A least-squares implementation of the field integrated method to solve time domain electromagnetic problems Zhifeng Sheng*, Rob Remist and Patrick Dewilde*, * CAS group, EEMCS, TUDelft Mekelweg 4, 2628 CD Delft, The Netherlands Email: {Z.Sheng, P.Dewilde}@ewi.tudelft.nl tEMLAB, EEMCS, TUDelft Mekelweg 4, 2628 CD Delft, The Netherlands Email: R.F.Remis@ewi.tudelft.nl Abstract-This paper presents the application of the least- squares field integrated method based on hybrid linear finite elements in time-domain electromagnetic (EM) problems with high contrast interfaces. The method proposes the use of edge based linear finite elements over nodal elements and edge elements of Whitney form. It shows how the equations have to be accommodated to yield a correct solution. It proposes a general strategy to combine edge linear finite elements and nodal linear finite elements. Numerical experiments show that the resulting algorithms are stable and achieve high quality field interpolation even in the presence of very high contrasts. Index Terms-least-squares, Field Integrated Equations, time domain computational electromagnetism, hybrid finite element, high contrast I. INTRODUCTION In strongly heterogeneous media, the constitutive parame- ters can jump by large amounts upon crossing the mater- ial interfaces. On a global scale, the EM field components are, therefore, not differentiable and Maxwell's equations in differential form cannot be used: one has to resort to the original integral form of the EM field relations as the basis for the computational method. The appropriate integral form is provided by the classical interrelations between the curl of the electric/magnetic field strength along a closed curve and the time rate of change of the magnetic/electric flux passing through a surface with the circulation loop as boundary. For these to hold, only integrability of the field is needed, which condition we impose in accordance with the physical condition of boundedness of the field quantities. To satisfy the constitutive relations (that are representative of physical volume effects), a fitting continuation of the boundary representations of the field components of an element into its interior is needed. We construct a consistent algorithm that meets all of these requirements, using a simplicial geometrical discretization combined with piecewise linear representation of the electric and magnetic field components along the edges of the elements, piecewise linear extrapolations into the interior of the elements and taking constant values of the constitutive coefficients in these interiors. Furthermore, we use NETGEN[8] to discretize the computational domain with a 2D(triangular)/3D(tetrahedra) mesh. We use simple boundary conditions (PEC, PMC) to truncate the computational domain. After properly assembling the local matrices, we obtain a symmetric positive definite system of algebraic equations, which we solve with a preconditioned iterative method. We test the accuracy of the method by implementing the four domain problem treated analytically in [4] in the time domain using our method. This experiment also documents the stability of our approach. II. FIELD REPRESENTATION In this paper we consider a 2D situation (the method applies perfectly well to 3D but the main effects can be demonstrated effectively in 2D), we use 'finite elements' consisting of triangles, and approximate the fields by linear interpolation inside the elements. Due to the nature of the interface condi- tions, a straight-forward application of the linear expansion function across boundaries would lead to large numerical error or excessive mesh refinement. Applying these interfaces conditions as constraints would result in semi-positive definite system matrices which are difficult to solve (see [7], [1]). It is advantageous to take them directly into account when discretizing the field quantities. The key point we propose is to approximate the field quantities, which are known to be continuous, with nodal linear finite elements, and the discontinuous ones are approximated with edge based finite elements. A. Geometrical quantities Before introducing the linear expansion functions (shape functions), we define a few geometrical quantities (see Fig. 1). Let A (ri, rj, rk) or A(i, j, k) be the triangle delimited by three vertexes with coordinates ri, rj, rk; A(i, j, k) the area in triangle A (i, j, k); ri the spatial coordinate of the vertex i; eij= rj rI the unit vector pointing from vertex i to j; ak = iz x Cij the normal unit vector perpendicular to the edge delimited by vertex i and j; ¢b(r) IA (r,rrk)j, Vr C A (ri, rj, rk) the linear shape function, which, by definition, is equal to 1 on vertex i and 0 on other vertexes in A (ri, rj, rk). 1-4244-1170-X/07/$25.00 c 2007 IEEE Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 4, 2008 at 02:39 from IEEE Xplore. Restrictions apply.