Un couplage FEM-BEM pour la modélisation de composites magnétoélectriques

International audience-Un couplage FEM-BEM basé sur une formulation en potentiel scalaire magnétique réduit est appliqué à la modélisation des composites magnétoélectriques. Une telle approche permet de ne pas considérer la région d'air et d'utiliser un unique maillage pour les sous-problèmes magnétiques, mécaniques et électriques qui oeuvre à l'effet magnétoélectrique. Un solveur Gauss-Seidel par bloc est mise en oeuvre pour résoudre le problème global. Mots clés-multiphysique, couplage FEM-BEM, composite magnétoélectrique

A FEM-BEM coupling for the modeling of linear magnetoelectric effects in composite structures

International audienceThe aim of this study is to model magnetoelectric effects involved in laminate composites made of magnetostrictive and piezoelectric materials. We therefore have to consider an electro-magneto-mechanical problem derived by coupling equations describing active materials, as Tefenol-D for magnetostrictive layers and PZT for piezoelectric layers [1]. The reference method for the resolution of this type of problems is the finite element method. Although this method is general and largely proven, it can be computationally expensive, especially for low-frequency electromagnetic problems where the resolution may require meshing the air and considering an infinite box to simulate the decay of electromagnetic fields at infinity. In our case, this method is well adapted to the modeling of electro-mechanical coupling. Indeed, the large permittivity of piezoelectric materials makes the electric field leaks negligible. It may become expensive for the modeling of the magneto-mechanical coupling especially if the volume of the active magnetic materials is very small compared to the volume of the air leading to a huge mesh with a lot of air. To solve this type of problem without having to mesh the air, coupling between finite element method for the active material and boundary element method for the air region have already been used with excellent results [2]. We will therefore use this type of method in the modeling of the magnetic problem. Although the magnetostrictive phenomenon is strongly nonlinear, we will consider it as linear as a first approximation. This approximation makes it dual to the piezoelectric phenomenon. The electro-magneto-mechanical coupling is therefore expressed by a linear matrix block system with sparse matrices for the mechanical and electrical problem and full matrices for the magnetic problem. The resulting linear system is solved using the Gauss-Seidel method with linear solvers adapted to the type of matrix, i.e, MUMPS used for sparse matrices and GMRES used for full matrices

Un couplage FEM-BEM pour la modélisation de composites magnétoélectriques

International audiencen couplage FEM-BEM basé sur une formulation en potentiel scalaire magnétique réduit est appliqué à la modélisation des composites magnétoélectriques. Une telle approche permet de ne pas considérer la région d’air et d’utiliser un unique maillage pour les sous-problèmes magnétiques, mécaniques et électriques qui œuvre à l’effet magnétoélectrique. Un solveur Gauss-Seidel par bloc estmise en œuvre pour résoudre le problème global

A FEM-BEM multiphysics coupling for the modeling of magnetoelectric composite structures

International audienceThis study concerns the modeling of structures made of composite materials with magnetoelectric effects arising from the combination of magnetostrictive and piezoelectric materials. Modeling these effects requires consideringan electro-magneto-mechanical problem derived bycoupling equations describing active materialssuch asTerfenol-D for the magnetostrictive phaseand PZT for the piezoelectric phase. A typicalmethod used for the resolution of this kind of problemis the finite element method(FEM). Although this method is general and has proven to be effective in many instances, it can becomecomputationally expensive, particularlyfor electromagnetic problems whereactive materials and coils are distantfromeach otherthus necessitating a huge mesh of air. In our case, the FEMis well adapted to the modeling of electro-mechanical coupling. Indeed, the large permittivity of piezoelectric materials makes the electric field leaks negligible.For magneto-mechanical coupling involving a small volume of active magnetic materials compared to air, FEM becomes expensive. Coupling between finite element method and boundary element method offers the possibility to not have to mesh the air while still providing excellent results(G. Meunier, J. Coulomb, S. Salon, and L. Krahenbul, “Hybrid finite element boundary element solutions for three dimensional scalar potential problems,” IEEE Transactions on Magnetics, vol. 22, no. 5, pp. 1040–1042, Sep. 1986). In this work an approach coupling FEM, for the electric and mechanicalfields, and BEM, for the magnetic field, is proposed to solve problems involving a reduced volume of active materials.The magnetostrictive phenomenon is strongly nonlinear, but it will be consideredlinear as a first approximation.Two dual formulations of the magneto-mechanical problemwill be presented, based on total magnetic vector potential and reduced magnetic scalar potential formulations respectively.The electro-mechanical problem is solvedby a classical FEM formulation. The electro-magneto-mechanical coupling isthen expressed by a matrix block system with sparse matrices for the mechanical and electrical problemsand full matrices for the magnetic problem. The resulting system is solved using theblockGauss-Seidel methodwith linear solvers adapted to the type of matrix, i.e, MUMPS used for sparse matrices and GMRES used for full matrices.Results of the two formulations and their performance will be compared

FEM-BEM modeling of nonlinear magnetoelectric effects in heterogeneous composite structures

International audienceThis paper proposes a mathematical model for 3D nonlinear magnetoelectric effects in heterogeneous composite structures. Through the coupling of the Finite Element Method (FEM) with the Boundary Element Method (BEM), only the active material is explicitly considered, and thus a single mesh is used to support all phenomena. A mixed formulation is used to model the magnetic phenomena, a vector potential formulation in the volume and a scalar potential formulation in the free space domain. Material laws for the magnetostrictive composite phase are derived from partial derivatives of a scalar invariant’s formulation of the Helmholtz free energy. The coupled problem is solved by iteratively solving single-physics problems, and the full algorithm is applied to the modeling of a test case

Modeling of magnetoelectric effects in composite structures by FEM-BEM coupling

International audienceIn this paper, a coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behaviour of magnetoelectric effects in composite structures. This coupling of numerical methods makes it possible not to have to consider a free domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns which is accompanied by shorter computation times compared to a classical FEM approach. A mixed magnetic vector potentialreduced magnetic scalar potential formulation is used for the magnetic problem, and classical FEM formulations are used for electrical and mechanical problems. The resulting global algebraic system is solved by a block Gauss-Seidel solver

FEM-BEM modeling of nonlinear magnetoelectric effects in heterogeneous composite structures

This paper proposes a multiphysics multi-method model for 3D nonlinear magnetoelectric effects in heterogeneous composite structures made of the association of piezoelectric and magnetostrictive materials. Through the coupling of the Finite Element Method with the Boundary Element Method, only the active material is explicitly considered, and thus a single mesh is used for the resolution of all the physics. A mixed formulation combining the vector potential in the volume and a scalar potential in the free space is used to model magnetic phenomena. Non-linear constitutive laws for the magnetostrictive phase are derived from partial derivatives of a scalar invariant's formulation of the Helmholtz free energy, while linear relations are used to describe piezoelectric behavior. The coupled problem is solved by iteratively solving single-physics problems, and the full algorithm is used to model a rotating coilless ME device which can operate as an energy harvester or as an actuator

A FEM-BEM coupling strategy for the modeling of magnetoelectric effects in composite structures

This paper deals with the modeling of devices based on magnetoelectric composite materials. These heterogeneous structures are made of ferromagnetic and ferroelectric materials, the mechanical coupling of which allows obtaining magneto-electric effects exceeding by several orders of magnitude the response of single-phase components. A coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behavior of magnetic effects, while classical FEM formulations are used for the electrical and mechanical problems. This coupling of numerical methods allows to avoid considering a free space domain around the active domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns, which is accompanied by shorter computation times compared to a pure FEM approach. The global algebraic system is solved by a block Gauss-Seidel type solver, which allows a good convergence of the multiphysics