**The Cell Method A Purely Algebraic Computational Method in Physics and Engineering** Finite Element Boundary Element Finite Volume and Finite Difference Analysis are all commonly used in nearly all engineering disciplines to simplify complex problems of geometry and change But they all tend to oversimplify The Cell Method is a more recent computational approach developed initially for problems in solid mechanics and electro magnetic field analysis It is a more algebraic approach and it offers a more accurate representation of geometric and topological features This will

Publisher : Momentum Press

Author : Ferretti, Elena

ISBN : 9781606506042

Year : 2014

Language: en

File Size : 19.75 MB

Category : Engineering Transportation

THE CELL METHOD

THE CELL METHOD

A PURELY ALGEBRAIC

COMPUTATIONAL METHOD

IN PHYSICS AND ENGINEERING

ELENA FERRETTI

MOMENTUM PRESS, LLC, NEW YORK

The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering

Copyright © Momentum Press®, LLC, 2014.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,

or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or

any other—except for brief quotations, not to exceed 400 words, without the prior permission

of the publisher.

First published by Momentum Press®, LLC

222 East 46th Street, New York, NY 10017

www.momentumpress.net

ISBN-13: 978-1-60650-604-2 (hardcover)

ISBN-10: 1-60650-604-8 (hardcover)

ISBN-13: 978-1-60650-606-6 (e-book)

ISBN-10: 1-60650-606-4 (e-book)

DOI: 10.5643/9781606506066

Cover design by Jonathan Pennell

Interior design by Exeter Premedia Services Private Ltd.

Chennai, India

10 9 8 7 6 5 4 3 2 1

Printed in the United States of America

Contents

Acknowledgments

vii

Preface

ix

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

2

1.1.1 Some Basics of Calculus

2

1.1.2 The e − d Definition of a Limit

4

1.1.3 A Discussion on the Cancelation Rule for Limits

8

1.2 Field and Global Variables

15

1.3 Set Functions in Physics

20

1.4 A Comparison Between the Cell Method and the Discrete Methods

21

2 Algebra and the Geometric Interpretation of Vector Spaces

2.1 The Exterior Algebra

23

23

2.1.1 The Exterior Product in Vector Spaces

24

2.1.2 The Exterior Product in Dual Vector Spaces

25

2.1.3 Covariant and Contravariant Components

35

2.2 The Geometric Algebra

40

2.2.1 Inner and Outer Products Originated by the Geometric Product

43

2.2.2 The Features of p-vectors and the Orientations of Space Elements

2.2.2.1 Inner Orientation of Space Elements

2.2.2.2 Outer Orientation of Space Elements

52

52

58

3 Algebraic Topology as a Tool for Treating Global Variables with the CM

73

3.1 Some Notions of Algebraic Topology

74

3.2 Simplices and Simplicial Complexes

77

v

vi • CONTENTS

3.3 Faces and Cofaces

80

3.4 Some Notions of the Graph Theory

88

3.5 Boundaries, Coboundaries, and the Incidence Matrices

93

3.6 Chains and Cochains Complexes, Boundary and Coboundary Processes

97

3.7 Discrete p-forms103

3.8 Inner and Outer Orientations of Time Elements

4 Classification of the Global Variables and Their Relationships

105

113

4.1 Configuration, Source, and Energetic Variables

114

4.2 The Mathematical Structure of the Classification Diagram

127

4.3 The Incidence Matrices of the Two Cell Complexes in Space Domain

135

4.4 Primal and Dual Cell Complexes in Space/Time Domain

and Their Incidence Matrices

137

5 The Structure of the Governing Equations in the Cell Method

5.1 The Role of the Coboundary Process in the Algebraic Formulation

145

146

5.1.1 Performing the Coboundary Process on Discrete 0-forms in Space

Domain: Analogies Between Algebraic and Differential Operators

151

5.1.2 Performing the Coboundary Process on Discrete 0-forms in Time

Domain: Analogies Between Algebraic and Differential Operators

155

5.1.3 Performing the Coboundary Process on Discrete 1-forms in Space/Time

Domain: Analogies Between Algebraic and Differential Operators

158

5.1.4 Performing the Coboundary Process on Discrete 2-forms in Space/Time

Domain: Analogies Between Algebraic and Differential Operators

164

5.2 How to Compose the Fundamental Equation of a Physical Theory

167

5.3 Analogies in Physics

168

5.4 Physical Theories with Reversible Constitutive Laws

177

5.5 The Choice of Primal and Dual Cell Complexes in Computation

178

6 The Problem of the Spurious Solutions in Computational Physics

183

6.1 Stability and Instability of the Numerical Solution

184

6.2 The Need for Non-Local Models in Quantum Physics

198

6.3 Non-Local Computational Models in Differential Formulation

203

6.3.1 Continuum Mechanics

204

6.4 Algebraic Non-Locality of the CM

208

References

215

Index

227

Acknowledgments

The author wishes to acknowledge those who, through the years, contributed to the research

activity that has allowed the realization of this book.

Among the Professors of the School of Engineering and Architecture – Alma Mater Studiorum University of Bologna (Italy), grateful thanks go to Prof. Angelo Di Tommaso, for having

competently oriented my academic education, Prof. Antonio Di Leo, for his continuous valuable suggestions and encouragement since I was a Ph.D. student, and Prof. Erasmo Viola, for

sharing his vast knowledge and experience with me each day.

Prof. Enzo Tonti, with his passionate seminarial activity on the role of classification in

physics, sparked my interest for the fascinating topic of algebraic formulation. This led me to

seek and deepen the mathematical foundations of the Cell Method, the major factor motivating this book. A precious help in this process was the possibility to preview some drafts of the

papers and books that Prof. Tonti wanted to share with me, in confidence.

I am also grateful to Prof. Satya N. Atluri, founder and Editor-in-Chief of many journals,

for having appreciated my contributions to the development of the Cell Method for computational fracture mechanics, in many circumstances. His estimate has been a source of renewed

enthusiasm in approaching this innovative subject.

A special thanks goes to Joel Stein for having had trust in this project and to Millicent Treloar for her swift feedback and guidance to improve this book.

Elena Ferretti

Bologna, October 2013

vii

Preface

The computational methods currently used in physics are based on the discretization of the differential formulation, by using one of the many methods of discretization, such as the finite element method (FEM), the boundary element method (BEM), the finite volume method (FVM),

the finite difference method (FDM), and so forth. Infinitesimal analysis has without doubt

played a major role in the mathematical treatment of physics in the past, and will continue to

do so in the future, but, as discussed in Chapter 1, we must also be aware that several important

aspects of the phenomenon being described, such as its geometrical and topological features,

remain hidden, in using the differential formulation. This is a consequence not of performing

the limit, in itself, but rather of the numerical technique used for finding the limit. In Chapter

1, we analyze and compare the two most known techniques, the iterative technique and the

application of the Cancelation Rule for limits. It is shown how the first technique, leading to

the approximate solution of the algebraic formulation, preserves information on the trend of the

function in the neighborhood of the estimation point, while the second technique, leading to

the exact solution of the differential formulation, does not. Under the topological point of view,

this means that the algebraic formulation preserves information on the length scales associated

with the solution, while the differential formulation does not. On the basis of this observation,

it is also proposed to consider that the limit provided by the Cancelation Rule for limits is exact

only in the broad sense (i.e., the numerical sense), and not in the narrow sense (involving also

topological information). Moreover, applying the limit process introduces some limitations as

regularity conditions must be imposed on the field variables. These regularity conditions, in

particular those concerning differentiability, are the price we pay for using a formalism that is

both very advanced and easy to manipulate.

The Cancelation Rule for limits leads to point-wise field variables, while the iterative procedure leads to global variables (Section 1.2), which, being associated with elements provided

with an extent, are set functions (Section 1.3). The use of global variables instead of field variables allows us to obtain a purely algebraic approach to physical laws (Chapter 4, Chapter 5),

called the direct algebraic formulation. The term “direct” emphasizes that this formulation is

not induced by the differential formulation, as is the case for the so-called discrete formulations

that are often compared to it (Section 1.4). By performing densities and rates of the global variables, it is then always possible to obtain the differential formulation from the direct algebraic

formulation.

Since the algebraic formulation is developed before the differential formulation, and not

vice-versa, the direct algebraic formulation cannot use the tools of the differential formulation for describing physical variables and equations. Therefore, the need for new suitable tools

arises, which allows us to translate physical notions into mathematical notions through the

ix

x • PREFACE

intermediation of topology and geometry. The most convenient mathematical setting where to

formulate a geometrical approach of physics is algebraic topology, the branch of the mathematics that develops notions corresponding to those of the differential formulations, but based on

global variables instead of field variables. This approach leads us to use algebra instead of differential calculus. In order to provide a better understanding of what using algebra instead of

differential calculus means, Chapter 2 deals with exterior algebra (Section 2.1) and geometric

algebra (Section 2.2), the two fundamental settings for the geometric study of spaces not just of

geometric vectors, but of other vector-like objects such as vector fields or functions. Algebraic

topology and its features are then treated in Chapter 3.

The Cell Method (CM) is the computational method based on the direct algebraic formulation developed by Enzo Tonti.1 Tonti’s first papers on the direct algebraic formulation date

back to 1974 (see Reference Section). The main motivation of these early works is that physical integral variables are naturally associated with geometrical elements in space (points, lines,

surfaces, and volumes) and time elements (time instants and time intervals), an observation that

also allows us to answer the question on why analogies exist between different physical theories:

“Since in every physical theory there are integral variables associated with space and

time elements it follows that there is a correspondence between the quantities and

the equations of two physical theories in which the homologous quantities are those

associated with the same space-time elements.”

The CM was implemented starting from the late ‘90s. The first theory described by means of

the direct algebraic formulation was electromagnetism in 1995, followed by solid mechanics

and fluids.

The strength of the CM is that of associating any physical variable with the geometrical

and topological features (Chapter 1, Chapter 4), usually neglected by the differential formulation. This goal is achieved by abandoning the habit to discretize the differential equations. The

governing equations are derived in algebraic manner directly, by means of the global variables,

leading to a numerical method that is not simply a new numerical method among many others.

The CM offers an interdisciplinary approach, which can be applied to the various branches of

classical and relativistic physics. Moreover, giving an algebraic system of physical laws is not

only a mathematical expedient, needed in computational physics because computers can only

use a finite number of algebraic operators. The truly algebraic formulation also provides us with

a numerical analysis that is more adherent to the physical nature of the phenomenon under consideration (Chapter 1). Finally, differently from their variations, the global variables are always

continuous through the interface of two different media and in presence of discontinuities of

the domain or the sources of the problem (Section 1.2). Therefore, the CM can be usefully

employed in problems with domains made of several materials, geometrical discontinuities

(corners), and concentrated sources. It also allows an easy computation in contact problems.

Even if having shown the existence of a common mathematical structure underlying the

various branches of physics is one of the most relevant key-points of the direct algebraic formulation, the purpose of this book is not that of explaining the origin of this common structure,

1

Enzo Tonti (born October 30, 1935) is an Italian mathematical physicist, now emeritus professor at the University

of Trieste (Italy). He began his own scientific activity in 1962, working in the field of Mathematical Physics, the

development of mathematical methods for application to problems in physics.

PREFACE • xi

as already extensively done by Tonti, in his publications. Our focus will be above all on giving

the mathematical foundations of the CM, and highlighting some theoretical features of the CM,

not yet taken into account or adequately discussed previously. To this aim, the basics of the CM

will be exposed in this book only to the extent necessary to the understanding of the reader.

One of the contributions given in this book to the understanding of the CM theoretical

foundations is having emphasized that the Cancelation Rule for limits acts on the actual solution of a physical problem as a projection operator, as we have already pointed out. In Section

1.1.3, this new interpretation of the Cancelation Rule for limits is discussed in the light of the

findings of non-standard calculus, the modern application of infinitesimals, in the sense of nonstandard analysis, to differential and integral calculus. It is concluded that the direct algebraic

approach can be viewed as the algebraic version of non-standard calculus. In fact, the extension

of the real numbers with the hyperreal numbers, which is on the basis of non-standard analysis,

is equivalent to providing the space of reals with a supplementary structure of infinitesimal

lengths. In other words, it is an attempt to recover the loss of length scales due to the use of the

Cancelation Rule for limits, in differential formulation. For the same reasons, the CM can be

viewed as the numerical algebraic version of those numerical methods that incorporate some

length scales in their formulations. This incorporation is usually done, explicitly or implicitly,

in order to avoid numerical instabilities. Since the CM does not need to recover the length

scales, because the metric notions are preserved at each level of the direct algebraic formulation, the CM is a powerful numerical instrument that can be used to avoid some typical spurious

solutions of the differential formulation. The problem of the numerical instabilities is treated

in Chapter 6, with special reference to electromagnetics, electrodynamics, and continuum

mechanics. Particular emphasis is devoted to the associated topic of non-locality in continuum

mechanics, where the classical local continuum concept is not adequate for modeling heterogeneous materials in the context of the classical differential formulation, causing the ill-posedness

of boundary value problems with strain-softening constitutive models. Further possible uses of

the CM for the numerical stability in other physical theories are under study, at the moment.

Some other differences and improvements, with respect to the papers and books on the CM

by other Authors, include:

• The CM is viewed as a geometric algebra, which is an enrichment (or more precisely,

a quantization) of the exterior algebra (Section 2.2.1). Since the geometric algebra provides compact and intuitive descriptions in many areas, including quantum mechanics,

it is argued (Section 4.1) that the CM can be used even for applications to problems of

quantum mechanics, a field not yet explored, at the moment.

• The p-space elements and their inner and outer orientations are derived inductively, and

not deductively. They are obtained from the outer product of the geometric algebra and

the features of p-vectors (Section 2.2.2). It is shown that it is possible to establish an isomorphism between the orthogonal complement and the dual vector space of any subset

of vectors, which extends to the orientations. Some similarities with the general Banach

spaces are also highlighted. It is concluded that the notions of inner and outer orientations are implicit in geometric algebra.

• Each cell of a plane cell complex is viewed as a two-dimensional space, where the points

of the cell, with their labeling and inner orientation, play the role of a basis scalar, the

edges of the cell, with their labeling and inner orientation, play the role of basis vectors,

and the cell itself, with its inner orientation, plays the role of basis bivector (Section 3.5).

xii • PREFACE

• Space and time global variables are treated in a unified four-dimensional space/time cell

complex, whose elementary cell is the tesseract (Sections 3.8, 5.1.2-5.1.4). The resulting

approach shows several similarities with the four-dimensional Minkowski spacetime.

Moreover, the association between the geometrical elements of the tesseract and the

“space” and “time” global variables allows us to provide an explanation (Section 4.4) of

why the possible combinations between oriented space and oriented time elements are

in number of 32, as observed by Tonti and summarized in Section 4.1. It is also shown

how the coboundary process on the discrete p-forms, which is the tool for building the

topological equations in the CM, generalizes the spacetime gradient in spacetime algebra

(Section 5.1.2).

• The configuration variables with their topological equations, on the one hand, and the

source variables with their topological equations, on the other hand, are viewed as a

bialgebra and its dual algebra (Section 4.1). This new point of view allows us to give

an explanation of why the configuration variables are associated with space elements

endowed with a kind of orientation and the source variables are associated with space

elements endowed with the other kind orientation.

• The properties of the boundary and coboundary operators are used in order to find the

algebraic form of the virtual work theorem (Section 4.2).

• It is made a distinction between the three coboundary operators, dD, dC , and dG, which,

being tensors, are independent of the labeling, the three incidence matrices, D, C, and

G, whose incidence numbers depend on the particular choice of labeling, and the three

matrices, TD, TC, and TG, which represent the coboundary operators for the given labeling of the cell complex (Section 5.1). In the special case where all the 1-cells of the threedimensional cell complex are of unit length, all the 2-cells are of unit area, and all the

3-cells are of unit volume, TD, TC, and TG equal D, C, and G, respectively. If this is not

the case, TD, TC, and TG are obtained with a procedure of expansion and assembling of

local matrices, which is derived from the procedure of expansion and assembling of the

stiffness matrix. The rows of D, C, and G give the right operators in the expansion step.

• Possible developments of the CM are investigated for the representation of reality

through a purely algebraic unifying gravitational theory, theorized by Einstein during

the last decades of his life (Section 6.4).

Elena

Bologna, October 2013

KEYWORDS

Cell method, heterogeneous materials, non-local models, non-standard analysis, bialgebra,

Clifford algebra, discrete formulations, fracture mechanics, electromagnetics, electrodynamics,

solid mechanics, fluid mechanics, space-time continuum, numerical instabilities, topological

features of variables, graph theory, coboundary process, finite element method, boundary element

method, finite volume method, finite difference method.

CHAPTER 1

A Comparison Between Algebraic

and Differential Formulations

Under the Geometrical and

Topological Viewpoints

In this chapter, we analyze the difference between the algebraic and the differential formulation

from the mathematical point of view.

The basis of the differential formulation is discussed in Section 1.1. Particular attention

is devoted to the computation of limits—by highlighting how the numerical techniques used

for performing limits may imply a loss of information. The main motivation for the most commonly used numerical technique in differential formulation, the Cancelation Rule for limits, is

to avoid the iterative computation of limits, which is implicit in the definition itself of a limit

(the e - d definition of a limit). The reason for this is that iterations necessarily involve some

degree of approximation, while the purpose of the Cancelation Rule for limits is to provide a

direct exact solution. Nevertheless, this exact solution is only illusory, since we pay the direct

computation of the Cancelation Rule for limits by losing information on the trend of the function in the neighborhood of the estimation point. Conversely, by computing the limit iteratively,

with the dimension of the neighborhood that decreases at each iteration, leading also the error

on the solution to decrease, we conserve information on the trend of the function in the neighborhood of the estimation point. This second way to operate, where the dimension of the neighborhood approaches zero but is never equal to zero, follows from the e - d definition of a limit

directly and leads to the algebraic formulation. When the Cancelation Rule for limits is used for

finding densities and rates, we also lose information on the space and time extent of the geometrical and temporal objects associated with the variables we are computing, obtaining point- and

instant-wise variables. By using the algebraic formulation, on the contrary, we preserve both

the length and the time scales. Consequently, the physical variables of the algebraic formulation

maintain an association with the space and time multi-dimensional elements. In Section 1.1.3,

we discuss how the Cancelation Rule for limits acts on the actual solution of a physical problem

as a projection operator. The consequence is that the algebraic formulations is to the differential

1

2 • THE CELL METHOD

formulation as the actual solution of a physical problem is to the projection of the actual solution on the tangent space of degree 0, where each physical phenomenon is described in terms

of space elements of degree 0, the points, and time elements of degree 0, the time instants. In

other words, the differential solution is the shadow of the algebraic solution in the tangent space

of degree 0. In Section 1.1.3, we also discuss how using the algebraic formulation, instead of

the differential formulation, is similar to performing non-standard calculus, the modern application of infinitesimals to differential and integral calculus, instead of standard calculus. In

this sense, the derivative of a function can be viewed as the standard part, or the shadow, of

the difference quotient. The extension of real numbers, which leads to non-standard calculus,

is indeed an attempt to recover the loss of length scales. As for other techniques that will be

discussed in Chapter 6, the enrichment with a length scale has a regularization effect on the

solution.

In Section 1.2, the features of the algebraic variables (global variables) are compared with

those of the differential variables (field variables). Then, with reference to the spatial description, we introduce the association between the global physical variables and the four space

elements (point P, line L, surface S, and volume V ) and/or the two time elements (time instant

I and time interval T). It is also discussed how the association between global variables and

space elements in dimensions 0, 1, 2, and 3 requires a generalization of the coordinate systems

and time axes, in order for the global variables to be used in numerical modeling. The suitable

reference structures are cell complexes, whose elements, properly labeled, are endowed with

spatial or time extents. The algebraic formulation then uses notations of algebraic topology,

which develops notions corresponding to those of the differential formulations, but based on

global variables instead of field variables. This allows us to use algebra instead of differential

calculus, for modeling physics.

In Section 1.3, we give the definition of set functions and recognize in the global variables

a special case of set functions, due to the association between global variables and elements

provided with an extent.

Finally, in Section 1.4, we compare the cell method (CM) with other so-called discrete

methods. The comparison shows how the CM is actually the only numerical method being truly

algebraic, at the moment.

1.1 Relationship Between How to Compute Limits and

Numerical Formulations in Computational Physics

1.1.1 Some Basics of Calculus

In order to explain why the algebraic approach of the cell method (CM) is a winning strategy, if

compared to that of the differential formulation, let’s start with a brief excursus on the foundation of the differential formulation, calculus.

As is well known, calculus is the mathematical study of how things change and how

quickly they change. Calculus uses the concept of limit to consider end behavior in the infinitely large and to provide the behavior of the output of a function as the input of that function

gets closer and closer to a certain value. The second type of behavior analysis is similar to looking at the function through a microscope and increasing the power of the magnification so as to

zoom in on a very small portion of that function. This principle is known as local linearity and

© 2018-2019 uberlabel.com. All rights reserved