Filling polyhedral molds

In manufacturing industry, finding an orientation for a mold that eliminates surface defects and ensures a complete fill after termination of the gravity casting process is an important and difficult problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron) such that, when it is filled, no air pockets and ensuing surface defects arise. Given a polyhedron in a fixed orientation, we present a linear time algorithm that determines whether the mold can be filled from that orientation without forming air pockets. We also present an algorithm that determines the most favorable orientation for a polyhedral mold in O(n²) time. A reduction from a well-known problem indicates that improving the O(n²) bound is unlikely for general polyhedral molds. We relate fillability to some well known classes of polyhedra. For some of these classes of objects, an optimal direction of fillability can be determined in linear time. Finally, for molds that satisfy a local regularity condition, we give an improved algorithm that runs in time O(nklog² nlog log(n/k)), where k is the number of venting holes needed to avoid air pockets in an optimal orientation.