1st Edition
Computing with hp-ADAPTIVE FINITE ELEMENTS Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications
With a focus on 1D and 2D problems, the first volume of Computing with hp-ADAPTIVE FINITE ELEMENTS prepared readers for the concepts and logic governing 3D code and implementation. Taking the next step in hp technology, Volume II Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications presents the theoretical foundations of the 3D hp algorithm and provides numerical results using the 3Dhp code developed by the authors and their colleagues.
The first part of the book focuses on fundamentals of the 3D theory of hp methods as well as issues that arise when the code is implemented. After a review of boundary-value problems, the book examines exact hp sequences, projection-based interpolation, and De Rham diagrams. It also presents the 3D version of the automatic hp-adaptivity package, a two-grid solver for highly anisotropic hp meshes and goal-oriented Krylov iterations, and a parallel implementation of the 3D code.
The second part explores several recent projects in which the 3Dhp code was used and illustrates how these applications have greatly driven the development of 3D hp technology. It encompasses acoustic and electromagnetic (EM) scattering problems, an analysis of complex structures with thin-walled components, and challenging simulations of logging tools. The book concludes with a look at the future of hp methods.
Spearheaded by a key developer of this technology with more than 20 years of research in the field, this self-contained, comprehensive resource will help readers overcome the difficulties in coding hp-adaptive elements.
THEORY AND CODE DEVELOPMENT
BOUNDARY-VALUE PROBLEMS
Single Elliptic Equation
Linear Elasticity
Maxwell Equations
Elasticity Coupled with Acoustics
EXACT HP SEQUENCES, PROJECTION-BASED INTERPOLATION, DE RHAM DIAGRAMS
Exact Polynomial Sequences
H1-, H(curl)-, and H(div)-Conforming Projection-Based Interpolation
Shape Functions
3D HP FINITE ELEMENT METHOD
Construction of FE Basis Functions on Regular Meshes
Supported h-Refinements
p-Refinements and the Minimum Rule
Constrained Approximation
3DHP CODE
Organization of the 3Dhp Code
Data Structure in FORTRAN 90
Data Structure Supporting Algorithms
GEOMETRY MODELING
GMP Manifold: Compatible Parametrizations
Transfinite Interpolation
Interfacing with CUBIT
Exact Geometry and Parametric Elements: Mesh Generation
AUTOMATIC HP ADAPTIVITY IN THREE SPACE DIMENSIONS
The hp Algorithm
Goal-Oriented hp Adaptivity
Examples
HIGH-PERFORMANCE COMPUTATION
Fast Integration Algorithm
Telescopic Solver
Linear Solvers
TWO-GRID HP SOLVER
Formulation
Elementary Convergence Theory
Implementation Details
Numerical Examples
A DOMAIN DECOMPOSITION-BASED PARALLEL IMPLEMENTATION
Mesh Repartitioning. Interfacing with Zoltan
A Nested-Dissections Parallel Multi-Frontal Solver
Parallel Mesh Refinements and Mesh Reconciliation
Numerical Examples
APPLICATIONS
ACOUSTIC SCATTERING PROBLEMS
Infinite Element
Examples
ELECTROMAGNETIC SCATTERING PROBLEMS
Formulation of Scattering Problems
EM Infinite Element
A Domain Decomposition Approach
Calculation of Radar Cross Section
Adaptivity
Examples
3D ELASTICITY AND THIN-WALLED STRUCTURES
Introduction
Classical Shell Theory-Comparison
Solutions of Complex Thin-Walled Structures
SIMULATION OF RESISTIVITY LOGGING DEVICES
Description and Finite Element Modeling of Resistivity Logging Measurements
2D Numerical Simulations of Axisymmetric Problems
3D Numerical Simulations
CONCLUSIONS AND FUTURE WORK
APPENDICES
REFERENCES
INDEX
Biography
Leszek Demkowicz, Jason Kurtz, David Pardo, Maciek Paszenski, Waldemar Rachowicz, Adam Zdunek
"Together with the first volume, the second volume forms a unique, up-to-date, and self-contained presentation of the current status of hp-adaptive finite elements …This two-volume book is therefore strongly recommended to all mathematicians as well as engineers working on hp-adaptive finite element methods."
—Journal of Applied Mathematics and Mechanics"This is an elegant framework for the hp element with generalize classical elements of Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini . . . The authors provide here such a useful survey within 400 pages." – Dietrich Braess, in Mathematical Reviews, 2009e
