The analytical integration of influense functions for solving elastic and potential problems by the boundary element method

V. P. Fedotov ; L. F. Spevak

Matematičeskoe modelirovanie, Tome 19 (2007) no. 2, pp. 87-104

Résumé

Practical problems of mathematical physics (elasticity, plasticity, diffusion, heat conduction etc.) require both a high accuracy of the solution and a considerable computation speed, often on line. The paper presents developed parallel algorithms for solving problems of the kind by a technique enabling a solution to be obtained at a speed many times exceeding that of numerical finite-element computations of equivalent dimensionality, the accuracy being the same. The calculation technique is based on the boundary element method. The required functions (velocity, stress, concentration, temperature, flux etc.) are represented analytically inside the body in terms of the values on the boundary. The boundary is partitioned into elements ? straight line segments or circular arcs for plane. The values on the boundary elements those are not specified by the boundary conditions are found from a system of linear algebraic equations whose coefficients are the integrals of influence functions and their derivatives taken over the boundary elements. Analytical formulae have been derived to compute all the necessary integrals for the above-mentioned types of elements. This has enabled the formulae derived to be applied to a plane-strain area of any geometry and with any elastic (diffusion or heat) properties. The formulae obtained offer higher computation accuracy in comparison with numerical integration and significantly accelerate the process of system matrix filling. The algorithms for system matrix filling and system solution can be easily parallelized, and this, along with the advantages of analytical integration over numerical one, offers a considerable acceleration of the process. The values obtained on the boundary and the analytical integration formulae enable one to determine the process characteristics inside the area analytically. In this case, even quantities requiring differentiation, like stress or flux, can be determined to a high accuracy, which cannot be attained by classical computation.