Joseph D. Baum

A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids

A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods.

A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids

A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest.

Edge-based finite element scheme for the Euler equations

We describe the development, validation, and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well-documented configurations. A flow solution about a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm

An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids

Abstract An accurate, fast, matrix-free implicit method has been developed to solve the three-dimensional compressible unsteady flows on unstructured grids. A nonlinear system of equations as a result of a fully implicit temporal discretization is solved at each time step using a pseudo-time marching approach. A newly developed fast, matrix-free implicit method is then used to obtain the steady-state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady flow problems involving moving boundaries. The numerical results obtained indicate that the use of the present implicit method leads to a significant increase in performance over its explicit counterpart, while maintaining a similar memory requirement.

Pulsed instability in rocket motors - A comparison between predictions and experiments

A series of 18 pulsed motor tests was conducted in heavy-wall solid rocket motors having an internal case diameter of 8.38 cm. The motor length was varied from 0.61 -1.22 m. Three related reduced-smoke propellants and four different grain designs were tested. All of the motors were linearly stable and all were pulsed into nonlinear instability. The data from this test series were used to evaluate the validity of previously developed pulser models and to evaluate the ability of a previously developed nonlinear instability analysis to predict the observed trends in the data. In most cases, the combined pulser and chamber stability models were found to be capable of predicting measured pulse amplitudes to within 10%. The ejecta pulser model had to be modified to account for the effect of internal wave reflection when propellant grain extended up to the nozzle entrance plane. The nonlinear instability analysis demonstrated the ability to predict many of the observed wave amplitude and shift trends in the data as a function of propellant and grain design. This combined analytical/experimental investigation also provides additional insight into the nature of nonlinear pulse triggered instability and the factors that influence its occurrence and severity.

A hybrid Cartesian grid and gridless method for compressible flows

A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian grid is used as baseline mesh to cover the computational domain, while the boundary surfaces are addressed using a gridless method. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time-accurate solution of the compressible Euler equations. An overall speed-up factor of about eight and a saving in storage requirements about one order of magnitude for a typical three-dimensional problem in comparison with the unstructured grid method are demonstrated.

The numerical simulation of strongly unsteady flow with hundreds of moving bodies

A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured grid, high-order, monotonicity preserving, ALE solver with automatic refinement and remeshing capabilities was enhanced by adding equations of state for high explosives, deactivation techniques and optimal data structures to minimize CPU overheads, automatic recovery of CAD data from discrete data, two new remeshing options, and a number of visualization tools for the preprocessing phase of large runs. The combination of these improvements has enabled the simulation of strongly unsteady flows with hundreds of moving bodies. Several examples demonstrate the effectiveness of the proposed methodology

Extension of Harten-Lax-van Leer Scheme for Flows at All Speeds.

The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute flow problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range. ISTORICALLY, numerical algorithms for the solution of the Euler and Navier‐Stokes equations are classified as either pressure-based or density-based solution methods. The pressurebased methods, originally developed and well suited for incompressible flows, are typically based on the pressure correction techniques. They usually use a staggered grid and solve the governing equations in a segregated manner. The density-based methods, originally developed and robust for compressible flows, use time-arching procedures to solve the hyperbolic system of governing equations in a coupled manner. In general, density-based methods are not suitable for efficiently solving low Mach number or incompressible flow problems, because of large ratio of acoustic and convective timescales at the low-speed flow regimes. To alleviate this stiffness and associated convergence problems, time-derivative preconditioning techniques have been developed and used successfully for solving low-Machnumber and incompressible flows by many investigators, including Chorin, 1 Choi and Merkle, 2 Turkel, 3 Weiss and Smith, 4 and Dailey and Pletcher, 5 among others. Such methods seek to modify the time component of the governing equations so that the convergence can be made independent of Mach number. This is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order, and thus condition number remains bounded independent of the Mach number of the flows. Over the last two decades characteristic-based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible Euler equations. When these upwind methods are used to compute the numerical fluxes for the preconditioned Euler equations, solution accuracy at low speeds can be compromised, unless the numerical flux formulation is modified to take into account the eigensystem of the precondi

IMPLEMENTATION OF UNSTRUCTURED GRID GMRES+LU-SGS METHOD ON SHARED-MEMORY, CACHE-BASED PARALLEL COMPUTERS

The implementation of an unstructured grid matrix-free GMRES+LU-SGS scheme on shared-memory, cache-based parallel machines is described. A special grid renumbering technique is used for the parallelization rather than the traditional method of partitioning the computational domain. The renumbering technique helps to avoid inter-processor data dependencies, cache-misses, and cache-line overwrite while allowing pipelining. The resulting source code can be used with maximum efficiency and without modifications on traditional (scalar) computers, vector supercomputers, and shared-memory parallel systems. Special attention has been paid to develop an optimally parallelized preconditioner for the GMRES scheme.

High-Reynolds number viscous flow computations using an unstructured-grid method

An unstructured grid method is presented to compute three-dimensional compressible turbulent flows for complex geometries. The Navier‐Stokes equations, along with the one-equation turbulence model of Spalart‐Allmaras are solved by the use of a parallel, matrix-free implicit method on unstructured tetrahedral grids. The developed method has been used to predict drags in the transonic regime for both DLR-F4 and DLR-F6 configurations to assess the accuracy and efficiency of the method. The results obtained are in good agreement with experimental data, indicating that the present method provides an accurate, efficient, and robust algorithm for computing turbulent flows for complex geometries on unstructured grids.