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 ∫ fΓ(b) f(h,T(b,h))dΓ(b) where variation of the vector b modifies the shape of the domain Ω on which the potential problem, ▽2T=0, is defined. The vector h is dependent on non-linear boundary conditions that are defined on the boundary Γ. 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 Γ is described by a boundary element mesh, where nodal co-ordinates are used in the definition of b. Propositions are presented to show how design sensitivities for the modified functional ∫Γ(b) f (h, T (b, h)) dΓ(b) + ∫Ω(b) γ(b,h)▽2T(b, h) dΩ(b) can be derived more readily with knowledge of the form of the adjoint function γ 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. © 2002 John Wiley and Sons. Ltd.