A Polynomial Spectral Calculus for Analysis of DG Spectral Element Methods

We introduce a polynomial spectral calculus that follows from the summation by parts property of the Legendre-Gauss-Lobatto quadrature. We use the calculus to simplify the analysis of two multidimensional discontinuous Galerkin spectral element approximations

An Efficient Sliding Mesh Interface Method for High-Order Discontinuous Galerkin Schemes

Sliding meshes are a powerful method to treat deformed domains in computational fluid dynamics, where different parts of the domain are in relative motion. In this paper, we present an efficient implementation of a sliding mesh method into a discontinuous Galerkin compressible Navier-Stokes solver and its application to a large eddy simulation of a 1-1/2 stage turbine. The method is based on the mortar method and is high-order accurate. It can handle three-dimensional sliding mesh interfaces with various interface shapes. For plane interfaces, which are the most common case, conservativity and free-stream preservation are ensured. We put an emphasis on efficient parallel implementation. Our implementation generates little computational and storage overhead. Inter-node communication via MPI in a dynamically changing mesh topology is reduced to a bare minimum by ensuring a priori information about communication partners and data sorting. We provide performance and scaling results showing the capability of the implementation strategy. Apart from analytical validation computations and convergence results, we present a wall-resolved implicit LES of the 1-1/2 stage Aachen turbine test case as a large scale practical application example

A machine learning framework for LES closure terms

In the present work, we explore the capability of artificial neural networks (ANN) to predict the closure terms for large eddy simulations (LES) solely from coarse-scale data. To this end, we derive a consistent framework for LES closure models, with special emphasis laid upon the incorporation of implicit discretization-based filters and numerical approximation errors. We investigate implicit filter types, which are inspired by the solution representation of discontinuous Galerkin and finite volume schemes and mimic the behaviour of the discretization operator, and a global Fourier cutoff filter as a representative of a typical explicit LES filter. Within the perfect LES framework, we compute the exact closure terms for the different LES filter functions from direct numerical simulation results of decaying homogeneous isotropic turbulence. Multiple ANN with a multilayer perceptron (MLP) or a gated recurrent unit (GRU) architecture are trained to predict the computed closure terms solely from coarse-scale input data. For the given application, the GRU architecture clearly outperforms the MLP networks in terms of accuracy, whilst reaching up to 99.9% cross-correlation between the networks' predictions and the exact closure terms for all considered filter functions. The GRU networks are also shown to generalize well across different LES filters and resolutions. The present study can thus be seen as a starting point for the investigation of data-based modeling approaches for LES, which not only include the physical closure terms, but account for the discretization effects in implicitly filtered LES as well