A superconvergent isogeometric formulation is presented for the transient analysis of wave equations with particular reference to quadratic splines. This formulation is developed in the context of Newmark time integration schemes and superconvergent quadrature rules for isogeometric mass and stiffness matrices. A detailed analysis is carried out for the full-discrete isogeometric formulation of wave equations and an error measure for the full-discrete algorithm is established. It is shown that a desirable superconvergence regarding the isogeometric transient analysis of wave equations can be achieved by two ingredients, namely, the design of a superconvergent quadrature rule and the criteria to properly define the step size for temporal integration. It turns out that the semi-discrete and full-discrete isogeometric formulations of wave equations with quadratic splines share an identical quadrature rule for a sixth-order accurate superconvergent analysis. Meanwhile, the relationships between the time step size and the element size are presented for various typical Newmark time integration schemes, in order to ensure the sixth-order accuracy in transient analysis. Numerical results of the transient analysis of wave equations consistently reveal that the proposed superconvergent isogeometric formulation is sixth-order accurate with respect to spatial discretizations, in contrast to the fourth-order accuracy produced by the standard isogeometric approach with quadratic splines.

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波动方程的超收敛等几何瞬态分析

提出了一种超收敛等几何公式，用于波动方程的瞬态分析，特别是二次样条。该公式是在 Newmark 时间积分方案和等几何质量和刚度矩阵的超收敛正交规则的背景下开发的。对波动方程的全离散等几何公式进行了详细分析，并建立了全离散算法的误差度量。结果表明，关于波动方程的等几何瞬态分析的理想超收敛可以通过两个要素来实现，即超收敛正交规则的设计和正确定义时间积分步长的标准。事实证明，对于六阶精确超收敛分析，具有二次样条的波动方程的半离散和全离散等几何公式共享相同的求积法则。同时，给出了各种典型Newmark时间积分方案的时间步长与单元尺寸的关系，以保证瞬态分析的六阶精度。波动方程瞬态分析的数值结果一致表明，与使用二次样条的标准等几何方法产生的四阶精度相比，所提出的超收敛等几何公式在空间离散化方面是六阶精度。给出了各种典型Newmark时间积分方案的时间步长与单元尺寸的关系，以保证瞬态分析的六阶精度。波动方程瞬态分析的数值结果一致表明，与使用二次样条的标准等几何方法产生的四阶精度相比，所提出的超收敛等几何公式在空间离散化方面是六阶精度。给出了各种典型Newmark时间积分方案的时间步长与单元尺寸的关系，以保证瞬态分析的六阶精度。波动方程瞬态分析的数值结果一致表明，与使用二次样条的标准等几何方法产生的四阶精度相比，所提出的超收敛等几何公式在空间离散化方面是六阶精度。