Implementation of surface impedance boundary conditions in the cell

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering Implementation of surface impedance boundary conditions in the cell method via the vector fitting technique P. Alotto A. De Cian G. Molinari M. Rossi

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Implementation of surface impedance boundary conditions in the cell method via the vector fitting technique

Implementation of SIBCs

859


P. Alotto Department of Electrical Engineering, University of Padova, Padova, Italy, and

A. De Cian, G. Molinari and M. Rossi Department of Electrical Engineering, University of Genova, Genova, Italy Abstract Purpose – To show a possible implementation of surface impedance boundary conditions (SIBCs) in a time domain formulation based on the cell method (CM). Design/methodology/approach – The implementation is based on vector fitting (VF), a technique which allows a time domain representation of a rational approximation of the surface impedance to be found. Findings – It is shown that very little computational effort is needed to find a very good VF approximation of simple SIBCs and that such approximation is easily fitted into existing CM codes. Research limitations/implications – The extension to higher order SIBCs has not been taken into account. Practical implications – The proposed approach avoids the use of convolution integrals, is accurate and easy to implement. Originality/value – This paper introduces the use of VF for the approximate time domain representation of SIBCs. Keywords Electromagnetic fields, Simulation, Numerical analysis Paper type Research paper

1. Introduction In the recent past, the cell method (CM) has been gaining attention since it has a quite intuitive formulation and since it allows complete freedom of the mesh. Several applications have been explored, including the full-Maxwell case (Alotto et al., 2006), which is traditionally treated by, among many possible approaches, the boundary element method and the finite difference time domain technique (FDTD). For both these latter methods large efforts have been devoted to the implementation of surface impedance boundary conditions (SIBCs) (Barmada et al., 2004), in order to model, for instance, non-ideal conductors and lossy coatings, without the need of meshing them. In its simplest form, for a layer of conductivity s, permeability m and permittivity 1, the SIBC reads: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi · · · · jvm ~ n~ ¼ z_ðvÞðn~ £ HÞ ~ ¼ ~ E~ 2 ðn~ · EÞ ð1Þ ðn~ £ HÞ ðs þ jv1Þ ·

·

~ and H ~ are the fields incident to the surface where the SIBC is applied, n~ its normal where E and z_ðvÞ the surface impedance as a function of angular frequency (the “dot” superscript

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 26 No. 3, 2007 pp. 859-872 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640710751271

COMPEL 26,3


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indicates complex numbers). One of the main challenges is to translate the frequency domain SIBC to an expression suitable for a time domain code: several methods have been proposed, ranging from simple approximations to various implementations of the convolution integral. The aim of this work is to find a suitable implementation of SIBCs for a CM-based code. To reach this goal, we propose a method based on the so-called vector fitting (VF) (Gustavsen and Semlyen, 1999), a technique widely used in electronic and power system circuit analysis to find circuit models of components known only by measured or calculated frequency sampling of their impedance. Basically, via an iterative procedure, VF allows to approximate a generic transfer function with a sum of partial fractions over a defined frequency range. Thanks to this representation, an expression of type (1) in the frequency domain can be easily translated into a sum of simple differential relations in time domain. The paper is organized as follows. In Section 2, following (Alotto et al., 2006), the principles of the application of the CM to 3D full-Maxwell problems are recalled, highlighting the modifications needed to implement SIBC. In Section 3, the VF technique is introduced ant its application in our context is explained. Numeric results on a simple test case are presented in Section 4. Finally, in Section 5 some conclusive remarks are drawn. 2. The full-Maxwell algorithm The full-Maxwell algorithm can be summarized in the following set of equations, written with reference to a domain in which a pair of meshes (called primal mesh and dual mesh) is introduced: bn ¼ bn21 2 Ce_ 2 q_

ð2Þ m

h ¼ Nb ~ h_ þ q d n~ þ1 ¼ dn~ þ C _

ð3Þ ð4Þ e

e ¼ Qd

ð5Þ

qe ¼ Se

ð6Þ

qm ¼ Th

ð7Þ

where e is the e.m.f. on primal edges, d the electric flux through dual faces, h the m.m.f. on dual edge, b the magnetic flux through primal faces, qe the electric charge flow through dual faces and qm the magnetic charge flow through primal faces (introduced, as in FDTD codes, to allow the implementation of the so-called perfectly matched layers); underlined symbols refer to integrals over time intervals of the corresponding variables. The matrices appearing in the previous expression belong to two different ~ ¼ C T , the dual categories: C, the primal faces-primal edges incidence matrix, and C faces-edges incidence matrix are topological and express the link between faces and edges on primal and dual mashes, while N, Q, S and T, being, respectively, the magnetic constitutive equation matrix, the electric constitutive equation matrix Ohm’s Law matrix and Ohm’s magnetic Law matrix, link variables on different meshes via discrete forms of the constitutive equations. When implementing the SIBC, one more relation has to be considered, namely:


ez ¼ I ðZhz Þ

ð8Þ

where ez are the e.m.f. on the primal edges lying on the surface were the SIBC is applied and hz are the m.m.f. on dual edges at least partially contained in primal and dual elements intersections lying on the same surface. Z accounts for the “projection” of dual edge variables shape function on primal edges, while I represents an integral operator; its meaning will be clarified in the following. The previous relation can be quite easily used in an iterative algorithm very similar to that implemented in FDTD codes: . Enforce excitation and boundary conditions on e.m.f. . Enforce SIBCs by means of equation (8). . Compute electric and magnetic charge fluxes by Ohm’s laws (6) and (7). . Compute magnetic fluxes by Faraday-Neumann’s law (2). . Compute m.m.f. by the magnetic constitutive equation (3). . Compute electric fluxes at next dual instant by Ampere-Maxwell’s law (4). . Compute e.m.f. along primal edges by the electric constitutive equation (5). . Iterate. One of the main issues in the CM is the choice of proper expressions for the constitutive equations (Codecasa et al., 2004). In this paper, the approach presented in Alotto et al. (2006) is applied: a barycentric dual mesh is introduced and the constitutive equations are created starting form the interpolation used to reconstruct fields inside primal volumes: if the primal mesh is composed by tetrahedra, standard face and edge interpolating functions can be used for the variables defined on it, while in the case of hexahedral primal meshes and for variables defined on dual meshes in both cases, the interpolation presented in Codecasa et al. (2004) is applied: this latter is based on the intersection (always hexahedral) between primal and dual meshes. The expressions for the interpolating functions and the corresponding matrices are recalled in the following for convenience; see Alotto et al. (2006) and Codecasa et al. (2004) for a detailed derivation. 2.1 Magnetic constitutive equation Considering a tetrahedral primal mesh, the magnetic induction field inside a primal volume can be reconstructed from the fluxes using a face interpolation: 4 X ~i ~ ¼ bi W ð9Þ B i¼1

~ i are the Raviart-Thomas face where bi is the magnetic flux through the i-th face and W shape functions. In this case, the coefficient N Vij of the magnetic constitutive equation corresponding to the element V can be computed as the projection of each shape function on the portion of dual edge li belonging to V: Z ~ j · d~l N Vij ¼ n V W ð10Þ ~li >V

where n V is the element reluctivity.


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If the primal mesh consists of hexahedra, we reconstruct the magnetic induction inside the intersection U between a primal volume V and a dual volume V~ using the interpolation: 3 X ~ ~ ¼ bU ð11Þ B i Wi i¼1

bU i

is the magnetic flux through the intersection between the i-th face of V where ~ and: and V, ~j £ n ~k ~i¼ n ð12Þ W ~i £ n ~j · n ~k n ~ ~ j and n ~ k are normal to the faces of V that also belong to V, ~ i, n In equation (12), vectors n ~ and their magnitude is equal to the area of the intersection between a primal face and V. Defining: U bU i ¼ b i di ;

dUi ¼

~ areað f i > VÞ areað f i Þ

ð13Þ

where fi is the i-th primal face and bi the magnetic flux through the i-th face of V, the corresponding contribution to the constitutive equation coefficient is in this case: Z U U U ~ i ·W ~ j dV W NU ¼ n d d ð14Þ i j ij U

2.2 Electric constitutive equation ~ consists in expressing the e.m.f. along a primal ~ ¼ 1 21 D The discrete counterpart of E edge as a function of the electric fluxes di on dual faces. The electric induction field inside it can be reconstructed starting from the fluxes using the interpolation: 3 X ~ ~ ¼ D dU ð15Þ i Wi i¼1

where dU i is the electric flux through the portion of the i-th dual face contained in V, and: ~j £ n ~k ~i¼ n ð16Þ W ~i £ n ~j · n ~k n ~ i, n ~ j and n ~ k are normal to the faces of V~ that also belong to V, In equation (16), vectors n and their magnitude is equal to the area of the intersection between a primal face and V. Defining: U dU i ø d i ai

aUi ¼

areaðf~i > V Þ areaðf~i Þ

ð17Þ

~ i ·W ~ j dV W

ð18Þ

where f~i is the i-th dual face, we get: QU ij

¼1

21

aUi

aUj

Z U

2.3 Ohm’s Law To find a discrete relation qe ¼ Se which links the e.m.f. on primal edges to electric currents through dual faces, for a primal mesh consisting of tetrahedra, we reconstruct the electric field inside a primal volume V using an edge interpolation: 6 X ~i ~¼ ei W ð19Þ E


i¼1

~ i is in this case the classical edge shape function and the The shape functions W v coefficient S ij corresponding to the element V can be computed as the flux of each shape function on the portion of dual face f~i belonging to V: Z ~ ~ j · dS ð20Þ S vij ¼ s vW f~i >V

If the primal mesh consists of hexahedra, we use the interpolation: ~¼ E

3 X

~ eU i Wi

ð21Þ

i¼1

~ where eU i is the e.m.f. along the intersection between the i-th primal edge and V, and: ~ i ¼ a~ j £ a~ k W a~ i £ a~ j · a~ k

ð22Þ

In equation (22), vectors a~ i , a~ j and a~ k are tangent to the edges of V that also belong to ~ and their magnitude is equal to the length of the intersection between these edges V, ~ Following the same approach used for the previous relations, with: and V. U eU i ø e i bi

bUi ¼

~ lengthðl i > VÞ lengthðl i Þ

ð23Þ

~ i ·W ~ j dV W

ð24Þ

we get: S vij ¼ s v bvi bvj

Z U

2.4 Magnetic Ohm’s law The matrix T representing the discrete counterpart of Magnetic Ohm’s law ~ links the m.m.f. on dual edges to the magnetic current through primal J~ m ¼ sm H, faces, and can be obtained in a similar way we use for Q: introducing the interpolation: ~ ¼ H

3 X

~ hU i Wi

ð25Þ

i¼1

where hU i is the m.m.f. along the part of i-th dual edge contained in V , and the shape function: ~ i ¼ a~ j £ a~ k W a~ i £ a~ j · a~ k

ð26Þ


863

COMPEL 26,3

where a~ i , a~ j and a~ k are tangent to the edges of V~ that also belong to V, and their magnitude is equal to the length of the intersection between these edges and V, the corresponding contribution to the matrix coefficient is: Z U U U ~ i ·W ~ j dV W ð27Þ T ij ¼ sm g i g j

864

with the usual approximation:

U


U hU i ø hi g i

g Ui ¼

lengthð~li > V Þ lengthð~li Þ

ð28Þ

2.5 Matrix Z Let us consider an intersection U between primal and dual volumes, lying on a SIBC surface. In the frequency domain, the f.e.m e_ U i along the portion of the i-th primal edge in U can be computed as: Z Z · · h· i ~ · d~l ¼ ~ 2 ðn~ · EÞ ~ n~ · d~l e_ U E E ð29Þ ¼ i l i >U

l i >U

being n~ normal to d~l. Using equation (1) we have: Z · ~ · d~l ~ _ e_ U HÞ ¼ z ð v Þ ð n £ i l i >U

ð30Þ

·

~ we get: and finally, substituting the interpolation (25) for H Z 3 X U _ e_ U ¼ z ð v Þ h j i j¼1

~ j Þ · d~l ¼ z_ðvÞ ðn~ £ W l i >U

X

U Z ij h_ j

ð31Þ

This latter relation allows to build Z element by element, leading to an equation of the kind: e_ z ¼ z_ðvÞZ h_ z

ð32Þ

i.e. the frequency domain counterpart of equation (8). 3. The vector fitting technique VF is a technique allowing to approximate a generic transfer function H(s) in a sum of simple fractional functions of the kind: H ðsÞ