MultiBody System SIMulation

... users on the other side of the fence ... have long said that until we numerical analysts take time to write good software and get it out to the users, our ideas will not be put into action. -C.W. GEAR IN [AIKE85] This monograph is based on my doctoral thesis which I wrote dur­ ing my work at the Interdisciplinary Center for Scientific Computing (IWR) at the Ruprecht-Karls University of Heidelberg. One of my intentions was and still is to stress the practical aspects leading from the conception of mathematical methods to their effective and efficient realization as scientific software. In my own experience, I had always wished there had been something to guide me through this engineering process which accompanies the basic research for which there were nu­ merous treatises dealing, e.g., with mathematical theory for descriptor systems. Therefore, I felt that writing this monograph provided a good op­ portunity to try to fill this gap by looking at software engineering from a scientific computing angle. Thus, this monograph contains a chap­ ter on software engineering with numerous examples from the work on MBSSIM. This is meant as a beacon for those of us who really do want to produce scientific software instead of just hacking some code. On the other hand, for those more interested in the theory of differential-algebraic equations, many bibliographical references have been included where appropriate.

0 Introduction 1 Multibody Systems in Technical Mechanics 1.1 Multibody Systems 1.1.1 Topology of MBS 1.1.2 Typical Applications in Technical Mechanics 1.2 Equations of Motion of MBS in Descriptor Form 1.2.1 Types of Constraints 1.2.2 Hamiltons Principle 1.2.3 dAlemberts and Jourdains Principles 1.3 Mathematical Properties of the Descriptor Form 1.3.1 The Index of the Descriptor Form 1.3.2 Approaches for the Numerical Treatment of the Descriptor Form 1.3.3 Consistency 1.3.4 Existence and Uniqueness of Solutions 1.3.5 Structures of the Index 1 Equations 1.4 Practical Aspects of MBS 1.4.1 Non-Smooth Models 1.4.2 Multibody Formalisms 1.5 Advantages of the Descriptor Form 1.5.1 An Example from Engineering 1.5.2 Minimal Model 1.5.3 Descriptor Model 1.6 Choice of coordinates 1.6.1 Relative Coordinates 1.6.2 Reference Point Coordinates 1.6.3 Natural Coordinates 1.6.4 Mixed Coordinates 1.7 Interdependence of Modeling and Simulation 1.7.1 Standard Approach: Forward Dynamics Simulation 1.7.2 A New Approach: Inverse Dynamics Simulation 1.8 A New Technique for Modeling of Universal Joints 1.8.1 A Standard Model for Universal Joints 1.8.2 A New Model for Universal Joints 1.9 Summary of the Properties of MBS 2 Software Engineering in Scientific Computing 2.1 Application Oriented Scientific Software 2.1.1 The Software Crisis 2.1.2 Implications of the Research Factor 2.1.3 Implications of Application Orientedness 2.1.4 Scientific Software Products and Feasibility Engineering 2.2 Complex Systems 2.2.1 Characteristics of Complex Systems 2.2.2 Key Factors for Mastering Complexity 2.2.3 The Meaning of Software Engineering 2.3 Software Quality 2.3.1 Criteria Pertaining to Product Operation 2.3.2 Criteria Pertaining to Product Transition 2.3.3 Criteria Pertaining to Product Revision 2.3.4 Software Quality Assurance 2.4 Programming in the Small 2.4.1 Coding and Design 2.4.2 Testing 2.5 Programming in the Large 2.5.1 The Classic Sequential Life Cycle Model 2.5.2 Prototyping 2.5.3 A Prototyping Oriented Life Cycle Model for Feasibility Engineering 2.6 Summary: Peculiarities of Feasibility Engineering 2.7 Implementation: The Scientific Software MBSSIM 2.7.1 Module Structure of MBSSIM 2.7.2 The User Interface of MBSSIM 3 Mathematical Methods for MBS in Descriptor Form 3.1 Adaptive Adams methods 3.1.1 Basics 3.1.2 Computational Formulae for Adaptive Adams Methods 3.1.3 Solution of the Nonlinear Corrector Systems 3.2 A New Strategy for Controlling Adaptivity 3.2.1 Formulae for Constant Stepsize 3.2.2 Practical Error Estimation 3.2.3 Choosing a New Order 3.2.4 Choosing a New Stepsize 3.3 A Runge-Kutta-Starter for Adaptive Adams Methods 3.3.1 Goals of Construction 3.3.2 Construction of the Runge-Kutta-Starter 3.3.3 Error Estimation and Stepsize Selection 3.3.4 Summary 3.4 A Numerical Comparison 3.5 Inverse Dynamics Integration 3.5.1 A Local Complexity Analysis 3.5.2 Inverse Dynamics: Taking a Global Perspective 3.5.3 Conclusions for Descriptor Models: O(n) methods 3.5.4 Inverse Dynamics Multistep Methods for MBS in Descriptor Form 3.5.5 A Monitoring Strategy for Approximate Jacobians in Corrector Systems 3.6 Exploiting the Optimization Superstructure 3.6.1 The Schur Complement Method 3.6.2 The Range Space Method for Multibody Simulation 3.6.3 Null Space Methods for Multibody Simulation 3.6.4 A Unified View of RSM and NSM 3.6.5 The NSM Based on LQ-Factorization 3.6.6 The NSM Based on LU-Factorization 3.6.7 A Nonsymmetric NSM Based on LU-Factorization 3.6.8 A Comparison of Complexity for Dense Linear Algebra Solvers 3.6.9 A Numerical Comparison of Dense Linear Algebra Solvers 3.7 Topological Solution Algorithms 3.7.1 Graphs of MBS 3.7.2 Solution of Closed Loop Systems 3.7.3 Recursive Solution of the Open Chain System 3.7.4 Ingredients of the Recursion 3.7.5 A Topological Solver Based on NSM 3.7.6 A Numerical Study for the Topological Solver Using IWR-chain 3.8 Projection Methods for Constrained Multibody Systems 3.8.1 The Drift Phenomenon 3.8.2 Exploitation o