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.

中文翻译：

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细化正交超收敛等几何方法用于薄梁和板的屈曲分析

开发了一种超收敛等几何方法用于薄梁和薄板的屈曲分析，其中特别考虑了二次基函数。该方法是通过改进用于几何和材料刚度矩阵的数值积分的求积规则来制定的。正交细化的标准是在假设薄梁和板的谐波屈曲模式下优化屈曲载荷精度。方法开发从细梁屈曲分析开始，其中具有二次基函数的材料刚度矩阵不涉及数值积分，因此可以相对容易地获得几何刚度矩阵的精化求积法则。随后，这种细梁几何刚度矩阵的精化求积法则通过张量积运算方便地推广到薄板几何刚度矩阵。同时，通过最小化屈曲载荷误差，推导出了薄板材料刚度矩阵的精化求积法则。事实证明，薄板材料刚度矩阵的精化求积法则通常取决于屈曲模式的波数。屈曲载荷的理论误差分析表明，具有精细正交规则的等几何方法为屈曲载荷计算提供了四阶精确的超收敛算法，比标准等几何分析方法高两个数量级。