New identities for fundamental solutions and their applications to non-singular boundary element formulations.

*(English)*Zbl 0969.74073Summary: Based on a general operational approach, we establish two new integral identities for the fundamental solutions of potential and elastostatic problems. Non-singular forms of the conventional boundary integral equations (BIEs) are derived by employing these two identities for fundamental solutions, and by using the two-term subtraction technique. Both the strongly (Cauchy type) and weakly singular integrals existing in the conventional BIEs are removed from the BIE formulation. The existence of non-singular forms of conventional BIEs raises new questions about the smoothness requirement in the boundary element method, since the two-term subtraction requires, theoretically, \(C^1\) continuity of the density function, rather than the \(C^0\) continuity as required by the original singular or weakly singular forms of conventional BIEs. Implication of the non-singular BIEs on the smoothness requirement are discussed in this paper.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74B05 | Classical linear elasticity |

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |