Temperature field optimization for laser powder bed fusion as a traveling salesperson problem with history

Abstract

Laser Powder Bed Fusion (LPBF) is a form of metal additive manufacturing in which a laser traces a path on a metal powder bed to heat powder particles and progressively melt and/or fuse them together to form a part. Outcomes of LPBF are functions of the metal's temperature history and are strongly affected by the path traced by the laser. Using temperature field optimization for LPBF as a motivating application, we focus on a class of combinatorial problems where admissible control policies are a permutation of a set of control actions. We abstract this class of problems as an optimal control problem with the objective of identifying an admissible control policy that minimizes a cost function of a dynamical system's state. While, in principle, the effect of the control actions on the cost function can last an infinitely long time, we will consider systems in which the effects of these control actions can be considered to last short (finite) periods of time. In this paper, we formalize this class of combinatorial problems as a Traveling Salesperson Problem with History and prove its equivalence to an Equality Generalized Traveling Salesperson Problem (E‐GTSP), enabling the use of well‐developed E‐GTSP solvers. We demonstrate this equivalence by computing the solutions obtained using an E‐GTSP solver for an LPBF‐inspired application.