration of geometry description in IGA

Method (FE-HMM)

To solve the homogenization problems, analytic approaches such as in [9], [29] mogenized equations are derived However, the coefficients of these equations areonly computed explicitly in some special cases, such as when the medium follows

ho-IsoGeometric Analysis (IGA): an overview

Geometry is defined by Computer Aided Design (CAD) software.

Hughes, Cottrell, Bazilevs, CMAME, 2005

R V´ azquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, 2010 4 / 33

IsoGeometric Analysis (IGA): an overview

Geometry is defined by Computer Aided Design (CAD) software.

CAD is based on Non Uniform Rational B-Splines ( NURBS ).

CAD and FEM use different descriptions for the geometry.

CAD and IGA use the same geometry description

I Maintain the geometric description given by CAD (NURBS).

I Iso-parametric approach: PDEs are numerically solved with NURBS.

Hughes, Cottrell, Bazilevs, CMAME, 2005

R V´ azquez (IMATI-CNR Italy) Introduction to Isogeometric Analysis Santiago de Compostela, 2010 4 / 33

Figure 1.7: Communication with CAD: a comparision between FEA and IGA

(Hughes-Cottrell-Bazilevs, CMAME, 2005)

some periodic assumptions, and not explicitly available in general Furthermore,

fully computations with complicated scale interactions of the heterogeneous

sys-tem are very ineffective due to high computational cost, and can be prohibited

Thus, to solve these problems, so far advanced computational technologies have

been developed

Literature review of various multiscale approaches can be found in [17], [30]

In this work, we focus on the heterogeneous multiscale method (HMM), which

was proposed in [32] A review for HMM can be found in [2], [6] This method

is a general framework which allows ones to develop various approaches to

ho-mogenization problems The simplest one is the finite element heterogeneous

multiscale method (FE-HMM), which uses standard finite elements such as

sim-plicial or quadrilateral ones in both macroscopic and microscopic level Solving

the so-called micro problems (with a suitable set up) in sampling domains around

traditional Gauss integration points allows one to approximate missing effective

10

information for the macro solver, where standard finite element methods are used.Details on FE-HMM can be found in Chapter 3.

Mul-tiscale Method (IGA-HMM)

Although very popular, the standard FEM still has some shortcomings whichaffect the efficiency of the FE-HMM method Firstly, the discretized geometrythrough mesh generation is required This process often leads to the geometricalerror even using the higher-order FEM Also, the communication of geometrymodel and mesh generation during analysis process is always needed and thisconsumes much time [21], especially for industrial problems Secondly, lower-order finite elements often require fine meshes to produce the desired accuracy ofapproximate solution for complicated problems, such as FE-HMM based on four-node quadrilateral element (Q4) which will be indicated in this work Thirdly,high-order discretizations still have some restrictions on element topologies (forexample, the connection of different types of corner, center, or internal nodes)and C0 continuity These disadvantages lead to an increasing in the number ofmicro coupling problems and raise the computational cost in FE-HMM Hence,

we need to consider alternative methods to solve these issues

In this thesis, we introduce a new approach, which is the main contribution ofthe research: a so-called Isogeometric analysis heterogeneous multiscale method(IGA-HMM) which utilizes NURBS as basis functions for both exact geometricrepresentation and analysis The NURBS are used as basis functions for bothmacro and micro element spaces, where the former FE-HMM utilizes standardFEM basis This tremendously facilitates high-order macroscopic discretizations

by a flexibility of refinements and degree elevations with an arbitrary continuity

of basis functions Several numerical results show the reliability, effectiveness androbustness of the proposed method

12

mathemati-In this chapter we present some preliminary results on homogenization theory ofelliptic equations A more detail on this theory is out of the aim of this thesis.For references, we refer to [9] , [29], [16].

First, we introduce some definitions which will be used in the latter parts

Definition 2.0.1 Let O be an open set in RN, and let α, β ∈R such that0< α < β.

We define M(α, β, O) the set of the N × N matrices a = (aij)1≤i,j≤N ∈(L∞(O))N ×N

In this chapter, Y will denote the box in RN defined by

Y = (0, l 1)× ×(0, lN), (2.2)where l 1 , , lN are given positive numbers

Definition 2.0.2 Let Y be defined by (2.2) and f a function defined a.e onRN The function f is called Y − periodic iff

f(x+kliei) =f(x) a.e on RN, ∀k ∈Z, ∀i ∈ {1, , N }, (2.3)

where e1, , eN is the canonical basis of RN.

Now, we consider the model problem

Model problem

Let Ω be a domain in RN, N = 1,2,3, with Lipschitz boundary ∂Ω on which weimpose Dirichlet conditions Given f ∈ L 2(Ω) as a source term, we consider thefollowing highly oscillating coefficient second-order elliptic problem

with α, β ∈R such that 0< α < β and M(α, β, Y) given by Definition 2.0.1.

In the following, we present the main convergence result, which plays an portant role in the homogenization theory Then, we will give a proof for thisresult

Theorem 2.1.1 Let f ∈ L 2(Ω) and u is the solution of (2.4) with a defined by

where (e j)Nj=1 is the canonical basis of RN

The following proposition gives us the ellipticity of the homogenized matrix

Proposition 2.1.2 The homogenized matrix a0 given by (2.10) is uniformly elliptic and bounded Moreover, ones have

a0 ∈ M(α,β

2

We refer to [29] or [16] for the proof of this proposition

In this section we give a rigorous proof of Theorem 2.1.1, following a generalmethod proposed by Tartar (1977) This method relies on the construction of aclass of oscillating test functions obtained by periodizing the solution of a problemset in the reference cell Let uε be the solution of (2.4) Firstly, we will provethat there exists a subsequence (still denoted by ε), such that

i) uε * u0 weakly in H01(Ω),

ii) uε * u0 strongly in L2(Ω), iii) ξε * ξ0 weakly in L2(Ω)
N

This implies that the sequence{u ε }, indexed by a sequence ofε which goes to 0, isbounded in Sobolev spaceH01(Ω) Therefore, there exists a subsequence, denoted

denoted by ε the converging subsequences

Let ϕ be a smooth function with compact support in Ω, ϕ(x) ∈ C 0∞(Ω) Wedefine a so-called oscillating test function ϕε(x) by

where ˆχ i(y) are the solution of the dual cell problems

The variational formulation corresponding to this oscillating test function ϕε(x)is

On the other hand,

is the product of a weak convergence (dε) and a strong one (uε), thus

Due to the density of C0∞(Ω) in H01(Ω), (2.33) is valid for any test function ϕ ∈

H01(Ω) Since a0 satisfies the same coercivity condition as a (Proposition 2.1.2),

the application of Lax-Milgram lemma shows that (2.33) admits a unique solution

inH01(Ω) Thus any subsequence of (uε) converges to the same limitu0 Hence theentire sequences (uε) converges to the homogenized solution u0 This concludesthe proof of the Theorem 2.1.1

The following two lemmas can be found in many homogenization textbooks,e.g [16]

Lemma 2.2.2 Let w(x, y)be a continuous function in x, square integrable and Y −periodic

in y, i.e w(x, y) ∈ L2per(Y;C(Ω)) Then, the sequence w(x, x/ε) converges weakly in

L2(Ω)to R

Y w(x, y)dy.

In the next section, we consider the convergence of the energy

associated to the problem (2.4), namely of the quantity

Eε(uε) =

Z

Ω

The following result was originally proved by De Giorgi and Spagnolo (1973):

Proposition 2.3.1 Let uε be the solution of (2.4) Then,

Eε(uε) → E0 u0

=Z

Ω

where u0 and a0 are given by Theorem 2.1.1

20

Proof. From the variational formulation of (2.4) written for uε, one has

3.0.1 Model problems

Let Ω be a domain in Rd(d = 1,2,3) with Lipschitz boundary ∂Ω on which weimpose Dirichlet conditions Given f ∈ L 2(Ω) as a source term, we consider thefollowing classical highly oscillating coefficient second-order elliptic problem

• conductivity tensoraε ∈(L∞(Ω))d×d, and is uniformly elliptic and bounded,

• ε is a parameter which represents a fine scale characterizing the multiscalenature ofaε

In the previous chapter, we has showed that the sequence {u ε } weakly verges inH01(Ω) to an elementu0 ∈ H 1

con-0(Ω) We have also proved that the so-calledhomogenized function u0 satisfies the homogenization problem

Our goal is to find this upscale solution u0

In general, the homogenized tensora0(x) cannot be explicitly computed Whenthe tensoraε(x) =a(x, x/ε) is periodic in its second argument, as in previous chap-ter, explicit equations fora0(x) are available But even in this case, at each point

x ∈ Ω, d cell- problems must be solved and a0(x) is obtained by taking priate integration Therefore, analytic a0(x) cannot be obtained due to infinitenumber of problems we have to solve

appro-The FE-HMM method gives a numerical solution to the homogenization problemfrom a different viewpoint

(FE-HMM)

The so-called finite element heterogeneous multiscale method (FE-HMM) scribes the homogenized (discrete) solution u0without pre-calculating or knowing

de-explicitly the homogenized tensor a0(x)

The information of the oscillation data needed for the computation is only quired on sampling domains, which is usually much smaller than the whole do-main This approach is compatible with different fields such as material sciences

re-or geology, where due to high costs, to have infre-ormation about micro structures

we only investigate locally some parts of the computational domain

Before describing the FE-HMM method, we first introduce the so-called macrofinite element space and micro finite element space, in which we will work Thenotations in this section follow [2], [4]

Figure 3.1: [4] In every element of the FE-HMM, the contribution to the stiffnessmatrix of the macroelements (a) is given by the solutions of the microproblems(b), which are computed using numerical quadrature on every microelement (c)

3.1.1 Macro finite element space

Let T H be a triangulation of Ω , using triangular or quadrilateral elements We

triangulation is assumed to be conformal and shape regular (see [Ciarlet]) Themacro finite element space is then defined as

where R p(K) is the space of polynomials on K of total degree at most p if K

is a simplicial element, or the space of polynomial on K of degree at most p ineach variable if K is a quadrilateral element On each macro element K ∈ T H weconsider

• Gauss quadrature points xKl ∈ K

• Gauss quadrature weightsωKl

• Sampling domains Kδl =xKl +δI , where I = −12 ,12
d

, where δ >ε

3.1.2 Micro finite element space

On each sampling domain Kδl, we continue to construct a (micro) triangulation

T h consisting of simplicial or quadrilateral elementsT of diameter hT and denote

h:= maxK∈ThhT Now, the micro finite element space is defined as

macro and micro space functions we are choosing: periodic or Dirichlet :

• W(Kδl) =Wper1 (Kδl) =n

v ∈ H 1 per(Kδl) :R

Kδlvdx= 0o

,for periodic coupling;

• W(Kδl) =H01(Kδl). for Dirichlet coupling

Notice that the size of sampling domainKδl is of size comparable to ε So, thediameter of the triangulationT h must satisfyh < ε However, this fine scale is onlyneeded on sampling domains, which are small parts of the whole computationaldomain Ω

Next, we describe the Finite Element Heterogeneous Multiscale Method

3.1.3 The FE-HMM method

The solution uH of the homogenization problem (3.2) is given by:

Find uH ∈ S 0p(Ω, T H) such that

l at the Gauss quadrature point xKl which reads

vlin,KH

l =vKH

l(xKl) +∇vH(xKl)·(x − x K l).

The FE-HMM framework consists of two components, owing to the macro andmicro scale of the homogenized problem:

• Selection of a macroscopic solver,

• Estimating the missing macro scale data by solving locally the fine scaleproblem

26

For macroscopic solver, it is natural to use the standard Pk element Supposethat the homogenized tensor a0(x) is known Then, follow the standard FEM,the global stiffness matrix is A = (Aij) is computed as

Aij =Z

Ω

a0(x)∇ΦHi (x)· ∇ΦHj (x)dx.

where{ΦH

i (x)}Mmac

i=1 is the basis functions of the macro space, Mmac is the number

of nodes (or degree of freedoms)

ele-That is the way we usually do in standard FEM methods

But there is an issue here: the homogenized tensora0(x) is indeed not available.Thus, the quantities {f ij(xKl)} are still not known This is the missing data

The key idea of the FE-HMM is that: instead of pre-computing the tensor

a0(x) (analytically or numerically), this method solves the so-called micro (or iliary) problems which are properly reformulated and then estimates the missinginformation {f ij(xKl)} by



∈ S q(Kδl, T h), andZ

Kδl

aε(x)∇ϕhi,Kδl · ∇zhdx= 0, ∀zh ∈ Sq(Kδl, T h), (3.8)

There is a of methodology between the traditional ph h of homogenization and the theory of homogenization In the lit-

erature, the representative volume element (RVE) method. .. parts of the computational domain

Before describing the FE-HMM method, we first introduce the so-called macrofinite element space and micro finite element space, in which we will work Thenotations... approaches to

ho-mogenization problems The simplest one is the finite element heterogeneous

multiscale method (FE-HMM), which uses standard finite elements such as

sim-plicial