Non-singular boundary integral equations (BIEs) are derived and applied to shape design sensitivity analysis of potential, elastic and acoustic problems. Conventional BIEs generally contain singular integrands resulted from the singularities of fundamental solutions, and undesirable features such as instability, inaccuracy and discontinuity near the boundary are caused in numerical solution procedures.
In order to remove these shortcomings, integral identities that describe simple physical phenomena closely related to the problems are combined with the conventional BIEs. Because the integrands of the resulting equations are sufficiently regularized near smooth boundary, more accurate and continuous solutions can be obtained by using the standard Gaussian quadrature not only in the domain but also near the boundary. In case of obtaining a field solution near a sharp corner or an edge, the integrands of the equations are not non-singular but weakly singular. Thus the accuracy of derived equations near the corner or the edge would be of the same order as that of weakly singular expressions which can be obtained by combining uniform potential and rigid body representations with the conventional BIEs for potential and elastic problems, respectively.
In addition, boundary variables and their tangential derivatives can be evaluated by boundary integral in the present formulation, and the accuracy of these values can be related to those values obtained by interpolation and numerical differentiation. More accurate tangential derivatives can be obtained on the smooth boundary by using the relation and combining two solutions.
In order to prove the accuracy and stability of the proposed equations, several simple problems are solved and the results are compared with those obtained by using conventional strongly singular and weakly singular equations. The results show that the proposed equations describe internal solutions very accurately.
An inverse problem and shape optimization problems are solved as practical examples. For the inverse problem, the whole field can be properly reconstructed from the informations at internal points which are located very close to the unknown boundary. For the optimization problems, a shape design problem of a fillet under the stress constraints and a dimensioning problem of car interior sections for reducing noise level are adopted. The optimum designs are obtained without any numerical difficulties, which shows accuracy and practicality of the present approach.
