Gravity effects of polyhedral bodies with linearly varying density

Abstract

We extend a recent approach for computing the gravity effects of polyhedral bodies with uniform density by the case of bodies with linearly varying density and by consistently taking into account the relevant singularities. We show in particular that the potential and the gravity vector can be given an expression in which singularities are ruled out, thus avoiding the introduction of small positive numbers advocated by some authors in order to circumvent undefined operations. We also prove that the entries of the second derivative exhibit a singularity if and only if the observation point is aligned with an edge of a face of the polyhedron. The formulas presented in the paper have been numerically checked with alternative ones derived on the basis of different approaches, already established in the literature, and intensively tested by computing the gravity effects induced by real asteroids with arbitrarily assigned density variations.

Elastic strain field due to an inclusion of a polyhedral shape with a non-uniform lattice misfit

“…(12). It is important to note that integrals (13)- (15) were extensively studied in the context of geophysical [43][44][45][46][48][49][50][54][55][56][57][58][59][60][61] and astronomy/spacecraft 47,51,62,63 problems. But, as mentioned in Introduction, fully analytical expressions for polyhedral bodies were obtained only when the density f (r) is a constant (N =0), a linear (N =1) or a quadratic (N =2) function of coordinates.…”

“…One can find, however, in the literature, the explicit solutions for a related problemcalculation of the Newtonian potential and its derivatives for a massive body of a polyhedral shape with an inhomogeneously distributed mass. This problem, being mathematically equivalent to the elasticity problem (see Section II), was solved however only in the simplest cases of linear [43][44][45][46][47] and quadratic [48][49][50] dependence of the mass density on coordinates. Thus, the task of the present paper has not been solved yet.…”

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

“…To accurately evaluate the near-field integrals in the adaptive multilevel fast multipole algorithm, the analytical expressions or higher-order quadrature rules are needed. For a tetrahedral element with constant source distributions, analytical expressions can be derived by simplifying the analytical formula originally designed for a homogenous polyhedral element [Cady, 1980;Çavşak, 2012;Conway, 2015;D'Urso, 2013D'Urso, , 2014aD'Urso, , 2014bDasgupta, 1988;Li and Chouteau, 1998;Okabe, 1979;Paul, 1974;Tsoulis, 2012;Tsoulis and Petrović, 2001;Werner, 1994;Won and Bevis, 1987]. However, very few of them have delivered the closed-form solutions for the potential and the gradient field and the gradient tensor simultaneously.…”

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

“…To this end we shall make use of a generalized version of the Gauss theorem, recently exploited with success in the geodesy literature [9][10][11][12][13], i.e., a version which correctly takes into account the singularities of the integrand function.…”

“…The rationale of the approach outlined in the paper is based on a generalized version of Gauss theorem, recently applied in the geodesy literature by D'Urso [9][10][11][12][13][14]. Accordingly, the surface integrals expressing the mechanical fields are transformed into the sum of an integral extended to the boundary of the loaded region and of an additional term accounting for the singularity of the field to be integrated.…”