Adjoint-assisted Pareto front tracing in aerodynamic and conjugate heat transfer shape optimization

Abstract

In this paper, a prediction-correction algorithm, built on the method proposed in [1], uses the adjoint method to trace the Pareto front. The method is initialized by a point on the Pareto front obtained by considering one of the objectives only. During the prediction and correction steps, different systems of equations are derived and solved by treating the Karush-Kuhn-Tucker (KKT) optimality conditions in two different ways. The computation of second derivatives of the objective functions (Hessian matrix) which appear in the equations solved to update the design variables is avoided. Instead, two approaches are used: (a) the computation of Hessian-vector products driving a Krylov subspace solver and (b) the approximations of the Hessian via Quasi-Newton methods. Three different variants of the prediction-correction method are developed, applied to 2D aerodynamic shape optimization problems with geometrical constraints and compared in terms of computational cost. It is shown that the inclusion of the prediction step in the algorithm and the use of Quasi-Newton methods with Hessian approximations in both steps has the lowest computational cost. This method is, then, used to compute the Pareto front in a 3D conjugate heat transfer shape optimization problem, with the total pressure losses and max. solid temperature as the two contradicting objectives.