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The Cell Method : A Purely Algebraic Computational Method In Physics And Engineering
by Ferretti, Elena
ISBN-13 : 9781606506042
Publisher : Momentum Press
Format : Hardback
Pub Date : January 2014
Status : Published

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Price: S$138.00


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Table of Contents
Acknowledgments vii
Preface ix
1 A Comparison Between Algebraic and Differential Formulations Under the Geometrical and Topological Viewpoints
1(22)
1.1 Relationship Between How to Compute Limits and Numerical Formulations in Computational Physics
2(13)
1.1.1 Some Basics of Calculus
2(2)
1.1.2 The - Definition of a Limit
4(4)
1.1.3 A Discussion on the Cancelation Rule for Limits
8(7)
1.2 Field and Global Variables
15(5)
1.3 Set Functions in Physics
20(1)
1.4 A Comparison Between the Cell Method and the Discrete Methods
21(2)
2 Algebra and the Geometric Interpretation of Vector Spaces
23(50)
2.1 The Exterior Algebra
23(17)
2.1.1 The Exterior Product in Vector Spaces
24(1)
2.1.2 The Exterior Product in Dual Vector Spaces
25(10)
2.1.3 Covariant and Contravariant Components
35(5)
2.2 The Geometric Algebra
40(33)
2.2.1 Inner and Outer Products Originated by the Geometric Product
43(9)
2.2.2 The Features of p-vectors and the Orientations of Space Elements
52(1)
2.2.2.1 Inner Orientation of Space Elements
52(6)
2.2.2.2 Outer Orientation of Space Elements
58(15)
3 Algebraic Topology as a Tool for Treating Global Variables with the CM
73(40)
3.1 Some Notions of Algebraic Topology
74(3)
3.2 Simplices and Simplicial Complexes
77(3)
3.3 Faces and Cofaces
80(8)
3.4 Some Notions of the Graph Theory
88(5)
3.5 Boundaries, Coboundaries, and the Incidence Matrices
93(4)
3.6 Chains and Cochains Complexes, Boundary and Coboundary Processes
97(6)
3.7 Discrete p-forms
103(2)
3.8 Inner and Outer Orientations of Time Elements
105(8)
4 Classification of the Global Variables and Their Relationships
113(32)
4.1 Configuration, Source, and Energetic Variables
114(13)
4.2 The Mathematical Structure of the Classification Diagram
127(8)
4.3 The Incidence Matrices of the Two Cell Complexes in Space Domain
135(2)
4.4 Primal and Dual Cell Complexes in Space/Time Domain and Their Incidence Matrices
137(8)
5 The Structure of the Governing Equations in the Cell Method
145(38)
5.1 The Role of the Coboundary Process in the Algebraic Formulation
146(21)
5.1.1 Performing the Coboundary Process on Discrete 0-forms in Space Domain: Analogies Between Algebraic and Differential Operators
151(4)
5.1.2 Performing the Coboundary Process on Discrete 0-forms in Time Domain: Analogies Between Algebraic and Differential Operators
155(3)
5.1.3 Performing the Coboundary Process on Discrete 1-forms in Space/Time Domain: Analogies Between Algebraic and Differential Operators
158(6)
5.1.4 Performing the Coboundary Process on Discrete 2-forms in Space/Time Domain: Analogies Between Algebraic and Differential Operators
164(3)
5.2 How to Compose the Fundamental Equation of a Physical Theory
167(1)
5.3 Analogies in Physics
168(9)
5.4 Physical Theories with Reversible Constitutive Laws
177(1)
5.5 The Choice of Primal and Dual Cell Complexes in Computation
178(5)
6 The Problem of the Spurious Solutions in Computational Physics
183(32)
6.1 Stability and Instability of the Numerical Solution
184(14)
6.2 The Need for Non-Local Models in Quantum Physics
198(5)
6.3 Non-Local Computational Models in Differential Formulation
203(5)
6.3.1 Continuum Mechanics
204(4)
6.4 Algebraic Non-Locality of the CM
208(7)
References 215(12)
Index 227
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