A Superconvergent Isogeometric Method with Refined Quadrature for Buckling Analysis of Thin Beams and Plates

Abstract

A superconvergent isogeometric method is developed for the buckling analysis of thin beams and plates, in which the quadratic basis functions are particularly considered. This method is formulated through refining the quadrature rules used for the numerical integration of geometric and material stiffness matrices. The criterion for the quadrature refinement is the optimization of the buckling load accuracy under the assumption of harmonic buckling modes for thin beams and plates. The method development starts with the thin beam buckling analysis, where the material stiffness matrix with quadratic basis functions does not involve numerical integration and thus the refined quadrature rule for geometric stiffness matrix can be obtained in a relatively easy way. Subsequently, this refined quadrature rule for thin beam geometric stiffness matrix is conveniently generalized to the thin plate geometric stiffness matrix via the tensor product operation. Meanwhile, the refined quadrature rule for the thin plate material stiffness matrix is derived by minimizing the buckling load error. It turns out that the refined quadrature rule for the thin plate material stiffness matrix generally depends on the wave numbers of buckling modes. A theoretical error analysis for the buckling loads evinces that the isogeometric method with refined quadrature rules offers a fourth-order accurate superconvergent algorithm for buckling load computation, which is two orders higher than the standard isogeometric analysis approach. Numerical results well demonstrate the superconvergence of the proposed method for the buckling loads corresponding to harmonic buckling modes, and for those related with non-harmonic modes, the buckling loads given by the proposed method are also much more accurate than their counterparts produced by the conventional isogeometric analysis.

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

“…Among various techniques to alleviate or remove shear locking, the reduced or selective reduced integration methods have been frequently used owing to their simplicity in numerical implementation [1,4,5]. During recent years, the rapidly growing isogeometric analysis [6][7][8][9][10][11][12][13] stimulates the geometrically exact analysis of Mindlin-Reissner plates and their onedimensional degeneration, i.e., Timoshenko beams. In the context of isogeometric Mindlin-Reissner plate analysis, the shear locking issue has often been resolved by employing higher-order basis functions [14,15], collocation formulation [16,17], reduced and selective reduced integration [18,19], mixed formulation [20], etc.…”