Combining Metamodels with Rational Function Representations of Discretization Error for Uncertainty Quantification

A new method is presented for extending metamodeling techniques to include the effects of finite element model mesh discretization errors. The method employs a rational function representation of the discretization error rather than the power series representation used by methods such as Richardson extrapolation. Examples dealing with simple function estimation and estimation of the vibrational frequency of a one dimensional bar showed that when extrapolated to the continuum, the rational function model gave more accurate estimates using fewer terms than the Richardson extrapolation technique. This is an important consideration for computational reliability assessment of high consequence systems, as small biases in solutions can significantly affect the accuracy of small magnitude probability estimates. In general, the rational function form of the discretization error produces a nonlinear model requiring an iterative nonlinear least-squares solution technique. However, all the examples studied in this work proved to be close-to-linear, meaning that the linear least-squares estimate of the model coefficients could not be improved. In subsequent nondeterministic analyses, the rational function based metamodel also produced more accurate estimates of failure probabilities using fewer terms than the Richardson extrapolation method under very severe extrapolation conditions. Rational function representations of discretization error offer greater flexibility by allowing a user to accurately extrapolate to a continuum representation from numerical experiments performed outside the asymptotic region where the usual power series representation is not converging. This allows the use of coarser meshes in the numerical experiments, saving a significant amount of time and effort.