A third medium approach for contact using first and second order finite elements

Third medium contact can be applied in situations where large deformations occur and self-contact is possible. Starting with Wriggers et al. (2013), this approach has been further developed and often applied in the area of topology optimization. Lately approaches have been discussed which use the gradient of the deformation measure to enhance the performance of the algorithm. Such approaches, however, require finite elements with quadratic shape function. In this paper two new regularization techniques are introduced which on one hand reduce the complexity of the gradient computation of the deformation measure and on the other hand allow the use of finite elements with linear shape functions. The approaches will be critically evaluated and applied to different two-dimensional problems.