Computing with HP-Adaptive Finite Elements: One and Two Dimensional Elliptic and Maxwell Problems Volume 1

Computing with HP-Adaptive Finite Elements: One and Two Dimensional Elliptic and Maxwell Problems Volume 1

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Offering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementation of hp-adaptive FE methods, focusing on projection-based interpolation and the corresponding hp-adaptive strategy. The book is split into three parts, progressing from simple to more advanced problems. Part I examines the hp elements for the standard 1D model elliptic problem. The author develops the variational formulation and explains the construction of FE basis functions. The book then introduces the 1D code (1Dhp) and automatic hp adaptivity. This first part ends with a study of a 1D wave propagation problem. In Part II, the book proceeds to 2D elliptic problems, discussing two model problems that are slightly beyond standard-level examples: 3D axisymmetric antenna problem for Maxwell equations (example of a complex-valued, indefinite problem) and 2D elasticity (example of an elliptic system). The author concludes with a presentation on infinite elements - one of the possible tools to solve exterior boundary-value problems. Part III focuses on 2D time-harmonic Maxwell equations. The book explains the construction of the hp edge elements and the fundamental de Rham diagram for the whole family of hp discretizations. Next, it explores the differences between the elliptic and Maxwell versions of the 2D code, including automatic hp adaptivity. Finally, the book presents 2D exterior (radiation and scattering) problems and sample solutions using coupled hp finite/infinite elements. In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.show more

Product details

  • Hardback | 398 pages
  • 170 x 248 x 26mm | 819.99g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 114 black & white illustrations
  • 1584886714
  • 9781584886716

Table of contents

1D PROBLEMS 1D Model Elliptic Problem A Two-Point Boundary Value Problem Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Sobolev Space H1(0, l) Well Posedness of the Variational BVP Examples from Mechanics and Physics The Case with "Pure Neumann" BCs Exercises Galerkin Method Finite Dimensional Approximation of the VBVP Elementary Convergence Analysis Comments Exercises 1D hp Finite Element Method 1D hp Discretization Assembling Element Matrices into Global Matrices Computing the Element Matrices Accounting for the Dirichlet BC Summary Assignment 1: A Dry Run Exercises 1D hp Code Setting up the 1D hp Code Fundamentals Graphics Element Routine Assignment 2: Writing Your Own Processor Exercises Mesh Refinements in 1D The h-Extension Operator. Constrained Approximation Coefficients Projection-Based Interpolation in 1D Supporting Mesh Refinements Data-Structure-Supporting Routines Programming Bells and Whistles Interpolation Error Estimates Convergence Assignment 3: Studying Convergence Definition of a Finite Element Exercises Automatic hp Adaptivity in 1D The hp Algorithm Supporting the Optimal Mesh Selection Exponential Convergence. Comparing with h Adaptivity Discussion of the hp Algorithm Algebraic Complexity and Reliability of the Algorithm Exercises Wave Propagation Problems Convergence Analysis for Noncoercive Problems Wave Propagation Problems Asymptotic Optimality of the Galerkin Method Dispersion Error Analysis Exercises 2D ELLIPTIC PROBLEMS 2D Elliptic Boundary-Value Problem Classical Formulation Variational (Weak) Formulation Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Examples from Mechanics and Physics Exercises Sobolev Spaces Sobolev Space H1(O) Sobolev Spaces of an Arbitrary Order Density and Embedding Theorems Trace Theorem Well Posedness of the Variational BVP Exercises 2D hp Finite Element Method on Regular Meshes Quadrilateral Master Element Triangular Master Element Parametric Element Finite Element Space. Construction of Basis Functions Calculation of Element Matrices Modified Element. Imposing Dirichlet Boundary Conditions Postprocessing- Local Access to Element d.o.f Projection-Based Interpolation Exercises 2D hp Code Getting Started Data Structure in FORTRAN 90 Fundamentals The Element Routine Modified Element. Imposing Dirichlet Boundary Conditions Assignment 4: Assembly of Global Matrices The Case with "Pure Neumann" Boundary Conditions Geometric Modeling and Mesh Generation Manifold Representation Construction of Compatible Parametrizations Implicit Parametrization of a Rectangle Input File Preparation Initial Mesh Generation The hp Finite Element Method on h-Refined Meshes Introduction. The h Refinements 1-Irregular Mesh Refinement Algorithm Data Structure in Fortran 90 (Continued) Constrained Approximation for C0 Discretizations Reconstructing Element Nodal Connectivities Determining Neighbors for Midedge Nodes Additional Comments Automatic hp Adaptivity in 2D The Main Idea The 2D hp Algorithm Example: L-Shape Domain Problem Example: 2D "Shock" Problem Additional Remarks Examples of Applications A "Battery Problem" Linear Elasticity An Axisymmetric Maxwell Problem Exercises Exterior Boundary-Value Problems Variational Formulation. Infinite Element Discretization Selection of IE Radial Shape Functions Implementation Calculation of Echo Area Numerical Experiments Comments Exercises 2D MAXWELL PROBLEMS 2D Maxwell Equations Introduction to Maxwell's Equation Variational Formulation Exercises Edge Elements and the de Rham Diagram Exact Sequences Projection-Based Interpolation De Rham Diagram Shape Functions Exercises 2D Maxwell Code Directories. Data Structure The Element Routine Constrained Approximation. Modified Element Setting up a Maxwell Problem Exercises hp Adaptivity for Maxwell Equations Projection-Based Interpolation Revisited The hp Mesh Optimization Algorithm Example: The Screen Problem Exterior Maxwell Boundary-Value Problems Variational Formulation Infinite Element Discretization in 3D Infinite Element Discretization in 2D Stability Implementation Numerical Experiments Exercises A Quick Summary and Outlook Appendix Bibliography Indexshow more