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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS2005

Full Multigrid Flow Solver

FMG3D (full multigrid 3 dimensions) is a pilot computer program that solves equations of fluid flow using a finite difference representation on a structured grid. Infrastructure exists for three dimensions but the current implementation treats only two dimensions. Written in Fortran 90, FMG3D takes advantage of the recursive subroutine feature, dynamic memory allocation, and structured-programming constructs of that language. FMG3D supports multi-block grids with three types of block-to-block interfaces: periodic, C-zero, and C-infinity. For all three types, grid points must match at interfaces. For periodic and C-infinity types, derivatives of grid metrics must be continuous at interfaces. The available equation sets are as follows: scalar elliptic equations, scalar convection equations, and the pressure-Poisson formulation of the Navier-Stokes equations for an incompressible fluid. All the equation sets are implemented with nonzero forcing functions to enable the use of user-specified solutions to assist in verification and validation. The equations are solved with a full multigrid scheme using a full approximation scheme to converge the solution on each succeeding grid level. Restriction to the next coarser mesh uses direct injection for variables and full weighting for residual quantities; prolongation of the coarse grid correction from the coarse mesh to the fine mesh uses bilinear interpolation; and prolongation of the coarse grid solution uses bicubic interpolation.

Mineck, Raymond E.↗
DOE OSTI2016

Discretization-Accuracy Convergence for Full Algebraic Multigrid

Full multigrid (FMG) is well known for converging to the level of discretization accuracy in a single cycle for a wide class of partial differential equations when the multigrid hierarchy is derived from problem geometry. When applying an FMG cycle to a hierarchy generated by algebraic multigrid (AMG), however, this scalable convergence to discretization accuracy is usually lost. Furthermore, this paper examines the cause of this loss and explores some improvements to standard AMG interpolation that can restore single-cycle convergence to discretization accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

A Full Multigrid Method for the Self-Adjoint Angular Flux Equation Based on Cellwise Block Jacobi Iteration

Previous work by Densmore et al. has demonstrated that multigrid methods for the self-adjoint angular flux (SAAF) equation with cellwise block Jacobi smoothing can be extremely effective at solving for the angular flux in one dimension. The extension of that work to two dimensions demonstrated that while the smoother works well, the use of a direct coarse-grid approximation can severely limit the rate of convergence for optically thin cells. In that paper, this convergence limit was evaded through the use of a partially Galerkin coarse-grid approximation. However, relying on a Galerkin coarse-grid approximation is not ideal as constructing the coarse-grid discretization in this approach is complicated. The motivation for this summary is to find a method that can work without the need for a Galerkin coarse-grid approximation. The Full Multigrid (FMG) method is well known as a method that can theoretically solve a problem to its discretization accuracy with optimal linear computational complexity, that is in just O(N) operations for N unknowns. Indeed, we develop an FMG method for the 2-D SAAF equation that yields discretization accuracy and does not require the use of a Galerkin coarse-grid approximation. Until recently, aside from proofs about the order of the error for FMG, little work had been done in quantifying when and how FMG can obtain discretization accuracy. However, recent work by Rodrigo et al. has introduced a Fourier analysis technique called worst-case right-hand side analysis used to calculate the FMG accuracy measure. The FMG accuracy measure is the upper bound on the constant of proportionality between the error of the FMG solution and the discretization error. In this summary, we develop and implement an FMG method for the 2-D Cartesian SAAF equations. We then perform worst-case right-hand side analysis and give numerical examples that demonstrate under what circumstances the method can solve the 2-D SAAF equations to discretization accuracy, for optically thin cells, without the need for a Galerkin coarse-grid approximation. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1993

Application of the multigrid solution technique to hypersonic entry vehicles

A multigrid solution procedure has been incorporated in a version of the Langley Aerothermodynamic Upwind Relaxation Algorithm. The multigrid scheme is based on the Full Approximation Storage approach and uses Full Multigrid to obtain a well defined fine mesh starting solution. Predictions were obtained using standard transfer operators and a 'V-cycle' was used to control grid sequencing. Computed hypersonic flow solutions compared with experimental data for a 15 degree sphere cone, blended-wing body, and shuttle-like geometries are presented. It is shown that the algorithm accurately predicts heating rates, and when compared with the single grid algorithm computes solutions in one-third the computational time.

Greene, Francis A.↗
NASA NTRS1984

Vectorized multigrid Poisson solver for the CDC CYBER 205

The full multigrid (FMG) method is applied to the two dimensional Poisson equation with Dirichlet boundary conditions. This has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method. Data structure and programming considerations and techniques are discussed, accompanied by performance details.

Barkai, D.↗
NASA NTRS2004

Textbook Multigrid Efficiency for Leading Edge Stagnation

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading- edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (F.4S) cycle per grid. Asymptotic convergence rates of the F.4S cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

Diskin, Boris↗
NASA NTRS2004

Textbook Multigrid Efficiency for Leading Edge Stagnation

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading-edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (FAS) cycle per grid. Asymptotic convergence rates of the FAS cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

Diskin, Boris↗
NASA NTRS1996

Effective Boundary Treatment for the Biharmonic Dirichlet Problem

The biharmonic equation can be rewritten as a system of two Poisson equations. Multigrid solution of this system is expected to converge with the same amount of work as solving two Poisson equations, requiring less than 70 floating point operations (scalar multiply or addition) per fine grid point to reach a solution using an FMG algorithm. For periodic boundary conditions, this goal is attained by simple, straightforward application of multigrid. For Dirichlet boundary conditions, however, convergence is impeded by poor interaction with the boundaries. Attempts to overcome the slowness without specifically addressing the boundaries have resulted in multigrid algorithms not attaining the Poisson convergence rate. We present three methods of boundary treatment with which full multigrid efficiency can be obtained. All implement an approach described by Brandt, concentrating some additional effort near the boundary. The first approach simply adds a number of relaxation sweeps over points close to the boundary. The second uses joint relaxation on near-boundary points. The third method takes something from each of the first two methods, resulting in a solver more suitable for highly parallel applications.

Brandt, A.↗
DOE OSTI2016

Segmental Refinement: A Multigrid Technique for Data Locality

In this paper, we investigate a domain decomposed multigrid technique, termed segmental refinement, for solving general nonlinear elliptic boundary value problems. We extend the method first proposed in 1994 by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid is eliminated on fine grids, with modest amounts of extra work and storage, while maintaining the asymptotic exactness of full multigrid. We observe an accuracy dependence on the segmental refinement subdomain size, which was not considered in the original analysis. Finally, we present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores on a Cray XC30.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Prediction of Film Cooling on Gas Turbine Airfoils

A three-dimensional Navier-Stokes analysis tool has been developed in order to study the effect of film cooling on the flow and heat transfer characteristics of actual turbine airfoils. An existing code (Arnone et al., 1991) has been modified for the purpose. The code is an explicit, multigrid, cell-centered, finite volume code with an algebraic turbulence model. Eigenvalue scaled artificial dissipation and variable-coefficient implicit residual smoothing are used with a full-multigrid technique. Moreover, Mayle's transition criterion (Mayle, 1991) is used. The effects of film cooling have been incorporated into the code in the form of appropriate boundary conditions at the hole locations on the airfoil surface. Each hole exit is represented by several control volumes, thus providing an ability to study the effect of hole shape on the film-cooling characteristics. Comparison is fair with near mid-span experimental data for four and nine rows of cooling holes, five on the shower head, and two rows each on the pressure and suction surfaces. The computations, however, show a strong spanwise variation of the heat transfer coefficient on the airfoil surface, specially with shower-head cooling.

Garg, Vijay K.↗
NASA NTRS1981

Multigrid solvers on parallel computers

Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.

Brandt, A.↗
NASA NTRS1996

Multigrid Approach to Incompressible Viscous Cavity Flows

Two-dimensional incompressible viscous driven-cavity flows are computed for Reynolds numbers on the range 100-20,000 using a loosely coupled, implicit, second-order centrally-different scheme. Mesh sequencing and three-level V-cycle multigrid error smoothing are incorporated into the symmetric Gauss-Seidel time-integration algorithm. Parametrics on the numerical parameters are performed, achieving reductions in solution times by more than 60 percent with the full multigrid approach. Details of the circulation patterns are investigated in cavities of 2-to-1, 1-to-1, and 1-to-2 depth to width ratios.

Wood, William A.↗
NASA NTRS1993

Viscous analysis of three-dimensional rotor flows using a multigrid method

A three-dimensional code for rotating blade-row flow analysis was developed. The space discretization uses a cell-centered scheme with eigenvalues scaling for the artificial dissipation. The computational efficiency of a four-stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full-multigrid method. An application is presented for the NASA rotor 67 transonic fan. Due to the blade stagger and twist, a zonal, non-periodic H-type grid is used to minimize the mesh skewness. The calculation is validated by comparing it with experiments in the range from the maximum flow rate to a near-stall condition. A detailed study of the flow structure near peak efficiency and near stall is presented by means of pressure distribution and particle traces inside boundary layers.

Arnone, A.↗
NASA NTRS1991

Multigrid calculation of three-dimensional viscous cascade flows

A 3-D code for viscous cascade flow prediction was developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full multigrid method. The Baldwin-Lomax eddy viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Arnone, A.↗
NASA NTRS1991

Multigrid calculation of three-dimensional viscous cascade flows

A three-dimensional code for viscous cascade flow prediction has been developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four-stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full-multigrid method. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large-scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Arnone, A.↗
NASA NTRS1996

A Multigrid Algorithm for Immersed Interface Problems

Many physical problems involve interior interfaces across which the coefficients in the problem, the solution, its derivatives, the flux, or the source term may have jumps. These interior interfaces may or may not align with a underlying Cartesian grid. Zhilin Li, in his dissertation, showed how to discretize such elliptic problems using only a Cartesian grid and the known jump conditions to second order accuracy. In this paper, we describe how to apply the full multigrid algorithm in this context. In particular, the restriction, interpolation, and coarse grid problem will be described. Numerical results for several model problems are given to demonstrate that good rates can be obtained even when jumps in the coefficients are large and do not align with the grid.

Adams, Loyce↗
NASA NTRS1993

Looking for O(N) Navier-Stokes solutions on non-structured meshes

Multigrid methods are good candidates for the resolution of the system arising in numerical fluid dynamics. However, the question is to know if those algorithms which are efficient for the Poisson equation on structured meshes will still apply well to the Euler and Navier-Stokes equations on unstructured meshes. The study of elliptic problems leads us to define the conditions where a full multigrid strategy has O(N) complexity. The aim of this paper is to build a comparison between the elliptic theory and practical CFD problems. First, as an introduction, we will recall some basic definitions and theorems applied to a model problem. The goal of this section is to point out the different properties that we need to produce an FMG algorithm with O(N) complexity. Then, we will show how we can apply this theory to the fluid dynamics equations such as Euler and Navier-Stokes equations. At last, we present some results which are 2nd-order accurate and some explanations about the behavior of the FMG process.

Morano, Eric↗
NASA NTRS1993

Looking for O(N) Navier-Stokes solutions on non-structured meshes

Multigrid methods are good candidates for the resolution of the system arising in Numerical Fluid Dynamics. However, the question is to know if those algorithms which are efficient for the Poissan equation on structured meshes will still apply well to the Euler and Navier-Stokes equations on unstructured meshes. The study of elliptic problems leads us to define the conditions where a Full Multigrid strategy has O(N) complexity. The aim of this paper is to build a comparison between the elliptic theory and practical CFD problems. First, as an introduction, we will recall some basic definitions and theorems applied to a model problem. The goal of this section is to point out the different properties that we need to produce an FMG algorithm with O(N) complexity. Then, we will show how we can apply this theory to the fluid dynamics equations such as Euler and Navier-Stokes equations. At last, we present some results which are 2nd-order accurate and some explanations about the behavior of the FMG process.

Morano, Eric↗