We dedicate this paper to Frank Rizzo on the occasion of his retirement.

Free space Green’s functions are derived for graded materials in which the thermal conductivity varies exponentially in one coordinate. Closed-form expressions are obtained for the steady-state diffusion equation, in two and three dimensions. The corresponding boundary integral equation formulations for these problems are derived, and the three-dimensional case is solved numerically using a Galerkin approximation. The results of test calculations are in excellent agreement with exact solutions and finite element simulations.

SUMMARYThe paper presents two M y equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is CIsa continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at comers and edges. These new findings specifically focus on the comer problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity.The resulting regularized and fmite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to C',' continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for comers is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.

The paper gives a theory for the response of two periodically excited masses with circular bases attached to the surface of an elastic half space. Results for two geometrically identical cylindrical masses, excited by a vertical harmonic force applied to one of the masses, illustrate the effect of the presence of a second mass upon the response of the excited mass and the conditions for which the response of the second mass is significant.

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying ®eld is C 1Ya continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justi®cation for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a ®nite approximation to the standard twosided Hadamard ®nite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided ®nite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the ®eld which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C 1Ya interpolations. IntroductionHypersingular integral equations are formed from differentiation of singular integral equations and as a result, the order of the singularities in the kernels is increased in these equations. The higher order singularities in the kernels requires more smoothness of the densities for the given integral to be ®nite. A suf®cient condition for the existence of hypersingular integrals is C 1Ya continuity of the density at the source point. This is a condition which is not met by the use of standard isoparametric boundary elements which only provide a piece-wise C 1Ya interpolation. For this reason, approximate solution techniques for solving hypersingular integral equations using the boundary element method (BEM) require special consideration.In general, this special consideration takes one of several forms. The hypersigularity may be reduced through integration by parts (Guidera and Lardner 1975;Giroire and Ne Âde Âlec 1995). In addition to being rather mathematically tedious, approaches of this form have yet to produce numerical results of the same accuracy as have been obtained with the more conventional approaches.Each of several remaining approaches seeks to impose C 1Ya continuity at the source point on the particular density in the hypersingular integral which is used in the numerical integration, as is required mathematically. The ®rst class of algorithms is based on interpolation schemes which are C 1Ya continuous. One way in which this is done is to introduce nodal displacement gradients in the interpolation scheme. Polch et al. (Polch et al. 1987) used a standard isoparametric representation for displacement gradients. This continuous displacement gradient ®eld was then coupled to the standard interpolation for displacements through a least-squares method. Young (Young 1996) has used a hybrid in...