Adaptive Hierarchical Isogeometric Finite Element Methods [electronic resource] /
Vuong, Anh-Vu.author.
Isogeometric finite elements combine the numerical solution of partial differential equations and the description of the computational domain given by rational splines from computer aided geometric design. This work gives a well-founded introduction to this topic and then extends isogeometric finite elements by a local refinement technique, which is essential for an efficient adaptive simulation. Thereby a hierarchical approach is adapted to the numerical requirements and the relevant theoretical properties of the basis are ensured. The computational results suggest the increased efficiency and the potential of this local refinement method.
Description based upon print version of record.
Preface; Acknowledgment; Contents; List of Figures; List of Tables; List of Algorithms; Chapter 1 Introduction; Scope of this Thesis; State of the Art; Structure of the Thesis; Notational Conventions; Chapter 2 Prerequisites from Applied Geometry and Spline Theory; 2.1 Parametric Curves, Surfaces and Volumes; 2.2 Spline Functions; 2.2.1 Polynomial Spaces; 2.2.2 Spline Spaces; 2.2.3 B-splines; 2.2.4 Non-Uniform Rational B-splines; 2.3 Multivariate Tensor-product Splines; 2.4 Algorithmic Aspects; 2.4.1 Evaluations; 2.4.2 Refinement; Chapter 3 Mathematical Modelling and Finite Element Analysis
3.1 Mathematical Models3.1.1 Heat and Potential Flow; 3.1.2 Continuum Mechanics; 3.2 Variational Formulation; 3.2.1 Abstract Setting; 3.2.2 Application to Models; 3.3 Finite Element Foundations; 3.3.1 Galerkin Projection; 3.3.2 Finite Element Function Spaces; 3.3.3 Reference Elements; 3.3.4 Convergence Analysis; 3.4 Finite Elements in Use; 3.4.1 Some Finite Element Spaces; 3.4.2 Isoparametric Finite Elements; 3.4.3 Mesh Refinement; 3.4.4 Adaptive Finite Elements and Error Estimation; 3.5 Implementation Issues; 3.5.1 Data Representation; 3.5.2 Assembly of System Matrices
3.5.3 Boundary Conditions3.5.4 Postprocessing; Chapter 4 Isogeometric Analysis; 4.1 Basic Idea and Fundamentals; 4.2 Isogeometric Analysis in Detail; 4.2.1 B-splines and NURBS as Basis Functions; 4.2.2 Isogeometric Mesh and Relationship to Finite Element Methods; 4.2.3 Geometry Mapping - the Relation to CAGD; 4.2.4 Shared Concepts between the Components; 4.3 Convergence Analysis; 4.4 Isogeometric Element Concept; 4.4.1 Isogeometric Reference Element; 4.4.2 B ́ezier Extraction; 4.5 Uniform Refinement; 4.6 Implementation Issues; 4.6.1 Data Representation; 4.6.2 Element Structure and Enumeration
4.6.3 Matrix Assembly4.6.4 Boundary Conditions; 4.6.5 Visualization; 4.7 Simulation Examples; 4.7.1 Heat Conduction on a Half-Disk; 4.7.2 Heat Conduction on an L-shape; 4.7.3 Plate with a Hole; 4.7.4 Turbine Blade; 4.7.5 Wheel Disk; 4.8 Conclusion and Applications; Chapter 5 Local Refinement for Isogeometric Analysis; 5.1 Introduction to Adaptive Simulation; 5.1.1 Aspects of Refinement in Isogeometric Analysis; 5.1.2 Isogeometric Analysis Refinement Techniques; 5.2 Hierarchical B-Splines; 5.2.1 Basic Definitions; 5.2.2 Properties of Hierarchical B-Splines
5.3 Adaptive Hierarchical Isogeometric Analysis5.3.1 Hierarchical Structures; 5.3.2 Hierarchical Refinement; 5.3.3 Hierarchical Reference Element; 5.3.4 Refinement Criterium and A Posteriori Error Estimation; 5.4 Implementation Issues; 5.4.1 Data structures; 5.4.2 Hierarchical Refinement; 5.4.3 Matrix Assembly; 5.4.4 Treatment of Boundary Conditions; 5.4.5 Visualization; 5.5 Numerical Simulation Cases; 5.5.1 Advection-Diffusion; 5.5.2 Heat Conduction of a Half-Disk; 5.5.3 Plate with Hole; 5.5.4 Circular Plate; 5.5.5 Heat Conduction on an L-shape; Chapter 6 Conclusions; Appendix A Software
Appendix B Geometric Data
Includes bibliographical references.
Isogeometric finite elements combine the numerical solution of partial differential equations and the description of the computational domain given by rational splines from computer aided geometric design. This work gives a well-founded introduction to this topic and then extends isogeometric finite elements by a local refinement technique, which is essential for an efficient adaptive simulation. Thereby a hierarchical approach is adapted to the numerical requirements and the relevant theoretical properties of the basis are ensured. The computational results suggest the increased efficiency and the potential of this local refinement method.
Mathematics.
Mathematics, general.