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2009 | Book

Introduction Read chapter Read first chapter

Numerics of Unilateral Contacts and Friction

Modeling and Numerical Time Integration in Non-Smooth Dynamics

Author: Christian Studer

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Applied and Computational Mechanics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Mechanics provides the link between mathematics and practical engineering app- cations. It is one of the oldest sciences, and many famous scientists have left and will leave their mark in this fascinating ?eld of research. Perhaps one of the most prominentscientists in mechanics was Sir Isaac Newton, who with his “laws of - tion” initiated the description of mechanical systems by differential equations. And still today, more than 300 years after Newton, this mathematical concept is more actual than ever. The rising computer power and the development of numerical solvers for diff- ential equations allowed engineersall over the world to predict the behavior of their physical systems fast and easy in an numerical way. And the trend to computational simulation methods is still further increasing, not only in mechanics, but practically in all branches of science. Numerical simulation will probablynot solve the world’s engineering problems, but it will help for a better understanding of the mechanisms of our models.

Table of Contents

Frontmatter
Introduction
Abstract
Reality is neither smooth nor non-smooth. Models of reality are smooth or nonsmooth depending on the questions we ask. If a physical system has rapidly changing phases, then it can be advantageous to model the system in a non-smooth way. For mechanical systems the impact times are usually much smaller than the global motion which is of interest. This motivates the study of rigid multibody dynamics. The set-valued force laws which model the constitutive behaviour of unilateral contacts and of friction lead to non-smooth models. Usually, the positions are assumed to be absolutely continuous, while the velocities are allowed to undergo jumps and are taken to be of bounded variation. Jumps in the velocities can not be accomplished by finite but only by impulsive forces.
Christian Studer
Mathematical Preliminaries
Abstract
This chapter aims at giving a short introduction to some mathematical concepts which are used in this work. Vector and matrix norms are discussed in Sect. 2.1, some comments concerning derivatives are given in Sect. 2.2 and a short introduction to convex analysis is provided in Sect. 2.3. A proximal point and a distance vector function are introduced, which are associated with a special class of normal cone inclusion problems. A discussion on global and local representations of such problems is given in Sect. 2.4. Section 2.5 deals with differential algebraic equations.
Christian Studer
Non-Smooth Mechanics
Abstract
This chapter deals with the non-smoothmodeling ofmechanical systems. In Sect. 3.1 it is reviewed how a mechanical system subjected to bilateral constraints can be formulated as a differential algebraic equation (DAE). The theory is extended in Sect. 3.2 to non-impulsive non-smooth motion, and it is discussed how unilateral contacts, friction and other non-smooth interactions can be modeled by set-valued force laws. In Sect. 3.3 impacts and other impulsive interactions are added to the model by stating impact equations and Newton’s extended impact law. Both impulsive and non-impulsive motion can be gathered together in so-called equalities of measures, which are discussed in Sect. 3.4. This chapter deals with rigid multibody systems. However, the resulting equations are much more general and can also be used for the non-rigid case.
Christian Studer
Inclusion Problems
Abstract
In this chapter it is discussed how inclusion problems can be solved. The method is demonstrated on the example of inclusions which describe the impact-free motion on acceleration level, i.e. inclusions which describe dynamical take-off or slip-stick transitions.
Christian Studer
Time-Stepping
Abstract
In this chapter time-stepping schemes are investigated more closely. In Sect. 5.1 possible discretizations for differential algebraic systems are discussed. These discretizations build the fundament of the time-stepping schemes. Special attention is paid to the drift phenomenon of acceleration and velocity based constraint formulations. We discuss the discretization of index-1, index-2 and index-3 DAE’s. The latter are treated either by a GGL [86] or a preconditioning approach [19]. Furthermore, the nonlinear gap function of an index-3 problem is linearized which yields an index-2-like scheme with additional explicit drift stabilization. In Sect. 5.2, the behaviour of non-smooth systems is discussed. The time evolution of such systems is split into different smooth parts, which are described by an underlying DAE. Event-driven methods do not discretize the non-smooth system itself but the different smooth parts, and detect switching points at which the underlying DAE and thus the discretization changes. Time-stepping schemes directly discretize the nonsmooth system regardless of switching points, see Sect. 5.3.Moreau’s time-stepping method is discussed in detail. This prominentmethod will also serve as base method for chapter 6. In addition, a brief review on existing time-stepping methods is given. First, the methods of Jean [47] and Paoli/Schatzman [68] are discussed. Furthermore, the contributions of Stiegelmeyr [85], Funk [36], Foerg [33] and Pfeiffer [72], as well as the methods of Stewart/Trinkle [84], Anitescu/Potra [10, 11], Anitescu/ Hart [9], Potra et al [74] and Gavrea et al [37] are reviewed. In addition, some ideas for two new time-stepping schemes which incorporate the GGL or the preconditioning approach are given. A discussion on the classification of the different schemes is found in Sect. 5.3.9.
Christian Studer
Augmented Time-Stepping by Step Size Adjustment and Extrapolation
Abstract
This chapter investigates how the accuracy of Moreau’s midpoint rule can be increased by step size adjustment and extrapolation. The idea is to perform time steps which contain switching points with a minimal step size Δt min . Smooth time steps, i.e. time steps which do not contain switching points, are processed with larger steps sizes. Furthermore, extrapolation methods are used to increase the integration order in these time steps. The integration order of the classical Moreau midpoint rule is analyzed in Sect. 6.1. In Sect. 6.2 it is discussed how switching points can be localized, which yields a step size controlled time-stepping algorithm. Finally, in Sect. 6.3 extrapolation methods are applied to Moreau’s midpoint rule, which requests some restrictions on these methods. Some examples are given in Sect. 6.5. The chapter is based on the original publications [92, 94].
Christian Studer
The dynamY Software
Abstract
This chapter provides a short overview on the dynamY software package [88], which can be used to simulate non-smooth mechanical systems. The dynamY software package consists of various C++ classes, which allow for the modeling and the simulation of a non-smooth system. The software works on the basis of smooth mechanical systems, which can be connected by non-smooth elements. The chapter gives a short overview on the software and discusses some calculated examples[88].
Christian Studer
Summary
Abstract
During the last years, the interest in modeling mechanical systems with impacts and friction has increased enormously. New mathematical models have been developed, which account for the structure of such mechanical systems. It has to be distinguished between two main classes of models, the regularized models and the set-valued models. While the regularized models must be seen as an extension of classical smooth mechanics, the set-valued models use complete new formulations which incorporate classical mechanics. Set-valued models lead to non-smooth formulations, which have much less input parameters as regularized models and which are especially suited to understand the mechanisms of impacts and friction. In this book, the time integration of mechanical systems which consist of rigid bodies which can interact by set-valued laws has been discussed. The contacts and the friction elements have been modeled in a set-valued and thus non-smooth way. The set-valued force laws have been written as inclusions in a very general way which accounts for the various types of non-smooth interactions, like for example unilateral contacts, spatial friction, Coulomb-Contensou friction etc. It was shown how inclusion problems can be solved using an augmented Lagrangian approach. Timestepping schemes have been discussed, and some extensions have been proposed which allow for a higher integration order and a step size adjustment. Special attention has been paid to keep all formalisms short and simple. We tried to give the various existing approaches on howto discretize a non-smooth system and on howto solve the resulting inclusion problems a common structure. As already mentioned, non-smooth mechanics includes also classical mechanics as most simple case. This can be verified in the modeling (inclusions reduce to equations), in the discretization (time-stepping schemes reduce to DAE integrators) but also in the solution process of the resulting inclusion problems (JORprox/SORprox reduce to classical JOR/SOR schemes).
Christian Studer
Backmatter
Metadata
Title
Numerics of Unilateral Contacts and Friction
Author
Christian Studer
Copyright Year
2009
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-01100-9
Print ISBN
978-3-642-01099-6
DOI
https://doi.org/10.1007/978-3-642-01100-9

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