In this work we present a consistent reduction of the relaxed micromorphic model to its corresponding two-dimensional planar model, such that its capacity to capture discontinuous dilatation fields is preserved. As a direct consequence of our approach, new conforming finite elements for $H^\mathrm{dev}(\mathrm{Curl},A)$ become necessary. We present two novel $H^\mathrm{dev}(\mathrm{Curl},A)$-conforming finite element spaces, of which one is a macro element based on Clough--Tocher splits, as well as primal and mixed variational formulations of the planar relaxed micromorphic model. Finally, we demonstrate the effectiveness of our approach with two numerical examples.

翻译：本文提出了一种将松弛微形态模型一致约化为相应二维平面模型的方法，并保持了其捕捉不连续膨胀场的能力。作为本方法的直接结果，需要构造适用于$H^\mathrm{dev}(\mathrm{Curl},A)$的新型协调有限元。我们提出了两种新颖的$H^\mathrm{dev}(\mathrm{Curl},A)$-协调有限元空间，其中一种是基于Clough--Tocher分割的宏单元，同时给出了平面松弛微形态模型的基本变分形式与混合变分形式。最后，通过两个数值算例验证了本方法的有效性。