Abstract A novel and ecient approach based on the combination of isogeometric analysis (IGA) framework and the heterogeneous multiscale method (HMM) for elliptic homogenization problems is proposed. These problems possess highly oscillating coecients leading to the extremely expensive cost while using traditional nite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this work is regarded as a potential candidate for solv- ing such problems. The method utilizes non-uniform rational B-splines (NURBS) as basis functions for both exact geometric representation and analysis. It tremen- dously facilitates high-order macroscopic discretizations thanks to the exibility of re nement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Numerical results show the reliability and eciency of the proposed method. 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 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 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) . 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 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 Bibliography 80