Technical Evaluation Report for Symposium AVT-147: Computational Uncertainty in Military Vehicle Design

The complexity of modern military systems, as well as the cost and difficulty associated with experimentally verifying system and subsystem design makes the use of high-fidelity based simulation a future alternative for design and development. The predictive ability of such simulations such as computational fluid dynamics (CFD) and computational structural mechanics (CSM) have matured significantly. However, for numerical simulations to be used with confidence in design and development, quantitative measures of uncertainty must be available. The AVT 147 Symposium has been established to compile state-of-the art methods of assessing computational uncertainty, to identify future research and development needs associated with these methods, and to present examples of how these needs are being addressed and how the methods are being applied. Papers were solicited that address uncertainty estimation associated with high fidelity, physics-based simulations. The solicitation included papers that identify sources of error and uncertainty in numerical simulation from either the industry perspective or from the disciplinary or cross-disciplinary research perspective. Examples of the industry perspective were to include how computational uncertainty methods are used to reduce system risk in various stages of design or development

Application of Symplectic Integration on a Dynamical System

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic integrators, and often, these techniques are developed for well-understood Hamiltonian systems such as Hill’s lunar equation. In this presentation, we explore how well symplectic techniques developed for well-understood systems (specifically, Hill’s Lunar equation) address discretization errors in MD systems which fail for one or more reasons

Estimation of grid-induced errors in computational fluid dynamics solutions using a discrete error transport equation

Computational fluid dynamics (CFD) has become a widely used tool in research and engineering for the study of a wide variety of problems. However, confidence in CFD solutions is still dependent on comparisons with experimental data. In order for CFD to become a trusted resource, a quantitative measure of error must be provided for each generated solution. Although there are several sources of error, the effects of the resolution and quality of the computational grid are difficult to predict y priori. This grid-induced error is most often attenuated by performing a grid refinement study or using solution adaptive grid refinement. While these methods are effective, they can also be computationally expensive and even impractical for large, complex problems. This work presents a method for estimating the grid-induced error in CFD solutions of the Navier-Stokes and Euler equations using a single grid and solution or a series of increasingly finer grids and solutions. The method is based on the discrete error transport equation (DETE), which is derived directly from the discretized PDE and provides a value of the error at every cell in the computational grid. The DETE is developed for two-dimensional, laminar Navier-Stokes and Euler equations within a generalized unstructured finite volume scheme, such that an extension to three dimensions and turbulent flow would follow the same approach. The usefulness of the DETE depends on the accuracy with which the source term, the grid-induced residual, can be modeled. Three different models for the grid-induced residual were developed: the AME model, the PDE model, and the extrapolation model. The AME model consists of the leading terms of the remainder of a simplified modified equation. The PDE model creates a polynomial fit of the CFD solution and then uses the original PDE in differential form to calculate the residual. Both the AME and PDE are used with a single grid and solution. The extrapolation model uses a fine grid solution to calculate the grid-induced residual on the coarse grid and then extrapolates that residual back to the fine grid. The DETE and residual models were then evaluated for four flow problems: (1) steady flow past a circular cylinder; (2) steady, transonic flow past an airfoil; (3) unsteady flow of an isentropic vortex; (4) unsteady flow past a circular cylinder with vortex shedding. Results demonstrate the fidelity of the DETE with each residual model as well as usefulness of the DETE as a tool for predicting the grid-induced error in CFD solutions

Multiextrapolação de Richardson e esquemas de 1a e 2a ordens, mistos e Crank-Nicolson sobre as esquações 2D de advecção-difusão e Fourier

Resumo: A análise de erros é objeto de estudo de grande importância em Dinâmica dos Fluidos Computacional (CFD). A acurácia e a confiabilidade da solução são algumas das dificuldades relacionadas a tal investigação. Para atender essas condições, a análise assintótica de soluções numéricas provê o conhecimento do comportamento de técnicas numéricas aplicadas na solução de modelos matemáticos que descrevem problemas físicos comumente utilizados em Engenharia. O objetivo principal desse trabalho é verificar a influência de esquemas híbridos como o método de correção adiada (MCA) e o método de Crank-Nicolson, bem como o efeito de parâmetros numéricos e físicos (número de Péclet) sobre a redução do erro de discretização com multiextrapolações de Richardson (MER). Para tanto, são consideradas as equações de advecção-difusão e equação de Fourier, ambas bidimensionais com termo fonte e condições de contorno de Dirichlet. As simulações numéricas foram realizadas com base no conhecimento da solução analítica obtida com o método das soluções fabricadas (MSF). As aproximações são desenvolvidas por meio do método de diferenças finitas com esquemas de 1ª e 2ª ordens mistos (para a equação de advecção-difusão) e de Crank-Nicolson (para a equação de Fourier). Na simulação numérica é utilizada a precisão quádrupla e o critério de parada baseado na norma l1 média. Na solução do sistema de equações foi utilizado o método multigrid. Para a análise a posteriori com MER foram deduzidas as ordens verdadeiras a priori através da expansão da série de Taylor com até três termos para ambas as equações e todas as variáveis de interesse. Com base na estimativa do erro de discretização por meio de MER, malhas refinadas são criadas para alcançar a acurácia de resultados indicando assim as ordens verdadeiras dos mesmos. Dentre as conclusões, constata-se que as ordens do erro de discretização obtidas a posteriori com MER comprovam a sua utilidade e eficiência para a estimativa de erros de discretização. Assintoticamente, para esquemas híbridos MCA e Crank-Nicolson, o valor do módulo do erro fica entre os dos esquemas puros. A ordem assintótica do esquema híbrido MCA é igual à ordem assintótica do esquema puro de menor ordem, o que não ocorre para o caso em que o método de Crank-Nicolson é aplicado. Neste caso, a ordem assintótica é igual à ordem do esquema puro de maior ordem. Observa-se que o efeito de pequenos valores no número de Péclet sobre a magnitude do erro de discretização obtido com MER, apresentam os melhores resultados