An Introduction to Linear and Nonlinear Finite Element Analysis

by Kythe, Prem K.; Wei, Dongming

• ISBN13: 9780817643089
• ISBN10: 0817643087
• eBook ISBN(s): 9780817681609
• Format: Hardcover
• Copyright: 2003-11-01
• Publisher: Birkhauser

Summary

Although finite element courses have become more popular in the undergraduate and graduate engineering, science, and applied mathematics curricula, there are very few introductory textbooks geared toward students accustomed to using computers for everyday assignments and research. 'An Introduction to Linear and Nonlinear Finite Element Analysis' fills this gap, offering a concise, integrated presentation of methods, applications, computational software tools, and hands-on programming projects. Suitable for junior/senior undergraduate and first-year graduate courses, the book is aimed at students from a variety of disciplines: engineering, physics, geophysics, and applied mathematics. Unlike existing texts designed with specific applications to a particular field of mechanical, civil, or chemical engineering, the emphasis here is on interdisciplinary applications. One- and two-dimensional linear and nonlinear initial/boundary value problems are solved using finite element, Newton's, and conjugate gradient methods. Mathematical theory is kept to a minimum, making the text accessible to students with varied backgrounds. Features: * Software tools using Mathematica, Matlab, Fortran, and commercial finite element codes, such as Ansys, integrated throughout the text * Numerous examples and exercises with diverse applications to linear and nonlinear heat transfer, fluid flows, mechanical vibrations, electromagnetics, and structures * Supporting material and selected solutions to problems available at the authors' websites: http://www.math.uno.edu/fac/pkythe.html and http://www.math.uno.edu/fac/dwei.html * Minimal prerequisites: a course in calculus of several variables, differential equations and linear algebra, as well as some knowledge of computers Primarily a classroom resource, the book may also be used as a self-study reference for researchers and practitioners who need a quick introduction to finite element methods.

Table of Contents

Preface	ix
Notation	xvii
Introduction	1	(24)
Historical Sketch	1	(1)
Euler-Lagrange Equations	2	(1)
Functionals	2	(1)
Boundary Conditions	3	(1)
Weak Variational Form	3	(6)
Galerkin Method	9	(2)
Rayleigh-Ritz Method	11	(7)
Exercises	18	(7)
One-Dimensional Shape Functions	25	(12)
Local and Global Linear Shape Functions	25	(3)
Local and Global Quadratic Shape Functions	28	(3)
Parametric Coordinates	31	(1)
Hermite Shape Functions	31	(3)
Exercises	34	(3)
One-Dimensional Second-Order Equation	37	(38)
Galerkin Finite Element Method	37	(26)
Linear Elements for Second-Order Equation	38	(4)
Local and Global Matrices	42	(3)
Quadratic Elements for Second-Order Linear Problems	45	(2)
Mixing Two Different Elements	47	(16)
Two Dependent Variables	63	(3)
Exercises	66	(9)
One-Dimensional Fourth-Order Equation	75	(14)
Euler-Bernoulli Beam Equation	75	(11)
Finite Element Equation	75	(2)
Boundary Conditions	77	(9)
Exercises	86	(3)
Two-Dimensional Elements	89	(18)
Linear Three-Node Triangular Elements	89	(2)
Bilinear Four-Node Rectangular Elements	91	(3)
Global Shape Functions	94	(2)
Triangular Coordinates	96	(3)
Shape Functions on the Sides of a Triangle	99	(6)
Linear Shape Functions on the Sides of a Triangle	99	(2)
Quadratic Shape Functions on the Sides of a Triangle	101	(4)
Exercises	105	(2)
Two-Dimensional Problems	107	(30)
Single Dependent Variable Problems	107	(22)
Local Weak Formulation	108	(1)
Finite Element Equation	109	(1)
Evaluation of Stiffness Matrix and Load Vector	110	(1)
Evaluation of Boundary Integrals	111	(1)
Assembly of Element Matrices	112	(17)
Exercises	129	(8)
More Two-Dimensional Problems	137	(38)
Heat Transfer	137	(7)
Torsion	144	(2)
Seepage	146	(6)
Fluid Flows	152	(12)
Exercises	164	(11)
Axisymmetric Heat Transfer	175	(32)
Radial Symmetry	175	(7)
Linear Elements	182	(12)
Linear Elements for Heat Transfer in Fluids	194	(2)
Nonlinear Heat Transfer	196	(6)
Gauss-Legendre Integration Method	198	(4)
Exercises	202	(5)
Transient Problems	207	(34)
Classical Methods	207	(5)
One-Dimensional Transient Problems	212	(5)
Time-Dependent Heat Conduction	217	(11)
Derivation of Finite Element Equations	218	(3)
Numerical Time Integration	221	(7)
Two-Dimensional Transient Problems	228	(5)
Exercises	233	(8)
Single Nonlinear One-Dimensional Problems	241	(28)
Newton's method	241	(2)
Radiation Heat Transfer	243	(9)
Stress Analysis of Plastic Rods	252	(4)
Power-Law Pressure Driven Flow between Two Plates	256	(3)
Mixing-Length Equation for Turbulent Flow in Pipes	259	(2)
Rayleigh-Ritz and Nonlinear Gradient Methods	261	(5)
Exercises	266	(3)
Plane Elasticity	269	(28)
Stress-Strain Relations	269	(2)
Constant-Strain Triangular Element	271	(1)
Virtual Displacement Finite Element Model	272	(1)
Weak Form Finite Element Model	273	(2)
Stiffness Matrix and Load Vector	275	(16)
Linear CST Element	275	(1)
Assembly of CST Elements	276	(8)
Bilinear Rectangular Element	284	(4)
Two or More Bilinear Rectangular Elements	288	(3)
Exercises	291	(6)
Stokes Equations and Penalty Method	297	(26)
Equality-Constrained Programs and Lagrange Multipliers	297	(2)
Penalty Formulation for Linear Stokes Equation	299	(2)
Penalty Linear Triangular Stokes Element	301	(8)
Penalty Bilinear Rectangular Stokes Element	309	(2)
Penalty Linear Triangular Power-law Stokes Element	311	(2)
Solutions by Conjugate Gradient Methods	313	(8)
Exercises	321	(2)
Vibration Analysis	323	(28)
Hamiltonian Principle	323	(4)
Free Axial Vibrations of an Elastic Rod	327	(3)
Free Vibrations of a Euler Elastic Beam	330	(4)
Free In-Plane Vibrations of an Elastic Plate	334	(8)
Axial Vibrations of a Plastic Rod	342	(2)
Eigenvalue Problems	344	(3)
Hyperbolic Equations	344	(2)
Parabolic Equations	346	(1)
Helmholtz Equation	347	(1)
Exercises	347	(4)
Computer Codes	351	(42)
Mathematica Codes	351	(12)
Ansys Codes	363	(5)
Matlab Codes	368	(6)
Fortran Codes	374	(19)
A Integration Formulas	393	(4)
B Special Cases	397	(8)
C Temporal Approximations	405	(4)
First-Order Derivative	405	(1)
Second-Order Derivative	406	(3)
D Isoparametric Elements	409	(12)
Introduction	409	(5)
Local ξ-Syatem	409	(2)
Triangular Elements	411	(1)
Rectangular Elements	412	(2)
Curvilinear Coordinates	414	(4)
One-Dimensional Integrals	416	(1)
Two-Dimensional Integrals	417	(1)
Pascal Triangle	418	(3)
E Green's Identities	421	(2)
F Gaussian Quadrature	423	(6)
G Gradient-Based Methods	429	(4)
Method of Steepest Descent	429	(2)
Conjugate Gradient Method	431	(2)
Bibliography	433	(6)
Index	439