Michael J. Brazell

Solving Multilinear Systems via Tensor Inversion

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree

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

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3D Mixed Element Discontinuous Galerkin with Shock Capturing

. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.

A Multi-Solver Overset Mesh Approach for 3D Mixed Element Variable Order Discretizations

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 preserve 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.

A high-order discontinuous-Galerkin octree-based AMR solver for overset simulations

A parallel high-order Discontinuous Galerkin method is developed for mixed elements to solve the Navier-Stokes equations. A PDE-based artificial viscosity is implemented to smooth and stabilize shocks. To solve this system of non-linear equations a Newton solver is implemented and preconditioned flexible-GMRES is used to solve the linear system arising from the Jacobian matrix. The preconditioners that are implemented include Jacobi relaxation, Gauss-Seidel relaxation, and a line solver. A wide variety of simulations are performed to demonstrate the capabilities of the DG solver. The inviscid simulations include a p-adapted subsonic flow over a cylinder, a p = 0 h-adapted hypersonic flow over a sphere, and a large scale p = 2 simulation of an aircraft with artificial viscosity to stabilize the shock formed on the wing. Two hypersonic viscous flows of a cylinder and sphere are simulated and compared to the NASA code LAURA. The solution matches closely to LAURA and the shock becomes more resolved as the polynomial degree is increased. The heating rate on the surface matches closely to LAURA at p = 3. Finally, the parallel scalability is tested and good speed up is obtained using up to 2048 processor cores. As the polynomial degree increases the scalability improves. Although, an ideal speedup was not shown this was contributed to load balancing. These simulations demonstrate the capability of the DG solver to handle strong shocks, complex geometry, hp-adaption, and parallel scalability.

High-Order Discontinuous Galerkin Mesh Resolved Turbulent Flow Simulations of a NACA 0012 Airfoil (Invited)

The goal of this work is the development and demonstration of an overset mesh approach which enables arbitrary combinations of traditional second-order accurate finite-volume discretizations and variable high-order accurate discontinuous Galerkin discretizations on structured, unstructured and Cartesian meshes. The combinations of different solvers is enabled through the development of a flexible driver routine coupled with a topology independent overset grid assembly module. The approach is designed to support dynamic overset mesh problems, as well as applications with dynamic adaptive mesh refinement. The developed approach is demonstrated for sample problems running in parallel using combinations of unstructured finite-volume, unstructured discontinuous Galerkin, and Cartesian discontinuous Galerkin solvers including dynamically adaptive meshes.

Using LES in a Discontinuous Galerkin method with constant and dynamic SGS models

The goal of this work is to develop a highly efficient off-body solver for use in overset simulations. Overset meshes have been gaining traction in recent years and are being used increasingly to simulate very complex large-scale problems. In particular we focus on a dual-mesh, dual-solver overset approach that combines specialized flow solvers in different regions of the flow domain: near-body and off-body. The near-body flow solver is designed to handle complicated geometry, anisotropic elements, and unstructured meshes. In contrast, the off-body solver is designed to be high-order, Cartesian, and use adaptive mesh refinement (AMR). The high-order discretization used for the off-body solver is based on the discontinuous Galerkin (DG) method. To get the most efficiency out of the method, a Cartesian grid is employed and tensor product basis functions are used in the DG formulation. The dense computational kernels allow this solver to obtain a near-constant cost per degree of freedom for a wide range of p-orders of accuracy. To further enhance the capabilities of the off-body solver, the DG solver in linked to an octree-based AMR library called p4est; this gives the ability for h-adaptation via non-conforming elements. p-adaption is also implemented in which each cell can be assigned a different polynomial degree basis. Combined h and p refinement is necessary for overlapping mesh problems where the off-body solver mesh must connect to the low-order near-body solver, since both mesh resolution and order of accuracy must be matched in the overlapping regions. To demonstrate the efficiency, accuracy, and capabilities of the DG AMR flow solver we simulate Ringleb flow and the Taylor-Green vortex problem. Finally, to demonstrate the overset capabilities a NACA 0015 wing case and a NREL PhaseVI wind turbine case are simulated.

Mesh-Resolved Airfoil Simulations Using Finite Volume and Discontinuous Galerkin Solvers

A parallel high-order Discontinuous Galerkin (DG) method is used to simulate turbulent flow over a NACA 0012 airfoil. Using a family of high-density grids (available on the NASA turbulence modeling resource website) mesh resolved solutions are obtained. The flow is simulated by solving the Reynolds-Averaged Navier-Stokes equations closed by the negative Spalart-Allmaras turbulence model. The flow conditions for this case are: α = 10, M = .15, and Re = 6 × 10. Lift, drag, pitching moment, pressure, and skin friction coefficients are provided for multiple grids and discretization orders and compared against other simulation results from the CFL3D and FUN3D solvers. The DG simulations give very similar results to these solvers and which further verifies both the DG solver and the other methods as having mesh resolved solutions. Also, it is shown that p-refinement converges quicker to the mesh resolved solutions compared to h-refinement.

An Investigation of Continuous and Discontinuous Finite-Element Discretizations on Benchmark 3D Turbulent Flows (Invited)

The Discontinuous Galerkin (DG) method provides numerical solutions of the NavierStokes equations with high order of accuracy in complex geometries and allows for highly efficient parallelization algorithms. These attributes make the DG method highly attractive for large eddy simulation (LES). The main goal of this work is to investigate the feasibility of adopting an explicit filter to the numerical solution of the Navier-Stokes equations to increase the numerical stability of underresolved simulations such as LES and to use the explicit filter in dynamic subgrid scale (SGS) models for LES. The explicit filter takes advantage of DG’s framework where the solution is approximated using a polynomial basis. The higher modes of the solution correspond to a higher order polynomial basis, therefore by removing high order modes the filtered solution contains only lower frequency content. The explicit filter is successfully used here to remove the effects of aliasing in underresolved simulations of the Taylor-Green vortex case at a Reynolds number Re = 1600. The de-aliasing is achieved by evaluating a solution at a higher order polynomial (effectively increasing the number of quadrature points used for integration) and then projecting the solution down to a lower order polynomial. The SGS models investigated include the constant coefficient Smagorinsky Model (CCSM), Dynamic Smagorinsky Model (DSM), and Dynamic Heinz Model (DHM). The Taylor-Green Vortex case exhibits a laminar-turbulent transition and it is shown that the dynamic SGS models capture this transition more accurately than the CCSM when a sufficiently high polynomial order is used. The explicit test-filter operation for the dynamic models introduces a commutation error. A brief comparison of the effects of the commutative error that exists with this filter implementation is shown although further investigation is needed to determine the more appropriate order of operations.

Discontinuous Galerkin Turbulent Flow Simulations of NASA Turbulence Model Validation Cases and High Lift Prediction Workshop Test Case DLR-F11

A traditional second-order accurate finite volume unstructured mesh solver and a high-order discontinuous Galerkin solver are used to simulate turbulent flow over a NACA 0012 airfoil. Using a family of standardized high-density grids, mesh-resolved solutions are obtained. The flow is simulated by solving the Reynolds-averaged Navier–Stokes equations closed by the negative Spalart–Allmaras turbulence model. The flow conditions for this case are α=10, M=0.15, and Re=6×106. Lift, drag, pitching moment, pressure, and skin friction coefficients are provided for multiple grids and discretization orders and are compared against other simulation results from well known solvers. The current simulations give very similar results to these benchmark solvers, pointing toward fully mesh-resolved simulations and providing verification evidence of correct and consistent implementation of these discretizations. Results obtained using the high-order discontinuous Galerkin discretizations show higher accuracy using fewer de...