Jerry S. Brock

Combined space and time convergence analysis of a compressible flow algorithm

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.

Non-linear Error Ansatz Models in Space and Time for Solution Verification

In computational physics and engineering, partial differential equations that govern the evolution of state variables are discretized for implementation and solution using finite-digit computer arithmetic. Solution verification is the activity that verifies that computed solutions of the discretized equations converge to the exact solution of the continuous equations. The state-of-the-practice is to estimate convergence rates and numerical errors from several computed solutions obtained with a sequence of coarse-to-refined grids. In addition to being valid only in the asymptotic regime of convergence, this approach to verification relies on three unverified assumptions. First, convergence must be monotonic. Second, explicit coupling between grid resolution or element size and time increment is neglected. Third, higher-order terms of the error have no significant influence and can be neglected. The main contribution of this work is to demonstrate that statistical methods to design computer experiments, analyze the output variance, screen significant effects, and fit statistical models can improve our ability to analyze solution convergence. The approach proposed allows one to best-fit non-linear error Ansatz models in space and time. This proposed method is also general and does not rely upon the aforementioned three assumptions. Results are illustrated with four test problems, including the Noh problem that evaluates a converging shocked flow and the Sedov blast wave that evaluates a diverging shocked flow. Publication approved for unlimited, public release (LA-UR-05-8228, Unclassified).

CONSISTENT METRICS FOR CODE VERIFICATION

1

Enhanced Verification Test Suite for Physics Simulation Codes

This document discusses problems with which to augment, in quantity and in quality, the existing tri-laboratory suite of verification problems used by Los Alamos National Laboratory (LANL), Lawrence Livermore National Laboratory (LLNL), and Sandia National Laboratories (SNL). The purpose of verification analysis is demonstrate whether the numerical results of the discretization algorithms in physics and engineering simulation codes provide correct solutions of the corresponding continuum equations.

Discrete-expansions for linear interpolation functions

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 models consistency and accuracy, and logical-coordinate evaluation used to locate particles within a grid. This report presents discrete-expansions for two linear interpolation functions commonly used within triangular and tetrahedral cell geometries. Application of the linear discrete-expansions for numerical analysis and localization within particle methods is outlined and discussed.

The dynamic sphere test problem

In this manuscript we define the dynamic sphere problem as a spherical shell composed of a homogeneous, linearly elastic material. The material exhibits either isotropic or transverse isotropic symmetry. When the problem is formulated in material coordinates, the balance of mass equation is satisfied automatically. Also, the material is assumed to be kept at constant temperature, so the only relevant equation is the equation of motion. The shell has inner radius r{sub i} and outer radius r{sub o}. Initially, the shell is at rest. We assume that the interior of the shell is a void and we apply a time-varying radial stress on the outer surface.

A Finite-Difference Logical-Coordinate Evaluation Method for Particle Localization

Particle methods require robust and efficient advection and localization methods which include logical-coordinate evaluation. The ability to compute logical coordinates with existing methods is, however, not guaranteed within grids that contain nonlinear elements. This note presents a new logical-coordinate evaluation method, based on finite-differences, that provides a robust and efficient solution for coordinate transformation. The new methods enhancedcapabilities are d emonstratedon a simple test problem. Particle methods, computational models of particle dynamics, require robust and efficient advection and localization methods. Localization methods 1) - 6) combine cell-searching and logical-coordinate evaluation methods to define particle-grid connectivity. This connectivity data consist of the identity of the grid cell in which the particle resides and the particle’s position relative to that cell, its transformed or logical coordinates. Cell-searching or guessing methods typically use the particle’s logical coordinates to both direct and halt the search. Particle methods are, therefore, predicated on robust and efficient logical-coordinate evaluation methods. Existing logical-coordinate evaluation methods are generalized in Ref. 6). Solutions usingthis technique are g uaranteed, and the coordinate vector is bound between known transformation limits, if the particle resides within the guessed cell. In contrast, an unbound coordinate solution may fail to exist for nonlinear grid element transformations. The problem of interest, however, occurs when these coordinates are unbound because only then will the particle have exited the cell duringadvection. This note continues by presentinga new evaluation method that is less sensitive to coordinate transformation. A test problem concludes this note. §2. Finite-difference evaluation method

The Finite Deformation Dynamic Sphere Test Problem

In this manuscript we describe test cases for the dynamic sphere problem in presence of finite deformations. The spherical shell in exam is made of a homogeneous, isotropic or transverse isotropic material and elastic and elastic-plastic material behaviors are considered. Twenty cases, (a) to (t), are thus defined combining material types and boundary conditions. The inner surface radius, the outer surface radius and the materials density are kept constant for all the considered test cases and their values are ri = 10mm, ro = 20mm and p = 1000Kg/m3 respectively.

The jump-off velocity of an impulsively loaded spherical shell

We consider a constant temperature spherical shell of isotropic, homogeneous, linearly elastic material with density {rho} and Lame coefficients {lambda} and {mu}. The inner and outer radii of the shell are r{sub i} and r{sub o}, respectively. We assume that the inside of the shell is a void. On the outside of the shell, we apply a uniform, time-varying pressure p(t). We also assume that the shell is initially at rest. We want to compute the jump-off time and velocity of the pressure wave, which are the first time after t = 0 at which the pressure wave from the outer surface reaches the inner surface. This analysis computes the jump-off velocity and time for both compressible and incompressible materials. This differs substantially from [3], where only incompressible materials are considered. We will consider the behavior of an impulsively loaded, exponentially decaying pressure wave p(t) = P{sub 0{sup e}}{sup -{alpha}t}, where {alpha} {ge} 0. We notice that a constant pressure wave P(t) = P{sub 0} is a special case ({alpha} = 0) of a decaying pressure wave. Both of these boundary conditions are considered in [3].

Evaluating Convergence Rates for Calculation-Verification Analyses

Calculation verification combines an assessment of solution convergence with a discretization-error estimate. A key requirement of calculation-verification analyses is the evaluation of an observed convergence rate. When calculation verification is applied to monotonic simulation data that obey the standard power-law error model and exercised using non-uniform grid refinement, the analysis requires the numerical solution of a nonlinear expression, termed the governing equation, to obtain the observed convergence rate. This effort explores both rational and normalized forms of the governing equation, where each formulation includes terms with a grid-refinement ratio with a convergence rate exponent. The normalized form of the governing equation contains multiple roots where one of the roots is the correct value and the second root is zero. The slope of the normalized formulations can be positive, negative or zero, which may be problematic for some rootfinding methods. In contrast, the rational formulation contains only one root and its slope is always positive. One recommendation from this study would be to solve a rational formulation of the calculation-verification governing equation.