In this paper, we present recent results in numeri cal simulation of free-surface flows using the paral lel finite element method. In our approach, the gov erning equations are the Navier-Stokes equations written for two incompressible fluids. We solve these equations over a non-moving mesh. An inter face function with two distinct values serves as a marker identifying the location of the free-surface. This function is transported throughout the com putational domain with a time-dependent advec tion equation. The stabilized finite element method is used to discretize the governing equations. The finite element formulations are implemented in parallel using Message Passing Interface libraries. High performance computing tools and optimized techniques are utilized to solve applications on un structured meshes with more than one billion tetra hedral elements. Numerical examples include the simulation of sloshing in tanker trucks and ships hydrodynamics.
Corollary 1: Consider the system given by (2), (4), (26), and (27) were Y , , is defined by (25), A = A I where A > 0 is a constant, and r is assumed to be constant, symmetric, and positive definite. Then the equilibrium x = 0 is stable in the sense of Lyapunov. Proof By combining (4), (25), and (27) the closed-loop dynamic system H s + (AH + C) s = Y,,,a (28) is obtained. The system given by (26), (2), and (28) is identical to the system (1)-(3) with Y , , , replacing Y and AH replacing K u. Then since the inertia matrix H is uniformly positive definite and A > 0 is a constant, Proposition I applies, and the result follows. Remark 2: Lyapunov stability for the system in Corollary I cannot be shown with the proof in [2] since A'H is not constant in general (see Remark I). In Proposition 1 less restrictive assumptions are made, and Lyapunov stability can be established.
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