Comparing 10 Methods for Solution Verification, and Linking to Model Validation

Grid convergence is often assumed as a given during computational analyses involving discretization of an assumed continuum process. In practical use of finite difference and finite element analyses, perfect grid convergence is rarely achieved or assured, and this fact must be addressed to make statements about model validation or the use of models in risk analysis. We have previously provided a 4-step quantitative implementation for a quantitative V&V process. One of the steps in the 4-step process is that of Solution Verification. Solution Verification is the process of assuring that a model approximating a physical reality with a discretized continuum (e.g. finite element) code converges in each discretized domain to a converged answer on the quantity of subsequent validation interest. The modeling reality is that often we are modeling a problem with a discretized code because it is neither continuous spatially (e.g. contact and impact) nor smooth in relevant physics (e.g. shocks, melting, etc). The typical result is a non-monotonic convergence plot that can lead to spurious conclusions about the order of convergence, and a lack of means to estimate residual solution verification error or uncertainty at confidence. We compare ten techniques for grid convergence assessment, each formulated to enable a quantification of solution verification uncertainty at confidence and order of convergence for monotonic and nonmonotonic mesh convergence studies. The more rigorous of these methods require a minimum of four grids in a grid convergence study to quantify the grid convergence uncertainty. The methods supply the quantitative terms for solution verification error and uncertainty estimates needed for inclusion into subsequent model validation, confidence, and reliability analyses. Naturally, most such methodologies are still evolving, and this work represents the views of the authors and not necessarily the views of Lawrence Livermore National Laboratory. Nomenclature ANSI American National Standards Institute ASME American Society of Mechanical Engineers ∆$B Benefit, usually in $$$ ∆$B Benefit, usually in $$$ B Bias error estimate BCR Benefit / Cost Ratio (∆$B-∆$C)/∆$C ∆$C Cost, usually in $$$ C Confidence, a numerical value CFD Computational Fluid Dynamics E Bias error as percent of Ffg Ffg Quantity of Interest, discretized (computational) model solution, for finest grid Fi Quantity of Interest, discretized (computational) model solution, for i grid Fi* Quantity of Interest, response surface estimate solution, for i grid Fo Quantity of Interest, exact solution