标题： 修正的梯形法则在近奇异曲面积分在三维界面Stokes流中的应用 显示英文标题

标题： Corrected Trapezoidal Rules for Near-Singular Surface Integrals Applied to 3D Interfacial Stokes Flow

摘要： 界面 Stokes 流动可以通过边界积分方程方法高效计算。 在三维情况下，目标点处的流体速度由所有界面的二维表面积分给出，从而降低了问题的维度。 一个核心挑战在于，对于靠近但不在界面的目标点，表面积分接近奇异，标准的数值积分会失去精度。 本文提出了一种方法，用于准确计算三维椭圆边值问题中出现的近奇异积分。 该方法基于在表面上基点附近的被积函数的局部级数逼近，通过将目标点正交投影到表面上得到基点。 结果级数逼近中的初等函数可以在基点附近的邻域内使用递归算法以高精度进行积分。 剩余的积分则使用标准的数值积分规则进行评估，在这里选择的是四阶梯形法则。 该方法可以简化为标准积分加上修正项，并且在整个过程中始终具有四阶精度。 该方法被应用于解决几个椭球刚性物体周围的 Stokes 流动。 我们比较了在有无修正的情况下，物体附近的速度误差以及围绕物体运动的粒子的时间和位移误差。 显示英文摘要

摘要： Interfacial Stokes flow can be efficiently computed using the Boundary Integral Equation method. In 3D, the fluid velocity at a target point is given by a 2D surface integral over all interfaces, thus reducing the dimension of the problem. A core challenge is that for target points near, but not on, an interface, the surface integral is near-singular and standard quadratures lose accuracy. This paper presents a method to accurately compute the near-singular integrals arising in elliptic boundary value problems in 3D. It is based on a local series approximation of the integrand about a base point on the surface, obtained by orthogonal projection of the target point onto the surface. The elementary functions in the resulting series approximation can be integrated to high accuracy in a neighborhood of the base point using a recursive algorithm. The remaining integral is evaluated numerically using a standard quadrature rule, chosen here to be the 4th order Trapezoidal rule. The method is reduced to the standard quadrature plus a correction, and is uniformly of 4th order. The method is applied to resolve Stokes flow past several ellipsoidal rigid bodies. We compare the error in the velocity near the bodies, and in the time and displacement of particles traveling around the bodies, computed with and without the corrections.