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An immersed boundary method for fluids using the XFEM and the hydrodynamic Boltzmann transport equation

机译：使用XFEM和流体动力学Boltzmann输运方程的流体浸入边界方法

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摘要

This paper presents a stabilized finite element formulation of the hydrodynamic Boltzmann transport equation (HBTE) to predict nearly incompressible fluid flow. The HBTE is discretized with Hermite polynomials in the velocity variable, and a streamline upwind Petrov-Galerkin formulation is used to discretize the spatial variable. A nonlinear stabilization scheme is presented, from which a simple linear stabilization scheme is constructed. In contrast to the Navier-Stokes (NS) equations, the HBTE is a first order equation and allows for conveniently enforcing Dirichlet conditions along immersed boundaries. A simple and efficient formulation for enforcing Dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method (XFEM). Numerical experiments indicate that both the linear and non-linear stabilization methods are sufficiently accurate and stable, but the linear formulation reduces the computational cost significantly. The accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined. Augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints, but may degrade the accuracy of the global solution. Comparisons with results of a single relaxation time lattice Boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement.
机译：本文介绍了流体动力学玻尔兹曼输运方程（HBTE）的稳定有限元公式，以预测几乎不可压缩的流体流动。 HBTE在速度变量中使用Hermite多项式离散化，并且使用顺风向上的Petrov-Galerkin公式离散化空间变量。提出了一种非线性稳定方案，从中构造了一个简单的线性稳定方案。与Navier-Stokes（NS）方程相反，HBTE是一阶方程，可以方便地沿浸入边界强制执行Dirichlet条件。提出了一种简单有效的方法来执行Dirichlet边界条件，并研究了其精度的扩展有限元方法（XFEM）捕获的沉浸边界。数值实验表明，线性和非线性稳定方法都足够准确和稳定，但是线性公式大大降低了计算成本。强制执行边界条件的精度令人满意，并且随着网格的细化显示出二阶收敛性。用惩罚项扩展边界条件公式可提高执行边界条件约束的准确性，但可能会降低整体解的精度。与单张弛豫时间晶格Boltzmann方法的结果比较表明，所提出的有限元方法具有更高的鲁棒性，并且计算成本对网格细化程度的依赖性较小。

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