# Finite Element Analysis with Error Estimators

## An Introduction to the FEM and Adaptive Error Analysis for Engineering Students

This key text is written for senior undergraduate and graduate engineering students. It delivers a complete introduction to finite element methods and to automatic adaptation (error estimation) that will enable students to understand and use FEA as a true engineering tool. It has been specifically developed to be accessible to non-mathematics students and provides the only complete text for FEA with error estimators for non-mathematicians. Error estimation is taught on nearly half of all FEM courses for engineers at senior undergraduate and postgraduate level; no other existing textbook for this market covers this topic.

Audience
Senior undergraduate and masters level courses in engineering, computational science, and some applied mathematics programs. Most aerospace, chemical, civil & mechanical engineering programs, & senior level electrical engineering courses

Paperback, 512 Pages

Published: June 2005

Imprint: Butterworth Heinemann

ISBN: 978-0-7506-6722-7

## Contents

• PrefaceNotation1. Introduction1.1 Finite Element Methods1.2 Capabilities of FEA1.3 Outline of Finite Element Procedures1.4 Assembly into the System Equations1.5 Error Concepts1.6 Exercises1.7 Bibliography2. Mathematical Preliminaries2.1 Introduction2.2 Linear Spaces and Norms2.3 Sobolev Norms2.4 Dual Problems, Self-Adjointness2.5 Weighted Residuals2.6 Boundary Conditions Terms2.7 Adding More Unknowns2.8 Numerical Integration2.9 Integration By Parts2.10 Finite Element Model Problem2.11 Continuous Nodal Flux Recovery2.12 A One-Dimensional Example Error Analysis2.13 General Boundary Condition Choices2.14 General Matrix Partitions2.15 Elliptic Boundary Value Problems2.16 Initial Value Problems2.17 Equivalent Forms2.18 Exercises2.19 Bibliography3. Element Interpolation and Local Coordinates3.1 Introduction3.2 Linear Interpolation3.3 Quadratic Interpolation3.4 Lagrange Interpolation3.5 Hermitian Interpolation3.6 Hierarchal Interpolation3.7 Space-Time Interpolation3.8 Nodally Exact Interpolations3.9 Interpolation Error 3.10 Gradient Estimates3.11 Exercises3.12 Bibliography4. One-Dimensional Integration4.1 Introduction4.2 Local Coordinate Jacobian4.3 Exact Polynomial Integration4.4 Numerical Integration4.5 Variable Jacobians4.6 Exercises4.7 Bibliography5. Error Estimation for Elliptic Problems5.1 Introduction5.2 Error Estimates5.3 Hierarchical Error Indicator5.4 Flux Balancing Methods5.5 Element Adaptivity5.6 H Adaptivity5.7 P Adaptivity5.8 HP Adaptivity5.9 Exercises5.10 Bibliography6. Super-convergent Patch Recovery6.1 Patch Implementation Database6.2 SCP Nodal Flux Averaging6.3 Computing the SCP Element Error Estimate6.4 Hessian Matrix6.5 Bibliography7. Variational Methods7.1 Introduction7.2 Structural Mechanics7.3 Finite Element Analysis7.4 Continuous Elastic Bar7.5 Thermal Loads on a Bar7.6 Reaction Flux Recovery for an Element7.7 Heat Transfer in a Rod7.8 Element Validation7.9 Eulerâs Equations of Variational Calculus7.10 Exercises7.11 Bibliography8. Cylindrical Analysis Problems8.1 Introduction8.2 Heat Conduction in a Cylinder8.3 Cylindrical Stress Analysis8.4 Exercises8.4 Bibliography9. General Interpolation9.1 Introduction9.2 Unit Coordinate Interpolation9.3 Natural Coordinates9.4 Isoparametric and Subparametric Elements9.5 Hierarchical Interpolation9.6 Differential Geometry9.7 Mass Properties9.9 Interpolation Error9.9 Element Distortions9.10 Space-Time Interpolation9.11 Exercises9.12 Bibliography10. Integration Methods10.1 Introduction10.2 Unit Coordinate Integration10.3 Simplex Coordinate Integration10.4 Numerical Integration10.5 Typical Source Distribution Integrals10.6 Minimal, Optimal, Reduced and Selected Integration10.7 Exercises10.8 Bibliography11. Scalar Fields11.1 Introduction11.2 Variational Formulation11.3 Element and Boundary Matrices11.4 Linear Triangle Element11.5 Linear Triangle Applications11.6 Bilinear Rectangulars11.7 General 2-D Elements11.8 Numerically Integrated Arrays11.9 Strong Diagonal Gradient SCP Test Case11.10 Orthtropic Conduction11.11 Axisymmetric Formulations11.12 Torsion11.13 Introduction to Linear Flows11.14 Potential Flow11.15 Axisymmetric Plasma Equilibria11.16 Slider Bearing Lubrication11.17 Transient Scalar Fields11.18 Exercises11.19 Bibliography12. Vector Fields12.1 Introduction12.2 Displacement Based Stress Analysis12.3 Planar Models12.3.1 Minimum Total Potential Energy12.3.2 Displacement Interpolations12.3.3 Strain-Displacement Relations12.3.4 Stress-Strain Law12.4 Matrices for the Constant Strain Triangle12.5 Stress and Strain Transformations12.6 Axisymmetric Solid Stress12.7 General Solid Stress12.8 Anisotropic Materials12.9 Circular Hole in an Infinite Plate12.10 Exercises12.11 BibliographyINDEX