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Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

机译：不可压缩流ALE模拟的几何守恒律和有限元方法透视

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摘要

This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed mesh accuracy when applied to deforming meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming meshes and preserve its fixed mesh high order temporal accuracy. An appropriate construction of the divergence of the mesh velocity guarantees GCL compliance while a separate construction of the mesh velocity itself allows the time-integrator to deliver its fixed mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier-Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming mesh. In all cases studied, the proposed reconstructions of the mesh velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids. (C) 2008 Elsevier Inc. All rights reserved.
机译：本文从不可压缩流的有限元方法（FEM）的角度重新审视了几何守恒律（GCL）。 GCL在解决变形域问题的任意拉格朗日欧拉（ALE）公式的背景下自然产生。传统上，将GCL遵从性解释为用于应用非定常流动求解算法来精确模拟变形域上均匀流动的一致性标准。我们引入了一个附加要求：时间积分器在应用于变形网格时必须保持其固定的网格精度。文献回顾表明，尽管许多作者使用ALE FEM，但很少有人讨论GCL问题。我们展示了如何使用高阶时间积分器（最高时间为五阶）对固定网格非定常有限元进行调换，以解决变形网格问题并保持其固定网格的高阶时间精度。网格速度散度的适当构造可确保符合GCL，而网格速度本身的单独构造可让时间积分器在变形域上传递其固定的网格高阶时间精度。封闭式解决方案中的问题的分析误差分析提供了时间积分器行为的洞察力。这也解释了为什么使用不可压缩的Navier-Stokes方程的保守公式可以达到高阶时间精度，而此处使用非保守公式和所有时间积分器只能观察到一阶时间精度。我们针对封闭式解决方案中的简单问题以及变形网格上非平凡流动的制造商解决方案，提供了详尽的时间步和网格细化研究。在所研究的所有情况下，为保守的公式提出的网格速度及其散度的拟议重构都可以在变形网格上实现最佳时间精度。 （C）2008 Elsevier Inc.保留所有权利。

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