David Bilyeu

A two-dimensional fourth-order unstructured-meshed Euler solver based on the CESE method

In this paper, Chang?s one-dimensional high-order CESE method 1] is extended to a two-dimensional, unstructured-triangular-meshed Euler solver. This fourth-order CESE method retains all favorable attributes of the original second-order CESE method, including: (i) flux conservation in space and time without using an approximated Riemann solver, (ii) genuine multi-dimensional algorithm without dimensional splitting, (iii) the CFL constraint for stable calculation remains to be ≤1, (iv) the use of the most compact mesh stencil, involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought, and (v) an explicit, unified space-time integration procedure without using a quadrature integration procedure. To demonstrate the new algorithm, three numerical examples are presented: (i) a moving vortex, (ii) acoustic wave interaction, and (iii) supersonic flow over a blunt body. Case 1 shows fourth-order convergence through mesh refinement. In Case 2, the nonlinear Euler solver is applied to simulate linear waves. In Case 3, superb shock capturing capabilities of the new fourth-order method without the carbuncle effect is demonstrated.

SM/MURF: Current Capabilities and Verification as a Replacement of AFRL Plume Simulation Tool COLISEUM

Abstract : The Spacecraft Multi-Scale/Multi-Physics Universal Research Framework (SM/MURF) is currently in development at the in-space propulsion branch of the Air Force Research Laboratory (AFRL). This framework unifies multiple research codes developed independently at this branch and is intended to simulate plasma under a wide range of time and length scales relevant to the spacecraft electric propulsion (EP) systems. The current development effort focuses on the modernization of the AFRL plume simulation tool, COLISEUM. In this paper, all the operations for a plume simulation are reviewed along with the results from integration tests to verify the correctness of those operations. As the final verification test, SM/MURF and COLISEUM are used to perform the same baseline plume simulation, and the results from the two codes are compared. In general, the agreement between the two codes is very good. A slight discrepancy was caused by a mismatch of where the field data are computed; SM/MURF and COLISEUM compute field data on cell-centers and nodes, respectively. This results in a slightly different electric field, cascading through particle trajectories, field calculations, sputter rate, and redeposition rate onto spacecraft components.

SOLVCON: A Python-Based CFD Software Framework for Hybrid Parallelization

SOLVCON is a new, open-sourced software framework for high-fidelity solutions of linear and non-linear hyperbolic partial differential equations. SOLVCON emphasizes scalability, portability, and maintainability for supercomputing by using emerging multi-core architectures. The code development effort follows Extreme Programming practices, including version control, documentation, issue tracking, user support, and frequent code releases. In SOLVCON, the Python framework includes all supportive functionalities for the work flow. For pre-processing operations, the Python framework provides parallelized mesh data input and automatically sets up domain decomposition. In calculations, the Python framework provides light-weight memory management through extensive use of pointers. Computation-intensive operations are implemented by using C and FORTRAN for high performance. The default numerical algorithm employed is the space-time Conservation Element and Solution Element (CESE) method. The code uses general unstructured meshes with mixed elements, including tetrahedra, hexahedra, prisms, and pyramids for threedimensional calculations. Hybrid parallelism includes shared- and distributed-memory parallelization. The temporal loop and the spatial loop in modern finite-volume methods are implemented in a two-layered structure in SOLVCON. Distributed-memory parallelization by domain decomposition and MPI is performed in the temporal loop. Shared-memory parallel computing by using accelerator technologies, e.g., General-Purpose Graphic Processor Unit (GPGPU), is performed in the spatial loop. More than 99% of the execution time of SOLVCON is used for number-crunching in the solver as a part of the space loop. Written in C or FORTRAN, a typical solver contains only 10% of the code statements in SOLVCON. To demonstrate the capabilities of newly developed SOLVCON, we performed CFD calculations by using 23 million elements. The code was run on a 512-core cluster. SOLVCON delivers calculations of flow variables in 11.29 million elements per second. The parallel efficiency is 70%. In the open-sourced SOLVCON, two solvers are available: (i) the Euler equations solver for compressible flows, and (ii) the velocity-stress equations solver for waves in anisotropic elastic solids. SOLVCON can be easily extended for other applications, including viscous flows, aero-acoustics, nonlinear solid mechanics, and electromagnetism. The Python framework allows fast adaption to new heterogeneous, multi-core hardware as well as further development of the code for peta-scale supercomputing.

A 3D Unstructured Mesh Euler Solver Based on the Fourth-Order CESE Method

Abstract : In this paper, the CESE method is extended and employed to construct a fourth-order, three-dimensional, unstructured-mesh solver for hyperbolic Partial Differential Equations (PDEs). This new CESE method retains all favorable attributes of the original second-order CESE method, including: (i) flux conservation in space and time without using a one-dimensional Riemann solver, (ii) genuinely multi-dimensional treatment without dimensional splitting (iii) the CFL constraint remains to be less than or equal to 1, and (iv) the use of a compact mesh stencil involving only the immediate neighboring nodes surrounding the node where the solution is sought. Two validation cases are presented. First higher order convergence is demonstrated by the linear advection equation. Second supersonic flow over a spherical body is simulated to demonstrates the schemes ability to accurately resolve discontinuities.

A Two-Dimensional Fourth-Order CESE Method for the Euler Equations on Triangular Unstructured Meshes

Previously, Chang 1 reported a new high-order Conservation Element Solution Element (CESE) method for solving nonlinear, scalar, hyperbolic partial differential equations in one dimensional space. Bilyeu et al. 2 have extended Chang’s scheme for solving a onedimensional, coupled equations with an arbitrary order of accuracy. In the present paper, the one-dimensional, high-order CESE method is extended for two-dimensional unstructured meshes. A formulation is presented for solving the coupled equations with the fourthorder in accuracy. The new high-order CESE methods share many favorable attributes of the original second-order CESE method, including: (i) the use of compact mesh stencil involving only the immediate mesh nodes surrounding the node where the solution is sought, (ii) the CFL stability constraint remains to be � 1, and (iii) superb shock capturing capability without using an approximate Riemann solver. To demonstrate the formulation, four test cases are reported, including (i) solution of a two-dimensional, scalar convection equation, (ii) the solution of the linear acoustic equations, (iii) the solution of the Euler equations for waves of small amplitudes, and (iv) the solution of the Euler equations for expanding shock waves. In all calculations, unstructured triangular meshes are used. In the first three cases, convergence tests show the fourth-order accuracy of the solutions. In the last case, numerical results of the fourth-order scheme are superior than that obtained by the second-order CESE method. In this paper, we extend the one-dimensional, high-order CESE method for solving two-dimensional coupled, nonlinear hyperbolic Partial Differential Equations (PDEs) by using unstructured meshes. The new formulation is currently fourth-order in accuracy, but it can be easily extended to a higher-order scheme by including more terms in the Taylor series expansion. The tenet of the CESE method is treating space and time in a unified manner in integrating the model equations for flux conservation. In the CESE method, the space-time domain is divided into non-overlapping Conservation Elements (CEs), over which the spacetime integration is performed to enforce flux conservation. The integration is facilitated by using Solution Elements (SEs), which in general do not coincide with the CEs. The unknowns are discretized inside each SE by using a Taylor series expansion. Aided by the prescribed discretization schemes for the unknowns inside each SE, space-time flux over each surface of a CE can be calculated and the overall flux conservation over each CE can be enforced. The result is an explicit formulation for updating the unknowns for time-marching calculation. Special features of the CESE method include: (i) The unknowns are discretized by using the Taylor series expansion in both space and time. The order of the Taylor series is also the order of accuracy of the method. (ii) The method has the most compact mesh stencil, which involves only the immediate neighboring mesh points of the cell where the solution is sought. (iii) The method is an explicit scheme in time-marching calculation. The stability criterion is CFL � 1. (iv) No approximate Riemann solver is used and the scheme is simple and efficient. (v) The CESE method includes a suite of numerical algorithms,