Application of a simulated annealing algorithm in the optimal placement of the source points in the method of the fundamental solutions

Abstract

The placement of source points constitutes a key issue for the method of the fundamental solutions. In particular, for problems with singularities of any kind the determination of the optimal placement of source points becomes relevant, as no linear combination of arbitrarily located source points can guarantee a reasonable approximation to the solution. This paper investigates the use of a ''Simulated Annealing'' algorithm in the optimal placement of source points in singular problems. The algorithm is essentially an iterative random search with adaptive moves along the coordinate directions. It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered. The proposed methodology is employed with a variety of test problems. Results are compared to those of an analytical optimisation routine and their relatively merits and disadvantages discussed. Simulated annealing is shown to be an attractive option for the optimisation of singular problems, with a high rate of success, and able to solve problems for which analytical optimisation routines fail.

Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods

“…Fairweather and Karageorghis [18][19][20] proposed the adaptive scheme, in which the coefficients of the linear representation of the solution as well as the position of the sources, which are given as a fixed number, are chosen by a non-linear least-squares algorithm. In paper [21], a simulated annealing algorithm was used for the optimal placement of source points in singular problems. Nishimura [22][23][24] proposed the genetic algorithm for the optimal arrangement of the source positions.…”

A meshless method for solving the free boundary problem associated with unilateral obstacle

“…For the first question, we refer to [7,20] for the choice of points. And for the second question, we will describe the regularize methods for solving an ill-posed linear system in the next subsection.…”

Frequency response analyses in vibroacoustics using the method of fundamental solutions

“…However, the key issue for the MFS is how to collocate the sources. For the best choice of the positions of the equivalent sources, the residual between the approximate and the exact boundary can be minimized [15].…”