An Introduction to the FEM and Adaptive Error Analysis for Engineering Students To order this title, and for more information, click here
By J. Akin, Professor of Mechanical Engineering, Rice University, Houston, TX
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.
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
Contents Preface
Notation
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 Concepts
1.6 Exercises
1.7 Bibliography
2. Mathematical Preliminaries
2.1 Introduction
2.2 Linear
Spaces and Norms
2.3 Sobolev Norms
2.4 Dual Problems, Self-Adjointness
2.5 Weighted Residuals
2.6 Boundary Conditions 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 Equivalent Forms
2.18 Exercises
2.19 Bibliography
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 Hierarchal Interpolation
3.7 Space-Time Interpolation
3.8 Nodally Exact Interpolations
3.9 Interpolation Error
3.10 Gradient Estimates
3.11 Exercises
3.12 Bibliography
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
4.7 Bibliography
5. Error Estimation for Elliptic Problems
5.1 Introduction
5.2 Error Estimates
5.3 Hierarchical Error Indicator
5.4 Flux Balancing Methods
5.5 Element Adaptivity
5.6 H Adaptivity
5.7 P Adaptivity
5.8 HP Adaptivity
5.9 Exercises
5.10 Bibliography
6. Super-convergent Patch Recovery
6.1 Patch Implementation Database
6.2 SCP Nodal
Flux Averaging
6.3 Computing the SCP Element Error Estimate
6.4 Hessian Matrix
6.5 Bibliography
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
7.11 Bibliography
8. Cylindrical Analysis Problems
8.1 Introduction
8.2 Heat Conduction in a Cylinder
8.3 Cylindrical Stress Analysis
8.4 Exercises
8.4
Bibliography
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
9.9 Interpolation Error
9.9 Element
Distortions
9.10 Space-Time Interpolation
9.11 Exercises
9.12 Bibliography
10. Integration Methods
10.1 Introduction
10.2 Unit Coordinate
Integration
10.3 Simplex Coordinate Integration
10.4 Numerical Integration
10.5 Typical Source Distribution Integrals
10.6 Minimal, Optimal,
Reduced and Selected Integration
10.7 Exercises
10.8 Bibliography
11. Scalar Fields
11.1 Introduction
11.2 Variational Formulation
11.3
Element and Boundary Matrices
11.4 Linear Triangle Element
11.5 Linear Triangle Applications
11.6 Bilinear Rectangulars
11.7 General
2-D Elements
11.8 Numerically Integrated Arrays
11.9 Strong Diagonal Gradient SCP Test Case
11.10 Orthtropic Conduction
11.11 Axisymmetric
Formulations
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
11.19 Bibliography
12. Vector Fields
12.1 Introduction
12.2 Displacement
Based Stress Analysis
12.3 Planar Models
12.3.1 Minimum Total Potential Energy
12.3.2 Displacement Interpolations
12.3.3 Strain-Displacement
Relations
12.3.4 Stress-Strain Law
12.4 Matrices for the Constant Strain Triangle
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 Exercises
12.11 Bibliography
INDEX
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