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E-raamat: Finite Element Methods: A Practical Guide

  • Formaat: EPUB+DRM
  • Sari: Mathematical Engineering
  • Ilmumisaeg: 26-Jan-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319499710
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Finite Element Methods: A Practical GuideSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Mathematical Engineering
  • Ilmumisaeg: 26-Jan-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319499710
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This book presents practical applications of the finite element method to general differential equations. The underlying strategy of deriving the finite element solution is introduced using linear ordinary differential equations, thus allowing the basic concepts of the finite element solution to be introduced without being obscured by the additional mathematical detail required when applying this technique to partial differential equations. The author generalizes the presented approach to partial differential equations which include nonlinearities. The book also includes variations of the finite element method such as different classes of meshes and basic functions. Practical application of the theory is emphasised, with development of all concepts leading ultimately to a description of their computational implementation illustrated using Matlab functions. The target audience primarily comprises applied researchers and practitioners in engineering, but the book may also be benefic

ial for graduate students.

An overview of the finite element method.- A first example.- Linear boundary value problems.- Higher order basis functions.- Nonlinear boundary value problems.- Systems of ordinary differential equations.- Linear elliptic partial differential equations.- More general elliptic problems.- Quadrilateral elements.- Higher order basis functions.- Nonlinear elliptic partial differential equations.- Systems of elliptic equations.- Parabolic partial differential equations.
1 An Overview of the Finite Element Method
1(4)
1.1 Features of the Finite Element Method
2(1)
1.2 Using This Book
3(2)
2 A First Example
5(20)
2.1 Some Brief Mathematical Preliminaries
5(1)
2.2 A Model Differential Equation
6(1)
2.3 The Weak Formulation
6(2)
2.4 Elements and Nodes
8(1)
2.5 Basis Functions
9(1)
2.6 The Finite Element Solution
10(1)
2.7 Algebraic Equations Satisfied by the Finite Element Solution
11(2)
2.8 Assembling the Algebraic Equations
13(4)
2.8.1 Calculating the Local Contributions
15(1)
2.8.2 Assembling the Global Matrix
16(1)
2.9 A Summary of the Steps Required
17(1)
2.10 Computational Implementation
17(2)
2.11 Evaluating the Finite Element Solution at a Given Point
19(1)
2.12 Is the Finite Element Solution Correct?
20(1)
2.13 Exercises
21(4)
3 Linear Boundary Value Problems
25(30)
3.1 A General Boundary Value Problem
25(2)
3.1.1 A Note on the Existence and Uniqueness of Solutions
26(1)
3.2 The Weak Formulation
27(3)
3.2.1 Some Definitions
27(1)
3.2.2 Deriving the Weak Formulation
28(2)
3.3 The Finite Element Solution
30(7)
3.3.1 The Mesh
30(1)
3.3.2 Basis Functions
30(4)
3.3.3 Sets of Functions
34(1)
3.3.4 The Finite Element Solution
34(1)
3.3.5 The System of Algebraic Equations
35(2)
3.4 Assembling the Algebraic Equations
37(3)
3.4.1 The Contributions from Integrals
37(2)
3.4.2 Completing the Assembly of the Linear System
39(1)
3.5 Approximating Integrals Using Numerical Quadrature
40(1)
3.6 The Steps Required
41(1)
3.7 Computational Implementation
42(3)
3.8 Robin Boundary Conditions
45(2)
3.8.1 The Weak Formulation
46(1)
3.8.2 The Finite Element Solution
47(1)
3.9 A Bound on the Accuracy of the Solution
47(2)
3.10 Exercises
49(6)
4 Higher Order Basis Functions
55(26)
4.1 A Model Differential Equation
55(1)
4.2 Quadratic Basis Functions
56(13)
4.2.1 The Mesh
56(1)
4.2.2 The Definition of Quadratic Basis Functions
57(3)
4.2.3 The Finite Element Solution
60(1)
4.2.4 The System of Algebraic Equations
60(2)
4.2.5 Assembling the Algebraic Equations
62(3)
4.2.6 Computational Implementation
65(4)
4.2.7 A Note on Using Quadrature for Higher Order Basis Functions
69(1)
4.3 Cubic Basis Functions
69(6)
4.3.1 The Mesh
70(1)
4.3.2 The Definition of Cubic Basis Functions
71(2)
4.3.3 The Finite Element Solution
73(1)
4.3.4 Assembling the Algebraic Equations
74(1)
4.4 General Basis Functions
75(3)
4.4.1 The Mesh
76(1)
4.4.2 Basis Functions
76(1)
4.4.3 The Finite Element Solution
77(1)
4.5 Convergence of the Finite Element Solution
78(1)
4.6 Exercises
79(2)
5 Nonlinear Boundary Value Problems
81(22)
5.1 A First Example
81(16)
5.1.1 The Weak Formulation
82(1)
5.1.2 The Mesh, Basis Functions and Finite Element Solution
83(1)
5.1.3 The Algebraic Equations
83(2)
5.1.4 Assembling the Algebraic Equations
85(2)
5.1.5 Computational Implementation
87(3)
5.1.6 Calculation of the Jacobian Matrix
90(6)
5.1.7 Numerical Approximation of the Jacobian Matrix
96(1)
5.2 A General Nonlinear Boundary Value Problem
97(3)
5.2.1 The Weak Formulation
97(1)
5.2.2 The Nonlinear System of Algebraic Equations
98(2)
5.3 Exercises
100(3)
6 Systems of Ordinary Differential Equations
103(16)
6.1 A Model Problem
103(1)
6.2 The Weak Formulation
104(1)
6.3 The Mesh and Basis Functions
105(1)
6.4 The Finite Element Solution
106(1)
6.5 The Algebraic Equations
107(2)
6.6 Assembling the Algebraic Equations
109(1)
6.7 Computational Implementation
110(2)
6.8 More General Linear Problems
112(3)
6.8.1 The Weak Formulation
113(1)
6.8.2 The Finite Element Solution
114(1)
6.9 Nonlinear Problems
115(2)
6.10 Exercises
117(2)
7 Linear Elliptic Partial Differential Equations
119(24)
7.1 A First Model Problem
119(1)
7.2 The Weak Formulation
120(2)
7.2.1 Sobolev Spaces in Two Dimensions
120(1)
7.2.2 Deriving the Weak Formulation
121(1)
7.3 The Mesh and Basis Functions
122(6)
7.3.1 A Mesh of Triangular Elements
122(3)
7.3.2 Linear Basis Functions
125(3)
7.4 Sets of Functions
128(1)
7.5 The Finite Element Solution
129(1)
7.6 The System of Algebraic Equations
130(2)
7.6.1 Satisfying the Dirichlet Boundary Conditions
130(1)
7.6.2 Using Suitable Test Functions
130(1)
7.6.3 The Linear System
131(1)
7.7 Assembling the System of Algebraic Equations
132(3)
7.7.1 Assembling the Entries Defined by Integrals
132(3)
7.7.2 Setting the Entries Defined Explicitly
135(1)
7.8 Evaluating the Solution at a Given Point
135(1)
7.9 Computational Implementation
136(3)
7.10 More Complex Geometries
139(1)
7.11 Exercises
140(3)
8 More General Elliptic Problems
143(18)
8.1 A Model Problem
143(1)
8.2 The Weak Formulation
144(1)
8.3 The Mesh and Basis Functions
145(1)
8.4 The Finite Element Solution
145(1)
8.5 The Algebraic Equations
145(2)
8.5.1 Satisfying the Dirichlet Boundary Conditions
146(1)
8.5.2 Using Suitable Test Functions
146(1)
8.5.3 The Linear System
147(1)
8.6 Assembling the Algebraic Equations
147(4)
8.6.1 Evaluating Integrals Over Ω
148(1)
8.6.2 Evaluating Integrals Over ∂ΩN
149(1)
8.6.3 Setting Entries Defined Explicitly
150(1)
8.7 Quadrature Over Triangles
151(1)
8.8 Computational Implementation
152(6)
8.9 Exercises
158(3)
9 Quadrilateral Elements
161(14)
9.1 A Model Problem
161(1)
9.2 The Weak Formulation
162(1)
9.3 The Computational Mesh
162(2)
9.4 Basis Functions
164(2)
9.5 Sets of Functions
166(1)
9.6 The Finite Element Formulation
166(1)
9.7 The System of Algebraic Equations
167(1)
9.8 Assembling the System of Algebraic Equations
168(3)
9.8.1 Evaluating Integrals Over Ω
169(2)
9.8.2 Evaluating Integrals Over ΩN
171(1)
9.8.3 Setting Entries Defined Explicitly
171(1)
9.9 Quadrature
171(1)
9.10 Exercises
172(3)
10 Higher Order Basis Functions
175(14)
10.1 The Model Boundary Value Problem and Weak Formulation
175(1)
10.2 Quadratic Basis Functions on a Mesh of Triangular Elements
176(4)
10.3 The Finite Element Solution
180(1)
10.4 The Algebraic Equations
180(1)
10.5 Assembling the Algebraic Equations
181(3)
10.5.1 Evaluating Integrals Over Ω
181(2)
10.5.2 Evaluating Integrals Over ΩN
183(1)
10.5.3 Setting Entries Defined Explicitly
184(1)
10.6 Exercises
184(5)
11 Nonlinear Elliptic Partial Differential Equations
189(8)
11.1 A Model Problem
189(1)
11.2 The Weak Formulation
190(1)
11.3 The Mesh and Basis Functions
191(1)
11.4 The Finite Element Solution
191(1)
11.5 The Nonlinear System of Algebraic Equations
192(1)
11.5.1 Satisfying the Dirichlet Boundary Conditions
192(1)
11.5.2 Using Suitable Test Functions
192(1)
11.6 Assembling the Nonlinear System of Algebraic Equations
193(3)
11.6.1 Evaluating Integrals Over Ω
194(1)
11.6.2 Evaluating Integrals Over ∂ΩN
195(1)
11.6.3 Setting Entries Defined Explicitly
196(1)
11.7 Exercises
196(1)
12 Systems of Elliptic Equations
197(6)
12.1 A Model Problem
197(1)
12.2 The Weak Formulation
198(1)
12.3 The Mesh and Basis Functions
199(1)
12.4 The Finite Element Solution
199(1)
12.5 The Algebraic Equations
200(2)
12.6 Exercises
202(1)
13 Parabolic Partial Differential Equations
203(8)
13.1 A Linear Parabolic Partial Differential Equation
203(2)
13.2 A Nonlinear Parabolic Equation
205(2)
13.3 A Semi-implicit Discretisation in Time
207(1)
13.4 Exercises
208(3)
Appendix A Methods for Solving Linear and Nonlinear Systems of Algebraic Equations 211(16)
Appendix B Vector Calculus 227(2)
Further Reading 229(2)
Index 231
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