Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods

by Kaliakin; Victor N.

• ISBN13: 9780824706791
• ISBN10: 082470679X
• eBook ISBN(s): 9781351990905
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-09-25
• Publisher: CRC Press

Summary

Functions as a self-study guide for engineers and as a textbook for nonengineering students and engineering students, emphasizing generic forms of differential equations, applying approximate solution techniques to examples, and progressing to specific physical problems in modular, self-contained chapters that integrate into the text or can stand alone! This reference/text focuses on classical approximate solution techniques such as the finite difference method, the method of weighted residuals, and variation methods, culminating in an introduction to the finite element method (FEM). Discusses the general notion of approximate solutions and associated errors! With 1500 equations and more than 750 references, drawings, and tables, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods: Describes the approximate solution of ordinary and partial differential equations using the finite difference method Covers the method of weighted residuals, including specific weighting and trial functions Considers variational methods Highlights all aspects associated with the formulation of finite element equations Outlines meshing of the solution domain, nodal specifications, solution of global equations, solution refinement, and assessment of results Containing appendices that present concise overviews of topics and serve as rudimentary tutorials for professionals and students without a background in computational mechanics, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods is a blue-chip reference for civil, mechanical, structural, aerospace, and industrial engineers, and a practical text for upper-level undergraduate and graduate students studying approximate solution techniques and the FEM.

Table of Contents

Preface	v
Acknowledgements	vi
Glossary of Notations and Units	vii
Governing Equations and Their Approximate Solution	1	(22)
Introductory Remarks	1	(1)
Some Simple Governing Equations	2	(9)
Mathematical Preliminaries	11	(7)
General Comments	11	(1)
Classification of Physical Problems	11	(1)
The Solution Domain and its Boundary	12	(1)
General Form of Governing Equations	13	(1)
General Form of Boundary Conditions	14	(3)
Defining a Well-Posed problem	17	(1)
Comments on Approximate Solutions	18	(1)
The Role of Mathematical Modeling in Design	19	(3)
Concluding Remarks	22	(1)
Computer Storage and Manipulation of Numbers	23	(16)
Introductory Remarks	23	(1)
Storage of Characters	23	(1)
Storage of Numbers	24	(10)
Bytes	24	(1)
Integers (Fixed-Point Numbers)	25	(1)
Floating-Point Numbers	26	(5)
Round off Error	31	(3)
Approximation Error	34	(2)
Algorithmic Stability and Error Growth	36	(1)
Concluding Remarks	37	(1)
Exercises	38	(1)
The Finite Difference Method	39	(44)
Introductory Remarks	39	(1)
Historical Note	39	(2)
General Steps	41	(1)
Ordinary Differential Equations	42	(18)
Partial Differential Equations	60	(14)
Elliptic Partial Differential Equations	61	(4)
Parabolic Partial Differential Equations	65	(7)
Hyperbolic Partial Differential Equations	72	(2)
Concluding Remarks	74	(2)
Exercises	76	(7)
The Method of Weighted Residuals	83	(42)
Introductory Remarks	83	(2)
Residuals	85	(1)
General Considerations	85	(5)
Interior Methods	87	(1)
Boundary Methods	88	(1)
Mixed Methods	89	(1)
Choice of Trial Functions	90	(1)
Specific Weighting Functions	91	(25)
Collocation Method	92	(5)
Subdomain Method	97	(4)
Method of Least Squares	101	(5)
The Bubnov--Galerkin Method	106	(4)
Method of Moments	110	(3)
Comparison of Results	113	(3)
Continuity Requirements	116	(1)
Weak Form	116	(4)
Concluding Remarks	120	(1)
Exercises	121	(4)
Variational Methods	125	(28)
Introductory Remarks	125	(2)
Admissible Functions	127	(1)
Functionals	128	(2)
Existence of Functionals	128	(2)
Derivation of Differential Equations	130	(9)
Stationary Functional Method	139	(9)
Relation to Weighted Residual Method	148	(1)
Related Methods	148	(1)
Kantorovich Method	148	(1)
Concluding Remarks	149	(1)
Exercises	150	(3)
Introduction to the Finite Element Method	153	(36)
Introductory Remarks	153	(1)
The Notion of Nodes	154	(1)
The Notion of Elements	155	(1)
Piecewise Defined Approximations	156	(3)
Some Specifics	159	(22)
A Historical Note	181	(6)
Early Contributions of Applied Mathematicians	181	(1)
Early Contributions of Mathematical Physicists	182	(1)
Early Contributions of Engineers	182	(2)
Synthesis	184	(2)
Growth	186	(1)
Present State of the Method	186	(1)
Concluding Remarks	187	(2)
Development of Finite Element Equations	189	(54)
Introductory Remarks	189	(1)
Selection of Primary Dependent Variables	190	(5)
Definition of Constitutive Relations	195	(5)
Identification of the Element Equations	200	(1)
Selection of Element Interpolation Functions	200	(6)
Convergence Criteria	200	(4)
Spatial Isotropy	204	(1)
Rate of Convergence	205	(1)
Information Regarding Element Nodes	205	(1)
Specialization of Element Equations	206	(1)
Illustrative Examples	207	(31)
Concluding Remarks	238	(1)
Exercises	239	(4)
Steps in Performing Finite Element Analyses	243	(60)
Introductory Remarks	243	(1)
Discretization of the Domain	244	(12)
Domain Discretization Error	244	(2)
Common Element Types	246	(2)
Element Characteristics	248	(1)
Placement of Elements	249	(1)
Element Shapes	250	(2)
Mesh Generation	252	(1)
Meshless Methods	252	(2)
Computer Implementation Issues	254	(2)
Assembly of Element Equations	256	(7)
Insight into Node and Element Numbering	262	(1)
Errors Associated with Formation and Assembly of Element Arrays	262	(1)
Nodal Specifications	263	(11)
Elimination Approach at the Global Level	264	(1)
Elimination Approach at the Element Level	265	(1)
Penalty Approach at the Global Level	266	(2)
Penalty Approach at the Element Level	268	(3)
Computer Implementation Issues	271	(3)
Solution of Global Equations	274	(2)
Mesh Renumbering Schemes	274	(2)
Secondary Dependent Variables	276	(3)
Postprocessing of Results	279	(1)
Interpretation of Results	279	(5)
Validity of Finite Element Analyses	279	(1)
Refinement of the Finite Element Solution	280	(4)
Illustrative Examples	284	(10)
Concluding Remarks	294	(1)
Exercises	295	(8)
Element Interpolation Functions	303	(54)
Introductory Remarks	303	(6)
Lagrangian Elements	309	(13)
One-Dimensional Lagrangian Elements	309	(3)
Two-Dimensional Lagrangian Elements	312	(6)
Three-Dimensional Lagrangian Elements	318	(2)
Lagrangian Elements: A Summary	320	(2)
Serendipity Elements	322	(8)
Two-Dimensional Serendipity Elements	322	(5)
Three-Dimensional Serendipity Elements	327	(3)
Triangular and Tetrahedral Elements	330	(13)
Element ``Degeneration''	333	(2)
Triangular Elements	335	(4)
Tetrahedral Elements	339	(4)
Triangular Prism Elements	343	(3)
Transition Elements	346	(3)
Nodal Condensation	349	(3)
Concluding Remarks	352	(1)
Exercises	353	(4)
Element Mapping	357	(40)
Introductory Remarks	357	(2)
General Aspects of Mapping	359	(1)
Treatment of Derivatives	360	(2)
Treatment of Integrals	362	(4)
Parametric Mapping	366	(15)
Isoparametric Elements	369	(7)
Valuation of Element Arrays	376	(5)
Element Distortions	381	(10)
Linear Triangular Elements	381	(1)
Quadratic Triangular Elements	382	(1)
Higher-Order Triangular Elements	383	(1)
Tetrahedral Elements	384	(1)
Bilinear Quadrilateral Elements	384	(2)
Biquadratic Quadrilateral Elements	386	(1)
Hexahedral Elements	387	(1)
Further Insight into Element Distortion	388	(3)
Computer Implementation Issues	391	(3)
Concluding Remarks	394	(1)
Exercises	395	(2)
Finite Element Analysis of Scalar Field Problems	397	(40)
Introductory Remarks	397	(1)
General Gaverning Equations	397	(10)
Historical Note	407	(1)
Development of General Finite Element Equations	408	(6)
Torsion of Straight, Prismatic Bars	414	(13)
The Solution of Saint--Venant	414	(3)
The Solution of Prandtl	417	(2)
Finite Element Equations	419	(8)
Flow Through Porous Geologic Media	427	(4)
Exercises	431	(6)
Finite Element Analysis in Linear Elastostatics	437	(58)
Introductory Remarks	437	(1)
Development of General Finite Element Equations	437	(9)
Three-Dimensional Idealizations	446	(10)
Plane Stress Idealizations	456	(5)
Generalized Plane Strain Idealizations	461	(8)
Axisymmetric Idealizations	469	(6)
Computation of Equivalent Nodal Loads	475	(11)
Plane Stress and Plane Strain Idealizations	476	(9)
Axisymmetric Idealizations	485	(1)
Potential Errors along Curved Boundaries	486	(1)
Relations between Moduli	486	(1)
Exercises	487	(8)
Implementation, Modeling, and Related Issues	495	(20)
Introductory Remarks	495	(1)
Role of Modeling and Analysis in Engineering Design	496	(1)
Phases of a Finite Element Analysis	497	(1)
Outline of a Finite Element Program	498	(2)
Meshing Guidelines Revisited	500	(6)
Element Types	500	(1)
Placement of Elements	501	(1)
Individual Element Shapes	502	(2)
Element Combinations	504	(1)
Changes in Mesh Density	505	(1)
Mesh Generation	506	(6)
Types of Meshes	507	(2)
Overview of Structured Meshing Schemes	509	(1)
Overview of Unstructured Meshing Schemes	509	(2)
Overview of General Meshing Schemes	511	(1)
Mesh Renumbering Schemes	512	(1)
Sources of Error	513	(1)
Programming Errors	513	(1)
Errors in Input Data	513	(1)
Concluding Remarks	513	(2)
A Mathematical Potpourri	515	(10)
Indicial Notation	515	(2)
Classification of PDEs	517	(2)
Some Finite Difference Formulas	519	(3)
Forward Differences in One Dimension	520	(1)
Backward Differences in One Dimension	520	(1)
Central Differences in 0ne Dimension	521	(1)
Self-Adjoint Operators	522	(3)
B Some Notes on Heat Flow	525	(10)
Introductory Remarks	525	(1)
Fourier's Law of Heat Conduction	525	(2)
Heat Conduction Equations	527	(5)
Three-Dimensional Heat Conduction	527	(3)
Two-Dimensional Heat Conduction	530	(1)
One-Dimensional Heat Conduction	531	(1)
Boundary and Initial Conditions	532	(3)
C Local and Natural Coordinate Systems	535	(14)
Introductory Remarks	535	(1)
Local Coordinate Systems	535	(2)
Natural Coordinate Systems	537	(9)
One-Dimensional Natural Coordinate Systems	537	(1)
Natural Coordinates for Triangular Elements	538	(3)
Natural Coordinates for Tetrahedral Elements	541	(3)
Natural Coordinates for Rectangular Elements	544	(1)
Natural Coordinates for Rectangular Prisms	545	(1)
Exercises	546	(3)
D The Patch Test	549	(10)
Introductory Remarks	549	(1)
Test A	550	(1)
Test B	551	(1)
Test C	551	(6)
Concluding Remarks	557	(2)
E Solution of Linear Systems of Equations	559	(20)
Direct Methods	560	(9)
Basic Gauss Elimination	560	(2)
Formal Statement of Basic Gauss Elimination	562	(1)
Triangular Decomposition	563	(2)
Operation Counts and Storage Requirements	565	(1)
Constant Bandwidth Solvers	566	(1)
Skyline Solvers	567	(1)
Frontal Solvers	568	(1)
Iterative Methods	569	(5)
Stationary Methods	569	(3)
Non-Stationary Methods	572	(2)
Preconditioning	574	(1)
Solution Errors	574	(5)
Sources of Ill-Conditioning	576	(3)
F Notes on Integration of Finite Elements	579	(34)
Introductory Remarks	579	(1)
Interpolatory Quadrature	580	(4)
Closed Newton-Cotes Formulae	581	(2)
Open Newton-Cotes Formulae	583	(1)
Gaussian Quadrature	584	(6)
Gauss-Legendre Quadrature	587	(2)
Modifications to Gaussian Quadrature	589	(1)
Numerical Integration of Lines	590	(1)
Numerical Integration of Rectangles	590	(3)
Numerical Integration of Prisms	593	(1)
Right Prisms	593	(1)
Triangular Prisms	594	(1)
Integration of Triangles and Tetrahedra	594	(5)
Exact Integration	594	(1)
Numerical Integration	595	(4)
Order of Numerical Integration	599	(14)
General Observations	599	(1)
Minimum and Optimal Order of Integration	600	(5)
Reduced Integration	605	(3)
Evaluating Fluxes	608	(2)
Flux Recovery	610	(3)
References	613	(51)
Index	664