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Finite Element Method

Physics and Solution Methods

1st Edition - July 14, 2022

Author: Sinan Muftu

Paperback ISBN:

9 7 8 - 0 - 1 2 - 8 2 1 1 2 7 - 4

eBook ISBN:

9 7 8 - 0 - 1 2 - 8 2 3 2 0 0 - 2

Finite Element Method: Physics and Solution Methods aims to provide the reader a sound understanding of the physical systems and solution methods to enable effective use of the… Read more

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Finite Element Method: Physics and Solution Methods aims to provide the reader a sound understanding of the physical systems and solution methods to enable effective use of the finite element method. This book focuses on one- and two-dimensional elasticity and heat transfer problems with detailed derivations of the governing equations. The connections between the classical variational techniques and the finite element method are carefully explained. Following the chapter addressing the classical variational methods, the finite element method is developed as a natural outcome of these methods where the governing partial differential equation is defined over a subsegment (element) of the solution domain. As well as being a guide to thorough and effective use of the finite element method, this book also functions as a reference on theory of elasticity, heat transfer, and mechanics of beams.

Cover Image
Title Page
Copyright
Dedication
Table of Contents
Preface
Acknowledgments
Chapter 1 Introduction
1.1 Modeling and simulation
1.2 Solution methods
Chapter 2 Mathematical modeling of physical systems
2.1 Introduction
2.2 Governing equations of structural mechanics
2.3 Mechanics of a flexible beam
2.4 Heat transfer
2.5 Problems
References
Chapter 3 Integral formulations and variational methods
3.1 Introduction
3.2 Mathematical background
3.3 Calculus of variations
3.4 Weighted residual integral and the weak form of boundary value problems
3.5 Method of weighted residuals
3.6 Problems
References
Chapter 4 Finite element formulation of one-dimensional boundary value problems
4.1 Introduction
4.2 A second order, nonconstant coefficient ordinary differential equation over an element
4.3 One-dimensional interpolation for finite element method and shape functions
4.4 Equilibrium equations in finite element form
4.5 Recovering specific physics from the general finite element form
4.6 Element assembly
4.7 Boundary conditions
4.8 Computer implementation
4.9 Example problem
4.10 Problems
Chapter 5 Finite element analysis of planar bars and trusses
5.1 Introduction
5.2 Element equilibrium equation for a planar bar
5.3 Finite element equations for torsion of a bar
5.4 Coordinate transformations
5.5 Assembly of elements
5.6 Boundary conditions
5.7 Effects of initial stress or initial strain
5.8 Postprocessing: Computation of stresses and reaction forces
5.9 Error and convergence in finite element analysis
Problems
Reference
Chapter 6 Euler–Bernoulli beam element
6.1 Introduction
6.2 C1-Continuous interpolation function
6.3 Element equilibrium equation
6.4 General beam element with membrane and bending capabilities
6.5 Coordinate transformations
6.6 Assembly, boundary conditions, and reaction forces
6.7 Postprocessing and computation of stresses in members
Example 6.1
Problems
Reference
Chapter 7 Isoparametric elements for two-dimensional elastic solids
7.1 Introduction
7.2 Solution domain and its boundary
7.3 Equations of equilibrium for two-dimensional elastic solids
7.4 General finite element form of equilibrium equations for a two-dimensional element
7.5 Interpolation across a two-dimensional domain
7.6 Mapping between general quadrilateral and rectangular domains
7.7 Mapped isoparametric elements
7.8 Numerical integration using Gauss quadrature
7.9 Numerical evaluation of the element equilibrium equations
7.10 Global equilibrium equations and boundary conditions
7.11 Postprocessing of the solution
References
Chapter 8 Rectangular and triangular elements for two-dimensional elastic solids
8.1 Introduction
8.2 Two-dimensional interpolation functions
8.3 Bilinear rectangular element (Q4)
8.4 Constant strain triangle (CST) element
8.5 Element defects
8.6 Higher order elements
8.7 Assembly, boundary conditions, solution, and postprocessing
References
Chapter 9 Finite element analysis of one-dimensional heat transfer problems
9.1 Introduction
9.2 One-dimensional heat transfer
9.3 Finite element formulation of the one-dimensional, steady state, heat transfer problem
9.4 Element equilibrium equations: general ordinary differential equation
9.5 Element assembly
9.6 Boundary conditions
9.7 Computer implementation
Problems
Chapter 10 Heat transfer problems in two-dimensions
10.1 Introduction
10.2 Solution domain and its boundary
10.3 The heat equation and its boundary conditions
10.4 The weak form of heat transfer equation in two dimensions
10.5 The finite element form of the two-dimensional heat transfer problem
10.6 Natural boundary conditions
10.7 Summary of finite element form of the heat equation and natural boundary conditions
10.8 Numerical integration of element equilibrium equations
10.9 Element assembly
10.10 Imposing the Essential boundary conditions
Problems
Reference
Chapter 11 Transient thermal analysis
11.1 Introduction
11.2 Transient heat transfer equation
11.3 Finite difference approximations to derivatives
11.4 Direct time integration of the heat transfer equation
11.5 Solution algorithm
11.6 Convergence, stability, and accuracy of time integration methods
References
Chapter 12 Transient analysis of solids and structures
12.1 Introduction
12.2 Vibration of single degree of freedom systems
12.3 Initial/boundary value problems for deformable solids
12.4 Vibration response of an Euler–Bernoulli beam
12.5 Semidiscrete equations of motion
12.6 Mass matrix
12.7 Damping matrix
12.8 Global equation of motion
12.9 Analytical analysis of vibration of semidiscrete systems
12.10 Direct time integration of the equation of motion of a solid
12.11 Convergence, stability, and accuracy of time integration methods
Problems
References
Appendix A MATLAB
A.1 Arithmetic
A.2 Mathematical functions
A.3 Matrices
A.4 Relational operators and flow control
A.5 Scripts and functions
A.6 Reading and saving files
A.7 Plotting
References
Appendix B Guidelines for writing a finite element code in MATLAB
B.1 Structure of a finite element code
B.2 Finite element program for solution of second-order ODEs
B.3 Finite element program for a two-dimensional frame
Appendix C Finite element analysis with ANSYS
C.1 GUI-based analysis
C.2 APDL-based analysis
References
Appendix D ANSYS tutorial: beam and bar elements
D.1 Example: simply supported beam
D.2 Example: suspended bridge
References
Appendix E ANSYS tutorial: two-dimensional linear elastic analysis
Appendix F ANSYS tutorial: thermomechanical deformation
F.1 GUI-based solution of the thermomechanical deformation problem
F.2 APDL-based solution of the thermomechanical deformation problem
Index