Geometric modeling, including the computational tasks of cutter swept volume generation, in-process workpiece update, and cutter-workpiece engagement (CWE) extraction, is an essential component of virtual machining which simulates the use of machine tools for part machining. Geometric modeling not only verifies the machining tool paths but also provides the required geometric information for simulating the involved cutting mechanics and dynamics. CWE, the instantaneous engagement area between the cutter and in-process workpiece at a given cutter location, is the key input for cutting force prediction, chatter analysis, and process parameter optimization. Given that extracting such information accurately and efficiently is still a known challenge in multi-axis milling, this research aims to develop an efficient method for reliable CWE determination based on voxel modeling.

To accomplish the computational tasks of workpiece update and CWE extraction, the algorithm of direct voxel tracing is developed to efficiently trace a line or any quadratic curve. As the tracing process is implemented by simple incremental/decremental operations, the algorithm yields high efficiency. Based on this tracing algorithm, a 3D circle voxelization method is developed to directly calculate immersion angels along the cutter axis.

Direct voxel tracing is initially implemented for three-axis milling on a basic uniform-grid voxel modeling space. Since allsliced cutter profiles by voxel layer boundary planes are 2D circles, workpiece update and CWE determination can be accomplished simultaneously. The algorithm is then extended to five-axis milling, and the notion of a cutter internal voxel space boundary is proposed to capture the cutter internal space accurately and execute a minimum number of deactivation operations for workpiece model voxels. Due to the much-increased geometric complexity in five-axis milling, a 3D circle voxelization algorithm is needed for CWE extraction. To further increase the computational time and memory efficiency, such tracing algorithm is implemented on two-level voxel modeling grids. To apply the developed algorithms to all types of cutters, a versatile implicit function for the surface of a general milling tool has been formulated, and efficient workpiece update is achieved by forming an oriented cylindrical bounding surface around the general milling tool.