DOI： 10.1016/j.cma.2026.119211

Abstract：The third medium contact has recently emerged as an effective approach for simulating contact problems, particularly in cases involving complex self-contact problems and extended optimization applications. By introducing a third medium between potential contact surfaces, contact constraints can be enforced through the constitutive response of the auxiliary material, thereby avoiding the explicit treatment of contact conditions. However, existing third medium formulations typically rely on regularization terms involving higher-order derivatives of the deformation field, which complicates the numerical implementation and significantly increases the computational cost. The Virtual Element Method (VEM) offers greater flexibility in mesh design for the third medium contact problems. However, the projection operators in VEM become significantly more complex due to the presence of higher-order derivatives. In this work, a first-order virtual element method is proposed for third medium contact problems in finite elasticity. To make third-medium contact compatible with a first-order VEM discretization, the auxiliary-field-based first-order regularization is incorporated into the VEM framework, enabling the entire problem to be discretized using only first-order derivatives with the first-order VEM framework. The resulting method retains the geometric flexibility of VEM while achieving higher computational efficiency. Several numerical examples involving complex self-contact are presented to demonstrate the performance of the proposed approach, including self-contact in flexible structures, deformation of auxetic metamaterials, and simulations of pneumatic soft robotic actuators. The results show that the proposed method provides stable and accurate contact simulations while significantly reducing the computational complexity compared with conventional higher-order regularization approaches.