Shape optimization of radiant enclosures with specular-diffuse surfaces by means of a random search and gradient minimization

Abstract

A technique of the shape optimization of radiant enclosures with specular-diffuse surfaces is proposed. The shape optimization problem is formulated as an operator equation of the first kind with respect to a surface to be optimized. The operator equation is reduced to a minimization problem for a least-squares objective shape functional. The minimization problem is solved by a combination of the pure random (or blind) search (the simplest stochastic minimization method) and the conjugate gradient method. The random search is used to find a starting point for the gradient method. The latter needs the gradient of the objective functional. The shape gradient of the objective functional is derived by means of the shape sensitivity analysis and the adjoint problem method. Eventually, the shape gradient is obtained as a result of solving the direct and adjoint problems. If a surface to be optimized is given by a finite number of parameters, then the objective functional becomes a function in a finite-dimensional space and the shape gradient becomes an ordinary gradient. Numerical examples of the shape optimization of “two-dimensional” radiant enclosures with polyhedral specular or specular-diffuse surfaces are given.