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NASA NTRS1992

Mapping unstructured grid computations to massively parallel computers

Investigated here is this mapping problem: assign the tasks of a parallel program to the processors of a parallel computer such that the execution time is minimized. First, a taxonomy of objective functions and heuristics used to solve the mapping problem is presented. Next, we develop a highly parallel heuristic mapping algorithm, called Cyclic Pairwise Exchange (CPE), and discuss its place in the taxonomy. CPE uses local pairwise exchanges of processor assignments to iteratively improve an initial mapping. A variety of initial mapping schemes are tested and recursive spectral bipartitioning (RSB) followed by CPE is shown to result in the best mappings. For the test cases studied here, problems arising in computational fluid dynamics and structural mechanics on unstructured triangular and tetrahedral meshes, RSB and CPE outperform methods based on simulated annealing. Much less time is required to do the mapping and the results obtained are better. Compared with random and naive mappings, RSB and CPE reduce the communication time two fold for the test problems used. Finally, we use CPE in two applications on a CM-2. The first application is a data parallel mesh-vertex upwind finite volume scheme for solving the Euler equations on 2-D triangular unstructured meshes. CPE is used to map grid points to processors. The performance of this code is compared with a similar code on a Cray-YMP and an Intel iPSC/860. The second application is parallel sparse matrix-vector multiplication used in the iterative solution of large sparse linear systems of equations. We map rows of the matrix to processors and use an inner-product based matrix-vector multiplication. We demonstrate that this method is an order of magnitude faster than methods based on scan operations for our test cases.

Hammond, Steven Warren↗

NASA NTRS2003

High Energy Boundary Conditions for a Cartesian Mesh Euler Solver

Inlets and exhaust nozzles are common place in the world of flight. Yet, many aerodynamic simulation packages do not provide a method of modelling such high energy boundaries in the flow field. For the purposes of aerodynamic simulation, inlets and exhausts are often fared over and it is assumed that the flow differences resulting from this assumption are minimal. While this is an adequate assumption for the prediction of lift, the lack of a plume behind the aircraft creates an evacuated base region thus effecting both drag and pitching moment values. In addition, the flow in the base region is often mis-predicted resulting in incorrect base drag. In order to accurately predict these quantities, a method for specifying inlet and exhaust conditions needs to be available in aerodynamic simulation packages. A method for a first approximation of a plume without accounting for chemical reactions is added to the Cartesian mesh based aerodynamic simulation package CART3D. The method consists of 3 steps. In the first step, a components approach where each triangle is assigned a component number is used. Here, a method for marking the inlet or exhaust plane triangles as separate components is discussed. In step two, the flow solver is modified to accept a reference state for the components marked inlet or exhaust. In the third step, the flow solver uses these separated components and the reference state to compute the correct flow condition at that triangle. The present method is implemented in the CART3D package which consists of a set of tools for generating a Cartesian volume mesh from a set of component triangulations. The Euler equations are solved on the resulting unstructured Cartesian mesh. The present methods is implemented in this package and its usefulness is demonstrated with two validation cases. A generic missile body is also presented to show the usefulness of the method on a real world geometry.

Pandya, Shishir↗

NASA NTRS1991

Accuracy of an unstructured-grid upwind-Euler algorithm for the ONERA M6 wing

Improved algorithms for the solution of the three-dimensional, time-dependent Euler equations are presented for aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured-grid flow solvers. The spatial discretization involves a flux-split approach that is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves either an explicit time-integration scheme using a multistage Runge-Kutta procedure or an implicit time-integration scheme using a Gauss-Seidel relaxation procedure, which is computationally efficient for either steady or unsteady flow problems. With the implicit Gauss-Seidel procedure, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady flow results are presented for both the NACA 0012 airfoil and the Office National d'Etudes et de Recherches Aerospatiales M6 wing to demonstrate applications of the new Euler solvers. The paper presents a description of the Euler solvers along with results and comparisons that assess the capability.

Batina, John T.↗

NASA NTRS1998

Self-Avoiding Walks over Adaptive Triangular Grids

In this paper, we present a new approach to constructing a "self-avoiding" walk through a triangular mesh. Unlike the popular approach of visiting mesh elements using space-filling curves which is based on a geometric embedding, our approach is combinatorial in the sense that it uses the mesh connectivity only. We present an algorithm for constructing a self-avoiding walk which can be applied to any unstructured triangular mesh. The complexity of the algorithm is O(n x log(n)), where n is the number of triangles in the mesh. We show that for hierarchical adaptive meshes, the algorithm can be easily parallelized by taking advantage of the regularity of the refinement rules. The proposed approach should be very useful in the run-time partitioning and load balancing of adaptive unstructured grids.

Heber, Gerd↗

NASA NTRS2002

Application of Quaternions for Mesh Deformation

A new three-dimensional mesh deformation algorithm, based on quaternion algebra, is introduced. A brief overview of quaternion algebra is provided, along with some preliminary results for two-dimensional structured and unstructured viscous mesh deformation.

Samareh, Jamshid A.↗

NASA NTRS2002

Application of Quaternions for Mesh

A new three dimensional mesh deformation algorithm, based on quaternion algebra, is introduced. A brief overview of quaternion algebra is provided, along with some preliminary results for two-dimensional structured and unstructured viscous mesh deformation.

Samareh, Jamshid A.↗

NASA NTRS1994

Laminar and turbulent flow computations of Type 4 shock-shock interference aerothermal loads using unstructured grids

This report presents computations for the Type 4 shock-shock interference flow under laminar and turbulent conditions using unstructured grids. Mesh adaptation was accomplished by remeshing, refinement, and mesh movement. Two two-equation turbulence models were used to analyze turbulent flows. The mean flow governing equations and the turbulence governing equations are solved in a coupled manner. The solution algorithm and the details pertaining to its implementation on unstructured grids are described. Computations were performed at two different freestream Reynolds numbers at a freestream Mach number of 11. Effects of the variation in the impinging shock location are studied. The comparison of the results in terms of wall heat flux and wall pressure distributions is presented.

Vemaganti, Gururaja R.↗

NASA NTRS1993

Hypersonic flow computations around re-entry vehicles

The development of an algorithm for the solution of the compressible Euler equations at high Mach numbers on unstructured tetrahedral meshes is described. The basic algorithm is constructed in the form of a central difference scheme plus an explicit added artificial viscosity based upon fourth order differences of the solution. The stability of the solution in the vicinity of strong gradients is preserved by the incorporation of an additional artificial viscosity based upon a second order difference. Higher order accuracy is regained by using the ideas of flux corrected transport to limit the amount of added viscosity. The solution is advanced to steady state by means of an explicit multi-stage time-stepping method. The computational efficiency of the complete process is improved by incorporating an unstructured multigrid acceleration procedure. A number of flows of practical interest are analyzed to demonstrate the numerical performance of the proposed approach.

Peraire, J.↗

NASA NTRS1987

An adaptive finite element scheme for the Euler and Navier-Stokes equations

The implementation of two explicit finite element schemes for the solution of the compressible Euler and Navier-Stokes equations is presented. The schemes can be employed with general unstructured triangular meshes in two dimensions. Either scheme can therefore be used as the basic solver in a solution adaptive mesh procedure in a direct manner. The particular adaptive approach which is advocated here is intended for the solution of steady state problems only and involves an adaptive regeneration of the grid at prescribed stages during the false transient. The grid regeneration is accomplished by a mesh generator which has the capability of generating triangular grids over computational domains of arbitrary shape. The procedure is illustrated by solving transonic flows over multi-airfoil configurations and high speed flows, involving shock interactions, past circular cylinders.

Morgan, K.↗

NASA NTRS1989

A Navier-Stokes solver for high speed equilibrium flows and application to blunt bodies

This paper presents a finite element method for the solution of Navier-Stokes equations with the assumption of thermodynamic and chemical equilibrium. The method employs an upwind finite element technique with an implicit time-marching scheme for the solution, and uses an adaptively generated unstructured triangular mesh with several layers of quadrilateral elements near solid walls. The complexity associated with the assumption that the flow is in equilibrium is treated consistently, and the inviscid flux Jacobian matrices are derived. Several problems involving inviscid and viscous hypersonic flow past blunt are solved. Results are compared with other numerical results and experimental data, and generally good agreement is observed.

Prabhu, Ramadas K.↗

NASA NTRS1989

The upwind control volume scheme for unstructured triangular grids

A new algorithm for the numerical solution of the Euler equations is presented. This algorithm is particularly suited to the use of unstructured triangular meshes, allowing geometric flexibility. Solutions are second-order accurate in the steady state. Implementation of the algorithm requires minimal grid connectivity information, resulting in modest storage requirements, and should enhance the implementation of the scheme on massively parallel computers. A novel form of upwind differencing is developed, and is shown to yield sharp resolution of shocks. Two new artificial viscosity models are introduced that enhance the performance of the new scheme. Numerical results for transonic airfoil flows are presented, which demonstrate the performance of the algorithm.

Giles, Michael↗

NASA NTRS1991

Three dimensional unstructured multigrid for the Euler equations

The three dimensional Euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy. The driving algorithm consists of an explicit vertex-based finite element scheme, which employs an edge-based data structure to assemble the residuals. The multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady-state of the fine grid solution. Variables, residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm. The preprocessing operation is shown to require a negligible fraction of the CPU time required by the overall solution procedure, while gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350,000 vertices. Solutions using globally regenerated fine meshes as well as adaptively refined meshes are given.

Mavriplis, D. J.↗

NASA NTRS1991

Three dimensional unstructured multigrid for the Euler equations

The three-dimensional Euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy. The driving algorithm consists of an explicit vertex-based finite-element scheme, which employs an edge-based data-structure to assemble the residuals. The multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady-state of the fine grid solution. Variables, residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm. The preprocessing operation is shown to require a negligible fraction of the CPU time required by the overall solution procedure, while gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350,000 vertices. Solutions using globally regenerated fine meshes as well as adaptively refined meshes are given.

Mavriplis, D. J.↗

NASA NTRS1992

Two-dimensional unstructured triangular grid generation

The capability of generating 2-D unstructured triangular meshes about arbitrary geometries is demonstrated. This work uses a distribution of boundary points and triangulates the computational domain using a Delaunay triangulation algorithm. Typically, initial cells are added based on cell aspect ratios or cell areas. A resulting mesh can then be used along with the connectivity of the cells to solve either a Euler or Navier-Stokes flow problem.

Jorgenson, Philip C. E.↗

NASA NTRS1992

Implicit upwind-Euler solution algorithms for unstructured-grid applications

The development of implicit upwind algorithms for the solution of the three-dimensional, time-dependent Euler equations on unstructured tetrahedral meshes is described. The implicit temporal discretization involves either a two-sweep Gauss-Seide relaxation procedure, a two-sweep Point-Jacobi relaxation procedure, or a single-sweep Point-Implicit procedure; the upwind spatial discretization is based on the flux-difference splitting of Roe. Detailed descriptions of the three implicit solution algorithms are given, and calculations for the Boeing 747 transport configuration are presented to demonstrate the algorithms. Advantages and disadvantages of the implicit algorithms are discussed. A steady-state solution for the 747 configuration, obtained at transonic flow conditions using a mesh of over 100,000 cells, required less than one hour of CPU time on a Cray-2 computer, thus demonstrating the speed and robustness of the general capability.

Batina, John T.↗

NASA NTRS1992

Computational/experimental analysis of three low sonic boom configurations with design modifications

The Euler code, designated AIRPLANE, which uses an unstructured tetrahedral mesh was used to compute near-field sonic boom pressure signatures on three modern low sonic boom configurations: the Mach 2, Mach 3, and Haglund models. The TEAM code which uses a multi-zoned structured grid was used to calculate pressure signatures for the Mach 2 model. The computational pressure signatures for the Mach 2 and Mach 3 models are compared with recent experimental data. The computed pressure signatures were extracted at distances less than one body length below the configuration and extrapolated to the experimental distance. The Mach 2 model was found to have larger overpressures off-ground-track than on-ground-track in both computational and experimental results. The correlations with the experiment were acceptable where the signatures were not contaminated by instrumentation and model-support hardware. AIRPLANE was used to study selected modifications to improve the overpressures of the Mach 2 model.

Cliff, Susan E.↗

NASA NTRS1993

Three-dimensional unstructured grid Euler method applied to turbine blades

Flow through a turbine annular cascade is calculated using a three-dimensional Euler method based on unstructured tetrahedral meshes. The equations are integrated in time using an explicit Runge-Kutta time-stepping scheme. The inviscid flux terms are discretized using a cell-centered finite-volume formulation with upwind flux-difference splitting. The tetrahedral meshes around the turbine blade are generated using an advancing-front technique with forced geometric periodicity between the blades. Good agreement is obtained between the present calculation and the experiment for both surface pressure distribution and flow behavior in the passage between the blades, demonstrating the capability of the present methodology for turbomachinery flow applications.

Kwon, Oh J.↗

NASA NTRS1993

A new procedure for dynamic adaption of three-dimensional unstructured grids

A new procedure is presented for the simultaneous coarsening and refinement of three-dimensional unstructured tetrahedral meshes. This algorithm allows for localized grid adaption that is used to capture aerodynamic flow features such as vortices and shock waves in helicopter flowfield simulations. The mesh-adaption algorithm is implemented in the C programming language and uses a data structure consisting of a series of dynamically-allocated linked lists. These lists allow the mesh connectivity to be rapidly reconstructed when individual mesh points are added and/or deleted. The algorithm allows the mesh to change in an anisotropic manner in order to efficiently resolve directional flow features. The procedure has been successfully implemented on a single processor of a Cray Y-MP computer. Two sample cases are presented involving three-dimensional transonic flow. Computed results show good agreement with conventional structured-grid solutions for the Euler equations.

Biswas, Rupak↗