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NASA NTRS1988

Velocity-pressure integrated versus penalty finite element methods for high Reynolds number flows

Velocity-pressure integrated and consistent penalty finite element computations of high Reynolds number, laminar flows are presented. In both of the methods, the pressure has been interpolated using linear shape functions for a triangular element. The triangular element is contained inside the bi-quadratic isoparametric element. It has been reported previously that the pressure interpolation method, when used in the velocity-pressure integrated method, yielded accurate computational results for high Reynolds number flows. It is shown that use of the same pressure interpolation method in the consistent penalty finite element method yielded accurate velocity and pressure fields which were comparable to those obtained using the velocity-pressure integrated method. Accuracy of the two finite element methods has been demonstrated by comparing the computational results with available experimental data and/or fine-grid finite difference computational results. Advantages and disadvantages of the two methods are discussed on the basis of accuracy and convergence nature. Example problems considered include a lid-driven cavity flow for Reynolds number of 10,000, a laminar backward-facing step flow, a laminar flow through a nest of cylinders, and a channel flow with an internal blockage. A finite element computer program (NSFLOW/P) for the 2-D, incompressible Navier-Stokes equations is also presented.

Kim, Sang-Wook↗

NASA NTRS1988

Storage and computationally efficient permutations of factorized covariance and square-root information arrays

A unified method to permute vector stored Upper triangular Diagonal factorized covariance and vector stored upper triangular Square Root Information arrays is presented. The method involves cyclic permutation of the rows and columns of the arrays and retriangularization with fast (slow) Givens rotations (reflections). Minimal computation is performed, and a one dimensional scratch array is required. To make the method efficient for large arrays on a virtual memory machine, computations are arranged so as to avoid expensive paging faults. This method is potentially important for processing large volumes of radio metric data in the Deep Space Network.

Muellerschoen, R. J.↗

NASA NTRS1988

Discretization formulas for unstructured grids

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formula in cartesian coordinates for the Laplacian operator, first derivative operators and the function for unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator are shown to agree with the Taylor series approach on a global average. In addition, a simple algorithm is presented to determine the Voronoi (finite difference) area of an unstructured grid.

Baumeister, Kenneth J.↗

NASA NTRS1988

Krylov subspace methods on supercomputers

A short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers is presented. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three-dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel/vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. A few approaches consisting of implementing efficient forward and backward triangular solutions are described in detail. Polynomial preconditioning as an alternative to standard incomplete factorization techniques is also discussed. Another efficient approach is to reorder the equations so as to improve the structure of the matrix to achieve better parallelism or vectorization. An overview of these and other ideas and their effectiveness or potential for different types of architectures is given.

Saad, Youcef↗

NASA NTRS1988

Storage and computationally efficient permutations of factorized covariance and square-root information matrices

A unified method to permute vector-stored upper-triangular diagonal factorized covariance (UD) and vector stored upper-triangular square-root information filter (SRIF) arrays is presented. The method involves cyclical permutation of the rows and columns of the arrays and retriangularization with appropriate square-root-free fast Givens rotations or elementary slow Givens reflections. A minimal amount of computation is performed and only one scratch vector of size N is required, where N is the column dimension of the arrays. To make the method efficient for large SRIF arrays on a virtual memory machine, three additional scratch vectors each of size N are used to avoid expensive paging faults. The method discussed is compared with the methods and routines of Bierman's Estimation Subroutine Library (ESL).

Muellerschoen, R. J.↗

NASA NTRS1989

Parallel algorithms and architecture for computation of manipulator forward dynamics

Parallel computation of manipulator forward dynamics is investigated. Considering three classes of algorithms for the solution of the problem, that is, the O(n), the O(n exp 2), and the O(n exp 3) algorithms, parallelism in the problem is analyzed. It is shown that the problem belongs to the class of NC and that the time and processors bounds are of O(log2/2n) and O(n exp 4), respectively. However, the fastest stable parallel algorithms achieve the computation time of O(n) and can be derived by parallelization of the O(n exp 3) serial algorithms. Parallel computation of the O(n exp 3) algorithms requires the development of parallel algorithms for a set of fundamentally different problems, that is, the Newton-Euler formulation, the computation of the inertia matrix, decomposition of the symmetric, positive definite matrix, and the solution of triangular systems. Parallel algorithms for this set of problems are developed which can be efficiently implemented on a unique architecture, a triangular array of n(n+2)/2 processors with a simple nearest-neighbor interconnection. This architecture is particularly suitable for VLSI and WSI implementations. The developed parallel algorithm, compared to the best serial O(n) algorithm, achieves an asymptotic speedup of more than two orders-of-magnitude in the computation the forward dynamics.

Fijany, Amir↗

NASA NTRS1990

Upwind control volumes - A new upwind approach

This paper presents a new algorithm for the solution of the steady Euler equations in two-dimensional flow. It uses an upwind control volume approach in which a standard flux residual on triangular cells is distributed with a downwind bias. The resulting scheme is second order accurate on unstructured triangular meshes and produces sharp shocks without the addition of any shock smoothing. Results demonstrate its performance on a standard AGARD test case using both smooth and irregular unstructured meshes.

Giles, Michael↗

NASA NTRS1988

Discretization formulas for unstructured grids

Baumeister, Kenneth J.↗

NASA NTRS1991

Development of a solution adaptive unstructured scheme for quasi-3D inviscid flows through advanced turbomachinery cascades

The objective of the present research is to develop a general solution adaptive scheme for the accurate prediction of inviscid quasi-three-dimensional flow in advanced compressor and turbine designs. The adaptive solution scheme combines an explicit finite-volume time-marching scheme for unstructured triangular meshes and an advancing front triangular mesh scheme with a remeshing procedure for adapting the mesh as the solution evolves. The unstructured flow solver has been tested on a series of two-dimensional airfoil configurations including a three-element analytic test case presented here. Mesh adapted quasi-three-dimensional Euler solutions are presented for three spanwise stations of the NASA rotor 67 transonic fan. Computed solutions are compared with available experimental data.

Usab, William J., Jr.↗

NASA NTRS1991

Numerical aspects of computing high Reynolds number flows on unstructured meshes

An edge-data structure describing a mesh edge-wise given the vertices of each edge and neighboring cell information is used developing algorithms for the Navier-Stokes equations on triangular meshes. Edge formulas for the Galerkin and finite-element discretization of gradient, divergence, Hessian, and Laplacian operators are derived. A simple edge formula is derived for the discretization of the Laplacian operator, where precise theoretical conditions for a discrete maximum principle can be ascertained. Practical issues associated with solving the Navier-Stokes equations on unstructured meshes are addressed, along with issues concerning the generation of highly stretched triangular meshes and the modeling of turbulence on unstructured meshes. A turbulence modeling strategy is proposed, and numerical results for a high-Reynolds-number flow about single- and multielement airfoils are discussed.

Barth, Timothy J.↗

NASA NTRS1992

A finite element boundary integral formulation for radiation and scattering by cavity antennas using tetrahedral elements

A hybrid finite element boundary integral formulation is developed using tetrahedral and/or triangular elements for discretizing the cavity and/or aperture of microstrip antenna arrays. The tetrahedral elements with edge based linear expansion functions are chosen for modeling the volume region and triangular elements are used for discretizing the aperture. The edge based expansion functions are divergenceless thus removing the requirement to introduce a penalty term and the tetrahedral elements permit greater geometrical adaptability than the rectangular bricks. The underlying theory and resulting expressions are discussed in detail together with some numerical scattering examples for comparison and demonstration.

Gong, J.↗

NASA NTRS1992

Supersonic jet mixing enhancement by delta-tabs

The results of a continuing investigation of the effect of vortex generators, in the form of small tabs at the nozzle exit, on the evolution of a jet are reported. Primarily, tabs of triangular shape are considered, and the effect is studied up to an equivalent jet Mach number of 1.8. By changing the orientation of the tab with respect to the nozzle exit plane, streamwise vortex pairs of opposite sign were generated. This resulted in either an outward election of jet core fluid into the ambient or an inward indentation of the mixing layer into the core of the jet. A triangular shaped tab with its apex leaning downstream, referred to as a delta tab, was found to be the most effective in influencing the jet evolution. Two delta tabs, spaced 180 degrees apart, completely bifurcated the jet. Four delta tabs increased jet mixing substantially, more than by various other methods tried previously; the mass flux at fourteen jet diameters downstream from the nozzle increased by about 50 percent over that for the no tab case. The tabs were found to be effective in jets with laminar or turbulent boundary layers as well as in jets with low or high core turbulence intensities.

Zaman, K. B. M. Q.↗

NASA NTRS1992

Unstructured grid methods for compressible flows

The implementation of the finite element method on unstructured triangular grids is described and the development of centered finite element schemes for the solution of the compressible Euler equation on general triangular and tetrahedral grids is discussed. Explicit and implicit Lax-Wendroff type methods and a method based upon the use of explicit multistep timestepping are considered. In the latter case, the convergence behavior of the method is accelerated by the incorporation of a fully unstructured multigrid procedure. The advancing front method for generating unstructured grids of triangles and tetrahedra is described and the application of adaptive mesh techniques to both steady and transient flow analysis is illustrated.

Morgan, K.↗

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 NTRS1992

Supersonic jet mixing enhancement by 'delta-tabs'

The results of a continuing investigation of the effect of vortex generators, in the form of small tabs at the nozzle exit, on the evolution of a jet are reported. Primarily, tabs of triangular shape are considered, and the effect is studied up to an equivalent jet Mach number of 1.8. By changing the orientation of the tab with respect to the nozzle exit plane, streamwise vortex pairs of opposite sign were generated. This resulted in either an outward selection of jet core fluid into the ambient or an inward indentation of the mixing layer into the core of the jet. A triangular shaped tab with its apex leaning downstream, referred to as a delta tab, was found to be the most effective in influencing the jet evolution. Two delta tabs, spaced 180 degrees apart, completely bifurcated the jet. Four delta tabs increased jet mixing substantially, more than by various other methods tried previously; the mass flux at fourteen jet diameters downstream from the nozzle increased by about 50 percent over that for the no tab case. The tabs were found to be effective in jets with laminar or turbulent boundary layers as well as in jets with low or high core turbulence intensities.

Zaman, K. B. M. Q.↗

NASA NTRS1992

Orthogonal canonical forms for second-order systems

It is shown that a linear second-order system with arbitrary damping cannot be reduced to Hessenberg-triangular form by means of orthogonal transformations. However, it is also shown that such an orthogonal reduction is always possible for the modal damping commonly assumed for models of flexible structures. It is shown that modally damped models can be orthogonally reduced to a new triangular second-order Schur form.

Williams, Trevor↗

NASA NTRS1992

Adaptive explicit and implicit finite element methods for transient thermal analysis

The application of adaptive finite element methods to the solution of transient heat conduction problems in two dimensions is investigated. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Two alternative solution procedures are considered. In the first procedure, the solution is advanced by explicit timestepping, with domain decomposition being used to improve the computational efficiency of the method. In the second procedure, an algorithm for constructing continuous lines which pass only once through each node of the mesh is employed. The lines are used as the basis of a fully implicit method, in which the equation system is solved by line relaxation using a block tridiagonal equation solver. The numerical performance of the two procedures is compared for the analysis of a problem involving a moving heat source applied to a convectively cooled cylindrical leading edge.

Probert, E. J.↗

NASA NTRS1993

Streamwise vorticity generation and mixing enhancement in free jets by delta-tabs

The effect of triangular tabs, placed at the nozzle exit, on the evolution of free jets is investigated. The effect, a large distortion of the jet cross section and a resultant increase in mixing downstream, has been inferred before to be due to a pair of streamwise vortices originating from each tab. In this paper, the generation mechanism of the stream wise vorticity (omega sub x) is considered first. Two sources are postulated. One is the upstream 'pressure hill', produced by the tab, which appears to be the dominant source. Another is due to vortex filaments shed from the sides of the tab and reoriented downstream by the mean shear of the mixing layer. In the case of a 'delta-tab', a triangular tab with its apex leaning downstream, vorticity from the two sources explain the stronger effect in that configuration. Data on the vorticity evolution for the effect of two delta-tabs are presented, up to twelve jet diameters from the exit, which show that the streamwise vortices persist even at the farthest measurement station. The magnitude of omega sub x-maximum decays continually with distance from the nozzle, its ratio to azimuthal vorticity maximum is found to be about 1/5 everywhere. The relative effect of a delta-tab on jets from an axisymmetric nozzle and a 8:1 rectangular nozzle is also studied. The mixing layer distortion is found to be less pronounced in the rectangular case. The jet mixing, as manifested by the mass flux measured at a downstream station, is increased in the axisymmetric jet but it is decreased in the rectangular jet under consideration by the delta-tab.

Zaman, K. B. M. Q.↗