Continuous collision detection for ellipsoids

A mathematical framework for the analysis and comparison of contact detection methods for ellipses and ellipsoids

The purpose of this research is to provide a framework for the analysis and comparison of contact detection algorithms for pairs of ellipses and ellipsoids. This work focuses primarily on the category of algorithms that are the most computationally efficient and can produce estimates of the separation and the penetration distance between ellipses and ellipsoids. Specifically, only analytic representations of the ellipses and ellipsoids are considered and contact detection for moving pairs of ellipsoids is not treated. The first contribution is a mathematical framework for the study of these algorithms, most notably with existence and uniqueness proofs for classes of contact detection algorithms, formal descriptions of the asymptotics of pairs of ellipses in close contact (or overlap), and a global analysis of constraints on the normals. The framework highlights the key role played by the different definitions of contact found in the literature, independent of the numerical strategies deployed to estimate the separation/penetration distance. Specifically, it is shown that all the studied algorithms can be expressed as minimization problems, with or without non-binding constraints on the normal(s) at the contact point(s), and that the constraints can be used to identify the global minima among the critical points in the minimization problem. Another contribution of this research, based on the mathematical framework introduced, is a better classification of the known algorithms. These algorithms are compared on established test problems, and their strengths and weaknesses are highlighted and explained in terms of their classification. Furthermore, this research provides comparisons in speed and stability between the most efficient algorithms in each category over a large sample size of test problems. Among the other contributions, this research describes inexpensive but effective initial estimates of the contact to be used in iterative algorithms. Finally, the usefulness of the new framework is illustrated with the introduction of a fast algorithm combining some new and old ideas.

Computing the Closest Approach Distance of Two Ellipsoids

This paper presents two practical methods for computing the closest approach distance of two ellipsoids in their inter-center direction. The closest approach distance is crucial for collision handling in the dynamic simulation of rigid and deformable bodies approximated with ellipsoids. To find the closest approach distance, we formulate a set of equations for two ellipsoids contacting each other externally in terms of the inter-center distance, contact point, and normal vector. The equations are solved robustly and efficiently using a hybrid of the fixed-point iteration method and bisection method with root bracketing, and a hybrid of Newton’s method and the bisection method. In addition to a stopping criterion expressed with the progress of the solution, we introduce a novel criterion expressed in terms of the error in distance. This criterion can be effectively employed in real-time applications such as computer games by allowing an unnoticeable error. Experimental results demonstrate the robustness and efficiency of the proposed methods in various experiments.

Effective Properties of Composites with Periodic Random Packing of Ellipsoids

The aim of this paper is to evaluate the effective properties of composite materials with periodic random packing of ellipsoids of different volume fractions and aspect ratios. Therefore, we employ computational homogenization. A very efficient MD-based method is applied to generate the periodic random packing of the ellipsoids. The method is applicable even for extremely high volume fractions up to 60%. The influences of the volume fraction and aspect ratio on the effective properties of the composite materials are studied in several numerical examples.

Self assembly of Janus ellipsoids

We propose a primitive model of Janus ellipsoids that represents particles with an ellipsoidal core and two semisurfaces coded with dissimilar properties, for example, hydrophobicity and hydrophilicity, respectively. We investigate the effects of the aspect ratio on the self-assembly morphology and aggregation processes using Monte Carlo simulations. We also discuss certain differences between our results and those of earlier results for Janus spheres. In particular, we find that the size and structure of the aggregate can be controlled by the aspect ratio.