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The Cell Method

Book Description

The Cell Method (CM) is a computational tool that maintains critical multidimensional attributes of physical phenomena in analysis. This information is neglected in the differential formulations of the classical approaches of finite element, boundary element, finite volume, and finite difference analysis, often leading to numerical instabilities and spurious results. This book highlights the central theoretical concepts of the CM that preserve a more accurate and precise representation of the geometric and topological features of variables for practical problem solving. Important applications occur in fields such as electromagnetics, electrodynamics, solid mechanics and fluids. CM addresses non-locality in continuum mechanics, an especially important circumstance in modeling heterogeneous materials. Professional engineers and scientists, as well as graduate students, are offered: • A general overview of physics and its mathematical descriptions; • Guidance on how to build direct, discrete formulations; • Coverage of the governing equations of the CM, including nonlocality; • Explanations of the use of Tonti diagrams; and • References for further reading.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Contents
  6. ACKNOWLEDGMENTS
  7. PREFACE
  8. 1 A COMPARISON BETWEEN ALGEBRAIC AND DIFFERENTIAL FORMULATIONS UNDER THE GEOMETRICAL AND TOPOLOGICAL VIEWPOINTS
    1. 1.1 Relationship Between How to Compute Limits and Numerical Formulations in Computational Physics
      1. 1.1.1 Some Basics of Calculus
    2. 1.1.2 The ε – δ Definition of a Limit
    3. 1.1.3 A Discussion on the Cancelation Rule for Limits
    4. 1.2 Field and Global Variables
    5. 1.3 Set Functions in Physics
    6. 1.4 A Comparison Between the Cell Method and the Discrete Methods
  9. 2 ALGEBRA AND THE GEOMETRIC INTERPRETATION OF VECTOR SPACES
    1. 2.1 The Exterior Algebra
      1. 2.1.1 The Exterior Product in Vector Spaces
      2. 2.1.2 The Exterior Product in Dual Vector Spaces
      3. 2.1.3 Covariant and Contravariant Components
    2. 2.2 The Geometric Algebra
      1. 2.2.1 Inner and Outer Products Originated by the Geometric Product
        1. 2.2.2 The Features of p-vectors and the Orientations of Space Elements
        2. 2.2.2.1 Inner Orientation of Space Elements
        3. 2.2.2.2 Outer Orientation of Space Elements
  10. 3 ALGEBRAIC TOPOLOGY AS A TOOL FOR TREATING GLOBAL VARIABLES WITH THE CM
    1. 3.1 Some Notions of Algebraic Topology
    2. 3.2 Simplices and Simplicial Complexes
    3. 3.3 Faces and Cofaces
    4. 3.4 Some Notions of the Graph Theory
    5. 3.5 Boundaries, Coboundaries, and the Incidence Matrices
    6. 3.6 Chains and Cochains Complexes, Boundary and Coboundary Processes
    7. 3.7 Discrete p-forms
    8. 3.8 Inner and Outer Orientations of Time Elements
  11. 4 CLASSIFICATION OF THE GLOBAL VARIABLES AND THEIR RELATIONSHIPS
    1. 4.1 Configuration, Source, and Energetic Variables
    2. 4.2 The Mathematical Structure of the Classification Diagram
    3. 4.3 The Incidence Matrices of the Two Cell Complexes in Space Domain
    4. 4.4 Primal and Dual Cell Complexes in Space/Time Domain and Their Incidence Matrices
  12. 5 THE STRUCTURE OF THE GOVERNING EQUATIONS IN THE CELL METHOD
    1. 5.1 The Role of the Coboundary Process in the Algebraic Formulation
      1. 5.1.1 Performing the Coboundary Process on Discrete 0-forms in Space Domain: Analogies Between Algebraic and Differential Operators
      2. 5.1.2 Performing the Coboundary Process on Discrete 0-forms in Time Domain: Analogies Between Algebraic and Differential Operators
      3. 5.1.3 Performing the Coboundary Process on Discrete 1-forms in Space/Time Domain: Analogies Between Algebraic and Differential Operators
      4. 5.1.4 Performing the Coboundary Process on Discrete 2-forms in Space/Time Domain: Analogies Between Algebraic and Differential Operators
    2. 5.2 How to Compose the Fundamental Equation of a Physical Theory
    3. 5.3 Analogies in Physics
    4. 5.4 Physical Theories with Reversible Constitutive Laws
    5. 5.5 The Choice of Primal and Dual Cell Complexes in Computation
  13. 6 THE PROBLEM OF THE SPURIOUS SOLUTIONS IN COMPUTATIONAL PHYSICS
    1. 6.1 Stability and Instability of the Numerical Solution
    2. 6.2 The Need for Non-Local Models in Quantum Physics
    3. 6.3 Non-Local Computational Models in Differential Formulation
      1. 6.3.1 Continuum Mechanics
    4. 6.4 Algebraic Non-Locality of the CM
  14. REFERENCE
  15. INDEX
  16. Ad Page
  17. Backcover