Session: 2.1 - Recent development in CFD and Verification and Validation methods

Paper Number: 158275

158275 - Robust Solution Verification Experiments on Nonuniform Meshes 

Abstract: 

Solution verification of computational analyses often utilizes Richardson extrapolation to model the convergence behavior of the studied model. The premise of Richardson extrapolation is that a power series expansion of an infinitely differentiable numerical solution can describe the discretization error of the numerical scheme when solved in the asymptotic regime. Indeed, when Richardson extrapolation accurately models the convergence behavior of the scheme, the analyst can back-compute the discretization error by fitting the power law model to the solution data points obtained from a family of uniformly refined meshes. In theory, three solutions should be enough to compute the unknown order of the power law. However, in practical settings, these solutions are noisy since they come from simulations that do not conform perfectly to the assumptions of the power law model. Commonly broken assumptions include not solving the computational model in the asymptotic regime due to high computational costs, discontinuities (e.g., sharp geometry changes or shock waves) breaking the assumption of an infinitely differentiable solution, and non-uniform mesh refinement. These broken assumptions add uncertainty to the solution data points. A common solution to this issue is to obtain solutions on more than three meshes and fit the model with a least-squares approach; in the ideal case, the observed order of convergence of the least-squares fit is the order of accuracy of the numerical scheme. Unfortunately, on top of the existing imperfections, least-square fits require enough data points to prevent over-fitting. The general guideline is to collect two to ten times as many data points as model parameters, totaling six to thirty solutions for standard Richardson extrapolation. Again, computational costs prohibit the analyst from collecting enough data points per this general guideline. Of interest to this work is how to prevent over-fitting given noisy solution data points and a low number of solution data points.

This work presents regularization strategies to weakly constrain the Richardson extrapolation model based on the prior information on the model convergence behavior. This viewpoint aligns with the Bayesian perspective; the prior is the assumed convergence behavior, and the observations are the individual solutions. Previous work has investigated Bayesian approaches for solution verification and showed promising improvement in the robustness of Richardson extrapolation. However, further research and development are needed to achieve the desired robustness. Robust solution verification is complex and conflates two sources of error in the model fitting process: (1) uncertainty from imperfect mesh refinements and (2) additional, non-modeled error terms in the power series due to non-ideal convergence. An ideal robust method will simply and adequately constrain the Richardson extrapolation model to add uncertainty to the prediction for the first type of error while providing a metric to evaluate the likelihood of the second type of error. If this method exists, an analyst can use a single modeling approach to handle additional uncertainty caused by imperfect mesh families and obtain guidance on when to account for additional terms in the power series expansion. This work will explore different regularization strategies and identify the most effective strategy for handling the first type of error to establish a robust method for solution verification.

Presenting Author: Justin Weinmeister Oak Ridge National Laboratory

Presenting Author Biography: Justin Weinmeister is an associate nuclear fluid mechanics engineer in the Energy Systems Development group of the Nuclear Energy and Fuel Cycle Division at Oak Ridge National Laboratory. His expertise includes computational fluid dynamics (CFD), heat transfer, optimization, and uncertainty quantification. At ORNL, his work covers CFD analysis, model validation, high heat flux design and optimization, and experimental fluid dynamics for multiphase flows. His work has included 2 MW target design for the First Target Station of the Spallation Neutron Source (SNS), coolant channel design for the Transformational Challenge Reactor, and component design for the Materials Plasma Exposure eXperiment (MPEX).