等几何分析中的复杂物理区域投影算子及误差分析

针对等几何分析中复杂物理区域上的偏微分方程求解问题,提出了多片参数域上双三次样条投影映射的方法.首先基于多片参数化,构造了复杂物理域上的一个投影映射;其次对于物理域上的光滑函数,讨论了该投影映射的逼近误差,理论分析表明该投影映射可达到最优逼近阶;最后基于投影映射的思想,给出了一类适用于基于等几何分析在复杂物理区域上二阶椭圆方程求解的样条空间.数值算例的结果表明,该方法求解的逼近误差阶可达到最优.

Projection Operators and Error Analysis of Complex Physical Domains in Isogeometric Analysis

For solving partial differential equations(PDEs) on complex physical regions in isogeometric analysis(IGA), a method of bicubic spline projection mapping on multi-parameter domains is presented. Firstly, a projection mapping is constructed for complex physical domains based on multi-patch parameterization. Secondly, the approximation error of the projection mapping is discussed for the smooth functions defined over the physical domain, and the theoretical analysis shows that the projection mapping can reach the optimal approximation order.Finally, an IGA-suitable spline space is provided for solving the second order elliptic PDEs on the complex physical domain in IGA based on the idea of projection mapping. Numerical results show that the method based on projection operator reaches optimal approximation order.