Asynchronous Variational Integrators

Abstract

We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton's variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly.To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated.Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.

A scalable architecture for ordered parallelism

“…First, they are common in graph analytics, especially in search problems [33,55]. Second, they are important in simulating systems whose state evolves over time, such as circuits [47], computers [12,59], networks [37,72], healthcare systems [39], and systems of partial differential equations [32,44]. Third, they are needed in systems that must maintain externally-imposed order constraints, such as geo-replicated databases where transactions must appear to execute in timestamp order [14], or deterministic architectures [17,45] and record-andreplay systems [36,77] that constrain the schedule of parallel programs to ensure deterministic execution.…”

Structure-preserving model reduction for mechanical systems

“…This approach gives rise to structure-preserving integration algorithms. In fact, the ideas of structure preservation used in this paper are also tied to the numerical techniques we employ; we make use of the Newmark integrator, which has recently been shown to be a variational integrator by Kane et al [13]; see also [20] for an asynchronous generalization to the PDE context and with applications to nonlinear elastodynamic simulations. As such, the model reduction gives rise to a time-discretized evolution equation which exactly preserves momentum and the symplectic form, and approximately preserves energy.…”

“…It would be quite interesting to put the theory here into a more hierarchical and adaptive context by making use of the recent developments of PDE asynchronous, multisymplectic integrators given in [20] and the CHARMs (conforming hierarchical adaptive refinement methods) methodology given in [15,16]. This would hopefully lead to a SPAHMR (structure-preserving adaptive hierarchical model reduction; pronounced "Spammer") methodology that would also automatically deal with any symmetries that are present in a given problem.…”

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

“…First, we quickly review, from Marsden et al [82] and Lew et al [80,81], some facts about multisymplectic variational integrators for smooth unconstrained problems. Then, we develop our approach for two different classes of constrained problems, using the generalized Lagrange multiplier approach.…”

“…, A}. Discrete (nonzero) traction boundary conditions can be easily handled by considering a slight modification of the discrete variational principle, as explained in Lew et al [80].…”