Sensitivity and optimization for shape and non-linear boundary conditions in thermal boundary elements

This paper is concerned with the minimization of functionals of the form (b) f(h; T(b; h)) d (b) where variation of the vector b modi es the shape of the domain on which the potential problem, ∇2T =0, is de ned. The vector h is dependent on non-linear boundary conditions that are de ned on the boundary . The method proposed is founded on the material derivative adjoint variable method traditionally used for shape optimization. Attention is restricted to problems where the shape of is described by a boundary element mesh, where nodal co-ordinates are used in the de nition of b. Propositions are presented to show how design sensitivities for the modi ed functional (b) f(h; T (b; h)) d (b) + (b) (b; h)∇2T(b; h) d (b) can be derived more readily with knowledge of the form of the adjoint function determined via non-shape variations. The methods developed in the paper are applied to a problem in pressure die casting, where the objective is the determination of cooling channel shapes for optimum cooling. The results of the method are shown to be highly convergent