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 value for 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 @math , in order to generate the discrete action. Then, the step (b) yields a variational time integration scheme that reduces to a known explicit one if @math 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 consider artificial viscous as well as boundary interaction effects and simulate a dam breaking set up. We compare the explicit and implicit SPH solutions and analyze momentum conservation of the dam breaking simulations.
The key elements in variational time integrators are the discrete action sum, the discrete Euler-Lagrange equations and the discrete Noether's theorem that were clearly understood due to early works (see @cite_9 and references therein). Numerical aspects and convergence properties were specifically considered in @cite_16 @cite_12 .
@cite_9 @cite_16 @cite_12
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The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speed-ups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact path-independent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the J-integral at the tip of a crack in a finite element mesh. The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior. This analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d'Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run. This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented. We present a technique to implement scalable variational integrators for generic mechanical systems in generalized coordinates. Systems are represented by a tree-based structure that provides efficient means to algorithmically calculate values (position, velocities, and derivatives) needed for variational integration without the need to resort to explicit equations of motion. The variational integrator handles closed kinematic chains, holonomic constraints, dissipation, and external forcing without modification. To avoid the full equations of motion, this method uses recursive equations, and caches calculated values, to scale to large systems by the use of generalized coordinates. An example of a closed-kinematic-chain system is included along with a comparison with the open-dynamics engine (ODE) to illustrate the scalability and desirable energetic properties of the technique. A second example demonstrates an application to an actuated mechanical system.