A face-smoothed cell method for static and dynamic piezoelectric coupled problems on polyhedral meshes

Low-order discretization schemes are suitable for modeling 3-D multiphysics problems since a huge number of degrees of freedom (DoFs) is typically required by standard high-order Finite Element Method (FEM). On the other hand, polyhedral meshes ensure a great flexibility in the domain discretization and are thus suitable for complex model geometries. These features are useful for the multiphysics simulation of micro piezoelectric devices with a thin multi-layered and multi-material structure. The Cell Method (CM) is a low-order discretization scheme which has been mainly adopted up to now for electromagnetic problems but has not yet been used for mechanical problems with polyhedral discretization. This work extends the CM to piezo-elasticity by reformulating local constitutive relationships in terms of displacement gradient. In such a way, piecewise uniform edge basis functions defined on arbitrary polyhedral meshes can be used for discretizing local constitutive relationships. With the CM matrix assembly is completely Jacobian-free and do not require Gaussian integration, reducing code complexity. The smoothing technique, firstly introduced for FEM, is here extended to CM in order to avoid shear locking arising when low-order discretization is used for thin cantilevered beams under bending. The smoothed CM is validated for static and dynamic problems on a real test case by comparison with both second-order FEM and experimental data. Numerical results show that accuracy is retained even if a much lower number of DoFs is required compared to FEM.