Solution Verification Linked to Model Validation, Reliability, and Confidence

The implementation of Verification and Validation (V&V) of a computational model of a physical system can be simply described as a 4-step 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 validation interest. The modeling reality is that often we are modeling a problem with a discretized code because it is neither smooth nor continuous spatially (e.g. contact and impact) or 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 error or uncertainty. We offer one emerging technique that enables a quantification of solution verification uncertainty at confidence and order of convergence for monotonic and non-monotonic mesh convergence studies. The method offers insight into code development (convergence order versus that expected), and supplies the quantitative terms needed for inclusion into subsequent model validation, confidence, and reliability analyses.

We show an example of a mesh convergence study where the use of this new method gives results essentially identical to those of the Richardson Extrapolation method in the ideal case. We show a second example of how this method can offer a quantitative estimate of solution verification error and uncertainty in a case of unusually severe non-monotonic convergence.

We are currently developing this uncertainty estimate into a predictive capability estimate as a topic for the future. However, even the procedure we will demonstrate offers a way to quantify the ever present and sometimes severe non-monotonicity observed during mesh convergence studies. The resulting term enters directly into subsequent analyses of model+system assessments of reliability at confidence.