Handling periodic boundary conditions on arbitrary mesh

To evaluate the electromagnetic properties of heterogeneous materials using the finite element (FE) method, the appropriate boundary conditions should be defined. The periodic boundary condition is one of the most efficient in terms of convergence rate. To impose these boundary conditions, the classical method (CM) requires a periodic mesh, where the opposite faces are meshed identically. This work presents a new accurate method to handle periodic boundary conditions on the arbitrary mesh. The first step of this method consists of establishing the periodic relation which relates an unknown on a face as a function of the unknowns of the associated triangle on the opposite face. The second step consists of introducing the periodicity relations in the FE system. The proposed method has been applied in the case of multi-scale homogenisation problem by computing the effective constitutive parameters of periodic structures. Numerical results of the present method are compared with those of CM which consists of using a periodic mesh.