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INTEGRATING A BILINEAR INTERPOLATION FUNCTION ACROSS QUADRILATERAL CELL BOUNDARIES

Description: Computational models of particle dynamics often exchange solution data with discretized continuum-fields using interpolation functions. These particle methods require a series expansion of the interpolation function for two purposes: numerical analyses used to establish the models consistency and accuracy, and logical-coordinate evaluation used to locate particles within a grid. This report presents a new method of developing discrete-expansions for interpolation; they are similar to multi-variable expansions but, unlike a Taylor's series, discrete-expansions are valid throughout a discretized domain. Discrete-expansions are developed herein by parametrically integrating the interpolation function's total-differential between two particles located within separate, non-contiguous cells. Discrete-expansions are valid for numerical analyses since they acknowledge the functional dependence of interpolation and account for mapping discontinuities across cell boundaries. The use of discrete-expansions for logical-coordinate evaluation provides an algorithmically robust and computationally efficient particle localization method. Verification of this new method is demonstrated herein on a simple test problem.
Date: January 8, 2001
Creator: BROCK, J. S.
Partner: UNT Libraries Government Documents Department

INTEGRATING A LINEAR INTERPOLATION FUNCTION ACROSS TRIANGULAR CELL BOUNDARIES

Description: Computational models of particle dynamics often exchange solution data with discretized continuum-fields using interpolation functions. These particle methods require a series expansion of the interpolation function for two purposes: numerical analysis used to establish the model's consistency and accuracy, and logical-coordinate evaluation used to locate particles within a grid. This report presents discrete-expansions for a linear interpolation function commonly used within triangular cell geometries. Discrete-expansions, unlike a Taylor's series, account for interpolation discontinuities across cell boundaries and, therefore, are valid throughout a discretized domain. Verification of linear discrete-expansions is demonstrated on a simple test problem.
Date: April 1, 2000
Creator: WISEMAN, J. R. & BROCK, J. S.
Partner: UNT Libraries Government Documents Department

Combined space and time convergence analysis of a compressible flow algorithm

Description: In this study, we quantify both the spatial and temporal convergence behavior simultaneously for various algorithms for the two-dimensional Euler equations of gasdynamics. Such an analysis falls under the rubric of verification, which is the process of determining whether a simulation code accurately represents the code developers description of the model (e.g., equations, boundary conditions, etc.). The recognition that verification analysis is a necessary and valuable activity continues to increase among computational fluid dynamics practicioners. Using computed results and a known solution, one can estimate the effective convergence rates of a specific software implementation of a given algorithm and gauge those results relative to the design properties of the algorithm. In the aerodynamics community, such analyses are typically performed to evaluate the performance of spatial integrators; analogous convergence analysis for temporal integrators can also be performed. Our approach combines these two usually separate activities into the same analysis framework. To accomplish this task, we outline a procedure in which a known solution together with a set of computed results, obtained for a number of different spatial and temporal discretizations, are employed to determine the complete convergence properties of the combined spatio-temporal algorithm. Such an approach is of particular interest for Lax-Wendroff-type integration schemes, where the specific impact of either the spatial or temporal integrators alone cannot be easily deconvolved from computed results. Unlike the more common spatial convergence analysis, the combined spatial and temporal analysis leads to a set of nonlinear equations that must be solved numerically. The unknowns in this set of equations are various parameters, including the asymptotic convergence rates, that quantify the basic performance of the software implementation of the algorithm.
Date: January 1, 2002
Creator: Kamm, J. R. (James R.); Rider, William & Brock, J. S. (Jerry S.)
Partner: UNT Libraries Government Documents Department

Volume tracking of interfaces having surface tension in two and three dimensions

Description: Solution algorithms are presented for tracking interfaces with piecewise linear (PLIC) volume-of-fluid (VOF) methods on fixed (Eulerian) two-dimensional (2-D) structured and three-dimensional (3-D) structured and unstructured grids. We review the theory of volume tracking methods, derive appropriate volume evolution equations, identify and present solutions to the basic geometric functions needed for interface reconstruction and volume fluxing, and provide detailed algorithm templates for modern 2-D and 3-D PLIC VOF interface tracking methods. We discuss some key outstanding issues for PLIC VOF methods, namely the method used for time integration of fluid volumes (operator splitting, unsplit, Runge-Kutta, etc.) and the estimation of interface normals. We also present our latest developments in the continuum surface force (CSF) model for surface tension, namely extension to 3-D and variable surface tension effects. We identify and focus on key outstanding CSF model issues that become especially critical on fine meshes with high density ratio interfacial flows, namely the surface delta function approximation, the estimation of interfacial curvature, and the continuum surface force scaling and/or smoothing model. Numerical results in two and three dimensions are used to illustrate the properties of these methods.
Date: March 1, 1996
Creator: Kothe, D.B.; Rider, W.J.; Mosso, S.J.; Brock, J.S. & Hochstein, J.I.
Partner: UNT Libraries Government Documents Department