A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

Multiple reciprocity boundary element analysis of two-dimensional anisotropic thermoelasticity involving an internal arbitrary non-uniform volume heat source

“…It concerns the terms containing logðzÞ in the integrands, as it may not be analytic everywhere in the domain. This issue has been explained in detail in Zhang et al (1996b), and Shiah and Tan (1999a); thus, it will only be briefly discussed here. If one defines the principal value of z as Àp < argðzÞ 6 p, the quantity logðzÞ is not analytic along the negative f 1 -axis.…”

Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity

“…It concerns the terms containing logz in the functions above, as it may not be analytic everywhere in the domain. This issue has been explained in great detail in Zhang, Tan and Afagh (1996b), thus it will only be brie¯y discussed here. If one writes z g 1 ig 2 and de®nes the principal value of argz as Àp`argz p, then logz is evidently not analytic along the negative g 1 -axis.…”

“…If rays from any point along an inner boundary are projected in arbitrary directions, they will cut through the domain. To resolve this problem, Zhang, Tan and Afagh (1996b), in their work on body-force loading, removed an in®nitesimal strip containing the negative f 1 -axis from the domain and carried out the limiting process as this strip vanishes. The result of this is a series of extra line integrals over the intervals along the negative f 1 -axis where it cuts the domain, such as l 0 Y l 1 Y l 2 Y l 3 , from the source point on the internal surface of the region.…”

“…Dif®culties arising from the fact that the temperature distribution in the domain is not governed by Poisson's equation are addressed and overcome in the study. The process of analytical transformation, otherwise, follows the same vein as the procedures for body-force loading described in Zhang, Tan and Afagh (1996b). It has been successfully implemented into the BEM computer code based on the quadratic isoparametric element formulation used in that study.…”

The modified Shanks transform for the solution of elastic problems by boundary integral equation (BIE) method

“…If the problem includes mass forces both mixed methods [22,23] and direct methods [24][25][26] are used. This direct method deals with the mass potential forces only and use physical variables (for instance, displacements) and not variables of potential density's type.…”