Publications

This report presents a detailed multi-methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. The errors are reported in terms of non-dimensional phase and group speeds, discrete diffusivity, artificial diffusivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis (aka von Neumann analysis) provides an automatic process for separating the spectral behavior of the discrete advective operator into its symmetric dissipative and skew-symmetric advective components. Further it is demonstrated that streamline upwind Petrov-Galerkin and its control-volume finite element analogue, streamline upwind control-volume, produce both an artificial diffusivity and an artificial phase speed in addition to the usual semi-discrete artifacts observed in the discrete phase speed, group speed and diffusivity. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super-convergent behavior in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behavior. While this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework.

ALEGRA is an arbitrary Lagrangian-Eulerian finite element code that emphasizes large distortion and shock propagation. This document describes the user input language for the code.

This report describes research and development of the large eddy simulation (LES) turbulence modeling approach conducted as part of Sandia's laboratory directed research and development (LDRD) program. The emphasis of the work described here has been toward developing the capability to perform accurate and computationally affordable LES calculations of engineering problems using unstructured-grid codes, in wall-bounded geometries and for problems with coupled physics. Specific contributions documented here include (1) the implementation and testing of LES models in Sandia codes, including tests of a new conserved scalar--laminar flamelet SGS combustion model that does not assume statistical independence between the mixture fraction and the scalar dissipation rate, (2) the development and testing of statistical analysis and visualization utility software developed for Exodus II unstructured grid LES, and (3) the development and testing of a novel new LES near-wall subgrid model based on the one-dimensional Turbulence (ODT) model.

Abstract not provided.

The Reproducing Kernel Particle Method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either ``mesh-free'' or ``mesh-full'' methods. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (first and second-order wave) equations. Numerical diffusivity for the former and phase speed for the later are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero. Group speed is also presented for the hyperbolic problems. Excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter.

The Reproducing Kernel Particle Method (RKPM) has many attractive properties that make it ideal for treating a broad class of physical problems. RKPM may be implemented in a mesh-full or a mesh-free manner and provides the ability to tune the method, via the selection of a dilation parameter and window function, in order to achieve the requisite numerical performance. RKPM also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi-scale problems. Although RKPM has many appealing attributes, the method is quite new and its numerical performance is still being quantified with respect to more traditional discretization methods. In order to assess the numerical performance of RKPM, detailed studies of RKPM on a series of model partial differential equations has been undertaken. The results of von Neumann analyses for RKPM semi-discretizations of one and two-dimensional, first and second-order wave equations are presented in the form of phase and group errors. Excellent dispersion characteristics are found for the consistent mass matrix with the proper choice of dilation parameter. In contrast, the influence of row-sum lumping the mass matrix is shown to introduce severe lagging phase errors. A higher-order mass matrix improves the dispersion characteristics relative to the lumped mass matrix but delivers severe lagging phase errors relative to the fully integrated, consistent mass matrix.