An asynchronous, variational method for simulating elastica in complex contact and impact scenarios is developed. Asynchronous Variational Integrators [1] (AVIs) are extended to handle contact forces by associating dierent time steps to forces instead of to spatial elements. By discretizing a barrier potential by an innite sum of nested quadratic potentials, these extended AVIs are used to resolve contact while obeying momentum- and energy-conservation laws. A series of two- and three-dimensional examples illustrate the robustness and good energy behavior of the method.
The simplest contact models for finite element simulation follow the early analytical work of Hertz @cite_21 in assuming frictionless contact of planar (or nearly planar) surfaces with small strain. In this regime, several approaches have been explored to arrive at a weak formulation of contact; for a high-level survey of these approaches, see for example the overview by Belytschko al @cite_25 or Wriggers @cite_12 . The first of these are the use of penalty forces, described for instance by Oden @cite_13 and Kikuchi and Oden @cite_17 . The penalty approach results in a contact force proportional to an arbitrary parameter and to the rate of interpenetration, or in more general formulations to an arbitrary function of rate of interpenetration and interpenetration depth; Belytschko and Neal @cite_0 discuss the choosing of this parameter in Section 8. Recent work by Belytschko al @cite_39 uses moving least squares to construct an implicit smooth contact surface, from which the interpenetration distance is evaluated. Peric and Owen @cite_30 describe how to equip penalty forces with a Coulomb friction model.
@cite_30 @cite_21 @cite_39 @cite_0 @cite_13 @cite_25 @cite_12 @cite_17
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The friction forces are assumed to follow the Coulomb law, with a slip criterion treated in the context of a standard return mapping algorithm. Consistent linearization of the field equations is performed which leads to a fully implicit scheme with non-symmetric tangent stiffness which preserves asymptotic quadratic convergence of the Newton-Raphson method. A new algorithm has been developed for smoothing the surfaces in finite element formulations of contact-impact. A key feature of this method is that the smoothing is done implicitly by constructing smooth signed distance functions for the bodies. These functions are then employed for the computation of the gap and other variables needed for implementation of contact-impact. The smoothed signed distance functions are constructed by a moving least-squares approximation with a polynomial basis. Results show that when nodes are placed on a surface, the surface can be reproduced with an error of about one per cent or less with either a quadratic or a linear basis. With a quadratic basis, the method exactly reproduces a circle or a sphere even for coarse meshes. Results are presented for contact problems involving the contact of circular bodies. Copyright (C) 2002 John Wiley Sons, Ltd. Contact-impact algorithms, which are sometimes called slideline algorithms, are a computationally time-consuming part of many explicit simulations of non-linear problems because they involve many branches, so they are not amenable to vectorization, which is essential for speed on supercomputers. The pinball algorithm is a simplified slideline algorithm which is readily vectorized. Its major idea is to embed pinballs in surface elements and to enforce the impenetrability condition only to pinballs. It can be implemented in either a Lagrange multiplier or penalty method. It is shown that, in any Lagrange multiplier method, no iterations are needed to define the contact surface. Examples of solutions and running times are given. This paper reviews recent results of Oden, Kikuchi, and Song [4] on the use of exterior penalty methods as a basis for finite element approximations of contact problems in linear elasticity. Preface. List of Boxes. Introduction. Lagrangian and Eulerian Finite Elements in One Dimension. Continuum Mechanics. Lagrangian Meshes. Constitutive Models Solution Methods and Stability. Arbitrary Lagrangian Eulerian Formulations. Element Technology. Beams and Shells. Contact--Impact. Appendix 1: Voigt Notation. Appendix 2: Norms. Appendix 3: Element Shape Functions. Glossary. References. Index. The numerical treatment of contact problems involves the formulation of the geometry, the statement of interface laws, the variational formulation and the development of algorithms. In this paper we give an overview with regard to the different topics which are involved when contact problems have to be simulated. To be most general we will derive a geometrical model for contact which is valid for large deformations. Furthermore interface laws will be discussed for the normal and tangential stress components in the contact area. Different variational formulations can be applied to treat the variational inequalities due to contact. Several of these different techniques will be presented. Furthermore the discretization of a contact problem in time and space is of great importance and has to be chosen with regard to the nature of the contact problem. Thus the standard discretization schemes will be discussed as well as techiques to search for contact in case of large deformations. Introduction Signorini's problem Minimization methods and their variants Finite element approximations Orderings, Trace Theorems, Green's Formulas and korn's Inequalities Signorini's problem revisited Signorini's problem for incompressible materials Alternate variational principles for Signorini's problem Contact problems for large deflections of elastic plates Some special contact problems with friction Contact problems with nonclassical friction laws Contact problems involving deformations and nonlinear materials Dynamic friction problems Rolling contact problems Concluding comments.