Optimal transport in non-convex geometries and its application in shrinkage porosity prediction

Abstract

Rooted in the Monge and Kantorovich problems, optimal transport seeks to minimize transportation costs, as quantified by Wasserstein distances, facilitating the transformation of one distribution into another distribution. However, its application faces challenges in addressing non-convex geometries, especially in interpolating mid-way distributions along transportation paths. In such scenarios, particles may breach the original shape boundaries, creating infeasible solutions. The article addresses this by proposing three innovative solutions: geometry mapping, trajectory planning and graph planning. These solutions aim to overcome the inherent limitations of optimal transport in non-convex spaces by offering mathematical formulations and implementation strategies. Importantly, the article goes beyond theoretical considerations by applying these solutions to predict shrinkage porosity in aluminum casting. This application demonstrates the broader relevance and effectiveness of the proposed solutions.