Copyrighted Material Dan Braha and Oded Maimon, A Mathematical Theory of Design: Foundations, Algorithms, and Applications, Springer, 1998, 708 p., Hardcover, ISBN: 0-7923-5079-0. PREFACE Part One THE DESIGN PROCESS: PROPERTIES, PARADIGMS AND THE EVOLUTIONARY STRUCTURE xv 1 1 INTRODUCTION AND OVERVIEW 3 1.1 Scope and Objectives 3 1.2 Common Properties of Design 4 1.3 Design Theories 5 1.4 The Mathematical Method of General Design Systems 6 1.5 Difficulties of the Application of the Mathematical Method 8 1.6 Preview of the Book 9 1.6.1 Modeling the Attribute Space (Chapter 4) 11 1.6.2 The Idealized Design Process (Chapter 5) 11 1.6.3 The 'Real' Design Process (Chapter 6) 12 1.6.4 Computational Analysis of Design (Chapters 5-7) 13 1.6.5 The Measurement of A Design Structural and Functional 14 Complexity (Chapters 8-9) 1.6.6 Algorithmic Methods and Design Applications (Parts III 14 & IV) 1.7 Concluding Remarks 14 References 15 2 DESIGN AS SCIENTIFIC PROBLEM-SOLVING 19 2.1 Introduction 19 2.1.1 Motivation and Objectives 19 2.1.2 Overview of the Chapter 20 2.2 Properties of the Design Problem 22 2.2.1 The Ubiquity of Design 22 2.2.2 Design as A Purposeful Activity 23 2.2.3 Design is A Transformation Between Descriptions 23 2.2.4 Categories of Design Requirements 23 2.2.5 Bounded Rationality and Impreciseness of Design 24
vi Problems 2.2.6 The Satisficing Nature of Design Problems 25 2.2.7 The Intractability of Design Problems 25 2.2.8 The Form of Design 25 2.3 Properties of the Design Process 27 2.3.1 Sequential and Iterative Natures of Design 27 2.3.2 The Evolutionary Nature of the Design Process 29 2.3.3 Design Process Categories 32 2.3.4 The Diagonalized Nature of Design 33 2.4 Survey of Design Paradigms 38 2.4.1 Defining A Design Paradigm 38 2.4.2 Design Paradigms 39 2.4.3 The Analysis-Synthesis-Evaluation (ASE) Design 40 Paradigm 2.4.4 Case-Based Design Paradigm 44 2.4.5 The Cognitive Design Paradigm 45 2.4.6 The Creative Design Paradigm and the SIT Method 46 2.4.7 The Algorithmic Design Paradigm 57 2.4.8 The Artificial Intelligence Design Paradigm 58 2.4.9 Design as A Social Process 61 2.5 Scientific Study of Design Activities 65 2.5.1 The Axiomatic Theory of Design 66 2.5.2 Design as Scientific Problem-Solving 67 2.6 A General Design Methodology 77 2.7 Summary 79 References 79 3 INTRODUCTORY CASE STUDIES 85 3.1 Electrical Design 85 3.1.1 Design of A "Data Tag" 85 3.1.2 Control Logic of Flexible Manufacturing Systems 90 3.1.3 Serial Binary Adder Unit Design 92 3.2 Mechanical Design 93 3.2.1 Mechanical Fasteners Design 93 3.2.2 Supercritical Fluid Chromatography (SFC) Design 96 3.2.3 Gear Box Design (Wormgear Reducer) 97 3.3 Flexible Manufacturing System Design 101 3.3.1 What is A Flexible Manufacturing System? 101 3.3.2 FMS Configuration Design Issues 102 3.4 Discussion 104 References 106 Part Two FORMAL DESIGN THEORY (FDT) 107 4 REPRESENTATION OF DESIGN ARTIFACTS 109 4.1 Introduction 109
vii 4.2 Modeling the Artifact Space 111 4.2.1 The Basic Modeling of Design Artifacts 111 4.2.2 Examples 115 4.2.3 Properties of the Design Space 124 4.3 Summary 134 Appendix - A (Proofs) 134 Appendix - B (Basic Notions of Set Theory) 139 References 141 5 THE IDEALIZED DESIGN PROCESS 143 5.1 Introduction 143 5.2 Motivating Scenarios 147 5.2.1 Mechanical Fasteners 147 5.2.2 The Car Horn 149 5.3 Preliminaries 152 5.3.1 The Attribute and Function Spaces 152 5.3.2 Proximity in Function and Attribute Spaces 154 5.3.3 Transformation Between Function and Attribute Spaces 160 5.3.4 Decomposition of Design Specification 163 5.3.5 Convergence of the Function Decomposition Stage 165 5.3.6 Order Relation for Attribute and Function Spaces 165 5.4 Idealized Design Process Axioms 167 5.4.1 Continuity in Function and Attribute Spaces 167 5.4.2 The Complexity of the Homeomorphism and 171 Continuity Problems 5.5 Basis for Function and Attribute 172 5.5.1 Definition and Properties 172 5.5.2 Space Character as A Descriptive Complexity Measure 174 5.6 Concluding Remarks 175 Appendix A - Basic Notions of Topology, and Language Theory 176 Appendix B - Bounded Post Correspondence Problem (BPCP) 180 Appendix C - Graph Isomorphism 180 Appendix D - Proofs of Theorems 180 References 185 6 MODELING THE EVOLUTIONARY DESIGN 187 PROCESS 6.1 Introduction 187 6.2 Preview of the Models 190 6.2.1 Type-1 Design Process 190 6.2.2 Type-2 Design Process 197 6.3 Detailed Modeling 199 6.3.1 Type-0 Design Process L, Q, P, TA, TS, S0, F 200 6.3.2 Type-1 Design Process L, Q, P, TA, TS, S0, F 203
viii 6.3.3 Type-2 Design Process L, Q, P, T, S0, F 203 6.4 Correctness and Complexity of the Design Process 205 6.4.1 Correctness of the Design Process 205 6.4.2 Computational Complexity of the Design Process Problem 207 6.5 Summary 214 Appendix A - Basic Notions of Automata Theory [Adopted from 3] 215 References 216 7 GUIDED HEURISTICS IN ENGINEERING DESIGN 217 7.1 Introduction 217 7.2 The Basic Synthesis Problem (BSP) 218 7.2.1 Problem Formulation 219 7.2.2 The Intractability of the BSP 222 7.3 The Constrained Basic Synthesis Problem (CBSP) 225 7.3.1 Problem Formulation 225 7.3.2 Universal Upper Bound on I 225 7.4 Refined Upper Bound on I 227 7.4.1 Probabilistic Design Selection 227 7.4.2 The Asymptotic Equipartition Property (AEP) 228 7.4.3 Consequences of the AEP on the CBSP 230 7.5 Design Heuristics for Feature Recognition 232 7.5.1 Geometric Modeling 232 7.5.2 Wireframe Feature Recognition 233 7.5.3 Combinatorial Analysis of the Connectivity Problem 235 7.5.4 Combinatorial Analysis of the Feature Recognition 236 Problem 7.6 Summary 237 Appendix A - The Satisfiability Problem 238 References 238 8 THE MEASUREMENT OF A DESIGN 241 STRUCTURAL AND FUNCTIONAL COMPLEXITY 8.1 Introduction 241 8.1.1 Complexity Judgment of Artifacts and Design Processes 241 8.1.2 Two Definitions of Design Complexity 243 8.1.3 Organization of the Chapter 245 8.2 Structural Design Complexity Measures 245 8.2.1 Description of the Valuation Measures 245 8.2.2 Basic Measures 247 8.2.3 Composite Measures 249 8.3 Evaluating the Total Assembly Time of A Product 255 8.3.1 Total Assembly Time and Assembly Time Measure 255 8.3.2 Assembly Defect Rates and Assembly Time Measure 261 8.3.3 Design Assembly Efficiency and Assembly Time Measure 262
ix 8.4 Thermodynamics and the Design Process 267 8.4.1 Natural Science and Engineering Design 267 8.4.2 The Balloon Model 268 8.5 Functional Design Complexity Measure 273 8.6 Summary 276 References 277 9 STATISTICAL ANALYSIS OF THE TIME 279 COMPLEXITY MEASURE 9.1 Introduction 279 9.2 Other Methods for Design for Assembly (DFA) 280 9.3 Results and Discussion of the Time Complexity Measure 283 9.4 The Barkan and Hinckley Estimation Method 285 9.5 Conclusions 287 Appendix A - Time Complexity Measure of A Motor Drive Assembly 289 References 290 Part Three ALGORITHMIC AND HEURISTIC METHODS FOR DESIGN DECISION SUPPORT 291 10 INTELLIGENT ADVISORY TOOL FOR DESIGN 293 DECOMPOSITION 10.1 Introduction 293 10.2 AND/OR Tree Representation of Design 294 10.3 Guiding the AND/OR Search Tree 297 10.4 A Prototype System to Implement the Design Search Algorithm 300 10.4.1 Case-1 Overview 300 10.4.2 Basic Case-1 Definitions 302 10.4.3 The Case Builder Interface 305 10.4.4 The Analyzer Interface 312 10.5 Summary 318 References 319 11 PHYSICAL DESIGN OF PRINTED CIRCUIT 321 BOARDS: GROUP TECHNOLOGY APPROACH 11.1 Introduction 321 11.1.1 The Role of Clustering (Grouping) in Design 321 11.1.2 The Circuit Partitioning Problem 324 11.2 Mathematical Formulation 330 11.3 Properties of the Circuit-Partitioning Problem 332 11.4 A Grouping Heuristic for the Circuit-Partitioning Problem 334 11.5 A Branch and Bound Algorithm 336 11.6 Computational Results Using the Branch and Bound Algorithm 339 11.7 Summary 345 Appendix A - A Brief Overview of Microelectronics Circuits and 345
x their Design Appendix B - (Proof of Theorem 11.1) 348 Appendix C - (Bounds on the Number of Times Net Type i Must be 349 Packed) References 351 12 PHYSICAL DESIGN OF PRINTED CIRCUIT 353 BOARDS: GENETIC ALGORITHM APPROACH 12.1 Introduction 353 12.2 The Genetic Algorithm Approach 354 12.3 A Genetic Algorithm for the Circuit-Partitioning Problem 356 12.4 Computational Results 358 12.5 Other Applications of Genetic Algorithm 361 12.5.1 The Catalogue Selection Problem 361 12.5.2 Outline of the Genetic Algorithm 362 12.6 Summary 363 References 363 13 ADAPTIVE LEARNING FOR SUCCESSFUL 365 DESIGN 13.1 Introduction 365 13.1.1 Managing the Intricate Correspondence Between Function 365 and Structure 13.1.2 The Applicability of the Methodology 367 13.2 Problem Formulation 368 13.3 Adaptive Learning of Successful Design 369 13.3.1 The Probabilistic Nature of the Design Process 369 13.3.2 Preliminaries 371 13.3.3 The P-Learning Algorithm 374 13.4 Illustrative Example 376 13.5 A Catalogue Structure for the P-Learning Algorithm 381 13.6 Summary 382 Appendix A - Computation of the Experimental Success Probabilities 383 Appendix B - Bayes Theorem 384 References 385 14 MAINTAINING CONSISTENCY IN THE DESIGN 387 PROCESS 14.1 Introduction 387 14.1.1 Variational Design 387 14.1.2 Design Consistency in Variational Design 388 14.1.3 Chapter Outline 389 14.2 Previous Efforts 389 14.2.1 Geometric Reasoning 389 14.2.2 Numerical Techniques based on Continuation Methods 390
xi 14.2.3 Other Numerical Techniques 391 14.2.4 Discussion 393 14.3 Design Consistency 394 14.4 Design Evolution in Variational Design Systems 396 14.5 Design Consistency Through Solution Trajectories 400 14.5.1 Definitions in Design Consistency 400 14.5.2 Theorems in Design Consistency 403 14.6 COAST Algorithm for Design Consistency 405 14.6.1 Mean Value Theorem 405 14.6.2 Rigorous Sensitivity Analysis Algorithm 406 14.6.3 Bifurcations and Infeasible Regions 408 14.7 Design of Cantilever Beam 409 14.7.1 Design Execution 410 14.7.2 Comparison with Other Methods 414 14.8 Summary 416 Appendix A - Interval Analysis Techniques 417 A.1 Solving Systems of Interval Equations 418 A.2 Existence and Uniqueness of Solutions 419 Appendix B - Constraint Model of Beam 419 References 421 15 CONSTRAINT-BASED DESIGN OF FAIRED 423 PARAMETRIC CURVES 15.1 Constraint-Based Curve Design 423 15.2 Previous Work 424 15.3 Maintaining Design Consistency in Constraint-Based Curve 425 Design 15.3.1 Distance Constraints 426 15.3.2 Arc Length 428 15.3.3 Consistency in Curve Fairing 428 15.3.4 COAST Methodology for Design Consistency 430 15.4 Examples 431 15.4.1 Bezier Curve from Distance Constraints 431 15.4.2 Apparel Design 435 15.5 Discussion 443 References 443 16 CREATING A CONSISTENT 3-D VIRTUAL LAST 445 FOR PROBLEMS IN THE SHOE INDUSTRY 16.1 Problems in Shoe Design Industry 445 16.2 Creation of A Virtual Last 448 16.2.1 Constraint Definitions 452 16.2.2 Curve Fairing 455 16.3 Results 456 16.4 Summary 459
xii References 459 Part Four DETAILED DESIGN APPLICATIONS 461 17 DESIGN OF A WORMGEAR REDUCER: A CASE 463 STUDY 17.1 Introduction 463 17.2 Conceptual Design of A Wormgear Reducer (Gear Box) 464 17.2.1 Confrontation 464 17.2.2 Problem Formulation 465 17.2.3 Design Concepts 466 17.3 Detailed Synthesis of the Gear Box 468 17.3.1 Motor Design 470 17.3.2 The Design of the Transmission Parts and the Outline of 471 Their Relative Position 17.3.3 Testing the Current Design Against the Wormgear Load 475 and Strength Constraints 17.3.4 Initial Design of the Casing (Box) 477 17.3.5 The Design of the Wormgear Shaft Set 479 17.3.6 Calculation and Check of the Shaft Set Parts 481 17.3.7 Strength and Wear-Resistance Constraints 486 17.3.8 Detailed Design of the Casing 487 17.3.9 Accessories Design 487 17.3.10 Casing Heat Balance Constraints 488 17.4 Discussion 491 17.4.1 Design Description ( L) 493 17.4.2 Transformation ( T) 494 17.5 A Methodology for Variational Design 495 17.5.1 The General Methodology 495 17.5.2 Demonstration 496 17.5.3 Design Execution 498 References 498 18 ADAPTIVE LEARNING FOR SUCCESSFUL 499 FLEXIBLE MANUFACTURING CELL DESIGN: A CASE STUDY 18.1 Introduction 499 18.2 Physical Configuration 500 18.3 Parameters and Performance Measures 503 18.3.1 Performance Measures 503 18.3.2 Parameters and Structural Assumptions 504 18.3.3 Evaluation of the Responses Through Simulation 506 18.4 Solving the Design Problem Using the P-Learning Algorithm 506 18.5 Concluding Remarks 512
xiii References 512 19 MAINTAINING DESIGN CONSISTENCY: 513 EXAMPLES 19.1 Wormgear Assembly Problem 514 19.1.1 Dimensions - Wormgear Assembly 514 19.1.2 Design Execution 514 19.2 Helical Compression Spring Problem 521 19.3 Other Design Areas 524 19.4 Point at A Distance from Two Points 528 19.5 Line Tangent to Two Circles 533 19.6 Helical Compression Spring (Continued) 539 Appendix A - Constraint Model of Wormgear 548 20 CASES IN EVOLUTIONARY DESIGN PROCESSES 551 20.1 Automobile Design Example 551 20.1.1 The Specification and Design Description Properties 551 20.1.2 The Production Rules 553 20.1.3 Car Synthesis Using the Design Search Algorithm (see 562 Chapter 10.3) 20.2 Forklift Design Example 564 20.2.1 The Specification and Design Description Properties 565 20.2.2 The Production Rules 568 20.2.3 Forklift Truck Synthesis Using the Design Search 576 Algorithm (see Chapter 10.3) 20.3 Computer Classroom Design Example 578 20.3.1 The Specification and Design Description Properties 578 20.3.2 The Production Rules 584 20.3.3 Computer Classroom Synthesis Using the Design Search 601 Algorithm (see Chapter 10.3) 20.4 Tire Design Example 604 20.4.1 The Specification and Design Description Properties 606 20.4.2 The Production Rules 608 20.4.3 Tire Synthesis Using the Design Search Algorithm (see 612 Chapter 10.3) 20.5 Fastener Design Example 614 20.5.1 The Specification and Design Description Properties 614 20.5.2 The Production Rules 615 20.5.3 Fastener Synthesis Using the Design Search Algorithm 620 (see Chapter 10.3) 20.6 Fastener Design Example (Continued) 621 20.6.1 The Specification and Design Description Properties 621 20.6.2 The Production Rules 624 Appendix A - Automobile Design 632 A.1 Engines 632
xiv A.2 The Body 638 A.3 Steering System 639 A.4 Suspension System 640 A.5 The Brake System 642 A.6 Power Train 644 A.7 Other Design Consideration 645 Part Five PRACTICAL CONSIDERATIONS 649 21 CONCLUDING REFLECTIONS 651 21.1 General Purpose Guidelines 651 21.1.1 Representation of Design Knowledge 651 21.1.2 Design Process 652 21.2 Algorithmic Design Guidelines 662 21.2.1 Logic Decomposition and Case Based Reasoning (Chapter 662 10) 21.2.2 Group Technology and Clustering Analysis (Chapter 11) 665 21.2.3 Solving Design Problems with Genetic Algorithms 666 (Chapter 12) 21.2.4 Probabilistic Selection Methods for System Design 667 (Chapter 13) 21.2.5 Maintaining Consistency in the Design Process (Chapters 667 14, 15, 16) 21.3 Summary 670 INDEX 671