Computation of the attractive force of an ellipsoid

• Published in: Computational mathematics and mathematical physics Vol. 53; no. 12; pp. 1882 - 1890
• Main Author: Savchenko, A. O.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2013; Springer Nature B.V
• Subjects: Approximation; Computation; Computational mathematics; Computational Mathematics and Numerical Analysis; Graph theory; Integrals; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical analysis; Studies; attractive force; quadrature rules; method for evaluation of triple integrals; potential of an ellipsoid
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542513120117

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.