On evaluation of local fields by fast multipole method employing smooth equivalent/check surfaces

Abstract

The work aims to increase the efficiency of finding local fields in problems, like those of fracture mechanics, for which extreme, rather than average, values of fields are of prime significance. We focus on using kernel independent fast multipole methods (KI-FMM) when a problem is solved by a boundary element method. To accurately calculate local fields, we employ smooth equivalent surfaces (SES), used in translations of a KI-FMM, instead of non-smooth surfaces having singular lines (12 edges of a cube) and/or points (8 vertices of a cube in 3D; 4 vertices of a square in 2D). We develop and employ exact representations of circular (in 2D) and spherical (in 3D) equivalent surfaces and higher-order approximations of equivalent densities on them. It is established that the accuracy and stability of a KI-FMM with the SES suggested are notably higher than those of a similar KI-FMM employing non-smooth (square in 2D, cubic in 3D) equivalent surfaces. Theoretical considerations presented and numerical tests performed show that, when using the SES, the complexity of a KI-FMM does not exceed that of an analytical FMM of the same accuracy. Numerical examples illustrate the exposition. The subroutines developed are of immediate use in any code of the KI-FMM to increase the accuracy of translations without increasing their time expense.