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# Finite Element Method

## Physics and Solution Methods

- 1st Edition - July 14, 2022
- Author: Sinan Muftu
- Language: English
- 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 fi… Read more

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Request a sales quoteFinite 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.

- Covers the detailed physics governing the physical systems and the computational methods that provide engineering solutions in one place, encouraging the reader to conduct fully informed finite element analysis
- Addresses the methodology for modeling heat transfer, elasticity, and structural mechanics problems
- Extensive worked examples are provided to help the reader to understand how to apply these methods in practice

- 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

- No. of pages: 540
- Language: English
- Edition: 1
- Published: July 14, 2022
- Imprint: Academic Press
- Paperback ISBN: 9780128211274
- eBook ISBN: 9780128232002

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### Sinan Muftu

*Finite Element Method*on ScienceDirect