text: 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.
  @cite_14: 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.
  @cite_24: 
  @cite_23: 
  @cite_16: 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.
  @cite_20: This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler–Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.
  @cite_11: Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.
summary: Variational integrators in mechanical systems start by considering mechanics from a variational point of view, following remarkable works of Lagrange and Hamilton @cite_24 . The Hamilton's principle or the least action principle allows to cast the Newton's framework into a geometric viewpoint in which the path followed by the physical system in the configuration space has optimal geometric properties analogously to the notion of geodesics on curved surfaces @cite_20 . Therefore, we can design numerical integrators that exploit the geometric structure behind mechanical systems, which are named geometric integrators @cite_11 @cite_20 . A special class of geometric integrators, called variational integrators, discretized the variational formulation of mechanics generating iterative schemes to compute an approximation for the path of the physical system with any order of accuracy. Besides, this discrete geometric framework can handle constraints, external and dissipative forces making variational integrators both versatile and powerful @cite_16 @cite_23 @cite_14 .