A fast and robust third medium contact approach using the neighbored element method

In several engineering applications, self-contact is a major effect or even desired feature, e.g., energy absorption in severe structural deformations, gripper in (soft-)robotics, and the optimization of those. Usual contact approaches require extensive calculation effort, but a fast and robust simulation approach could lead to remarkable improvements in product design. The third medium contact is a very promising approach, which can fulfill the requirements mentioned above. This method was also already applied in topology optimization approaches. However, the most concurrent regularization approaches require at least quadratic shape functions for hexahedron finite elements and at least cubic shape functions for tetrahedrons. In recent works, a new regularization technique was developed: herein, the gradients of deformation measures are computed by additional nodal degrees of freedom in the finite elements, i.e., as additional fields of unknowns. This is accompanied by a reduction of the order of the shape function. In this work, we derive a special discretization of the additional unknowns which enables self-contact based on the neighbored element method. This approach leads to a fast algorithm for the computation of the regularized set of partial differential equations: the displacements are the only nodal degrees of freedom and can be computed with linear finite element shape functions while the additional unknowns are discretized in the quadrature points of the mesh and solved in a staggered manner. The results are critically analyzed and illustrated for several two-dimensional problems.