THE ISOGEOMETRIC MULTISCALE FINITE ELEMENT METHOD FOR HOMOGENIZATION PROBLEMS

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


Acknowledgement
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, encouragement, and enthusiasm. Without his guidance, this thesis would not have been
completed.
I would like to thank Prof. Dang Duc Trong - Dean of the Faculty of Mathematics and Computer Science, and Dr. Nguyen Thanh Long for introducing
me to Partial Differential Equations (PDEs). I will never forget the wonderful
lectures 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 in
the 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 learning
will always inspires me.
I would like to thank Mr. Thai Hoang Chien, Mr. Tran Vinh Loc and all the
members in Division of Computational Mechanics, Ton Duc Thang University
for the helpful discussion of Isogeometric Analysis (IGA) when I first begin my
journey doing the research.
Last but not the least, I would like to thank my parents for giving birth to me
and continuously supporting me spiritually throughout my life.

Ho Chi Minh city, September, 2012

Hoang Tuong

iv


Contents
Abstract

ii

Publications

iii

List of Figures

xi

List of Tables

xiii

Notations

xiv

1

Introduction

1

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

1.6

The Finite Element Heterogeneous Multiscale Method (FE-HMM)

9

v


1.7

The Isogeometric Analysis Heterogeneous Multiscale Method (IGAHMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3

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

The Finite Element Heterogeneous Multiscale Method (FE-HMM)

22

3.0.1

Model problems . . . . . . . . . . . . . . . . . . . . . . . . .

22

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

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

3.1

3.4

4

11

The Isogeometric Analysis Heterogeneous Multiscale Method (IGAHMM)

41

vi


4.1

4.2

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

An isogeometric analysis heterogeneous multiscale method (IGAHMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

Priori Error Estimates . . . . . . . . . . . . . . . . . . . . .

51

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-

4.2.4
4.3

gree elevation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4

63

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 . . . . . . . . . . . . . . . . . . . . . .
Bibliography

75
80

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 () . . . . . . . . . .

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) . . . . . . . . . . . . .

3.1

10

[4] In every element of the FE-HMM, the contribution to the stiffness matrix of the macroelements (a) is given by the solutions of the
microproblems (b), which are computed using numerical quadrature 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) with ε = 0.1 30

3.4

H 1 (energy) error between Dirichlet coupling conditions for δ =

1.1ε, δ = 35 ε and Periodic coupling condition for δ = ε. . . . . . . .

viii

33


3.5

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

and Periodic coupling condition for δ = ε. . . . . . . . . . . . . . .
3.6

H 1 (energy) norm between Dirichlet coupling conditions for δ =

1.1ε, δ = 35 ε and Periodic coupling condition for δ = ε. . . . . . . .
3.7

34

H 1 error when using periodic coupling conditions for δ = 1.1ε, δ =
5
3ε

3.9

34

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

and Periodic coupling condition for δ = ε. . . . . . . . . . . . . . .
3.8

33

and δ = ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

L2 error when using periodic coupling conditions for δ = 1.1ε, δ =
5
3ε

and δ = ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.10 H 1 (energy) norm when using periodic coupling conditions for δ =
1.1ε, δ = 35 ε and δ = ε. . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.11 L2 norm when using periodic coupling conditions for δ = 1.1ε, δ =
5
3ε

and δ = ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.12 The convergence in H 1 (energy) norm of the solution of problem
(3.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.13 The convergence in L2 norm of the solution of problem (3.14). . .

38

3.14 The convergence rates of H 1 error match the result. . . . . . . . .

39

3.15 The convergence rates of L2 errors match the predicted result. . .

39

3.16 The convergence rates of the error in H 1 and L2 norms match the
predicted result when using H 1 micro refinement strategy. . . . . .

40

4.1

This figure illustrate cubic shape basis of B-spline . . . . . . . . .

43

4.2

2D quadratic, cubic and quartic 2D B-spline basis functions . . .

44

4.3

Physical mesh and control mesh with quadratic NURBS surface .

45

ix


4.4

Solution of the problem 4.3.1 . . . . . . . . . . . . . . . . . . . . .

4.5

L2 error of the thermal square problem, using L2 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6

57

L2 error of the thermal square problem, using H 1 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9

57

H 1 error of the thermal square problem, using H 1 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8

56

H 1 error of the thermal square problem, using L2 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7

55

59

Coarse mesh of the domain described in problem 4.3.2 and its control net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.10 Meshes for the quarter circular annulus test cases. . . . . . . . . .

62

4.11 Solution of the problem 4.3.2 . . . . . . . . . . . . . . . . . . . . .

62

4.12 H 1 error of the thermal quarter annulus problem, using H 1 micro
refinement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.13 L2 error of the thermal quarter annulus problem, using H 1 micro
refinement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.14 Coarse mesh of the geometry domain Ω and its control net with
degree p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.15 Solution distribution of the problem 4.3.3 . . . . . . . . . . . . . .

66

4.17 Convergence of the energy norm and max norm, purely using degree elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.16 Left: IGA-HMM solutions of the problem 4.3.3 on semi circle

boundary for various degrees. Right: zoom into the solutions. . .
x

68


4.18 H 1 error of the thermal quarter annulus problem, using NURBS of
degree 5 in micro space . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.19 L2 error of the thermal quarter annulus problem, using NURBS of
degree 5 in micro space . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.20 CPU time solving thermal quarter annulus problem, using NURBS
of degree 5 in micro space . . . . . . . . . . . . . . . . . . . . . . .

xi

72


List of Tables
1.1

Significant of u and f in application . . . . . . . . . . . . . . . . .

3.1

The independence on ε of the HMM solution (δ = ε, periodic micro
constraint, Nmac fixed.) . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

. . . . . . . . . . . . . . .

56

H 1 error of the thermal square problem, using H 1 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4

55

H 1 error of the thermal square problem using L2 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3

32

L2 error of the thermal square problem, using L2 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2

31

Periodic micro boundary condition performs better than the Dirichlet one ( δ = 35 ε, ε = 0.005, Nmac fixed).

4.1

6

58

L2 error of the thermal square problem, using H 1 micro refinement

strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.5

Performance comparision between FE-HMM and IGA-HMM . . .

60

4.6

H 1 error of the thermal quarter annulus problem, using H 1 micro

refinement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

63

4.7

L2 error of the thermal quarter annulus problem, using H 1 micro

refinement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8

Max norm, energy norm of the solution of problem 4.3.3 and its
CPU-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

64

67

H 1 error of the thermal quarter annulus problem, using NURBS of

degree 5 in micro space . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.10 L2 error of the thermal quarter annulus problem, using NURBS of
degree 5 in micro space . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.11 CPU time solving the thermal quarter annulus problem, using

NURBS of degree 5 in micro space . . . . . . . . . . . . . . . . . .

72

A.1 Control points and weights for problem 4.3.2 . . . . . . . . . . . .

75

A.2 Control points and weights for problem 4.3.3 . . . . . . . . . . . .

76

xiii


Notations
Rn , RN , Rd : Euclidean spaces
Ω : open subset of Rn
α = (α1 , ..., αn ) ∈ Nn : multi - index of order |α| = α1 + · · · + αn

∂u
: partial derivative of u with respect to xi
∂xi
∂ |α| u(x)
Dα u(x) =
: α - th derivative of u, for multi - index α
∂xα1 1 · · · ∂xαnn

uxi ≡

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

∆u : Laplacian of u, ∆u =

n

i=1

uxi xi

C (Ω) ≡ C 0 (Ω) = {u : Ω → R | u continuous}
C Ω = {u ∈ C (Ω) | u is uniformly continuous}
C k (Ω) = {u : Ω → R | u is k - times continuously differentiable}
C k Ω = u ∈ C k (Ω) | Dα u is uniformly continuous for all |α|
C ∞ (Ω) = {u : Ω → R | u is infinitely differentiable} =

∞

k=1

k

C k (Ω)

Cc (Ω) = {u ∈ C (Ω) | u has compact support}
Cck (Ω) = u ∈ C k (Ω) | u has compact support
Cc∞ (Ω) = {u ∈ C ∞ (Ω) | u has compact support} (sometimes also denoted by D (Ω))
Lp (Ω) =

u : Ω → R | u is Lebesgue measureable,

xiv

Ω

|u|p dx < ∞ , 1

p<∞


u

=

Lp (Ω)

Ω

|u|p dx, 1

p<∞

L∞ (Ω) = {u : Ω → R | u is Lebesgue measureable, esssupΩ |u| < ∞}
u

L∞ (Ω)

= esssup |u|
Ω

Lploc (Ω) = u : Ω → R | u ∈ Lp (U ) for each compact set U ⊂ Ω , 1
Du

Lp (Ω)

≡ ∇u

Lp (Ω)

= |∇u|

Lp (Ω)

W k,p (Ω) , H k (Ω) , etc. (k = 0, 1, 2, ..., 1

p < ∞) denote Sobolev spaces

Cper (Y ) = v ∈ C (Rd ) | v is Y - periodic function
∞
Cper
(Y ) = v ∈ C ∞ Rd | v is Y - periodic function

Lpper (Y ) = v ∈ Lp (Y ) | v is Y - periodic function
1
Hper
(Y )

=

1

Wper
(Y ) =

A

F

=

∞ (Y )
Cper

H 1 (Ω)

1
v ∈ Hper
(Y ) :

trace (AT A) =

p<∞

vdx = 0
Y

i,j

a2ij , A ∈ Rd×d

xv

Chapter 1
Introduction
In this chapter, we introduce some basic knowledge to prepare for the next chapters.

1.1

Heterogeneous material

Homogeneity and heterogeneity are concepts which describe the uniformity and
nonuniformity of a substance. A material that is homogeneous is uniform in
composition or character; one that is heterogeneous lacks uniformity in one of
these qualities [19, 18].

Figure 1.1: Heterogeneous material (www.advancedproductslab.com)
1


Heterogeneity can be the variation in compositions, such as in composites
material or porous media, see Fig. 1.1. These heterogeneities lead to a nonuniform
distribution of local materials properties. Heterogeneity is also present at various
length scales and in different forms. For example, polycrystalline materials are
composed of many crystalities of varying size and orientation.

1.2

Multiscale modeling

When modeling phenomena, traditional models usually focus on one scale. If we

interested in the macro behavior of a system, we model the effect of the smaller
scales by some (simple) constitutive relations. On the other hand, if the micro
effect is our interest, we usually presume that there is no change in the macro
scales. However, for a more complex problem which has important features at
multiple (spatial or temporal) scales, we need to include into the models the
information 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 system on one level using the information from different levels. Each level addresses
a phenomenon over a specific window of length and time. On each level particular
approaches are used for description of the system, see Fig. 1.3. Multiscale modeling is especially important in material engineering because we usually need to
predict material properties or system behavior based on knowledge of the micro
structure and properties of elementary processes. We refer to [15, 30] for more
details.

Figure 1.3: Multiscale modeling ()

1.3

Homogenization theory

In various fields of science and technology such as Mechanics, Physics, Chemistry
and Engineering, when studying composite materials, macroscopic properties of
crystalline or polymer structures, nuclear reactor design, etc ones have to deal
with boundary value problems with heterogeneous media. If the period of the
3

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 the
period of the structure to a typical length in the region. Starting from the microscopic description of the problem, we want to find the macroscopic or effective
behavior of the system. This process of seeking an average formulation is called
homogenization.
The theory of homogenization has been studied for many years. Some mathematicians in this field include: A. Benssoussan, J.L.Lions, G. Papanicolaou [9],
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 different
phases, one being the matrix and the other is the inclusions, see figure Fig. 1.4).



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í ệểì é ẵ ặ ỉ ì ỉ ì ễệ ệ é

unit periodic cell Y = (0, 1)d . The conductivity in Ω is not constant but varies
periodically 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. there
α > 0 such that for any vector ξ ∈ Rd , and at any

exist two positive constant β
point y ∈ Y

d

2

aij (y)ξi ξj

α|ξ|

β|ξ|2

i,j =1

At this point, the matrix A is not necessary symmetric (such as the case when
some drift is taken intro account in the diffusion process). The matrix a(y) is a
periodic function of y, with period Y , and it may be discontinuous in y (to model
the discontinuity of conductivities from one phase to the other).
Denoting by f (x) the source term (a scalar function defined in Ω), and enforced

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


 −∇ · (aε (x)∇uε (x)) = f (x) in Ω,

uε (x) = 0 on ∂ Ω,

(1.1)

where aε (x) = a ( xε ) , x = (x1 , .., xd ) ∈ Ω.

Equation (1.1) is popular in many application problems such as continuum
mechanics, electrostatics,... where u is displacement field or electric field,... respectively, see Table 1.1.

5


Table 1.1: Significant of u and f in application
Application Problem

u

f

Structures and solid mechanics Displacement

Mechanical force

Heat conduction

Temperature

Heat flux

Acoustic fluid

Displacement potential Particle velocity

Potential flows

Pressure

Particle velocity

General flows

Velocity

Fluxes

Electrostatics

Electric potential

Charge density

Magnetostatics

Magnetic potential

Magnetic intensity

In this thesis, we restrict our research on static heat problems, where u is the
temperature field. We emphasize that the applications for other problems are
similar.
To solve the problem (1.1), due to the requirement on regularities of the solution and in the numerical approaches, we consider the corresponding weak form,
where (1.1) is replaced by a variational formulation, namely
Find u ∈ H01 (Ω) such that
Ω

aε (x)∇uε (x) · ∇v (x)dx =

f (x)v (x)dx,
Ω

∀v ∈ H01 (Ω) ,

Observe that there are two different scales characterized our model problem
(1.1), the macroscopic scale x and the microscopic scale x/ε, which describes the
micro-oscillations.
The domain Ω with its conductivity a (x/ε) is highly-oscillating with periodic
heterogeneities of length scale ε. Usually one does not need the full details of
the variations of the potential or temperature, but rather some global averaged
behavior of 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 means
that 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 homogenization These questions are very important in the applications since, if one can give
positive answers, then the limit coefficients, as it is well known from engineers
and physicists, are good approximations of the global characteristics of the composite material, when regarded as an homogeneous one. Moreover, replacing the
problem by the limit one allows us to make easy numerical computations.

1.4

Finite Element Analysis (FEA)

Partial differential equations model a wide range of problems in biological and
physical sciences. Many interesting (and realistic) PDE models are complicated
and cannot be solved analytically. Finite Element Method (FEM) is one of the
most 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
7


numerical analysis and minimization of variational calculus to obtain approximate
solutions to vibration systems. Since then, FEA has been used and developed by
many people all over the world from many different fields, from mathematics to
engineering: 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].

1.5

Isogeometric Analysis (IGA)

The isogeometric analysis was first proposed by Hughes and co-workers [21] and
now has attracted the attention of academic as well as industrial engineering
community all over the world. The IGA allows to closely link the gap between
Computer Aided Design (CAD) and Finite Element Analysis (FEA). It means
that the IGA uses the same basis functions to describe both the geometry of domain (CAD) and the approximate solution. Being different from basis functions
of the standard FEM based on Lagrange polynomial, isogeometric approach utilizes more general basis functions such as B-splines and Non-Uniform Rational
B-splines (NURBS) that are common in CAD geometry. The exact geometry is
therefore maintained at the coarsest level of discretization and the re-meshing is
performed on this coarsest level without any communication with CAD geometry,
see Fig. 1.7. Furthermore, B-splines (or NURBS) provide a flexible way to make
refinement, de-refinement, and degree elevation [21]. They allow us to achieve
easily the smoothness of arbitrary continuity in comparison with the traditional
FEM. For a reference on IGA, we recommend the excellent book [11] and we refer
to the NURBS book [27] for geometric description.

8


T-splines
Sederberg et al., 2003&2004

IGA: geometry description and mesh refinement
Figure 1.5: An illustration of CAD objects using NURBS (www.tsplines.com)

First refinement
“Patch”, Ω = (0, 1)2

:

ocal refinement

Physical domain Ω
F

NURBS to T-spline conversion

lume meshing (hopefully...)
wi B i
Ri = wwi Bi
◦ F−1
w
i=1,...,N1
i=1,...,N1
Figure 1.6: An illustration of geometry description in IGA
The geometrical map F and the weight w are fixed at the coarsest level of
discretization!
R. V´
azquez (IMATI-CNR Italy)

1.6

Introduction to Isogeometric Analysis

Santiago de Compostela, 2010

7 / 33

The Finite Element Heterogeneous Multiscale
Method (FE-HMM)

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


Geometry is defined by Computer Aided Design (CAD) software.
CAD is based on Non Uniform Rational B-Splines (NURBS).
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.

CAD and FEM
use different descriptions for the geometry.
Maintain the geometric description given by CAD (NURBS).
Iso-parametric approach: PDEs are numerically solved with NURBS.
CAD and IGA
use the same geometry description.
Hughes, Cottrell, Bazilevs, CMAME, 2005

Maintain the geometric description given by CAD (NURBS).

Figure 1.7:Iso-parametric
Communicationapproach:
with CAD: PDEs
a comparision
between FEA
and with
IGA NURBS.
are numerically
solved
R. V´
azquez (IMATI-CNR Italy)

Introduction to Isogeometric Analysis

Santiago de Compostela, 2010

4 / 33

(Hughes-Cottrell-Bazilevs,
CMAME,
2005)
Hughes,
Cottrell, Bazilevs, CMAME,
2005
R. V´
azquez (IMATI-CNR Italy)

Introduction to Isogeometric Analysis

Santiago de Compostela, 2010

some periodic assumptions, and not explicitly available in general. Furthermore,
fully computations with complicated scale interactions of the heterogeneous system 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 homogenization problems. The simplest one is the finite element heterogeneous
multiscale method (FE-HMM), which uses standard finite elements such as simplicial 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
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information for the macro solver, where standard finite element methods are used.
Details on FE-HMM can be found in Chapter 3.

1.7

The Isogeometric Analysis Heterogeneous Multiscale Method (IGA-HMM)

Although very popular, the standard FEM still has some shortcomings which
affect the efficiency of the FE-HMM method. Firstly, the discretized geometry
through mesh generation is required. This process often leads to the geometrical
error even using the higher-order FEM. Also, the communication of geometry
model and mesh generation during analysis process is always needed and this
consumes much time [21], especially for industrial problems. Secondly, lowerorder finite elements often require fine meshes to produce the desired accuracy of
approximate solution for complicated problems, such as FE-HMM based on fournode quadrilateral element (Q4) which will be indicated in this work. Thirdly,

high-order discretizations still have some restrictions on element topologies (for
example, the connection of different types of corner, center, or internal nodes)
and C 0 continuity. These disadvantages lead to an increasing in the number of
micro 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 of
the research: a so-called Isogeometric analysis heterogeneous multiscale method
(IGA-HMM) which utilizes NURBS as basis functions for both exact geometric
representation and analysis. The NURBS are used as basis functions for both
macro and micro element spaces, where the former FE-HMM utilizes standard
FEM basis. This tremendously facilitates high-order macroscopic discretizations
11


by a flexibility of refinements and degree elevations with an arbitrary continuity
of basis functions. Several numerical results show the reliability, effectiveness and
robustness of the proposed method.

12