Analytical Solutions of Gravity Vector and Gravity Gradient Tensor Caused by a 2D Polygonal Body with a 2D Polynomial Density Contrast

摘要

In this paper, analytical solutions are presented for the gravity vector and gravity gradient tensor at any point produced by a 2D body whose cross-section is an arbitrary polygon and the density contrast is a 2D arbitrary-order polynomial function varying in both horizontal and vertical directions. In addition, we analyze the singularity of our expressions. For the gravity vector, the singularity points only exist at the vertices of the polygon. But for the gravity gradient tensor, there are two situations: (1) if every side of the polygon is not parallel to z-axis, the singularity points will only exist at the vertices of the polygon; (2) if there is any side parallel to z-axis in the polygon, all the points on the line passing through the side parallel to z-axis will become singularity points. To avoid this singularity, observation points can be moved from the singularity points by a minimal distance. Besides, the analytic expressions are validated compared with conventional method that sums up the gravity effects of a series of units with uniform densities, with the numerical stability also being evaluated through numerical tests. What is more, applications with some numerical examples and effective models show that our analytical solution within the range of numerical stability is superior in computational accuracy and efficiency to the conventional method that sums up the gravity effects of a series of units with uniform densities. In a word, our expressions provide an effective method for computing the gravity vector and gravity gradient tensor of an irregular 2D body with complicated density variation.