Trajectory Optimization for Spatial Microstructure Control in Electron Beam Metal Additive Manufacturing

TL;DR

This work addresses spatial control of diffusion-driven microstructure in metal AM by coupling a discretized heat equation for temperature $T$ with a nonlinear microstructure model based on the Johnson–Mehl–Avrami–Kolmogorov framework. An augmented Lagrangian differential dynamic programming (AL-DDP) solver, accelerated on GPUs, computes time-varying Gaussian beam power fields to shape hardness distributions, and an approximate beam-motion realization enables hardware deployment in EB-PBF. System identification identifies Avrami parameters $(n,A,E)$ from ER70S-6 data, which are refined against measured hardness to improve predictive accuracy. Experimental validation demonstrates that optimized power trajectories can closely reproduce target hardness distributions, with a 51.1% RMSE reduction after parameter updates, highlighting the approach’s potential for spatially targeted microstructure control in additive manufacturing and its applicability to hybrid manufacturing and real-time estimation scenarios.

Abstract

Metal additive manufacturing (AM) opens the possibility for spatial control of as-fabricated microstructure and properties. However, since the solid state diffusional transformations that drive microstructure outcomes are governed by nonlinear ODEs in terms of temperature, which is itself governed by PDEs over the entire part domain, solving for the system inputs needed to achieve desired microstructure distributions has proven difficult. In this work, we present a trajectory optimization approach for spatial control of microstructure in metal AM, which we demonstrate by controlling the hardness of a low-alloy steel in electron beam powder bed fusion (EB-PBF). To this end, we present models for thermal and microstructural dynamics. Next, we use experimental data to identify the parameters of the microstructure transformation dynamics. We then pose spatial microstructure control as a finite-horizon optimal control problem. The optimal power field trajectory is computed using an augmented Lagrangian differential dynamic programming (AL-DDP) method with GPU acceleration. The resulting time-varying power fields are then realized on an EB-PBF machine through an approximation scheme. Measurements of the resultant hardness shows that the optimized power field trajectory is able to closely produce the desired hardness distribution.

Figures (6)

• Figure 1: Fitted data resulting from a linear least squares regression with isothermal assumption and $n=0.05$.
• Figure 2: Correspondence between predicted and measured hardness for the samples using both the isothermal fit with linear regression, and nonlinear regression for nonisothermal data, plotted against the perfect prediction line.
• Figure 3: Target hardness distribution for demonstration, a downsampled version of Carnegie Mellon University's "Scotty" mascot. The hardness range is set by the empirically determined maximum hardness of 420 HV (fully martensitic) and minimum hardness of 135 HV (fully tempered).
• Figure 4: Naive and optimized power fields over time. Optimized 1 is the optimal power field for the parameters identified in Section \ref{['sec:system-identification']}, while Optimized 2 is the same for the updated parameters from Section \ref{['sec:experimental-validation']}.
• Figure 5: Temperature fields resulting from the naive and two optimized power field trajectories over time.