Results for “Poisson solver” — Page 4

NASA NTRS1996

MGLab: An Interactive Multigrid Environment

MGLab is a set of Matlab functions that defines an interactive environment for experimenting with multigrid algorithms. The package solves two-dimensional elliptic partial differential equations discretized using either finite differences or finite volumes, depending on the problem. Built-in problems include the Poisson equation, the Helmholtz equation, a convection-diffusion problem, and a discontinuous coefficient problem. A number of parameters controlling the multigrid V-cycle can be set using a point-and-click mechanism. The menu-based user interface also allows a choice of several Krylov subspace methods, including CG, GMRES(k), and Bi-CGSTAB, which can be used either as stand-alone solvers or as multigrid acceleration schemes. The package exploits Matlab's visualization and sparse matrix features and has been structured to be easily extensible.

Bordner, James↗

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.↗

NASA NTRS1996

A Hybrid Boundary Element-Finite Volume Method for Unsteady Transonic Airfoil Flows

A hybrid boundary element finite volume method for unsteady transonic flow computation has been developed. In this method, the unsteady Euler equations in a moving frame of reference are solved in a small embedded domain (inner domain) around the airfoil using an implicit finite volume scheme. The unsteady full-potential equation, written in the same frame of reference and in the form of the Poisson equation. is solved in the outer domain using the integral equation boundary element method to provide the boundary conditions for the inner Euler domain. The solution procedure is a time-accurate stepping procedure, where the outer boundary conditions for the inner domain are updated using the integral equation -- boundary element solution over the outer domain. The method is applied to unsteady transonic flows around the NACA0012 airfoil undergoing pitching oscillation and ramp motion. The results are compared with those of an implicit Euler equation solver, which is used throughout a large computational domain, and experimental data.

Hu, Hong↗

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.↗

NASA NTRS1995

A comparison of two algorithms for simulating collisionless systems

Two completely different simulation algorithms are compared by applying them to the same stellar dynamical problems: one is a self-consistent field (SCF) method for solving Poisson's equation and the other is a phase-space method for integrating the collisionless Boltzmann equation. We consider simulations of spherical stellar systems which are initially far from equilibrium and relax to their final states by gravitational collapse. The initial conditions consist of either uniform-density spheres or nonequilibrium models having Plummer density profiles, in which velocity dispersions are assigned according to given virial ratios. If a few tens of radial expansion terms with hundreds of thousands of particles are used in the SCF code, excellent agreement is found between the results it generates and those obtained with the phase-space solver, provided that a sufficiently large number of grid cells are employed with the latter. These findings imply that for simulating collisionless systmes over many dynamical times, the SCF approach based on sampling phase space is competitive with the approach treating phase space as a continuous fluid. The results of our tests make it possible to estimate the number of particles and basis functions required in situations like those modeled. Limitations of the SCF method and the choice of an optimal set of basis functions are also discussed.

Hozumi, Shunsuke↗

NASA NTRSDate not supplied

Material Characterization and Modeling of Room Temperature Vulcanizing Silicone

Room Temperature Vulcanizing silicone (RTV) is a high-temperature adhesive that has successfully been used as a gap-filler between Thermal Protection System (TPS) tiles for heatshields on numerous missions. It is also used to bond instrumentation plugs such as temperature and pressure sensors into the heatshields. While RTV has been traditionally assumed to be a non-porous and non-ablating material, numerous experiments have shown that RTV pyrolyzes and becomes highly porous as it is heated. Heating RTV has also shown swelling, or intumescence, which can pose unique problems that lead to roughness induced boundary-layer transition, surface oxide formation and contamination of heat shield sensors. Therefore, it is crucial to understand and model the intumescence phenomenon of RTV. As data for RTV material properties is limited, the first step in modeling RTV is to collect material properties such as pyrolysis mass-loss, microstructure change, virgin and char porosity, etc. which was performed in our initial study. Additionally, thermomechanical properties such as Young’s modulus and Poisson ratio are required for modeling the intumescence of RTV, which were taken from literature and the coefficient of thermal expansion was collected using in-situ heating and Micro Computed Tomography (µ-CT) in previous studies. Finally, numerous other properties such as pyrolysis gas properties, virgin and char thermal conductivity and specific heat were compiled from previous experiments and literature into a material database that can be used for simulations. In Porous Material Analysis Toolbox based on OpenFOAM (PATO) [4], structural mechanics coupled with material response was used for simulating the intumescence of RTV as it is heated. However, since the permeability of the material is very low, the pyrolysis gas creates an internal pressure build-up as the material is being heated, significantly contributing to the deformation of the material. To correctly characterize this phenomenon, additional physics models were implemented into PATO's stress analysis solver, and results were compared with RTV dilatometry test data as a preliminary verification case. Future work will include experiments of RTV at the Plasmatron X facility and the in-situ heating cell with µ-CT, and improvement of simulation tools to more accurately model RTV intumescence.

TPS↗

NASA NTRSDate not supplied

Material Properties and Modeling of Room Temperature Vulcanizing Silicone

Room Temperature Vulcanizing silicone (RTV) is a high-temperature adhesive that has successfully been used as a gap-filler between Thermal Protection System (TPS) tiles for heatshields on numerous missions. It is also used to bond instrumentation plugs such as temperature and pressure sensors into the heatshields. While RTV has been traditionally assumed to be a non-porous and non-ablating material, numerous experiments have shown that RTV pyrolyzes and becomes highly porous as it is heated. Heating RTV has also shown swelling, or intumescence, which can pose unique problems that lead to roughness induced boundary-layer transition, surface oxide formation and contamination of heat shield sensors. Therefore, it is crucial to understand and model the intumescence phenomenon of RTV. As data for RTV material properties is limited, the first step in modeling RTV is to collect material properties such as pyrolysis mass-loss, microstructure change, virgin and char porosity, etc. which was performed in our initial study. Additionally, thermomechanical properties such as Young’s modulus and Poisson ratio are required for modeling the intumescence of RTV, which were taken from literature and the coefficient of thermal expansion was collected using in-situ heating and Micro Computed Tomography (µ-CT) in previous studies. Finally, numerous other properties such as pyrolysis gas properties, virgin and char thermal conductivity and specific heat were compiled from previous experiments and literature into a material database that can be used for simulations. In Porous Material Analysis Toolbox based on OpenFOAM (PATO) [4], structural mechanics coupled with material response was used for simulating the intumescence of RTV as it is heated. However, since the permeability of the material is very low, the pyrolysis gas creates an internal pressure build-up as the material is being heated, significantly contributing to the deformation of the material. To correctly characterize this phenomenon, additional physics models were implemented into PATO's stress analysis solver, and results were compared with RTV dilatometry test data as a preliminary verification case. Future work will include experiments of RTV at the Plasmatron X facility and the in-situ heating cell with µ-CT, and improvement of simulation tools to more accurately model RTV intumescence.

PATO↗