Mathematical Modeling and Optimization for Powder-Based Additive Manufacturing

Abstract In powder-bed additive manufacturing, a laser beam melts a thin layer of metallic powder, which solidifies during cooling and bonds to the already processed solid material. After finishing one layer, the surface is covered with a new thin powder layer, and the melting process iterates, until an entire three-dimensional object emerges. Current scanning strategies are not optimized with respect to certain optimality criteria such as minimizing temperature gradients, which may lead to considerable residual stresses in the part. This work deals with the ordering in which the laser processes a single layer. Finding a laser trajectory is formulated as a mathematical optimization problem. The goal is to determine a path that minimizes the temperature gradients during the manufacturing process and thus reduces internal stresses in the material. The mathematical model combines discrete decisions on the trajectory with the continuous heat equation. After discretizing the heat equation, the resulting model turns out to be a mixed-integer linear programming problem, which is solved using a simplex-based branch-and-cut approach. Numerical results for a test-set of sample objects demonstrate the potential of this approach.