Muutke küpsiste eelistusi

Mathematical Theory of Subdivision: Finite Element and Wavelet Methods [Kõva köide]

(Dr. Sandeep Kumar is serving as Professor in the Department of Mechanical Engineering at Indian Institute of Technology (Banaras Hindu University), Varanasi.), , (Dr. Ashish Pathak is serving as an Assistant Professor in the Department o)
  • Formaat: Hardback, 230 pages, kõrgus x laius: 254x178 mm, kaal: 630 g, 1 Tables, black and white; 47 Illustrations, black and white
  • Ilmumisaeg: 12-Jul-2019
  • Kirjastus: CRC Press
  • ISBN-10: 1138051586
  • ISBN-13: 9781138051584
Teised raamatud teemal:
Mathematical Theory of Subdivision: Finite Element and Wavelet MethodsSuurem pilt
  • Formaat: Hardback, 230 pages, kõrgus x laius: 254x178 mm, kaal: 630 g, 1 Tables, black and white; 47 Illustrations, black and white
  • Ilmumisaeg: 12-Jul-2019
  • Kirjastus: CRC Press
  • ISBN-10: 1138051586
  • ISBN-13: 9781138051584
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781138051584.html
Märksõnad:
This book provides good coverage of the powerful numerical techniques namely, finite element and wavelets, for the solution of partial differential equation to the scientists and engineers with a modest mathematical background. The objective of the book is to provide the necessary mathematical foundation for the advanced level applications of these numerical techniques. The book begins with the description of the steps involved in finite element and wavelets-Galerkin methods. The knowledge of Hilbert and Sobolev spaces is needed to understand the theory of finite element and wavelet-based methods. Therefore, an overview of essential content such as vector spaces, norm, inner product, linear operators, spectral theory, dual space, and distribution theory, etc. with relevant theorems are presented in a coherent and accessible manner. For the graduate students and researchers with diverse educational background, the authors have focused on the applications of numerical techniques which are developed in the last few decades. This includes the wavelet-Galerkin method, lifting scheme, and error estimation technique, etc.

Features:

Computer programs in Mathematica/Matlab are incorporated for easy understanding of wavelets.

Presents a range of workout examples for better comprehension of spaces and operators.

Algorithms are presented to facilitate computer programming.

Contains the error estimation techniques necessary for adaptive finite element method.

This book is structured to transform in step by step manner the students without any knowledge of finite element, wavelet and functional analysis to the students of strong theoretical understanding who will be ready to take many challenging research problems in this area.
Preface xi
Authors xv
Chapter 1 Overview of Finite Element Method
1(26)
1.1 Some Common Governing Differential Equations
1(2)
1.2 Basic Steps of Finite Element Method
3(10)
1.3 Element Stiffness Matrix for a Bar
13(5)
1.3.1 Direct Method
13(1)
1.3.2 Potential Energy Method
13(1)
1.3.3 Higher Order Bar Element via Interpolation Functions
14(4)
1.4 Element Stiffness Matrix for Single Variable 2D Element
18(4)
1.5 Element Stiffness Matrix for a Beam Element
22(5)
References
25(2)
Chapter 2 Introduction to Wavelets for Solution of Differential Equations
27(26)
2.1 Wavelet Basis Functions
27(5)
2.2 Wavelet-Galerkin Method
32(2)
2.3 Daubechies Wavelets for Boundary and Initial Value Problems
34(19)
References
51(2)
Chapter 3 Fundamentals of Vector Spaces
53(22)
3.1 Introduction
53(1)
3.2 Vector Spaces
54(6)
3.3 Normed Linear Spaces
60(1)
3.4 Inner Product Spaces
61(3)
3.5 Banach Spaces
64(1)
3.6 Hubert Space
65(2)
3.7 Projection on Finite Dimensional Space
67(4)
3.8 Change of Basis---Gram Schmidt Orthogonalization Process
71(1)
3.9 Riesz Bases and Frame Conditions
72(3)
References
73(2)
Chapter 4 Operators
75(24)
4.1 General Concept of Functions
75(1)
4.2 Operators
76(4)
4.3 Linear and Adjoint Operators
80(3)
4.4 Functionals and Dual Space
83(3)
4.5 Spectrum of Bounded Linear Self-Adjoint Operator
86(4)
4.6 Classification of Differential Operator
90(1)
4.7 Existence, Uniqueness, and Regularity of Solution
91(8)
References
98(1)
Chapter 5 Theoretical Foundations of the Finite Element Method
99(22)
5.1 Distribution Theory
100(5)
5.2 Sobolev Spaces
105(5)
5.2.1 IP(Ω)Spaces
106(1)
5.2.2 Some Common Sobolev Spaces, Norms, and Inner Products
107(1)
5.2.3 Relation Between Cm(Ω) and Hm(Ω)
108(2)
5.3 Variational Method
110(7)
5.4 Nonconforming Elements and the Patch Test
117(4)
References
120(1)
Chapter 6 Wavelet Based Methods for Differential Equations
121(50)
6.1 Fundamentals of Continuous and Discrete Wavelets
122(5)
6.2 Multiscaling
127(2)
6.3 Classification of Wavelet Basis Functions
129(2)
6.3.1 Orthogonal Wavelets
129(2)
6.3.2 Semi-orthogonal Wavelets
131(1)
6.3.3 Bi-orthogonal Wavelets
131(1)
6.4 Discrete Wavelet Transform
131(6)
6.5 Lifting Scheme for Discrete Wavelet Transform
137(2)
6.6 Lifting Scheme to Customize Wavelets
139(21)
6.7 Non-Standard form of Matrix and Its Solution
160(7)
6.8 Multigrid Method
167(4)
References
169(2)
Chapter 7 Error Estimation
171(18)
7.1 Introduction
171(2)
7.2 A-Priori Error Estimation
173(1)
7.3 Recovery Based Method
174(1)
7.4 Residual Based Error Estimators
175(2)
7.5 Goal Oriented Error Estimators
177(2)
7.6 Hierarchical and Wavelet Based Error Estimators
179(1)
7.7 Model Problem
180(9)
7.7.1 Error by Recovery Based Method
182(1)
7.7.2 Error by Explicit Residual Estimator
183(2)
7.7.3 Error by Implicit Residual Estimator
185(1)
7.7.4 Error by Goal Oriented Estimator
186(1)
7.7.5 Error by Hierarchical and Wavelet Based Estimator
187(1)
References
188(1)
Appendix A Sets 189(6)
Appendix B Fields 195(4)
Appendix C Cm and Lp Spaces 199(10)
Appendix D Daubechies Filter and Connection Coefficients 209(10)
Appendix E Fourier Transform 219(8)
Index 227
Dr. Sandeep Kumar is serving as Professor in the Department of Mechanical Engineering at Indian Institute of Technology (Banaras Hindu University), Varanasi. He received his Ph.D. degree from Applied Mechanics Department, Indian Institute of Technology Delhi in the year 1999. His field of interests is computational mechanics: wavelets, finite element method, and meshless method, etc.



Dr. Ashish Pathak is serving as an Assistant Professor in the Department of Mathematics, Institute of Science (Banaras Hindu University). He received his Ph.D. degree from Department of Mathematics, Banaras Hindu University in the year 2009. His research interests include wavelet analysis, functional analysis, and distribution theory.



Dr. Debashis Khan received his Ph.D. degree in Mechanical Engineering from Indian Institute of Technology Kharagpur in the year 2007. Just after completing his Ph. D. he joined as an Assistant Professor in the Department of Mechanical Engineering at Indian Institute of Technology (Banaras Hindu University) Varanasi and presently he is serving as associate professor in the same department. His research interests include solid mechanics, fracture mechanics, continuum mechanics, finite deformation plasticity, finite element method.