论文标题
平面区域的无光谱网状正交正交曲线界定的平面区域
Spectral mesh-free quadrature for planar regions bounded by rational parametric curves
论文作者
论文摘要
这项工作介绍了光谱，无网格，Green基于定理的数值正交方案，用于整合受合理参数曲线界定的平面区域的功能。我们的算法分为两个步骤：（1）我们首先找到与Green定理相对应的区域边界曲线的线积分的中间正交规则。 （2）然后，我们使用高阶正规规则来计算沿坐标轴的积分的数值抗体，该轴用于评估绿色定理线积分。我们提出了两种计算中间正交规则的方法。第一个是频谱准确的（相对于正交点的数量，它比任何代数顺序比任何代数顺序更快），并且相对易于实现，但不能保证多项式精确性。第二个保证了多项式集成的精确性，直至预先指定的度k，具有先验数量的正交点，并保留了第一个的收敛性能，但更复杂。正交方案具有用于计算几何矩，沉浸几何分析，高阶网格之间的保守现场转移以及具有合理几何形状的模拟初始化的应用。我们将使用我们方法制成的正交方案与文献中的其他方法进行了比较，并表明它们在正交点的数量和计算时间方面都更有效。
This work presents spectral, mesh-free, Green's theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region's boundary curves corresponding to Green's theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green's theorem line integral. We present two methods to compute the intermediate quadrature rule. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of simulations with rational geometry. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time.
