A quadrature-based superconvergent isogeometric frequency analysis with macro-integration cells and quadratic splines

Abstract

A quadrature-based superconvergent isogeometric method is presented for efficient frequency analysis with macro-integration cells and quadratic splines. The present development particularly refers to the isogeometric analysis of scalar wave equations. The single element, two-element and three-element macro-integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy, in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix. The 1D quadrature rules with various macro-integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix. Consequently, 3-point, 5-point and 7-point quadrature rules with identical precision are established for the single element, two-element and three-element macro-integration cells, respectively. By construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1D case. The 2D and 3D superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1D integration algorithms. It is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem, which needs special treatments for the multidimensional higher order mass matrix formulation. The proposed approach is featured by its simplicity for numerical implementation and efficiency using macro-integration cells. Numerical examples confirm the efficacy of the present methodology.