非结构化网格的静态运输问题的动态变分多尺度方法,International Journal for Numerical Methods in Fluids

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非结构化网格的静态运输问题的动态变分多尺度方法
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2023-02-07 , DOI: 10.1002/fld.5184
Zelu Xu ₁ , Onkar Sahni ₁

Affiliation

本文介绍了一种基于变分多尺度 (VMS) 的有限元方法，其中动态计算稳定参数。当前的动态过程采用具有未知系数的稳定参数的一般结构/形式，并以局部方式动态计算它们，从而产生基于动态 VMS 的有限元方法。因此，不需要具有预定义系数的静态稳定参数。开发了适用于非结构化网格的基于变分 Germano 恒等式 (VGI) 的本地过程，以本地方式执行动态计算。基于局部 VGI 的过程应用于网格中的每个内部顶点，并且首先在每个顶点处局部确定未知系数，随后，对于每个元素，取元素的顶点的最大值。为了使当前过程实用，从初级粗尺度解决方案构建了一个较粗略的二级解决方案，这是在每个内部顶点周围的一片元素上局部完成的。此外，采用平均步骤使局部动态过程具有鲁棒性。目前，新的动态 VMS 公式应用于一维和二维中由平流扩散和不可压缩纳维-斯托克斯方程控制的稳态问题，以证明其有效性。

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A dynamic variational multiscale method on unstructured meshes for stationary transport problems

This paper presents a variational multiscale (VMS) based finite element method where the stabilization parameter is computed dynamically. The current dynamic procedure takes in a general structure/form of the stabilization parameter with unknown coefficients and computes them dynamically in a local fashion resulting in a dynamic VMS-based finite element method. Thus, a static stabilization parameter with pre-defined coefficients is not needed. A variational Germano identity (VGI) based local procedure suitable for unstructured meshes is developed to perform the dynamic computation in a local fashion. The local VGI based procedure is applied for each interior vertex in the mesh and unknown coefficients are first determined locally at each vertex, and subsequently, for each element a maximum value is taken over the vertices of the element. To make the current procedure practical, a coarser secondary solution is constructed from the primary coarse-scale solution, which is done locally over a patch of elements around each interior vertex. Further, averaging steps are employed to make the local dynamic procedure robust. Currently, the new dynamic VMS formulation is applied to steady problems governed by the advection-diffusion and incompressible Navier-Stokes equations in both 1D and 2D to demonstrate its efficacy and effectiveness.

更新日期：2023-02-07

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