A non-iterative derivation of the common plane for contact detection of polyhedral blocks

A non-iterative derivation for finding the common plane between two polyhedral blocks is presented. By exploiting geometric relations between the normal of a plane and the closest vertex on a block, the common plane can be resolved without resorting to an iterative method. To facilitate derivations, normals in half-space are decomposed into finite subsets in which each subset corresponds to the same closest vertex on a block. The gap function, originally dependent on the normal and the two closest vertices, becomes a function of the normal only. To compute the gap for a given normal subset, the maximum theorem and the maximum projection theorem are introduced. The maximum theorem reduces finding the maximum in a subset to its boundary. Calculating the gap in 2D in a given subset thus reduces to checking two inner products. The maximum projection theorem further reduces finding the maximum on a 3D boundary to an explicit form. Three numerical examples are used to demonstrate the accuracy and efficiency of the proposed scheme. The example in which the blocks are in contact further shows the existence of a local maximum while calculating the gap and illustrates the potential deficiencies in using the Cundall's iterative scheme. Copyright (c) 2007 John Wiley & Sons, Ltd.