Shape Optimization and Identification of Solid Geometries Considering Discontinuities

Shape sensitivity analysis of heat-conducting bodies is performed in general terms incorporating interface conditions and boundary singularities. Adjoint variables and the material derivative concept are utilized to obtain the material derivatives of volume and surface integrals of temperature and heat flux. Two illustrative examples are then analyzed by iterative numerical techniques incorporating the boundary element method of discretization. In the first problem, the interface position in a nonhomogeneous material is optimized for a minimum of total surface heat flow. The second problem involves the determination of the solidification interface shape in the so-called steady-state one-phase Stefan problem. Numerical results, checked by exact solutions, where available, indicate that the proposed solution procedure is suitable for free boundary problems in heat transfer.