Finite Element Analysis with Error Estimators

Finite Element Analysis with Error Estimators

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

1st Edition - June 22, 2005

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  • Author: J. Akin
  • eBook ISBN: 9780080472751

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Description

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.

Key Features

  • The only introductory FEA text with error estimation for students of engineering, scientific computing and applied mathematics
  • Includes source code for creating and proving FEA error estimators

Readership

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

Table of Contents

  • Preface

    Features of the text and accompanying resources

    Notation

    Chapter 1: Introduction

    1.1 Finite element methods

    1.2 Capabilities of FEA

    1.3 Outline of finite element procedures

    1.4 Assembly into the system equations

    1.5 Error section-titles

    1.6 Exercises

    Chapter 2: Mathematical preliminaries

    2.1 Introduction

    2.2 Linear spaces and norms

    2.3 Sobolev norms *

    2.4 Dual problem, self-adjointness

    2.5 Weighted residuals

    2.6 Boundary condition terms

    2.7 Adding more unknowns

    2.8 Numerical integration

    2.9 Integration by parts

    2.10 Finite element model problem

    2.11 Continuous nodal flux recovery

    2.12 A one-dimensional example error analysis

    2.13 General boundary condition choices

    2.14 General matrix partitions

    2.15 Elliptic boundary value problems

    2.16 Initial value problems

    2.17 Eigen-problems

    2.18 Equivalent forms *

    2.19 Exercises

    Chapter 3: Element interpolation and local coordinates

    3.1 Introduction

    3.2 Linear interpolation

    3.3 Quadratic interpolation

    3.4 Lagrange interpolation

    3.5 Hermitian interpolation

    3.6 Hierarchical interpolation

    3.7 Space-time interpolations*

    3.8 Nodally exact interpolations *

    3.9 Interpolation error *

    3.10 Gradient estimates *

    3.11 Exercises

    Chapter 4: One-dimensional integration

    4.1 Introduction

    4.2 Local coordinate Jacobian

    4.3 Exact polynomial integration *

    4.4 Numerical integration

    4.5 Variable Jacobians

    4.6 Exercises

    Chapter 5: Error estimates for elliptic problems

    5.1 Introduction

    5.2 Error estimates

    5.3 Hierarchical error indicator

    5.4 Flux balancing error estimates

    5.5 Element adaptivity

    5.6 H-adaptivity

    5.7 P-adaptivity

    5.8 HP-adaptivity

    5.9 Exercises

    Chapter 6: Super-convergent patch recovery

    6.1 Patch implementation database

    6.2 SCP nodal flux averaging

    6.3 Computing the SCP element error estimates

    6.4 Hessian matrix *

    6.5 Exercises

    Chapter 7: Variational methods

    7.1 Introduction

    7.2 Structural mechanics

    7.3 Finite element analysis

    7.4 Continuous elastic bar

    7.5 Thermal loads on a bar *

    7.6 Reaction flux recovery for an element

    7.7 Heat transfer in a rod

    7.8 Element validation *

    7.9 Euler’s equations of variational calculus *

    7.10 Exercises

    Chapter 8: Cylindrical analysis problems

    8.1 Introduction

    8.2 Heat conduction in a cylinder

    8.3 Cylindrical stress analysis

    8.4 Exercises

    Chapter 9: General interpolation

    9.1 Introduction

    9.2 Unit coordinate interpolation

    9.3 Natural coordinates

    9.4 Isoparametric and subparametric elements

    9.5 Hierarchical interpolation

    9.6 Differential geometry *

    9.7 Mass properties *

    9.8 Interpolation error *

    9.9 Element distortion*

    9.10 Space-time interpolation *

    9.11 Exercises

    Chapter 10: Integration methods

    10.1 Introduction

    10.2 Unit coordinate integration

    10.3 Simplex coordinate integration

    10.4 Numerical integration

    10.4.1 Unit coordinate quadrature

    10.4.2 Natural coordinate quadrature

    10.5 Typical source distribution integrals *

    10.6 Minimal, optimal, reduced and selected integration

    10.7 Exercises

    Chapter 11: Scalar fields

    11.1 Introduction

    11.2 Variational formulation

    11.3 Element and boundary matrices

    11.4 Linear triangular element

    11.5 Linear triangle applications

    11.5.1 Internal source

    11.6 Bilinear rectangles *

    11.7 General 2-d elements

    11.8 Numerically integrated arrays

    11.9 Strong diagonal gradient SCP test case

    11.10 Orthotropic conduction

    11.11 Axisymmetric conductions

    11.12 Torsion

    11.13 Introduction to linear flows

    11.14 Potential flow

    11.15 Axisymmetric plasma equilibria

    11.16 Slider bearing lubrication

    11.17 Transient scalar fields

    11.18 Exercises

    Chapter 12: Vector fields

    12.1 Introduction

    12.2 Displacement based stress analysis summary

    12.3 Planar models

    12.4 Matrices for the constant strain triangle (CST)

    12.5 Stress and strain transformations *

    12.6 Axisymmetric solid stress *

    12.7 General solid stress *

    12.8 Anisotropic materials *

    12.9 Circular hole in an infinite plate

    12.10 Dynamics of solids

    12.11 Exercises

    Index

Product details

  • No. of pages: 512
  • Language: English
  • Copyright: © Butterworth-Heinemann 2005
  • Published: June 22, 2005
  • Imprint: Butterworth-Heinemann
  • eBook ISBN: 9780080472751

About the Author

J. Akin

Affiliations and Expertise

Professor of Mechanical Engineering, Rice University, Houston, TX

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