An overset mesh approach for 3D mixed element high-order discretizations

Michael J. Brazell, Jayanarayanan Sitaraman, Dimitri J. Mavriplis
2016 Journal of Computational Physics  
7 A parallel high-order Discontinuous Galerkin (DG) method is used to solve the compressible Navier-Stokes equations in an overset mesh framework. The DG solver has many capabilities including: hp-adaption, curved cells, support for hybrid, mixed-element meshes, and moving meshes. Combining these capabilities with overset grids allows the DG solver to be used in problems with bodies in relative motion and in a near-body off-body solver strategy. The overset implementation is constructed to
more » ... rve the design accuracy of the baseline DG discretization. Multiple simulations are carried out to validate the accuracy and performance of the overset DG solver. These simulations demonstrate the capability of the high-order DG solver to handle complex geometry and large scale parallel simulations in an overset framework. The goal of this work is to devise an accurate, efficient and robust three-15 dimensional high-order method based on DG discretizations [23] for simu-16 lating a wide variety of compressible flows in an overset grid framework. 17 Although the DG solver can handle unstructured meshes there are some sit-18 uations where even an unstructured solver could benefit from an overset grid 19 framework. Bodies in relative motion such as helicopters or wind turbines are 20 difficult to simulate with a single grid. Typically these simulations require 21 mesh movement or re-meshing every time step. Overset grids solve this issue 22 by allowing multiple grids to move relative to each other. Another advantage 23 to overset grids is the ability to combine a near-body unstructured DG solver 24 with an efficient cartesian mesh off-body solver, this can greatly increase the 25 overall efficiency and capabilities of the solver. This has been demonstrated 26 with success in [24], which combines a second-order accurate, unstructured, 27 finite-volume solver [25] with an AMR finite difference solver [26, 27]. Also, 28 previous work by our group combined a hexahedral DG solver with a second-29 order, finite-volume, unstructured solver [28, 29]. These approaches have two 30 drawbacks: firstly they depend on low-order methods for the unstructured 31 solvers and secondly many fringe cells are needed in the overlap region of the 32 grids to maintain accuracy. 33 To overcome these issues high-order methods with compact stencils can 34 be used. Recently, DG has been used in an overset framework to solve the 35 two-dimensional, Euler equations on abutting and overset grids [30]. Flux Re-36 construction (FR) has been used to solve the two-dimensional, compressible 37 Navier-Stokes equations on sliding meshes [31]. Streamline/upwind Petrov-38 Galerkin (SUPG) has been used to solve the two-dimensional, compressible, 39 Navier-Stokes equations along with the Reynolds Averaged Navier-Stokes 40 (RANS) equations on steady and unsteady moving mesh problems [32]. Spec-41 tral Element Method (SEM) has been used to solve the three-dimensional, 42 incompressible Navier-Stokes equations on moving, overlapping, hexahedral 43 grids [33]. This has demonstrated the feasibility and high level of accuracy 44 that can be obtained using compact-stencil, high-order methods (DG, FR, 45 SUPG, and SEM) in an overset framework. 46 In this paper, our previous work has been extended to the new DG solver 47 using a newly developed high-order overset mesh framework called TIOGA 48 (Topology Independent Overset Grid Assembler). The high-order, parallel 49 DG solver [23] is capable of solving the compressible Navier-Stokes equations 50 and the RANS equations closed by the Spalart-Allmaras turbulence model 51 (negative-SA variant) [34, 35]. It also can handle hybrid, mixed-element 52 2 meshes (tetrahedra, pyramids, prisms, and hexahedra), moving meshes, curved 53 elements, and incorporates both p-enrichment and h-refinement capabili-54 ties using non-conforming elements (hanging nodes). The novelty in this 55 approach is the capability of using a 3D high-order method with mixed-56 elements, moving mesh, and turbulent flow in an overset framework [36] . 57 This gives the ability to solve complicated relative motion problems at high-58 order and be combined with an efficient off-body solver. 59 In the following sections, the governing equations are described, fol-60 lowed by the DG discretization and its implementation for three-dimensional 61 problems. The solution methodology is described next which discusses the 62 strongly coupled, implicit, non-linear solver within the overset framework. 63 Lastly, a variety of simulations are used to demonstrate the order of ac-64 curacy, temporal accuracy, implicit solver performance, and moving mesh 65 accuracy of the overset solver. This is followed by two large scale simulations 66 demonstrating the use of mixed element meshes and turbulent flow over a 67 wing. 68 P α,β m−2 (ξ) m > 1 4 where α and β are parameters that give Jacobi polynomials their orthogo-79 nality properties with respect to the weight (1 − ξ) α (1 + ξ) β . Similar to the 80 segment a basis is created for tetrahedral, pyramidical, prismatic, and hexa-81 hedral element types. Also, the basis is used to represent both the solution 82 and geometrical mapping. The polynomial degree can be chosen indepen-83 dently for each, for example the solution can be represented by a p = 1 basis 84 and the geometrical mapping can be represented by a p = 2 basis. 85 The residual consists of volume and face integrals which are approximated 86 using numerical quadrature. For hexahedral elements and quadrilateral ele-87 ments tensor products of Gaussian quadrature points are used. For the other 88 element types a set of efficient numerical quadrature rules are used [37]. The 89 quadrature rules are chosen to exactly integrate polynomials within the vol-90 ume integrals to 2p accuracy and the face integrals to 2p + 1 accuracy. The 91 integrals over faces Γ require special treatment for the fluxes in these terms. 92 The advective fluxes are calculated using a Riemann solver. Implemented 93 Riemann solvers include: Lax-Friedrichs [38], Roe [39], and artificially up-94 stream flux vector splitting scheme (AUFS) [40]. In this work, only the Lax-95 Friedrichs flux is used. The diffusive fluxes are handled using a symmetric 96 interior penalty (SIP) method [41-43].
doi:10.1016/j.jcp.2016.06.031 fatcat:rzwfvd4eezhzje72cipv4jbdfa