Variational Time Integrators Research Articles

A class of structure-preserving discontinuous Galerkin variational time integrators for Birkhoffian systems

Accurate time integrators that preserving Birkhoffian structure are of great practical use for Birkhoffian systems. In this paper, a class of structure-preserving discontinuous Galerkin variational integrators (DGVIs) is presented. Start from the Pfaff action functional, the technique of variational integrators combined with discontinuous Galerkin time discretization is used to derive numerical schemes for Birkhoffian systems. For the derived DGVIs, symplecticity is proved rigorously through the preserving of particular 2-forms induced by these integrators. Linear stability and order of accuracy of the DGVIs are illustrated considering the example of linear damped oscillators. The order of accuracy and the property of preserving conserved quantities of the developed DGVIs are also confirmed by numerical examples. Comparisons are made with several numerical schemes such as backward/forward Euler, Runge–Kutta and RBF methods to show the advantages of DGVIs in preserving the Birkhoffians.

Fixed point implementation of a variational time integrator approach for smoothed particle hydrodynamics simulation of fluids

Variational time integrators are derived in the context of discrete mechanical systems. In this area, the governing equations for the motion of the mechanical system are built following two steps: (a) Postulating a discrete action; (b) Computing the stationary point of the discrete action. The former is formulated by considering Lagrangian (or Hamiltonian) systems with the discrete action being constructed through numerical approximations of the action integral. The latter derives the discrete Euler–Lagrange equations whose solutions give the variational time integrator. In this paper, we build variational time integrators in the context of smoothed particle hydrodynamics (SPH). So, we start with a variational formulation of SPH for fluids. Then, we apply the generalized midpoint rule, which depends on a parameter α, in order to generate the discrete action. Then, the step (b) yields a variational time integration scheme that reduces to an explicit approach if α∈{0,1} but it is implicit otherwise. Hence, we design a fixed point iterative method to approximate the solution and prove its convergence condition. Besides, we show that the obtained discrete Euler–Lagrange equations preserve linear momentum. In the experimental results, we simulate a bubble flow and a dam breaking set up and consider viscosity as well as boundary interaction effects. We compare standard and implicit SPH solutions. We analyze linear momentum conservation and other benchmark quantities to conclude that the proposed algorithm is accurate and preserves the linear momentum better than the counterpart one for dam breaking set up.

On variational and symplectic time integrators for Hamiltonian systems

Various systems in nature have a Hamiltonian structure and therefore accurate time integrators for those systems are of great practical use. In this paper, a finite element method will be explored to derive symplectic time stepping schemes for (non-)autonomous systems in a systematic way. The technique used is a variational discontinuous Galerkin finite element method in time. This approach provides a unified framework to derive known and new symplectic time integrators. An extended analysis for the new time integrators will be provided. The analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties. These new time stepping schemes are necessary to get accurate and stable simulations of (forced) water waves and other non-autonomous variational systems, which we illustrate in our numerical results.

Discontinuous variational time integrators for complex multibody collisions

SUMMARYThe objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non‐classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one‐sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one‐sided constraints, whereas the corrector configuration is obtained by a closest‐point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non‐classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics. Copyright © 2014 John Wiley & Sons, Ltd.

Hamiltonian Dynamics of Several Rigid Bodies Interacting with Point Vortices

We derive the dynamics of several rigid bodies of arbitrary shape in a 2-dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We adopt the idea of Vankerschaver et al. (2009) to derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system. The reduced system is described by a non-canonical symplectic form, which has previously been derived for a single, circular disk using heavy differential-geometric machinery in an infinite-dimensional setting. In contrast, our derivation makes use of the fact that the dynamics of the fluid, and thus the point vortex dynamics, is determined from first principles. Using this knowledge we can directly determine the dynamics on the reduced, finite-dimensional phase space, using only classical mechanics. Furthermore, our approach easily handles several bodies of arbitrary shapes. From the Hamiltonian description we derive a Lagrangian formulation, which enables the system for variational time integrators. We briefly describe how to implement such a numerical scheme and simulate different configurations for validation.

Nonlinear manifold learning for model reduction in finite elastodynamics

Model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as Principal Components Analysis (PCA). Hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold, we explore here reduced order modeling based on nonlinear dimensionality reduction methods. Such methods are gaining popularity in diverse fields of science and technology, such as machine perception or molecular simulation. We consider finite deformation elastodynamics as a model problem, and identify the manifold where the dynamics essentially take place – the slow manifold – by nonlinear dimensionality reduction methods applied to a database of snapshots. Contrary to linear dimensionality reduction, the smooth parametrization of the slow manifold needs special techniques, and we use local maximum entropy approximants. We then formulate the Lagrangian mechanics on these data-based generalized coordinates, and develop variational time-integrators. Our proof-of-concept example shows that a few nonlinear collective variables provide similar accuracy to tens of PCA modes, suggesting that the proposed method may be very attractive in control or optimization applications. Furthermore, the reduced number of variables brings insight into the mechanics of the system under scrutiny. Our simulations also highlight the need of modeling the net effect of the disregarded degrees of freedom on the reduced dynamics at long times.

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

We present the asynchronous multi-domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary-value problem in hyperelasticity. Variational time integration schemes, based on the principle of minimal action within the Lagrangian framework, are constructed for the equation of motion and implemented into a variational finite element framework, which is systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly-incompressible materials. For efficient parallel computing, we use the dual domain decomposition method with local Lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains. The α-method for time discretization and the multi-domain spatial decomposition enable us to use different types of integrators (explicit vs. implicit) and different time steps on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. The energy conservation of our nonlinear, midpoint, asynchronous integration scheme is investigated using the Energy method, and both local and the global energy error estimates are derived. We illustrate the performance of proposed variational multi-domain time integrators by means of three examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we investigate energy conservation and stability. Finally, we apply the method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly with disparate material data of individual parts.