Mathematical Foundations of Computational Engineering

Springer-Verlag Berlin Heidelberg GmbH

EngineeringONLINE L1BRARY

http://www.springer.de/engine/

Peter [an Pahl- Rudolf Damrath

MathematicalFoundations ofComputationaIEngineeringAHandbook

Springer

.

Prof. Dr. Peter [an PahITechnische Universităt BerlinInstitut fur Allgemeine BauingenieurmethodenStraBe des 17. Juni 13510623 BerlinGermanypahI@ifb.bv.tu -berlin.de

Prof. Dr.-Ing. Rudoif DamrathUniversit ăt HannoverFG Angewandte Informatik im BauingenieurwesenAppelstraBe 9A30167 HannoverGermany

Transiation:Dr. Felix PahISchopenhauerstraBe 6314129 BerlinGermanyfpahI@web.de

Die Deutsche Bibliothek - CIP-Einheit saufnahme

Pahi, Peter [an:Mathematical found ations of computational engineering : a handbook 1Peter[an Pahi ; Rudolf Damrath . Transl. from the Germa n by Felix PahI. - Berlin ;Heidelberg ; New York; Barcelona ; Hongkong ; London ; Mailand ; Paris;Singapu r ; Tokio : Spr inger, 2001

Dt. Ausg. u.d.T.: Pahi, Peter [an: Mathematische Gru ndlagen derIngenieurinform atik

ISBN 978-3-642-63238-9

ISBN 978-3-642-63238-9 ISBN 978-3-642-56893-0 (eBook)DOI 10.1007/978-3-642-56893-0

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PREFACE

Mathematics is one of the foundations of engineering. Because of the great impor­tance of the physical behavior of engineering products, calculus usually lies at thecenter of the mathematical education of engineers; it is employed in the mathemat­ical formulation of physical problems. This formulat ion has contributed significant­ly to the systematization of engineering and the mastering of engineering tasks.

Before computers were introduced into engineering, numerical solutions of themathematical formulations of engineering problems involving irregular geometry,varying material properties, multiple influences and complex production pro­cesses were difficult to determine. Nowadays, computers amplify human mentalcapacities by a factor of 109 with respect to speed of calculation, storage capacityand speed of communication; this has created entirely new possibilities for solvingmathematically formulated physical problems. New fields of science, such as com­putational mechanics, and widely applied new computational methods, such asthe finite element method, have emerged.

While computers were being introduced, the character of engineering changedprofoundly. While the key to competitiveness once lay in using better materials, de­veloping new methods of construction and designing new engineering systems,success now depends just as much on organization and management. The rea­sons for these changes include a holisitic view of the market, the product, the econ­omy and society, the importance of organization and management in global com­petition as well as the increased complexity of technology, the environment andthe interactions among those participating in planning and production.

Given the new character of engineering, the traditional mathematical foundationsno longer suffice. Branches of mathematics which are highly developed but haveso far been of little importance to engineers now prove to be important tools in acomputer-oriented treatment of engineering problems. These fields are, however,not readily accessible to many engineers, since frequently even fundamental con­cepts are not treated systematically in their education. Thus, there is no sound ba­sis for a productive dialog between engineers and mathematicians. To make mat­ters more difficult , many of the hitherto neglected fields are based directly on thefoundations of mathematics and hence exhibit a degree of abstraction which engi­neers are not accustomed to.

VI Preface

The developments in the use of computers in engineering have shown that an in­adequate education in mathematics may have grave consequences. In the areasof planning , organization and management, in particular, the potential of classicalgraph theory was not brought to bear on the abstraction of computer models andthe systematization of the methods of solution. Numerous laws and methodswhich, with a sufficient background in mathematics, could have been taken fromthe literature, were reinvented with much effort. The foundations of topology fur­nish an example of this phenomenon.

Due to the rapid development of information and communication technology, withperformance increasing by a factor of 100 per decade, new areas of applicationare constantly emerging. This makes it particularly difficult to determine that partof the abundant repertoire of mathematics which will form a solid basis for the prop­er utilization of computers in engineering in the coming decades. In this book, wetry to compile these essentials. We have arranged the material so that it can belearned in the order of the chapters of the book. We assume that traditional mathe­matics for engineers is treated in addition : Essential branches of mathematics arenot addressed in this book , since there is a vast literature on them.

The treatment of foundat ions begins with logic in Chapter 1. There are various rea­sons for this. For one, logic is a tool for the development of the other chapters ofthe book. Also, the creation of models and processes requires a systematic ap­proach, which relies on a consistent application of the laws of logic. An exampleof the systematic use of logic is furnished by a correct treatment of implications andequivalences.

Set theory, treated in Chapter 2, forms the basis for the mathematical structurestreated in the subsequent chapters. Set operations are of fundamental importancein all areas of computer applicat ion. Set theory leads to concepts like relation andmapping , which are fundamental for the classification and ordering of informationand hence for approaches like object-oriented modelling .

Mathematics contains basic algebraic, ordina l and topological structures. All otherbranches of mathematics rest on these basic structures, which are treated inChapters 3 to 5.

Algebraic structures describe operations on elements of sets. In contrast to tradi­tional mathematics for engineers, the restriction to real numbers is lifted in orderto lay a systematic foundation for general operations on values of different types,for instance logical variables, sets, vectors and matrices. These foundations areapplied in all subsequent chapters.

Preface VII

Ordinal structures are of paramount importance for many computer-based algo­rithms. Reliable algorithms require a precise treatment of the properties of orderrelations and a systematic distinction between comparable and incomparable ele­ments of a set. Many data structures cannot be designed or implemented withoutan understanding of order relations and their properties. The study of the conver­gence properties of iterative and sorting algorithms also relies on ordinal struc­tures.

If a system of subsets is singled out in a set, the set acquires a topological struc­ture. Topological structures form the basis for determining connectedness andseparation of sets, convergence of sequences , nets and filters , compactness ofspaces and continuity of functions. The convergence of approximation methodsfor solving the mathematical formulation of physical problems cannot be studiedwithout an understanding of topological spaces. The description of geometricshapes on the computer also depends on a reliable analysis of the associatedproblems in topology.

Quantification in engineering relies on the natural , whole, rational, real and com­plex number systems and the quaternions. These number systems exhibit differ­ent algebraic , ordinal and topological structures, which are treated in Chapter 6.Knowledge of the properties of the number systems is essential for constructingreliable numerical algorithms.

Groups, treated in Chapter 7, have played an important role in the developmentof mathematics. Group theory deals with an operation on two elements of a set,the result of which is again an element of the set. Two of these three values areknown, and the third value (an operand or the result) is to be determined. Thestructure of groups proves to be extraordinarily rich. It allows a systematic treat­ment of many fundamental mathematical problems. For example , Galois used itto prove that a circle cannot be squared with compass and straightedge alone. Asystematic treatment of geometry can also be carried out on the basis of grouptheory. The practical applications of group theory include the systematic analysisof the topology of triangulated bodies.

In designing models and algorithms , the description of the relations between theelements of sets is of fundamental importance. This is the subject of graph theory,treated in Chapter 8. Graph theory relies on the algebra of relations. This algebra,in which graphs are described by matrices, leads to a set of theories and methodswhich allow the properties of graphs to be determined algebraically. Many practicalproblems in engineering can be solved using graphs, including problems in man­agement and organization . Among these are the determination of paths in trafficnetworks, of reliability in complex systems and of the optimal order of processingsteps.

VIII Preface

Tensor theory, treated in Chapter 9, forms the basis for a reliable formulation ofphysical engineering problems. Tensor formulations have the special property ofbeing independent of the chosen coordinate system . This aids the understand ingof the essential characteristics of the formulated problems and thus facilitates thesystematic development of algorithms. It is on this basis that complex physical pro­cesses in solids , liquids and multi-phase systems can be rendered susceptible toa universally valid implementation.

Engineers deal with events that depend on chance : The repetition of an experi­ment under seemingly identical conditions yields different results . Random eventsare studied using stochastical methods, treated in Chapter 10. These methods as­sign probabilities to the different outcomes of an experiment. There are typicalprobability distributions in engineering, for example for the reliability of a systemof components and for the behavior at the nodes of a traffic network. Random pro­cesses for time-dependent random variables are of great practical importance.Their description using Markov chains forms the basis of the theory of queues,which is applied in many computer simulations of processes in engineering.

The chapters of the book are structured uniformly. Each chapter begins with anintroduction, which highlights the main points of the chapter. It uses concepts andmentions properties which are defined and explained in subsequent parts of thechapter. The sections also begin with introductions, which are similarly structured.Every paragraph of the text begins with a term which appears in boldface for em­phasis. This term is explained in the paragraph. The highlighted terms are in­tended to aid the reader in grasping the structure of the sections of the book withlittle effort . Proofs are included in the text , in particular where they significantly aidcomprehension or form the basis for the development of algorithms for computerimplementations.

The desire to write this book emerged during our long-standing cooperation at theTechnische Universitat Berlin in the area of "Theoretische Methoden der Bau- undVerkehrstechnik" (theoret ical methods in civil engineering). While developing thisfield together, we realized that the topics covered in educat ion and the informationtechnology employed are short-lived compared to the content of other areas of en­gineering . Yet the application of computer science in engineering needs a stablebasis. Out of this realization grew the desire to compile the mathematical founda­tions which are independent of the rapid developments and incessant changes ina book and thus to create a durable basis for future developments. The book differssignificantly from our lecture notes, which deal with current information and com­munication technologies, including development environments and their applica­tions in engineering.

Preface IX

Dr. Felix Pahlplayedan importantrole in shapingthe book as a whole. He repeat­edlyproofread thechapterswithgreatcareandusedhisbackground asa physicistto makevaluablesuggestions for structuring the material.Particularlyin the chap­ters on topology and group theory, Dr. Pahl contributedto most of the proofs andprovidedinvaluableassistancein formulating themconcisely. His specialcommit­ment to this book deserves our personal thanks.

The contentof the presentbook imposesstrongdemandson the graphicaldesignof the text, the figures and the formulas. With admirable intuition, Mrs. ElizabethMaue has given the book an attractiveappearance. As the book took shape overan extendedperiodof time,duringwhichall chapterswerethoroughlyrevisedsev­eral times,her patiencehasbeenput to a severetest. Mrs.Maue'scommitted par­ticipation, whichresultedin the particularlyappealingpresentation of thisbook,de­serves our grateful recognition.

This book took shape over the course of more than seven years. Duringthis timeour wives, Irmgard Pahl and Heidemarie Damrath, showed great understandingfor our extraordinary workload. By their great patience, they gave us the freedomand supportwithoutwhich this book could not have beencompleted in its presentform. We thank them with all our heart.

Berlin, May 2000

Peter Jan PahlRudolf Damrath

CONTENTS

1 LOGIC1.1 Representation of Thought 1

1.2 Elementary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21.3 Propositional Logic 5

1.3.1 Logical Variables and Connectives 51.3.2 Logical Expressions 111.3.3 Logical Normal Form 141.3.4 Logical Rules of Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18

1.4 Predicate Logic 201.5 Proofs and Axioms 26

2 SET THEORY2.1 Sets 312.2 Algebra of Sets 342.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Types of Relations 402.5 Mappings 442.6 Types of Mappings 462.7 Cardinality and Countability 512.8 Structures 56

3 ALGEBRAIC STRUCTURES

3.1 Introduction 593.2 Inner Operations 603.3 Sets with One Operation 633.4 Sets with Two Operations 68

3.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.2 Additive and Multiplicative Domains 693.4.3 Dual Domains 76

3.5 Vector Spaces 863.5.1 General Vector Spaces 863.5.2 Real Vector Spaces 94

3.6 Linear Mappings 993.7 Vector and Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109

3.7.1 Definitions 1093.7.2 Elementary Vector Operations 1123.7.3 Elementary Matrix Operations 1163.7.4 Derived Scalars " .. " ., , 1233.7.5 Complex Vectors and Matrices " '" 127

XII Contents

4 ORDINAL STRUCTURES

4.1 Introduction 131

4.2 Ordered Sets 132

4.3 Extreme Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138

4.4 Ordered Sets with Extremality Properties . . . . . . . . . . . . . . . . . . . . . . . 141

4.5 Mappings of Ordered Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.6 Properties of Ordered Sets 151

4.7 Ordered Cardinal Numbers 158

5 TOPOLOGICAL STRUCTURES

5.1 Introduction 161

5.2 Topological Spaces 163

5.3 Bases and Generating Sets 167

5.4 Metric Spaces 172

5.5 Point Sets in Topological Spaces 179

5.6 Topological Mappings 184

5.7 Construction of Topologies 189

5.7.1 Final and Initial Topologies 189

5.7.2 Subspaces 195

5.7.3 Product Spaces 199

5.8 Connectedness of Sets 201

5.8.1 Disconnections and Connectedness 201

5.8.2 Connectedness of Constructed Sets . . . . . . . . . . . . . . . . . . . . 207

5.8.3 Components and Paths 212

5.9 Separation Properties 219

5.10 Convergence 227

5.10.1 Sequences 227

5.10.2 Subsequences 237

5.10.3 Series 241

5.10.4 Nets 249

5.10.5 Filters 255

5.11 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

5.11.1 Compact Spaces 259

5.11.2 Compact Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 268

5.11 .3 Locally Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

5.12 Continuity of Real Functions 276

Contents XIII

6 NUMBER SYSTEM

6.1 Introduction 285

6.2 Natural Numbers 286

6.3 Integers 289

6.4 Rational Numbers 293

6.5 Real Numbers 296

6.6 Complex Numbers 303

6.7 Quaternions 307

7 GROUPS

7.1 Introduction 309

7.1.1 Group Theory 309

7.1 .2 Outline 312

7.2 Groups and Subgroups 314

7.3 Types of Groups 319

7.3.1 Permutation Groups 320

7.3.2 Symmetry Groups 323

7.3.3 Generated Groups 327

7.3.4 Cyclic Groups 330

7.3.5 Groups of Integers 333

7.3.6 Cyclic SUbgroups 338

7.4 Class Structure 342

7.4.1 Classes 342

7.4.2 Cosets and Normal Subgroups 344

7.4.3 Groups of Residue Classes 350

7.4.4 Conjugate Elements and Sets 352

7.5 Group Structure 357

7.5.1 Introduction 357

7.5.2 Homomorphism 359

7.5.3 Isomorphism 366

7.5.4 Isomorphic Types of Groups 373

7.5.5 Automorphisms 376

7.6 Abelian Groups 382

7.6.1 Introduction 382

7.6.2 Classification of Abelian Groups 383

7.6.3 Linear Combinations 388

7.6.4 Direct Sums 394

XIV Contents

7.6.5 Constructions of Abelian Groups 402

7.6.6 Decompositions of Abelian Groups 411

7.7 Permutations 417

7.7.1 Introduction 417

7.7.2 Symmetric Groups 418

7.7.3 Cycles 422

7.7.4 Conjugate Permutations , . . 428

7.7.5 Transpositions 431

7.7.6 Subgroups of a Symmetric Group 434

7.7.7 Group Structure of the Symmetric Group S4 439

7.7.8 Class Structure of the Symmetric Group S4 450

7.8 General Groups 455

7.8.1 Introduction 455

7.8 .2 Classes in General Groups 456

7.8.3 Groups of Prime-power Order 464

7.8.4 Normal Series 473

7.9 Unique Decomposition of Abelian Groups 482

8 GRAPHS

8.1 Introduction 489

8.2 Algebra of Relations 491

8.2.1 Introduction 491

8.2.2 Unary Relations 492

8.2 .3 Homogeneous Binary Relations 496

8.2.4 Heterogeneous Binary Relations 504

8.2.5 Unary and Binary Relations 509

8.2.6 Closures 512

8.3 Classificat ion of Graphs 517

8.3.1 Introduction 517

8.3.2 Directed Graphs 518

8.3.3 Bipartite Graphs 524

8.3.4 Multigraphs 530

8.3.5 Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

8.4 Structure of Graphs 538

8.4.1 Introduct ion 538

8.4.2 Paths and Cycles in Directed Graphs 539

8.4.3 Connectedness of Directed Graphs 546

Contents XV

8.4.4 Cuts in Directed Graphs 552

8.4.5 Paths and Cycles in Simple Graphs 563

8.4.6 Connectedness of Simple Graphs 567

8.4.7 Cuts in Simple Graphs 569

8.4.8 Acyclic Graphs 574

8.4.9 Rooted Graphs and Rooted Trees 580

8.5 Paths in Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 584

8.5.1 Introduction 584

8.5.2 Path Algebra 586

8.5.3 Boolean Path Algebra 600

8.5.4 Real Path Algebra 602

8.5.4.1 Minimal Path Length 602

8.5.4.2 Maximal Path Length 604

8.5.4.3 Maximal Path Reliability 606

8.5.4.4 Maximal Path Capacity 607

8.5.5 Literal Path Algebra 609

8.5.5.1 Path Edges 609

8.5.5.2 Common Path Edges 611

8.5.5.3 Simple Paths " 613

8.5.5.4 Extreme Simple Paths " 615

8.5.5.5 Literal Vertex Labels 616

8.5.5.6 Literal Edge Labels for Simple Graphs 619

8.5.5.7 Applications in Structural Analysis 620

8.5.6 Properties of Path Algebras 621

8.5.7 Systems of Equations 626

8.5.7.1 Solutions of Systems of Equations 626

8.5.7.2 Direct Methods of Solution 630

8.5.7.3 Iterative Methods of Solution 640

8.6 Network Flows 645

8.6.1 Introduction 645

8.6.2 Networks and Flows 647

8.6.3 Unrestricted Flow 649

8.6.4 Restricted Flow 653

8.6.5 Maximal Flow 657

8.6.6 Maximal Flow and Minimal Cost 662

8.6.7 Circulation 665

XVI Contents

9 TENSORS

9.1 Introduction 671

9.2 Vector Algebra 672

9.2.1 Vector Spaces 672

9.2.2 Bases 675

9.2.3 Coordinates 679

9.2.4 Metrics 682

9.2.5 Construction of Bases 686

9.2.6 Transformation of Bases 692

9.2.7 Orientation and Volume 700

9.3 Tensor Algebra 702

9.3.1 Introduction 702

9.3.2 Tensors 704

9.3.3 Transformation of Tensor Coordinates 710

9.3.4 Operations on Tensors 712

9.3.5 Antisymmetric Tensors 716

9.3.6 Tensors of First and Second Rank 730

9.3.7 Properties of Dyads 739

9.3.8 Tensor Mappings 752

9.4 Tensor Analysis 764

9.4.1 Introduction 764

9.4.2 Point Spaces 766

9.4.3 Rectilinear Coordinates 768

9.4.4 Derivatives with Respect to Global Coordinates 770

9.4.5 Curvilinear Coordinates 775

9.4.6 Christoffel Symbols 781

9.4.7 Derivatives with Respect to Local Coordinates 787

9.4.8 Tensor Integrals 796

9.4.9 Field Operations 806

9.4.10 Nabla Calculus 819

9.4.11 Special Vector Fields 825

9.4.12 Integral Theorems 829

Contents XVII

10 STOCHASTICS

10.1 Introduction 841

10.2 Random Events 843

10.2.1 Introduction 843

10.2.2 Elementary Combinatorics 843

10.2.3 Algebra of Events 846

10.2.4 Probability 848

10.2.5 Reliability 853

10.3 Random Variables " 858

10.3.1 Introduction 858

10.3.2 Probability Distributions 860

10.3.3 Moments 865

10.3.4 Functions of One Random Variable 868

10.3.5 Functions of Several Random Variables 872

10.3.6 Discrete Distributions 878

10.3.6.1 Bernoulli Distribution 878

10.3.6.2 Binomial Distribution 879

10.3.6.3 Pascal Distribution " 881

10.3.6.4 Poisson Distribution 884

10.3.7 Continuous Distributions " 887

10.3.7.1 Gamma Distribution 887

10.3.7.2 Normal Distribution 890

10.3.7.3 Logarithmic Normal Distribution 895

10.3.7.4 Maximum Distributions 898

10.3.7.5 Minimum Distributions 905

10.4 Random Vectors 906

10.4.1 Introduction 906

10.4.2 Probability Distributions 907

10.4.3 Moments 912

10.4.4 Functions of a Random Vector 916

10.4.5 Multinomial Distribution 918

10.4.6 Multinormal Distribution 920

10.5 Random Processes 922

10.5.1 Introduction 922

10.5.2 Finite Markov Processes in Discrete Time 926

10.5.2.1 Introduction 926

10.5.2.2 States and Transitions 926

XVIII Contents

10.5.2.3 Structural Analysis 932

10.5.2.4 Spectral Analysis 936

10.5.2.5 First Passage 940

10.5.2.6 Processes of Higher Order 948

10.5.3 Finite Markov Processes in Continuous Time 949

10.5.3.1 Introduction 949

10.5.3.2 States and Transition Rates 949

10.5.3.3 First Passage 954

10.5.3.4 Queues 959

10.5.3.5 Queue Systems 970

10.5.4 Stationary Processes 976

10.5.4.1 Introduction 976

10.5.4.2 Probability Distributions and Moments 976

10.5.4.3 Stationary Processes in Discrete Time 979

10.5.4.4 Stationary Processes in Continuous Time 986

Index 991