An isoparametric approach to high-order curvilinear boundary-layer meshing

This is the final version of the article. Available from Elsevier via the DOI in this record.The generation of high-order curvilinear meshes for complex three-dimensional geometries is presently a challenging topic, particularly for meshes used in simulations at high Reynolds numbers where a thin boundary layer exists near walls and elements are highly stretched in the direction normal to flow. In this paper, we present a conceptually simple but very effective and modular method to address this issue. We propose an isoparametric approach, whereby a mesh containing a valid coarse discretization comprising of high-order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary-layer mesh. The validity of the prismatic mesh provides a suitable mapping that allows one to obtain very fine mesh resolutions across the thickness of the boundary layer. We describe the method in detail for a high-order approximation using modal basis functions, discuss the requirements for the splitting method to produce valid prismatic and tetrahedral meshes and provide a sufficient criterion of validity in both cases. By considering two complex aeronautical configurations, we demonstrate how highly stretched meshes with sufficient resolution within the laminar sublayer can be generated to enable the simulation of flows with Reynolds numbers of 106 and above.This work was partly supported by EU Grant No. 265780 as part of the EU FP7 project “IDIHOM: Industrialization of High-Order Methods — A Top-Down Approach”. We would like to thank Dr. Tobias Leicht of DLR for asking a very pertinent question concerning the validity of the generated high-order mesh that we believe to have answered in this article. We also thank Jean-Eloi Lombard for his assistance in generating the mesh for Fig. 15

On the Impact of Triangle Shapes for Boundary Layer Problems Using High-Order Finite Element Discretization

The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. Large angles are also found to be not problematic for the conditioning of the linear systems arising from the discretizations. Further, when choosing preconditioning strategies, coupling strengths among elements rather than element angle sizes should be taken into account. With an appropriate preconditioner, solutions on meshes with and without large angles can be achieved within a comparable time

Issures in Discontinuous High-Order Methods: Broadband Wave Computation and Viscous Boundary Layer Resolution

A new discontinuous formulation named Correction Procedure via Reconstruction (CPR) was developed for conservation laws. CPR is an efficient nodal differential formulation unifying the discontinuous Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods, is easy to implement. In this thesis, we deal with two issues: the efficient computation of broadband waves, and the proper resolution of a viscous boundary layer with the high-order CPR method. A hybrid discontinuous space including polynomial and Fourier bases is employed in the CPR formulation in order to compute broad-band waves. The polynomial bases are used to achieve a certain order of accuracy, while the Fourier bases are able to exactly resolve waves at a certain frequency. Free-parameters introduced in the Fourier bases are optimized in order to minimize both dispersion and dissipation errors by mimicking the dispersion-relation-preserving (DRP) method for a one-dimensional wave problem. For the one-dimensional wave problem, the dispersion and dissipation properties and the optimization procedure are investigated through a wave propagation analysis. The optimization procedure is verified with a wave propagation analysis and several numerical tests. The two-dimensional wave behavior is investigated through a wave propagation analysis and the wave propagation properties are verified with a numerical test of the two-dimensional acoustic wave equation. In order to understand the mesh size requirement to resolve a viscous boundary layer using CPR method, grid resolution studies are performed. . It is well known that the mesh size, which is defined from non-dimensional wall distance y^+=1, gives accepted results to simulate viscous boundary layer problem for 2nd order finite volume method. For high-order CPR formulation, what grid size y^+ is required to match the results from the 2nd order finite volume method with y^+=1. 1D and 2D burger\u27s equation are used as the viscous boundary layer model problem. Skin friction is used as the indicator of accuracy for the resolution of a boundary layer. Keywords: (Correction Procedure via Reconstruction), A Hybrid Discontinuous Space, Wave Propagation Analysis, Grid Resolution Study, Method of Manufactured Solution