A superconvergent isogeometric formulation for eigenvalue computation of three dimensional wave equation

Abstract

A superconvergent isogeometric formulation is presented to compute the eigenvalues for three dimensional wave equation. This three dimensional superconvergent isogeometric formulation is characterized by a higher order mass matrix formulation with particular reference to the quadratic basis functions. The three dimensional higher order mass matrix is built upon an optimal combination of the reduced bandwidth mass matrix and the consistent mass matrix. The frequency error associated with the isogeometric discretization of three dimensional wave equation is derived in detail. In particular, the optimal mass combination parameter for higher order mass matrix is devised as a function of the two spatial wave propagation angles, which enables that arbitrary frequency corresponding to a given wave propagation direction can be computed in a superconvergent way. Two extra orders of accuracy, i.e., 6th order of accuracy, are attained by the proposed higher order mass matrix than the consistent mass matrix for the frequency computation of three dimensional wave equation. The dispersion property of the present three dimensional higher order mass matrix formulation is examined as well. The accuracy of the proposed three dimensional superconvergent isogeometric formulation is testified by several numerical examples.