Topology optimization for problems with internal or external contact represents a seeming paradox: if the topology is initially unknown, how can one identify contact boundaries? And further: even if boundaries are known and well-described, it is complicated and logistically complex to administrate contact points or surfaces and their associated Lagrange Multipliers. Both challenges can be alleviated using so-called third approaches[1], where contact forces are transferred through “void” domains with tailored elastic properties that generate high stiffness when directionally compressed but zero stiffness in shearing. In this talk, we discuss and demonstrate recent advances in applying third medium approaches to topology optimization. The initial works[2, 3], are improved for: 1) computational stability through better finite element discretizations; 2) extensions to three dimensions and 3) extensions to multiphysics settings including thermostructural and electric contacts. Altogether, our studies indicate that the third medium approach to contact problems provides an efficient and viable alternative to much more complex and geometry-dependent “classical” approaches. REFERENCES [1] P. Wriggers, J. Schroder, and A. Schwarz. A finite element method for contact using a third medium. Computational Mechanics, 52(4):837–847, 2013. [2] G.L. Bluhm, O. Sigmund, and K. Poulios. Internal contact modeling for finite strain topology optimization. Computational Mechanics, 67:1099–1114, 2021. [3] AH Frederiksen, O Sigmund, and K Poulios. Topology optimization of self-contacting structures. Computational Mechanics, 2023. online.