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.
Seeking to exactly enforce non-penetration along the contact surface leads to generalizations of the method of Lagrange multipliers. Hughes al @cite_9 and Nour-Omid and Wriggers @cite_42 provide an overview of this approach in the context of contact response. Such contraint enforcement can be viewed as a penalty force in the limit of infinite stiffness, impossible to attain in practice since the system becomes ill-conditioned. Taylor and Papadopoulos @cite_32 considers persistent contact by extending Newmark to treat jump conditions in kinematic fields, thus reducing undesirable oscillatory modes. However, the effects of these modifications on numerical dissipation and long-time energy behavior is not considered.
@cite_9 @cite_42 @cite_32
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Abstract We present a finite element method for a class of contact-impact problems. Theoretical background and numerical implementation features are discussed. In particular, we consider the basic ideas of contact-impact, the assumptions which define the class of problems we deal with, spatial and temporal discretizations of the bodies involved, special problems concerning the contact of bodies of different dimensions, discrete impact and release conditions, and solution of the nonlinear algebraic problem. Several sample problems are presented which demonstrate the accuracy and versatility of the algorithm. Abstract The merits and limitations of some existing procedures for the solution of contact problems, modeled by the finite element method, are examined. Based on the Lagrangian multiplier method, a partitioning scheme can be used to obtain a small system of equation for the Lagrange multipliers which is then solved by the conjugate gradient method. A two-level contact algorithm is employed which first linearizes the nonlinear contact problem to obtain a linear contact problem that is in turn solved by the Newton method. The performance of the algorithm compared to some existing procedures is demonstrated on some test problems. This paper addresses the formulation and discrete approximation of dynamic contact impact initial-value problems. The continuous problem is presented in the context of non-linear kinematics. Standard semi-discrete time integrators are introduced and are shown to be unsuccessful in modelling the kinematic constraints imposed on the interacting bodies during persistent contact. A procedure that bypasses the aforementioned difficulty is proposed by means of a novel variational formulation. Numerical simulations are conducted and the results are reported and discussed.