Thạc Sĩ The isogeometric multiscale finite element method for homogenization problems

HO CHI MINH CITY UNIVERSITY OF SCIENCE
MSc THESIS IN MATHEMATICS
2012

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
1.7 The Isogeometric Analysis Heterogeneous Multiscale Method (IGA-HMM) 11


2 Preliminary results on homogenization theory


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 nite element heterogeneous multiscale method (FE-HMM) . 23
3.1.1 Macro nite element space . 24
3.1.2 Micro nite 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 renement 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 The Isogeometric Analysis Heterogeneous Multiscale Method (IGA-HMM) 41


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 Renement 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 eciency of IGA-HMM with a exible degree elevation 63
4.3.4 An higher order of IGA-HMM in both macro and micropatch space . 67


Conclusions and future work 73
Appendix 75
Control data for NURBS objects 75
Bibliography 80