The third medium contact method has recently come into popularity as an alternative to traditional contact methods in contexts where search for contact boundaries is problematic, i.e. topology optimization. To enforce the contact constraints, it relies on a fictitious compliant material occupying the void space. In finite strain setting, this necessitates regularization techniques to improve the behavior of the third medium material. A number of existing models rely on penalization of locally computed second gradients of displacements, either through direct calculation on second-order elements or through additional degrees of freedom. Here we propose an alternative approach using element-wise deformation gradient averaging to effectively penalize spatial variations of the deformation gradient, together with a linear elastic term enforcing constant third medium stiffness. Our approach enables the use of first-order finite element formulations without any additional degrees of freedom and is therefore easy to implement. We demonstrate the robustness of the proposed method on several well-established benchmarks.

翻译：第三介质接触方法作为传统接触方法的一种替代方案，在接触边界搜索存在问题的场景（如拓扑优化）中近来受到广泛关注。该方法通过在空隙空间填充虚构的柔性材料来施加接触约束。在有限应变框架下，这需要采用正则化技术以改善第三介质材料的行为。现有多种模型依赖于对局部计算的位移二阶梯度进行惩罚，其实现方式包括在二阶单元上直接计算或引入额外自由度。本文提出一种基于单元级变形梯度平均的替代方法，该方法能有效惩罚变形梯度的空间变化，并结合线性弹性项以保持第三介质刚度恒定。我们的方法支持采用一阶有限元公式且无需任何额外自由度，因而易于实现。我们在多个经典基准测试中验证了所提方法的鲁棒性。