# MDC-ELLIPSOIDs

MDC-ELLIPSOIDs is a MATLAB® code to calculate the minimum distance between two ellipsoidal surfaces in a non-conformal contact situation. The implemented methodology is specifically designed for rigid contact detection where the surfaces are in a convex-convex interaction and do not deform or alter there local geometric properties whenever there is contact. Therefore, in rigid contact, surfaces may share a common geometric locus, i.e., surfaces may overlap. Minimum distance calculation is performed by numerically solving a system on non-linear equations with the Newton-Raphson method with analytical Jacobians. The system of non-linear equations corresponds to a set of necessary, although not sufficient, geometric constraints whose solution consists of the pair of points that have a common normal direction. To restrict the solution space to a unique pair of points, different types of constraints are considered: collinear constraint (vector cross-product); orthogonal constraint (vector inner-product); and locus constraint (point belongs to surface). To guarantee unicity, an additional non-equality condition, that evaluates the distances of all the candidate solutions, is considered. All constraints have a well-defined analytical expression that relate the geometric entities envolved in the common normal concept, i.e., the normal, tangent, binormal and distance vectors which are deducible given only the implicit surface functional. Contact detection is performed by simply evaluating the sign of the minimum distance between the surfaces. The three types of contact detection statuses are: (i) no contact (surfaces appart ⇒ minimum distance greater than zero); (ii) external contact (contact at a single point ⇒ minimum distance equal to zero); and (iii) contact (surfaces overlap ⇒ minimum distance lesser than zero). Householder transformation proved to be useful to deduce the tangential and binormal vectors given the analytical expression of the normal vector which, by vector calculus, is derived as the gradient of the implicit function. The Householder transformation is used to provide a base of three orthogonal vectors, in which one of them is collinear to the surface normal. Proximity queries were also implemented to test for coarse (with the traditional bounding spheres) and narrow (with the separate condition) contact detection. Note that the proximity queries do not calculate the minimum distance between ellipsoids. Applications The MDC-ELLIPSOIDs code was originally developed to perform computational simulations for contact analysis of multibody systems. However, the proposed methodology has important applications in many other areas of applied science such as, molecule simulation in computational physics, modeling discontinuous mechanical systems (discrete element method) in geomechanics, humanoid design in biomechanics, virtual reality simulation and computer animation.

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## References in zbMATH (referenced in 11 articles , 1 standard article )

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