Computational Geometric Learning High Quality Surface Mesh Generation for Swept Volumes

We present a novel, efficient and flexible scheme to generate a high quality mesh that approximates the outer boundary of a swept volume. Our approach comes with two guarantees. First, the approximation is conservative, i.e., the swept volume is enclosed by the generated mesh. Second, the one-sided Hausdorff distance of the generated mesh to the swept volume is upper bounded by a user defined tolerance. Exploiting this tolerance the algorithm generates a mesh that is adapted to the local complexity of the swept volume boundary, keeping the overall output complexity remarkably low. The algorithm is two-phased: the actual sweep and the mesh generation. In the sweeping phase we introduce a general framework to compute a compressed voxelization. The phase is tailored for an easy application of parallelization techniques. We show this for our exemplary implementation and provide a multi-core solution as well as a GPU based solution using CUDA. For the meshing phase we utilize and extend the well known Delaunay refinement such that it generates an adaptive conservative approximation that obeys the user defined upper bound on the one-sided Hausdorff distance to the swept volume. The approach is able to handle inputs of high complexity and compute an approximation with a very high precision, which we demonstrate on real industrial data sets. Note to Practitioners — This work is motivated by the following problem we were posed by a car manufacturer: To measure the movement of an engine during test drives, sensors were placed onto the motor compartment of a car. With this setup the sensors recorded the position and orientation of the engine every 5 ms. The reason for these test drives were clearance checks between parts of the engine and other components, fixated on the chassis, e.g., the oil pan and a neighboring component, cf. Figure 1. We are now interested in the following question: Which volume in space does the vibrating part of the engine occupy? We here present a method that approximatively computes the outer boundary of this volume that is also called the swept volume. The model that generates the volume (hence generator) will be a CAD model, that is swept (hence swept volume) along the sequence of motions (trajectory). A trajectory can of course also come from CAD. The scenario depicted in Figure 2 shows a user-created path during maintainability analysis. The swept volume (or rather its outer boundary) can be used to verify that the path is collision free and that spatial tolerances are not violated. In contrast This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827, CGL, Computational Geometry Learning. Andreas von Dziegielewski is with the faculty of Physics, Mathematics and Computer Science at the Johannes Gutenberg-university of Mainz [email protected] Michael Hemmer is with the School for Computer Science, Tel Aviv University [email protected] Elmar Schömer is with the faculty of Physics, Mathematics and Computer Science at the Johannes Gutenberg-university of Mainz [email protected] Fig. 2. Example of a CAD user generated swept volume for maintainability analysis. Left: transparent swept volume boundary (red) with engine (green) and chassis (blue). Right: non-transparent swept volume boundary (red) and chassis (blue). to previous methods that compute the swept volume, our swept volume approximation is reliable, meaning, that we give guarantees on the quality of our approximation in terms of how close we are to the actual swept volume. Future work would be the extension of our approach to kinematic chains, e.g., the swept volume of a car seat under all possible configurations.