Data-driven variational multiscale modeling of subgrid-scale effects in discontinuous Galerkin methods

对于基于Galerkin公式的大涡模拟，变分多尺度(VMS)方法为闭合模型的构造提供了严格的数学基础，将解空间分解为粗尺度(有限元)近似和无限维细尺度互补。不连续Galerkin(DG)方法因其稳健性、守恒性和高阶精度而特别适合于流动模拟。然而，由于其不连续的基础和相关的变分通量项，DG方法与迄今建立的VMS框架不兼容。我们最近介绍了一个新的VMS-DG框架，它基于一个特定的VMS精细闭包函数，协调了间断Galerkin方法和变分多尺度方法。每个细尺度闭合函数都是作为一个变分细尺度问题的解出现的，但与已建立的细尺度格林函数相比，格林函数自然地考虑了跨不连续面的贡献，并避免了繁琐的卷积。此外，对于对流扩散方程，我们证明了与连续Galerkin方法不同，DG离散中的细尺度闭包函数具有高度的局部化支持。在这个项目中，我们利用这一基础在DG框架中开发了一种新的数据驱动子网格规模的建模方法。它基于三个中心假设：(1)由于细尺度闭包函数的局部化，DG方法能够将其精确的单元局部模拟作为变分细尺度问题的近似解，也可以用于(不可压缩的)Navier-Stokes方程。(2)通过现代数据驱动的模型降阶技术计算精细闭包解，可以解决在运行时解决大量此类细微规模问题的计算挑战。(3)由于采用了一致的(基于残差的)VMS闭包形式，数据驱动的方法自然地与控制方程(即物理方程)交织在一起，因此可以在大涡模拟中适当地表示次网格尺度的效应。我们的研究计划包括将我们的新的VMS-DG框架扩展到不可压缩的Navier-Stokes方程，在DG背景下推导和研究局部精细闭包函数的两个模拟变量，以及发展非线性Deim耦合约化基方法及其数据驱动的校准，以使它们能够通过非常有效的变分精细问题的解来进行实际计算。通过建立良好的基准问题，验证了数据驱动方法的可行性、计算效率以及通过单元-局部亚格子尺度模式实现的大涡模拟的准确性。

For large eddy simulation based on a Galerkin formulation, the variational multiscale (VMS) method provides a mathematically rigorous basis for the construction of closure models, decomposing the solution space into a coarse-scale (finite element) approximation and an infinite dimensional fine-scale complement. Discontinuous Galerkin (DG) methods are particularly suitable for flow simulations due to their robustness, conservation properties, and higher-order accuracy. DG methods, however, have been incompatible with the VMS framework established to date due to their discontinuous basis and associated variational flux terms. We recently introduced a new VMS-DG framework that reconciles the discontinuous Galerkin approach with the variational multiscale method, based on a specific VMS fine-scale closure function. Each fine-scale closure function emerges as the solution of a variational fine-scale problem, but in contrast to the established fine-scale Green’s function naturally accounts for contributions across discontinuities and avoids tedious convolution. Moreover, we demonstrated for the advection-diffusion equation that unlike in continuous Galerkin methods, fine-scale closure functions in DG discretizations exhibit a highly localized support. In this project, we leverage this foundation to develop a new data-driven subgrid-scale modeling methodology in a DG framework. It is based on three central hypotheses: (1) Due to the localization of fine-scale closure functions, DG methods enable their accurate element-local modeling as approximate solutions of the variational fine-scale problem, also for the (incompressible) Navier-Stokes equations. (2) The computational challenge of solving a very large number of such fine-scale problems during run-time can be tackled by computing fine-scale closure solutions via modern data-driven model order reduction technology. (3) Due to the consistent (residual-based) VMS closure formulation, the data-driven methodology remains naturally intertwined with the governing equations (i.e. the physics) and can thus appropriately represents subgrid-scale effects in large eddy simulations. Our research program involves the extension of our new VMS-DG framework to the incompressible Navier-Stokes equations, the derivation and investigation of two modeling variants for localized fine-scale closure functions in a DG context, and the development of a nonlinear DEIM-coupled reduced basis method and its data-driven calibration to enable their practical computation via extremely efficient solution of variational fine-scale problems. The feasibility of the data-driven approach, its computational efficiency, and the accuracy of large eddy simulations that can be achieved through the element-local subgrid-scale model are tested via well-established benchmark problems.