The elliptical harmonic method for gravity forward computation of two-dimensional (2D) bodies with constant, polynomial and exponential density distributions is presented. This paper gives the elliptical harmonic expansions for the gravity field of 2D bodies mainly including the gravitational attractions and gradients, and the conversions from the surface integral expressions of the elliptical harmonic coefficients and the surface integral of the density function into line integrals using Gauss divergence theorem for evaluation of the gravity field. Due to the requirements of the Gauss divergence and Stokes theorems that the boundary surface or curve of the body is piecewise smooth and the vector field as well as its first-order partial derivatives are continuous, the effects of the discontinuities of the elliptical coordinates on the integral conversion using Gauss divergence and Stokes theorems are considered. The integrands of the converted line integrals are the analytical expressions with respect to the coordinates of the internal point of the 2D body for the constant and polynomial density models, but are not elementary functions for the exponential density model that the integrands can be solved by approximating the density function with polynomial function. The numerical experiments with two tested bodies including a rectangular cylinder body with quadratic density contrast varying with depth and a 26-sided polygon body with quadratic density contrast varying in both horizontal and vertical directions show the convergence, accuracy and stability of the elliptical harmonic algorithm. Compared with the circular harmonic approach, the elliptical harmonic approach has smaller non-convergence region between the boundary of the 2D body and the reference ellipse and is convergent faster for most of the external observation points.

提出了一种椭圆形谐波方法，可以对具有恒定，多项式和指数密度分布的二维（2D）物体进行重力正向计算。本文给出了二维物体重力场的椭圆谐波展开，主要包括引力引力和梯度，并利用高斯发散定理将椭圆谐波系数的表面积分表达式和密度函数的表面积分转换成线积分。用于评估重力场。由于高斯散度和斯托克斯定理的要求，物体的边界表面或曲线是分段光滑的，并且矢量场及其一阶偏导数是连续的，考虑了使用高斯散度和斯托克斯定理的椭圆坐标的不连续性对积分转换的影响。转换后的线积分的被积是常数和多项式密度模型关于2D体内部点坐标的解析表达式，但不是指数密度模型的基本函数，因为积分可以通过近似来求解具有多项式函数的密度函数。在两个被测物体上进行的数值实验，包括一个矩形的圆柱体，其二次密度对比度随深度而变化；一个26面的多边形体，其二次密度对比度在水平和垂直方向上均发生变化，从而证明了椭圆谐波算法的收敛性，准确性和稳定性。