2014 Volume 6 Pages 29-32
This paper presents analytic solutions of the shape derivatives (Fréchet derivatives with respect to domain variation) for singular points of cost functions in shape-optimization problems for the domain in which the boundary value problem of a partial differential equation is defined. A design variable is given by a domain mapping. Cost functions are defined as functionals of the design variable and the solution to the boundary value problem. The analytic solutions for singular points such as crack tips and boundary points of the mixed boundary conditions on a smooth boundary are obtained by using the generalized $J$ integral.