First order finite element formulations for third medium contact

• Published in: Computational mechanics Vol. 76; no. 3; pp. 829 - 845
• Main Authors: Wriggers, P., Korelc, J., Junker, Ph
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2025; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Complexity; Computational Science and Engineering; Deformation; Energy; Engineering; Finite element method; Heat treating; Hexahedral finite elements; Lagrange multiplier; Original Paper; Quadrilaterals; Regularization; Shape functions; Theoretical and Applied Mechanics; Topology optimization; Finite elements; Frictionless contact; Finite deformations; Hyperelasticity
• ISSN: 0178-7675; 1432-0924; 1432-0924
• DOI: 10.1007/s00466-025-02628-y

Third medium contact can be applied in situations where large deformations occur and self-contact is possible. This specific discretization technique has the advantage that the inequality constraint, inherent in contact formulations, is circumvented. The approach has several applications, like soft robotic or topology optimization. Recent approaches have been explored, using the gradient of the deformation measure to improve algorithmic performance. However, these methods typically require quadrilateral or hexahedral finite elements with quadratic shape functions, adding to their complexity. Also, the computation of second order gradients using quadratic triangular or tetrahedral elements does not lead to reasonable results since these gradients are constant at element level. In this paper, we apply a new regularization technique to triangular and tetrahedral finite elements of lowest ansatz order that approximates the gradient computations and thus reduces computational complexity.