Periodic Boundary Conditions in the FEM using Arbitrary Meshes ...

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classical method to take into account the periodic boundary. conditions, the .... Cj+γjCl and Ck+γkCl respectively and to delete the column Cl. Thus the matrix ...
Periodic Boundary Conditions in the FEM using Arbitrary Meshes Ouail Ouchetto Laboratoire de Recherche et Innovation en Informatique FSAC & FSJES Aïn Chock, Université Hassan II Casablanca, Maroc [email protected]

Abstract— This paper presents a new approach to impose periodic boundary conditions on arbitrary mesh. This method is based on two steps; the first one consists in writing each unknown of the first side as function of the unknowns of the opposite side. The second step is to introduce the periodicity relations in the matrix system obtained by finite element discretization. The proposed method has been applied in the context of homogenization problem. Index Terms— Arbitrary mesh; Finite element method (FEM); Periodic boundary conditions.

I. INTRODUCTION The electromagnetic analysis of periodic composite materials has always been an important topic in computational electromagnetic. For the characterization of these materials, the Floquet theory is applied and it allows reducing the studied structure to elementary cell with periodic boundary conditions. The electromagnetic properties can be computed directly or by homogenization procedure if the period of the studied structure is small compared to incident wavelength [1-4]. A variety of numerical methods have been used for computing these properties. The finite element method remains the most used technique for electromagnetic properties analysis in radar scattering, antennas, RF and microwave engineering, high-speed/highfrequency circuits, wireless communication, electromagnetic compatibility, photonics, remote sensing, and space exploration. The success of FEM is due to its ability to model the heterogeneous material with complex shape and to incorporate the different types of boundary conditions. In the classical method to take into account the periodic boundary conditions, the mesh is created by the same way on the boundaries where these conditions are applied. We note that all existing mesh generators do not have all functionalities to generate the periodic mesh and in the case of complex structures and the existing methods are not always flexible. The proposed solution consists in imposing periodic boundary conditions on arbitrary or no periodic mesh. Some recent works was sudying this subject in various areas. In elasticity, Nguyena et al. have proposed a method using the Lagrange interpolation formulation or the cubic spline interpolation formulation [5]. In electromagnetism, Aubertin et al. have introduced the double Lagrange multipliers method using the

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Saad Zaamoun Laboratoire de Marketing - FSJES Aïn Chock, Université Hassan II Casablanca, Maroc

scalar and vector potential formulations [6]. In this paper, we present a new method easy to implement and it is not related to a specific context or problem. II. PROPOSED METHOD We consider a tri-periodic artificial structure composed by two or more distinct components. This structure is modeled as identical inclusions emerged in an infinite homogenous background. The periods along the axes (ox), (oy) and (oz) are respectively α, β and δ. By using the Floquet theory, the studied structure is reduced to an elementary cell Y with periodic boundary conditions. The proposed method can be applied for the computation of the different electromagnetic properties using nodal finite element method as: the effective constitutive parameters, the electromagnetic potential, electromagnetic field in 2D, etc. For this, we use the general matrix form (eq. 1) of the electromagnetic problem using nodal finite element method. ( )=( ) (1) where [A] and (b) represent, respectively, the matrix system and the second member. The vector (U) contains the unknowns associated to the mesh nodes noted N.

Fig.1. Two opposite faces are meshed differently.

In the case of the arbitrary mesh, two opposite faces are meshed differently (see Fig. 1). The idea of the proposed method consists to express each unknown on a face as function of the unknowns on the opposite face. For this purpose, we consider a node nl=(xl, β, zl )∈N of the face S(y= β). By using the translation (0, - β, 0), the associated point of nl is M =(xl, 0, zl)∉ N (see Fig. 1). In the first step, we determine the triangle = containing the point M. We note that the triangle verify the following relations: (



)⨯(



)≥0

(



)⨯(



)≥0

(



)⨯(



between the components bl, bi, bj and bk associated to nodes nl, ni, nj and nk has the same form: =

=

+

)≥0

where ⨯ is the scalar product and ⊗ is the vector product. In the second step, the unknown will be approached as function of the unknowns , and associated to the nodes , and .

(6)

To introduce this relation, in the first time, we replace bl by γibi+ γjbj + γkbk in the matrix system. In the second time, we replace the kth row Lk of [A] by γkLk - Ll and bk by -γibi - γjbj. After that, we remove the lth row of [A]. Then the matrix system becomes: ..., ,..., ..+ − ,…, , ,.. ( ) = …, ,…, ,…,−

(2)

+



,…,

,

,…

(7)

This process will be applied for all nodes of the face S(y=β). To introduce the periodic conditions in (ox) and (oz) directions, this method will applied for all nodes of S(x=α) and S(z=δ). The unit cell has 12 edges, for example E1, E2, E3 and E4 are the edges parallel to (ox), we take E1 as a reference edge and the unknowns of E2, E3 and E4 will be expressed as function of those of E1. The periodicity relation related the unknown ul of E1 and the unknowns of E2 has the following form: =

+

(8)

The introduction of this relation in the matrix system, we apply the same technique presented previously for the all nodes of E1. III. Fig. 2. Triangle

=

contains the point M and the point H represents

the intersection of the two lines (

) and (

).

Thus, we express as function of and , and as function of and . The final expression of is giving as follow: ( , ) ( , ) ( , ) , = + ( , ) ( , ) , , ( , ) (3) + ( , ) where d(:,:) is the Euclidean distance. The periodicity relation related the unknown ul of S(y=β) and ui, uj, uk of S(y=0) has the following form: =

=

+

+

(4)

The introduction of periodicity relation consists on replacing ul by γiui+ γjuj + γkuk in vector (U) of the matrix system (1). To remove the redundancy of ui, uj, uk in (U), the idea is to replace the columns Ci, Cj , Ck of [A] by Ci+γiCl, Cj+γjCl and Ck+γkCl respectively and to delete the column Cl. Thus the matrix system becomes: .., + +. . + + +. . + , ,.. .., + , ,… = ( ) (5) …, ,…, ,…, ,…, In order to make the matrix [A] square, we take into account the periodicity effect on the second member (b). The relation

NUMERICAL RESULTS

To validate the proposed method, we present the homogenization problem at fixed frequency using the concept of two-scale convergence and nodal finite element method. We consider a tri-periodic structure which composed by two distinct components modeled as an identical inclusions emerged in a homogenous background. The elementary cell Y is characterized by the permittivity ( ) and the permeability ( ). The local problem is about to determine the periodic subcorrectors χk and χkμ for (k = 1, 2, 3) [7]: . ( ( ).



. ( ( ).



( ) =0

(9)

( ) =0

where u denotes the gradient, u=(x, y, z) and ek is the kth vector of the canonical basis. From the solutions of these problems, the effective constitutive parameters are expressed as function χkϵ and χkμ and have the following form: = =

1 | | 1 | |

( ). (



( ))

( ). (



( ))

where I3 is the identity dyadic, and |Y| is the volume of Y. For the numerical validation, the studied periodic structure

contains the cubical inclusions suspended in the host medium. The relative permittivity and permeability of inclusions are i =1 and μi =1, and of the host medium are e = 40 and μe = 25. Figures 3 and 4 plot the effective relative permittivity and relative permeability respectively as a function of the volume fraction f. The proposed method is compared to the classical one [8-11] which uses a periodic mesh to resolve the problem (9). As can be observed from these Figures, the present method and the classical method produce exactly the same effective constitutive parameters for different values of f. The obtained results confirm the efficiency and the effectiveness of the proposed method.

IV.

A new method to impose the periodic boundary conditions on the arbitrary mesh is presented. It is based on two steps; the first one is to express each unknown on a face as function of the unknowns of the associated triangle on the opposite face. The second step is to introduce all periodic relations in the matrix system. This method is applied in homogenization problem context. In the numerical results section, the obtained results confirm the validity and the efficiency of the proposed method. REFERENCES [1]

Fig. 3. Effective relative permittivity plotted as function of the volume fraction f.

Fig. 4. Effective relative permeability plotted as function of the volume fraction f.

CONCLUSION

O. Ouchetto, B. Abou El Majd, H. Ouchetto, B. Essakhi, S. Zouhdi, “Homogenization of Periodic Structured Materials with Chiral Properties,” IEEE Trans. Antennas Propag., in press. [2] O. Ouchetto, H. Ouchetto, S. Zouhdi, A. Sekkaki, “Homogenization of Maxwell’s Equations in Lossy Biperiodic Metamaterials,” IEEE Trans. Antennas Propag., vol. 61, no 8, pp. 4214-4219, 2013. [3] Ouail Ouchetto, “Modélisation large bande de métamatériaux bianisotropes et de surfaces structurées,” Doctoral dissertation, Paris 11, 2006. [4] O. Ouchetto, S. Zouhdi, A. Bossavit, B. Miara, “Effective electromagnetic properties of structured chiral metamaterials,” Photonic Metamaterials: From Random to Periodic. Optical Society of America, June 2006. [5] V. D. Nguyen, E. Bchet, C. Geuzaine and L. Noels, “Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Computational Materials Science,” vol. 55, pp. 390-406, 2012. [6] M. Aubertin, T. Henneron, F. Piriou, P. Guerin, and J. C. Mipo. Periodic and anti-periodic boundary conditions with the Lagrange multipliers in the FEM. IEEE Transactions on Magnetics, vol. 46, no. 8, pp. 34173420, 2010. [7] G. Allaire, “Homogenization and two-scale convergence,” SIAM J. Math. Anal., vol. 23, no. 6, p. 1482-1518, 1992. [8] O. Ouchetto, S. Zouhdi, A. Razek, and B. Miara, “Effective constitutive parameters of structured chiral metamaterials,” Microw. Opt. Tech. Lett., vol. 48, pp. 1884-1886, Sep. 2006. [9] O. Ouchetto, S. Zouhdi, A. Bossavit, G. Griso, and B. Miara, “Effective constitutive parameters of periodic composites,” in Eur. Microw. Conf., Paris, France, Oct. 2005, p. 145. [10] O. Ouchetto, S. Zouhdi, A. Bossavit, G. Griso, B. Miara, and A. Razek, “A new approach for the homogenization of three-dimensional metallodielectric lattices: the periodic unfolding method,” PECS-VI, International Symposium on Photonic and Electromagnetic Crystal Structures, June 2005. [11] O. Ouchetto, S. Zouhdi, A. Bossavit, G. Griso, and B. Miara, “Homogenization of 3-D structured composites of complex shaped inclusions,” Progress Electromagn. Res. Symp., Hangzhou, China, pp. 112, Aug. 2005.