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Über dieses Buch

... 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.



Chapter 0. Introduction

Multibody Systems (MBS) play an important role in concurrent computer aided technical mechanics. Major areas of application include vehicle dynamics, robotics, biomechanics, heavy machinery and many more. The ability to design and optimize such technical multibody systems gives a company working in this area the competitive edge on the market. However, for a long time there were no generally applicable mathematical tools to this extent which was due in part to the simulation of MBS being either too slow or too inaccurate. Efficient simulation and the reliable generation of derivative information are at the heart of modern derivative based dynamic optimization methods like the optimization boundary value problem (B VP) approach developed by BOCK and coworkers [Bock81, Bock83, BP84, Sch187]. Presently, due to its applicability in many engineering disciplines, mathematical optimization may be considered a new key technology [BSS93, HJLS96].
Reinhold von Schwerin

Chapter 1. Multibody Systems in Technical Mechanics

The dynamics of multibody systems (MBS) is a focal point of multi-body modeling in mechanical engineering. Its central role was emphasized by the research program „Dynamik von Mehrkörpersystemen“ of the Deutsche Forschungsgemeinschaft (DFG). Within this research program, the goal of the MBSSIM project was to develop integrators for the differential-algebraic equations (DAE) of MBS in descriptor form as opposed to the traditional approach of treating MBS in minimal form for which standard ODE integrators had been used for a long time. The main motivation for the project at the outset was the potential presented by the straightforward inclusion of kinematic loops as well as the easy coupling of subsystems in descriptor models contrary to the situation with minimal models. Before we explain these types of equations and their derivation in more detail, we will first put the multibody system approach in perspective within the general framework of mechanical engineering and then describe in broad terms what this approach consists of as well as what the typical engineering problems in this area are. Having derived the equations of motion (EOM), we will highlight some practical aspects of modeling and simulation of MBS. As we do so, we will make a new contribution to the modeling of MBS in natural coordinates which further enhances the structures in such models.
Reinhold von Schwerin

Chapter 2. Software Engineering in Application Oriented Scientific Computing

The MBSSIM software project is one particular instance of software development for complex applications in interdisciplinary scientific computing. As such, the software product should be applicable in both science and industrial engineering resulting in a technology transfer from scientific computing to industry. In fact, such application oriented scientific computing must be viewed as a new key technology of its own [HJLS96]. However, scientific computing entails research, which means that the initial aim is to find feasible solutions. This aspect warrants special attention within the procedural model of software development. Hence, by detailing the specifics of software development in scientific computing it will be seen that it makes sense to distinguish the scientific software product from the industrial product in the sense that the former should be viewed as an advanced prototype of the latter. This is in agreement with a sophisticated software life cycle model based on prototyping.
Reinhold von Schwerin

Chapter 3. Mathematical Methods for MBS in Descriptor Form

Recall from section 1.3 that we want to consider IVP for the descriptor form of the equations of motion of MBS, where we assume that an index reduction to index one has been performed, e.g., by employing a multibody formalism like MBSNAT. This means that we deal with a semi-explicit DAE system with linearly-implicit algebraic equations in the variables p, v, a and λ:
$$\begin{array}{*{20}{c}} {\dot p = v}&;&{\dot v = a} \end{array}$$
$$\left( {\begin{array}{*{20}{c}} {M(p)}&{G{{(p)}^T}} \\ {G(p)}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} a \\ \lambda \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {f(t,p,v)} \\ {\gamma (p,v)} \end{array}} \right)$$
subject to the constraints
$$\begin{array}{*{20}{c}} {g(p) = 0}&{(position constraint)} \end{array}$$
$$\begin{array}{*{20}{c}} {G(p)v = 0}&{(velocity constraint)} \end{array}$$
with the consistent initial values
$${p_0}: = p({t_0}),{v_0}: = v({t_0}),{a_0}: = a({t_0}), and {\lambda _0}: = \lambda ({t_0}).$$
Reinhold von Schwerin

Chapter 4. Applications

In the following, utilizing our new modeling technique for rotationally symmetric bodies with spherical respectively universal joints as presented in section 1.8, we derive a natural coordinate model for a wheel suspension system for upper-class passenger cars as shown in figure 4.1. The modeling process will be described in detail in the present section followed by a comparison of different multibody integrators on this model as well as a relative coordinate one in the next section. The relative coordinate model is a modification due to Grupp and Simeon 1 [GS94] of the original proposal by HILLER and FRIK [HF91] in [KSP91]. We have already seen in section 1.7 that the new natural coordinate model has the potential to speed up simulation considerably in connection with an appropriate integration scheme based on inverse dynamics like MBSIDY, i.e. if modeling and simulation are optimally dovetailed.
Reinhold von Schwerin

Chapter 5. Putting It All Together: The MBSSIM Scientific Software Project

To conclude, we will summarize this monograph in which we have described in detail the mathematical methods, algorithms and software for the simulation and optimization of multibody systems in descriptor form. They were developed within the MBSSIM project which was embedded in two grant programs by the DFG and the BMBF, respectively. These mathematical techniques represent significant progress within the important application domain of technical mechanics, which could only be achieved through application oriented interdisciplinary scientific computing. As a result, the scientific software product MBSSIM has meanwhile reached maturity and is currently being integrated into the industrial strength software product AMIGOS, which allows graphical modeling and analysis of multibody systems, thus making the sophisticated numerical methods easily accessible to the practitioner who must solve real-life problems.
Reinhold von Schwerin


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