Removal of spurious outlier frequencies and modes from isogeometric discretizations of second- and fourth-order problems in one, two, and three dimensions

Abstract

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, a few spurious modes appear that possess inaccurate frequencies, denoted as “outliers”. The outlier frequencies and corresponding modes are at the root of several efficiency and robustness issues in isogeometric analysis. One example is explicit dynamics where outlier frequencies unnecessarily reduce the critical time step. Another example is wave propagation where the inaccurate outlier modes may participate in the solution. In this paper, we first investigate the spurious outlier frequencies and corresponding modes of isogeometric discretizations of second- and fourth-order model problems and provide a complete characterization. We then devise a new approach that removes all outliers modes without negatively affecting the accuracy of the discretizations. Our approach is variationally consistent and works for a range of common boundary conditions on tensor product domains. We finally demonstrate that our approach allows a much larger critical time step, irrespective of polynomial degree, providing a pathway towards efficient higher-order explicit dynamics.