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No-gap second-order optimality conditions for additive manufacturing

Hiba Hmede, Luc Paquet, Gerd Wachsmuth

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.

No-gap second-order optimality conditions for additive manufacturing

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.

Paper Structure

This paper contains 16 sections, 32 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setting and preliminaries
  3. Problem setting and standing assumptions
  4. Results concerning the state mapping
  5. Results obtained previously concerning our optimal control problem
  6. Higher regularity of the trajectories for local minimizers
  7. General constrained optimization problems in Banach spaces
  8. Cones and polyhedricity
  9. First-order optimality conditions and constraint qualifications
  10. Necessary and sufficient second-order optimality conditions
  11. Verification of the conditions for the application of the general theory
  12. A problem without tau-dependence
  13. The general case
  14. The second derivative is a Legendre form
  15. No-gap second order optimality conditions in additive manufacturing
  16. ...and 1 more sections

Key Result

Proposition 2.2

Under our general assumptions, for every given initial datum $y_{0} \in L^{2}(\Omega)$ and every $\gamma \in H^1(0,T;\mathbb{R}^2)$, problem eqn:linsystem has a unique weak solution $y_{\gamma}$ (we say that $y_{\gamma}$ is the state corresponding to $\gamma$).

Figures (1)

  • Figure 1: The powder bed layer.

Theorems & Definitions (64)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • Definition 2.7
  • ...and 54 more