No-gap second-order optimality conditions for additive manufacturing

TL;DR

This paper advances the mathematical understanding of optimizing laser trajectories in additive manufacturing by establishing gapless second-order optimality conditions, ensuring precise control of thermal processes.

Contribution

It introduces a novel framework for second-order optimality conditions in non-convex constraints, with reformulation in Banach spaces and higher regularity results for local minimizers.

Findings

No gap between necessary and sufficient second-order conditions.

Reformulation in Banach spaces enables advanced optimality analysis.

Higher regularity of trajectories and Lagrange multipliers for local minimizers.

Abstract

Additive manufacturing by laser fusion on a metal oxides powder bed has developed considerably in the last few years and allows to produce a wide range of complex parts. The mathematical models correspond to initial boundary value problems for the heat equation with moving heat sources according to the laser trajectories. The main questions concern the optimization of the trajectories scanned by the laser and of the thermal treatment time in order to melt the powder where it is desired to make the part and to minimize the thermal gradients. Our purpose in this current paper is to pursue the study of the optimization model that we have introduced in a previous paper. Here, we consider second-order optimality conditions for non-necessarily convex constraints on the laser paths. In particular, we obtain no gap between the second-order sufficient optimality condition and the necessary second-order optimality condition. To achieve this goal, we reformulate our optimal control problem in order to fit it in the framework of the abstract theory of optimization under constraints in Banach spaces. Higher regularity of the trajectories for local minimizers is also proved implying higher regularity of the corresponding Lagrange multipliers. The case of the regularity of the trajectories for stationary points is left open.