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NASA NTRS1990

Parameter estimation problems for distributed systems using a multigrid method

The problem of estimating spatially varying coefficients of partial differential equations is considered from observation of the solution and of the right hand side of the equation. It is assumed that the observations are distributed in the domain and that enough observations are given. A method of discretization and an efficient multigrid method for solving the resulting discrete systems are described. Numerical results are presented for estimation of coefficients in an elliptic and a parabolic partial differential equation.

Ta'asan, Shlomo↗

NASA NTRS1989

Algebraic multigrid methods applied to problems in computational structural mechanics

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Mccormick, Steve↗

NASA NTRS1990

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

Mavriplis, D. J.↗

NASA NTRS1990

Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver

A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.

Bates, J. R.↗

NASA NTRS1990

An investigation of design optimization using a 2-D viscous flow code with multigrid

Computational fluid dynamics (CFD) codes have advanced to the point where they are effective analytical tools for solving flow fields around complex geometries. There is also a need for their use as a design tool to find optimum aerodynamic shapes. In the area of design, however, a difficulty arises due to the large amount of computer resources required by these codes. It is desired to streamline the design process so that a large number of design options and constraints can be investigated without overloading the system. There are several techniques which have been proposed to help streamline the design process. The feasibility of one of these techniques is investigated. The technique under consideration is the interaction of the geometry change with the flow calculation. The problem of finding the value of camber which maximizes the ratio of lift over drag for a particular airfoil is considered. In order to test out this technique, a particular optimization problem was tried. A NACA 0012 airfoil was considered at free stream Mach number of 0.5 with a zero angle of attack. Camber was added to the mean line of the airfoil. The goal was to find the value of camber for which the ratio of lift over drag is a maximum. The flow code used was FLOMGE which is a two dimensional viscous flow solver which uses multigrid to speed up convergence. A hyperbolic grid generation program was used to construct the grid for each value of camber.

Doria, Michael L.↗

NASA NTRS1991

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The system of equations consisting of the full Navier-Stokes equations and two turbulence equations was solved for in the steady state using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time stepping scheme with a stability bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positively. Low Reynolds number modifications to the original two equation model are incorporated in a manner which results in well behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved for, initializing all quantities with uniform freestream values, and resulting in rapid and uniform convergence rates for the flow and turbulence equations.

Mavriplis, D. J.↗

NASA NTRS1990

A multiblock multigrid three-dimensional Euler equation solver

Current aerodynamic designs are often quite complex (geometrically). Flexible computational tools are needed for the analysis of a wide range of configurations with both internal and external flows. In the past, geometrically dissimilar configurations required different analysis codes with different grid topologies in each. The duplicity of codes can be avoided with the use of a general multiblock formulation which can handle any grid topology. Rather than hard wiring the grid topology into the program, it is instead dictated by input to the program. In this work, the compressible Euler equations, written in a body-fitted finite-volume formulation, are solved using a pseudo-time-marching approach. Two upwind methods (van Leer's flux-vector-splitting and Roe's flux-differencing) were investigated. Two types of explicit solvers (a two-step predictor-corrector and a modified multistage Runge-Kutta) were used with multigrid acceleration to enhance convergence. A multiblock strategy is used to allow greater geometric flexibility. A report on simple explicit upwind schemes for solving compressible flows is included.

Cannizzaro, Frank E.↗

NASA NTRS1991

An effective multigrid method for high-speed flows

The use is considered of a multigrid method with central differencing to solve the Navier-Stokes equations for high speed flows. The time dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for 2-D laminar flow over a circular cylinder and a 15 deg compression ramp.

Swanson, R. C.↗

NASA NTRS1991

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The system of equations consisting of the full Navier-Stokes equations and two turbulence equations has been solved for in the steady-state using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time stepping scheme with a stability bound local time-step, while the turbulence equations are advanced in a point-implicit scheme with a time-step which guarantees stability and positivity. Low Reynolds number modifications to the original two-equation model are incorporated in a manner which results in well behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved for, initializing all quantities with uniform freestream values, and resulting in rapid and uniform convergence rates for the flow and turbulence equations.

Mavriplis, D. J.↗

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 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

Development of a pressure based multigrid solution method for complex fluid flows

In order to reduce the computational difficulty associated with a single grid (SG) solution procedure, the multigrid (MG) technique was identified as a useful means for improving the convergence rate of iterative methods. A full MG full approximation storage (FMG/FAS) algorithm is used to solve the incompressible recirculating flow problems in complex geometries. The algorithm is implemented in conjunction with a pressure correction staggered grid type of technique using the curvilinear coordinates. In order to show the performance of the method, two flow configurations, one a square cavity and the other a channel, are used as test problems. Comparisons are made between the iterations, equivalent work units, and CPU time. Besides showing that the MG method can yield substantial speed-up with wide variations in Reynolds number, grid distributions, and geometry, issues such as the convergence characteristics of different grid levels, the choice of convection schemes, and the effectiveness of the basic iteration smoothers are studied. An adaptive grid scheme is also combined with the MG procedure to explore the effects of grid resolution on the MG convergence rate as well as the numerical accuracy.

Shyy, Wei↗

NASA NTRS1992

Development of a multigrid transonic potential flow code for cascades

Finite-volume methods for discretizing transonic potential flow equations have proven to be very flexible and accurate for both two and three dimensional problems. Since they only use local properties of the mapping, they allow decoupling of the grid generation from the rest of the problem. A very effective method for solving the discretized equations and converging to a solution is the multigrid-ADI technique. It has been successfully applied to airfoil problems where O type, C type and slit mappings have been used. Convergence rates for these cases are more than an order of magnitude faster than with relaxation techniques. In this report, we describe a method to extend the above methods, with the C type mappings, to airfoil cascade problems.

Steinhoff, John↗

NASA NTRS1992

Multigrid acceleration of a fractional-step solver in generalized curvilinear coordinate systems

The efficiency of the INS3D-fractional step (FS) code has been significantly enhanced by accelerating the Poisson solver with a multigrid (MG) method. This is apparently the first implementation of the MG method for solving the 3D discrete Poisson-like equation obtained in the derivation of FS solution methods for the incompressible Navier-Stokes equations in generalized nonorthogonal coordinate systems using a staggered arrangement of the variables. The MG solver is insensitive to the geometry, even in cases of highly nonorthogonal clustered meshes, as well as to the free parameters of MG methods.

Rosenfeld, Moshe↗

NASA NTRS1992

A 3D finite element multigrid solver for the Euler equations

A low storage, computationally efficient algorithm for the solution of the compressible Euler equations on unstructured tetrahedral meshes is developed. The algorithm takes the form of a centered scheme with the explicit addition of a high accuracy artificial viscosity and the solution is advanced to steady state by means of a multistage timestepping method. The side-based data structure which is employed enables a clear connection to be established between the proposed algorithm and upwind cell vertex schemes for unstructured meshes. The computational efficiency of the procedure 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 NTRS1992

Investigation of upwind, multigrid, multiblock numerical schemes for three dimensional flows. Volume 1: Runge-Kutta methods for a thin layer Navier-Stokes solver

A state-of-the-art computer code has been developed that incorporates a modified Runge-Kutta time integration scheme, upwind numerical techniques, multigrid acceleration, and multi-block capabilities (RUMM). A three-dimensional thin-layer formulation of the Navier-Stokes equations is employed. For turbulent flow cases, the Baldwin-Lomax algebraic turbulence model is used. Two different upwind techniques are available: van Leer's flux-vector splitting and Roe's flux-difference splitting. Full approximation multi-grid plus implicit residual and corrector smoothing were implemented to enhance the rate of convergence. Multi-block capabilities were developed to provide geometric flexibility. This feature allows the developed computer code to accommodate any grid topology or grid configuration with multiple topologies. The results shown in this dissertation were chosen to validate the computer code and display its geometric flexibility, which is provided by the multi-block structure.

Cannizzaro, Frank E.↗

NASA NTRS1992

Adaptive multigrid domain decomposition solutions for viscous interacting flows

Several viscous incompressible flows with strong pressure interaction and/or axial flow reversal are considered with an adaptive multigrid domain decomposition procedure. Specific examples include the triple deck structure surrounding the trailing edge of a flat plate, the flow recirculation in a trough geometry, and the flow in a rearward facing step channel. For the latter case, there are multiple recirculation zones, of different character, for laminar and turbulent flow conditions. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable, non-staggered grid computation.

Rubin, Stanley G.↗

NASA NTRS1992

An effective multigrid method for high-speed flows

Swanson, R. C.↗