We provide an analytical solution of the heat equation in the half-space subject to a moving Gaussian heat flux with piecewise constant parameters. The solution is of interest in powder bed fusion applications where these parameters can be used to control the conduction of heat due to a scanning beam of concentrated energy. The analytical solution is written in a dimensionless form as a sum of integrals over (dimensionless) time. For the numerical computation of these integrals we suggest a quadrature scheme that utilizes pre-calculated lookup tables for the required quadrature orders. Such a scheme is efficient because the required quadrature orders are strongly dependent on the parameters in the heat flux. The possibilities of using the obtained computational technique for the control and optimization of powder bed fusion processes are discussed.

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Powder-bed-based additive manufacturing involves melting of a powder bed using a moving laser or electron beam as a heat source. In this paper, we formulate an optimization scheme that aims to control this type of melting. The goal consists of tracking maximum temperatures on lines that run along the beam path. Time-dependent beam parameters (more specifically, beam power, spot size, and speed) act as control functions. The scheme is greedy in the sense that it exploits local properties of the melt pool in order to divide a large optimization problem into several small ones. As illustrated by numerical examples, the scheme can resolve heat conduction issues such as concentrated heat accumulation at turning points and non-uniform melt depths.