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 multi-material 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. We provide an open-source implementation of the algorithm in MATLAB.

这项工作提出了基于频谱的，无网格的，基于格林定理的数值正交方案，用于对以有理参数曲线为边界的平面区域上的函数进行积分。我们的算法分两步进行：（1）我们首先沿着与格林定理相对应的区域边界曲线找到线积分的中间正交规则。（2）然后，我们使用一个高阶正交规则来计算沿坐标轴的被积数的数值反导数，该数字反导数用于评估格林定理线积分。我们提出了两种计算中间正交规则的方法。第一个在频谱上是准确的（就正交点的数量而言，它的收敛速度快于任何代数阶），并且相对易于实现，但不能保证多项式的准确性。$ķ$具有先验数量的正交点，并保留第一个正交点的收敛特性，但稍微复杂一些。正交方案可应用于几何矩的计算，沉浸几何分析，高阶网格之间的保守场传递以及具有合理几何形状的多材料模拟的初始化。我们将使用我们的方法产生的正交方案与文献中的其他方法进行了比较，结果表明，在正交点数量和计算时间方面，它们都更加有效。我们在MATLAB中提供了该算法的开源实现。