An adaptive Fixed-Mesh ALE method for free surface flows

Abstract

In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian-Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

An oil sloshing study: adaptive fixed-mesh ALE analysis and comparison with experiments

“…In this section we summarize the key ingredients of the adaptive xed-mesh ALE approach used to solve the free surface problem. A more detailed description of the method can be found in [5].…”

“…Note that the L 2 (Ω) projection is not the only possibility (see [28] for other options of interpolators used in the ghost penalty method). A more detailed description of this stabilization technique can be found in [5]…”

“…As the level set interface evolves in time, it will intersect the elements of the nite element mesh in an arbitrary manner. To properly integrate the physical properties in the computational domain, a modied integration rule is used in the elements cut by the interface [6,5]. The use of the enhanced integration allows to impose the zero traction boundary condition correctly at the exact position of the interface.…”

“…Interface problems, either, free-surface [5,6], a thermally coupled analysis considering phase-change [7,8], or even, solid mechanics problems with stresses location, damage or plasticity [9,10,11], are promising cases where the use of adaptivity is relevant because thin layers of the full domain contains the most important information of the global behavior of the problem. In this work an adaptive xed-mesh ALE method is used to numerically approximate the sloshing of a rectangular tank partially lled with a domestic oil subjected to a controlled movement generated by a shake table.…”

“…This xed-mesh ALE approach was used satisfactorily both in free surface problems [17] as in the numerical simulation of oating solids [18], appearing as a powerful tool for the numerical approximation of moving domains. In [5], the xed-mesh ALE method was coupled with an adaptive mesh algorithm giving place to a highly ecient and robust method. In that work, several numerical aspects were discussed in detail, such as, the global stabilization method used to ensure bounded pressure and to solve convective dominated ows for equal interpolation spaces between velocity and pressure, and additionally, the stability terms designed to stabilize the ill-conditioning introduced by the cuts on the background nite element mesh.…”