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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

A method for generating 3-D finite difference grids about or within arbitrary shapes is presented. The 3-D Poisson equations are solved numerically, with values for the inhomogeneous terms found automatically by the algorithm. Those inhomogeneous terms have the effect near boundaries of reducing cell skewness and imposing arbitrary cell height. The method allows the region of interest to be divided into zones (blocks), allowing the method to be applicable to almost any physical domain. A FORTRAN program called 3DGRAPE has been written to implement the algorithm. Lastly, a method for redistributing grid points along lines normal to boundaries will be described.

Sorenson, Reese L.↗
NASA NTRS1988

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

A method for generating 3-D finite difference grids about or within arbitrary shapes is presented. The 3-D Poisson equations are solved numerically, with values for the inhomogeneous terms found automatically by the algorithm. Those inhomogeneous terms have the effect near boundaries of reducing cell skewness and imposing arbitrary cell height. The method allows the region of interest to be divided into zones (blocks), allowing the method to be applicable to almost any physical domain. A FORTRAN program called 3DGRAPE has been written to implement the algorithm. Lastly, a method for redistributing grid points along lines normal to boundaries will be described.

Sorenson, Reese L.↗
NASA NTRS1984

Two-dimensional computer simulation of EMVJ and grating solar cells under AMO illumination

A computer program, SCAP2D (Solar Cell Analysis Program in 2-Dimensions), is used to evaluate the Etched Multiple Vertical Junction (EMVJ) and grating solar cells. The aim is to demonstrate how SCAP2D can be used to evaluate cell designs. The cell designs studied are by no means optimal designs. The SCAP2D program solves the three coupled, nonlinear partial differential equations, Poisson's Equation and the hole and electron continuity equations, simultaneously in two-dimensions using finite differences to discretize the equations and Newton's Method to linearize them. The variables solved for are the electrostatic potential and the hole and electron concentrations. Each linear system of equations is solved directly by Gaussian Elimination. Convergence of the Newton Iteration is assumed when the largest correction to the electrostatic potential or hole or electron quasi-potential is less than some predetermined error. A typical problem involves 2000 nodes with a Jacobi matrix of order 6000 and a bandwidth of 243.

Gray, J. L.↗
NASA NTRS1999

Piezoelectric Field Enhanced Second-Order Nonlinear Optical Susceptibilities in Wurtzite GaN/AlGaN Quantum Wells

Second-order nonlinear optical processes including second-harmonic generation, optical rectification, and difference-frequency generation associated with intersubband transitions in wurtzite GaN/AlGaN quantum well (QW) are investigated theoretically. Taking into account the strain-induced piezoelectric (PZ) effects, we solve the electronic structure of the QW from coupled effective-mass Schrodinger equation and Poisson equation including the exchange-correlation effect under the local-density approximation. We show that the large PZ field in the QW breaks the symmetry of the confinement potential profile and leads to large second-order susceptibilities. We also show that the interband optical pump-induced electron-hole plasma results in an enhancement in the maximum value of the nonlinear coefficients and a redshift of the peak position in the nonlinear optical spectrum. By use of the difference-frequency generation, THz radiation can be generated from a GaN/Al(0.75)Ga(0.25)N with a pump laser of 1.55 micron.

Liu, Ansheng↗
NASA NTRS1979

The accurate solution of Poisson's equation by expansion in Chebyshev polynomials

A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.

Haidvogel, D. B.↗
NASA NTRS1980

A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson's equation

A method for generating two dimensional finite difference grids about airfoils and other shapes by the use of the Poisson differential equation is developed. The inhomogeneous terms are automatically chosen such that two important effects are imposed on the grid at both the inner and outer boundaries. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. A FORTRAN computer program has been written to use this method. A description of the program, a discussion of the control parameters, and a set of sample cases are included.

Sorenson, R. L.↗
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 NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗
NASA NTRS1984

A vectorized solution for incompressible flow

An algorithm is developed to obtain solutions to the unsteady Reynolds-averaged incompressible Navier-Stokes equations in general curvilinear coordinates on a vector processor. The governing equations are in nonconservative form with the velocity and pressure as dependent variables. Two momentum equations and the Poisson equation for pressure form a set of three governing equations for three flow field unknowns: u, v, and p. The governing equations and boundary conditions are expressed in terms of boundary-conforming curvilinear coordinates, and a checkerboard SOR iteration is used to solve the governing equations. Several possible sequences for a checkerboard SOR iteration are investigated for finding the best overall convergence rate. The efficiency and capability of the present algorithm was assessed using the example of an 18 percent thick NACA 66(3)018 airfoil at zero degree angle of attack for chord Reynolds number range 1000-40,000.

Patel, N. R.↗
NASA NTRS1985

The electrostatics of a dusty plasma

Voyager 1 and 2 observations of the Saturnian ring system have led to the discovery of several interesting phenomena associated with its fine dust component. The dust grains are immersed in a plasma. The present paper is concerned with the electrostatics of a dusty plasma, giving particular attention to the capacitance and charge of an individual grain in the presence of neighboring grains and the surrounding plasma. The Poisson equation and gauge considerations are discussed along with a solution of the Poisson equation for several models, taking into account impermeable grains, permeable grains, and a spherical capacitor model. A comparison of potential shapes for three models is conducted, and the effect of neighbors on charging currents is explored.

Whipple, E. C.↗
NASA NTRS1990

Analysis of energy states in modulation doped multiquantum well heterostructures

A precise and effective numerical procedure to model the band diagram of modulation doped multiquantum well heterostructures is presented. This method is based on a self-consistent iterative solution of the Schroedinger equation and the Poisson equation. It can be used rather easily in any arbitrary modulation-doped structure. In addition to confined energy subbands, the unconfined states can be calculated as well. Examples on realistic device structures are given to demonstrate capabilities of this procedure. The numerical results are in good agreement with experiments. With the aid of this method the transitions involving both the confined and unconfined conduction subbands in a modulation doped AlGaAs/GaAs superlattice, and in a strained layer InGaAs/GaAs superlattice are identified. These results represent the first observation of unconfined transitions in modulation doped multiquantum well structures.

Ji, G.↗
NASA NTRS1999

Hierarchical Approach to 'Atomistic' 3-D MOSFET Simulation

We present a hierarchical approach to the 'atomistic' simulation of aggressively scaled sub-0.1 micron MOSFET's. These devices are so small that their characteristics depend on the precise location of dopant atoms within them, not just on their average density. A full-scale three-dimensional drift-diffusion atomistic simulation approach is first described and used to verify more economical, but restricted, options. To reduce processor time and memory requirements at high drain voltage, we have developed a self-consistent option based on a solution of the current continuity equation restricted to a thin slab of the channel. This is coupled to the solution of the Poisson equation in the whole simulation domain in the Gummel iteration cycles. The accuracy of this approach is investigated in comparison to the full self-consistent solution. At low drain voltage, a single solution of the nonlinear Poisson equation is sufficient to extract the current with satisfactory accuracy. In this case, the current is calculated by solving the current continuity equation in a drift approximation only, also in a thin slab containing the MOSFET channel. The regions of applicability for the different components of this hierarchical approach are illustrated in example simulations covering the random dopant-induced threshold voltage fluctuations, threshold voltage lowering, threshold voltage asymmetry, and drain current fluctuations.

Asenov, Asen↗
NASA NTRS1992

Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations

A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.

Soh, Woo Y.↗
NASA NTRS1991

Simulations of curved turbulent boundary layers

The objective of this work is to develop a space-time accurate numerical method for the solution of incompressible Navier-Stokes equations in generalized coordinates. The resulting code is to be used for direct and large-eddy simulation of turbulence in complex geometries. In a previous paper, the system of Navier-Stokes equations in general curvilinear coordinates was solved by a second-order accurate finite-difference scheme. Satisfactory results were obtained for several flows in two and three dimensions. The system of Navier-Stokes for the fluxes are given in Orlandi (1989). The main deficiency of the numerical scheme was the large CPU time required for the solution of the Poisson equation for the 'pressure' field. The point SOR relaxation, in conjunction with a multigrid scheme, was used for the Poisson equation. In some cases, particularly with very fine grids, it was impossible to obtain a divergent-free flow. A preliminary attempt is made to compute the spatially evolving flow of Swearingen & Blackwelder. To reduce the streamwise distance, the inflow was at a distance x = 60 cm from the leading edge.

Orlandi, Paolo↗
NASA NTRS1998

Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.

Wilson, Robert V.↗