Homogenization of the Maxwell Equations at Fixed Frequency

Abstract

Abstract. The homogenization of the Maxwell equations at fixed frequency is addressed in this paper. The bulk (homogenized) electric and magnetic properties of a material with a periodic microstructure are found from the solution of a local problem on the unit cell by suitable averages. The material can be anisotropic and satisfies a coercivity condition. The exciting field is generated by an incident field from sources outside the material under investigation. A suitable sesquilinear form is defined for the interior problem, and the exterior Calderón operator is used to solve the exterior radiating fields. The concept of two-scale convergence is employed to solve the homogenization problem. A new a priori estimate is proved as well as a new result on the correctors.

Estimates of scattered electromagnetic fields

“…Note that the given incident fields E i , H i act as a bounded linear functional on H.curl, /. The weak formulation (2.8) has, for given material properties in the domain, a unique solution bounded in H.curl, / because of the Lax-Milgram theorem [10,14]. From the existence of the interior electric field, we construct the corresponding magnetic field H.x/ as in [10].…”

“…The weak formulation (2.8) has, for given material properties in the domain, a unique solution bounded in H.curl, / because of the Lax-Milgram theorem [10,14]. From the existence of the interior electric field, we construct the corresponding magnetic field H.x/ as in [10]. With the internal fields E.x/ and H.x/, we apply the trace operator from the inside, , and construct the tangential scattered electric field from the outside, C .E s / D .E E i /.…”

“…We are looking for solutions E and H of (2.1) in the space H.curl, /. A weak formulation of this problem is given in Equation (2.8), see also [10]. The boundary conditions on @ , that connect the solutions in the domains and e , are introduced in the formulation of the exterior problem, and the pertinent radiation conditions presented in Section 2.2.…”

Numerical upscaling for the eddy-current model with stochastic magnetic materials

“…Regarding Eq. (24) it is obvious that the three components of the weighting vector in a node y j , denoted by z 1 ; z 2 ; z 3 are coupled within the equations for the three components of X Ár . Considering a lexicographical numbering of the nodes we choose a point-wise gathering for the unknowns since it leads to smaller blocks compared to an equation-wise aggregation.…”

“…Heterogeneous composite materials have widely been analyzed, see e.g. [23,24]. A scaling theory for Maxwell's equation concerning composite material was developed in Ref.…”