High-order Discontinuous Galerkin Simulations on Moving Domains using ALE Formulations and Local Remeshing and Projections

Abstract

High-order methods such as the Discontinuous Galerkin (DG) method are considered promising alternatives for the numerical simulation of turbulent flows with complex vortical structures and non-linear interactions. Being based on high-order discretization of conservation laws, the DG methods are able to obtain stable solutions with low numerical dissipation for compressible flow problems. Since the schemes can easily be implemented on fully unstructured meshes, DG methods are well-suited for problems on complex geometries, which is an essential requirement for real-world applications. Many practical problems involve time-varying geometries, such as rotor-stator flows, flapping flight or fluid-structure interactions. For these deforming domain problems, a number of solutions have been proposed such as embedded domain methods, 5 space-time methods 8 and the Arbitrary Lagrangian-Eulerian (ALE) method. The popular ALE method can be viewed as a change of variable using a smooth mapping from a fixed reference domain to the moving physical domain, which allows the mesh to change in time and leads to a set of modified equations in the reference domain. The approach is efficient and easy to implement, and is one of the most widely used techniques in particular for CFD problems discretized using high-order accurate methods. However, a limitation with the ALE method is that in order for the deformation mapping to be smooth, the mesh topology must be fixed which manes that the initial element connectivities have to be kept unchanged throughout the time evolution. This restriction can be severe for large or complex deformations, where remeshing is required to maintain well-shaped elements. In order to transfer the solutions between the original and the recomputed mesh, careful treatment is needed to obtain accuracy and stability. Many interpolation techniques have been proposed, including standard L projections, but in general the accuracy is significantly reduced when frequent remeshing is employed. The projections are straight-forward to formulate and have many desirable properties, but the implementation is complicated and costly for high dimensional unstructured meshes, which limits their practical applications. In this work, we propose a simple combined approach for solving deforming domain problems with large deformation with high-order accuracy. The method is based on nodal discontinuous Galerkin formulation and an arbitrary Lagrangian-Eulerian framework. The mesh adjustment during the domain motion follows a spring-based technique with local element flipping. Since all the topological changes are local, the corresponding L projections are also local and can be precomputed and easily applied in both 2D and 3D. In this paper, we first demonstrate our framework on a 2D model problem of an inviscid Euler vortex, where we show that the scheme remains high-order accurate for complex mesh reconfigurations and frequent edge connectivity changes. We also present a 2D laminar flow problem to show the ability of our method to deal with complex domain motions. Finally, we carry out a convergence test in 3D based on a similar Euler-vortex model problem, which again demonstrates the high-order accuracy of the method.