Rasmus Tamstorf

Etienne Vouga, David Harmon, Rasmus Tamstorf et al.

An asynchronous, variational method for simulating elastica in complex contact and impact scenarios is developed. Asynchronous Variational Integrators (AVIs) are extended to handle contact forces by associating different time steps to forces instead of to spatial elements. By discretizing a barrier potential by an infinite 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.

Stephen F. McCormick, Rasmus Tamstorf

The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization formulation of symmetric positive definite linear equations arising from the discretization of elliptic partial differential equations. This focus provides an alternate viewpoint to other expositions and, more importantly, it enables a simplification of the development while clarifying the principles that lead to effective algorithms. The development begins with multigrid as an iterative solver exemplified by the so-called V-cycle. It then introduces the full multigrid method as a direct solver in the sense that it is aimed directly at the source of the matrix equations. In this way, full multigrid attempts to achieve discretization-level accuracy at a cost comparable to that of a few matrix multiplies on the finest level.

Etienne Vouga, David Harmon, Rasmus Tamstorf et al.

Asynchronous Variational Integrators (AVIs) have demonstrated long-time good energy behavior. It was previously conjectured that this remarkable property is due to their geometric nature: they preserve a discrete multisymplectic form. Previous proofs of AVIs' multisymplecticity assume that the potentials are of an elastic type, i.e., specified by volume integration over the material domain, an assumption violated by interaction-type potentials, such as penalty forces used to model mechanical contact. We extend the proof of AVI multisymplecticity, showing that AVIs remain multisymplectic under relaxed assumptions on the type of potential. The extended theory thus accommodates the simulation of mechanical contact in elastica (such as thin shells) and multibody systems (such as granular materials) with no drift of conserved quantities (energy, momentum) over long run times, using the algorithms in [3]. We present data from a numerical experiment measuring the long time energy behavior of simulated contact, comparing the method built on multisymplectic integration of interaction potentials to recently proposed methods for thin shell contact.