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Numerical methods with improved accuracy

Research Team: Dr. Vladimir Puzyrev, Dr. Quangling Deng, Prof. Victor Calo

All numerical methods involve approximation errors. When applied to wave propagation problems, these errors can be classified into amplitude and phase (dispersion) errors. For large-scale problems, the phase errors tend to accumulate, resulting in completely erroneous results. These errors could be reduced either by using a finer mesh or high-order elements. Both of these options come with a large increase in computational cost. The group developed a simple but effective technique for reducing dispersion errors in isogeometric analysis, a new numerical method, which was introduced in 2005 and received much attention since then.

The idea is based on blending different methods or quadrature rules to minimize the dispersion errors. This new optimally-blended isogeometric analysis significantly improves the phase accuracy and yields two extra orders of convergence (superconvergence) in wave propagation and structural vibration problems. To quantify these errors, the team generalized the Pythagorean eigenvalue error theorem to account for quadrature errors on the resulting weak forms. The proposed technique can be applied to arbitrary high-order isogeometric elements.