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

Abstract

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.