The cell method : a purely algebraic computational method in physics and engineering / Elena Ferretti.

Format:

Book

Author/Creator:

Ferretti, Elena, author.

Language:

English

Subjects (All):

Mathematical physics.

Engineering mathematics.

Physical Description:

1 online resource (246 p.)

Place of Publication:

New York, [New York] (222 East 46th Street, New York, NY 10017) : Momentum Press, 2014.

Language Note:

English

Summary:

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.

Contents:

1. A comparison between algebraic and differential formulations under the geometrical and topological viewpoints

1.1 Relationship between how to compute limits and numerical formulations in computational physics

1.2 Field and global variables

1.3 Set functions in physics

1.4 A comparison between the cell method and the discrete methods

2. Algebra and the geometric interpretation of vector spaces

2.1 The exterior algebra

2.2 The geometric algebra

3. Algebraic topology as a tool for treating global variables with the CM

3.1 Some notions of algebraic topology

3.2 Simplices and simplicial complexes

3.3 Faces and cofaces

3.4 Some notions of the graph theory

3.5 Boundaries, coboundaries, and the incidence matrices

3.6 Chains and cochains complexes, boundary and coboundary processes

3.7 Discrete p-forms

3.8 Inner and outer orientations of time elements

4. Classification of the global variables and their relationships

4.1 Configuration, source, and energetic variables

4.2 The mathematical structure of the classification diagram

4.3 The incidence matrices of the two cell complexes in space domain

4.4 Primal and dual cell complexes in space/time domain and their incidence matrices

5. The structure of the governing equations in the cell method

5.1 The role of the coboundary process in the algebraic formulation

5.2 How to compose the fundamental equation of a physical theory

5.3 Analogies in physics

5.4 Physical theories with reversible constitutive laws

5.5 The choice of primal and dual cell complexes in computation

6. The problem of the spurious solutions in computational physics

6.1 Stability and instability of the numerical solution

6.2 The need for non-local models in quantum physics

6.3 Non-local computational models in differential formulation

6.3.1 Continuum mechanics

6.4 Algebraic non-locality of the CM

References

Index.

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 215-226) and index.

Title from PDF title page (viewed on March 9, 2014).

ISBN:

1-60650-606-4

OCLC:

870904465

Publisher Number:

10.5643/9781606506066