A general formulation for the contact between superellipsoid surfaces and nodal points

The superellipsoids, part of the family of superquadratics, have geometric and mathematical properties that not only make them suitable for the representation of solids, ranging from spheres to relatively thin sheets, but also allow describing a very broad number of complex geometries. Their mathematical description favors a very efficient numerical treatment, in particular when used in the context of contact mechanics. Here, the contact between superellipsoids and nodal meshes, such as those used in finite elements, is analyzed. Based on the analytical properties of the implicit form of the superellipsoid equations, defined in a local coordinate frame, all the quantities necessary to handle a contact problem are derived. Recognizing that one of the most expensive tasks in contact mechanics is the contact detection, a three-stage contact search strategy is envisaged and implemented. Only in the close proximity between a node and the superellipsoid is a detailed contact search performed. In case of contact, the contact forces and their application points are evaluated using a continuous contact force model based on a penalty formulation. The existence of friction forces is considered. The methods developed and implemented for the contact between superellipsoids and nodal meshes are demonstrated and discussed with two contact scenarios comprising the impact of a superellipsoid on a hanging 'towel', and the contact of several superellipsoids with a flexible elastic platform.