Adjoint structure approach for shapesensitivity analysis using BEM

An adjoint approach for the shape design sensitivity analysis of plane elasticity problems with a direct boundary integral equation formulation is presented. The objective function or the constraint function is expressed in an integral form and augmented by incorporating the elasticity equations via adjoint functions. The variation of the augmented function is taken by performing its material derivative to obtain the sensitivity of the objective function or the constraint function. The adjoint functions are obtained through the solution of the adjoint problem resulting from setting the local variations of the displacement, velocity, and traction fields equal to zero. An approximate procedure for the application of concentrated adjoint load in the boundary element framework is suggested. This allows the computation of displacement, traction, and stress sensitivities at discrete nodal points. The requirement of choosing an averaging characteristic function in earlier formulations is obviated by this development. The present formulation includes the treatment of body forces in sensitivity calculations. A series of numerical examples is solved to demonstrate the validity of the present approach.