TICAM REPORT 97-02February, 1997

Modeling of Electromagnetic Absorption/ScatteringProblems Using hp-Adaptive Finite Elements

L. Demkowicz and L. Vardapetyan

MODELING OF ELECTROMAGNETICABSORPTION/SCATTERING PROBLEMSUSING hp-ADAPTIVE FINITE ELEMENTS

L. Demkowicz and L. VardapetyanThe Texas Institute for Computational and Applied Mathematics

The University of Texas at AustinTaylor Hall 2.400

Austin, Texas 78712, USA

Dedicated to Prof. J. Tinsley Odenon the occasion of his 60th birthday

January 1997

Abstract

A model problem for the steady-state form of Maxwell's equations is considered.The problem is formulated in a weak form using a Lagrange multiplier, laying down afoundation for a general class of novel hp-adaptive FE approximations. A convergenceproof for affine elements is presented. The proposed method is illustrated and veri-fied with a series of 2D experiments including elements with curved boundaries andnonhomogeneous media.

Key words: Maxwell's equations, hp finite elements, error estimates

AMS subject classification: 65N30, 35L15

1 Introduction

This work has been motivated by two classes of practical applications. The first one dealswith the design and analysis of mixed digital and analog signal packages. One of the ultimategoals of such an analysis is to determine the variation of the capacitance, resistance andinductance of the package with frequency w. As the noise from the digital part of the packagemay be picked up and amplified by the analog part, the modeling of the interaction between

1

different parts becomes of crucial importance. The frequency content of the "switching noise"of the digital components spreads from the DC to very high harmonics. Consequently, whensolved in the frequency domain, the problem calls for a methodology which would deliverresults with a quality independent of the frequency w, in particular for w ---t O.

One possible alternative is the "electric circuit approach" where the interaction betweenthe components is modeled with a simplified circuit model [36, 37]. This methodology pro-vides for a cost effective initial approximation of the electrical characteristics of the packageand frequently delivers sufficiently precise results. For complex geometries, however, an ac-curate determination of the field quantities through the solution of the Maxwell equationsbecomes indispensable. The global capacitance, inductance and resistance of the system arethen deduced from the electric and magnetic fields, see e.g. [2].

The second class of problems motivating this work deals with modeling of the absorptionand scattering of steady state electromagnetic waves by the human body, see e.g. [9]. Therethe main challenge comes from complex curvilinear geometries and high nonhomogenuityof the scatterer/absorber. Again, modeling through the solution of Maxwell's equations isnecessary.

In both classes of problems singular solutions are expected. These may result from bothgeometry and rapidly varying material constants. One of the most powerful methodologieswhich permits a succesful modeling of singular solutions is the hp-adaptive finite elements[1]. We emphasize that a true hp method allows locally varying both element size h and orderof approximation p. Only then are the exponential rates of convergence accessible for a wideclass of functions with singularities.

The purpose of the presented research has been to design an hp finite element methodfor steady-state Maxwell's equations. Summing up our discussion, we want the following:

• a formulation for a class of problems with discontinuous material properties,

• a formulation with approximation (stability) properties uniform with respect to fre-quency w,

• a discretization with the possibility of varying locally order of approximation p andelement size h,

• curved elements allowing one to model complex, curvilinear geometries.

We think we have found a method which satisfies all these conditions.

There is an extensive literature on the subject. We have benefited from a large numberof related contributions. The mathematical foundations and the necessary functional settingwere established in [10]. H( curl)-conforming FE approximations were introduced by Nedelec

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[27, 28, 29] and, independetly in the engineering literature among others by Lee [16, 17, 18,19, 20, 12]. A comprehensive analysis of various existing formulations was provided byMonk [21, 22, 23, 24, 25], see in particular [24] for the first application of hp methods toelectromagnetics (with constant p, however). For comprehensive reviews of existing work werefer to [35, 15]. For the theory on hp approximations we refer to [1] and for our relatedwork on hp discretizations to [4, 31, 5, 8]. Finally, for a number of recent innovative ideas,including the L2-residual formulations see [14, 11, 26].

The plan of the presentation is as follows. After a short review of basis electromagneticsin the next section, we present in section 3 a model problem reflecting all essential difficultiesconnected with the solution of Maxwell's equations, except for absorbing boundary condi-tions. We discuss existence, uniqueness and stability of the solutions. Section 4 presents thedefinition of a novel finite element with variable order of approximation for which we areable to prove the convergence, at least for elements with polynomial shape functions (affinefamilies of elements). Finally, in section 5 we present a number of 2D numerical experiments.

2 A Review of Basic Electromagnetics

The equations

As stated in [15], "the problem of electromagnetic analysis is actually a problem of solvingMaxwell's equations subject to given boundary conditions". More precisely, the followingfour fundamental laws are the cornerstones of electromagnetics [34].

• Faraday's law,

r EodT+dd r BodS=O (2.1)Jas t Js

Here S denotes an arbitrary oriented surface, as its boundary, oriented consistentlywith the orientation of surface S, E is the electric field intensity, and B denotes themagnetic flux density.

• Ampere's law,rHo dT = r J 0 dS + dd r D 0 dS (2.2)Jas Js t Js

Here H denotes the magnetic field intensity, D is the electric flux density, and Jdenotes the current density.

• Gauss' law for the electric field:

(2.3)

3

[ Quantity ISymbol ] SI units IElectric field intensity E = E(aJ, t) VimElectric flux density D = D(aJ, t) C/m2Magnetic field intensity H = H(aJ, t) A/mMagnetic flux density B = B(aJ, t) Wb/m2Current density J = J(aJ, t) A/m2Volume charge density p=p(aJ,t) C/m3

Table 1: Electromagnetic field vectors

where V is an arbitrary volume with surface S oriented outward, and P denotes thevolume charge density.

• Gauss' law for the magnetic field:

is BodS = 0 (2.4)

All field quantities E, D, H, B, J, p are functions of space location aJ E JB33 and time t. Noneof the them needs to be globally continuous. The charge density may even be a distribution, ifsurface charges in between different media occur. The corresponding SI units are summarizedin Table 1. The field quantities are related by the following consitutive laws for a simplemedium:

•

•

• Ohm's law:

D=EE

B=µH

J = aE

(2.5)

(2.6)

(2.7)

(2.8)

Here E is the permittivity, µ is the permeability, and a denotes the conductivity of the medium.Denoting by EO the permittivity of free space,

1 -9EO = 3671"10 F /m,

and by µo the permeability of free space,

(2.9)

4

aIr 0-=0antenna( copper) 0-= 5.8 107 S/mabsorber 0-= 0.3 - 7.2 S/m

Table 2: Values of conductivity 0- of interest

we introduce the dielectric constant Cr and relative permeability µr,

(2.10)

For the materials of interest the relative permeability µr practically equals one, and thedielectric constant will typically vary in the range of 1 - 70. Typical values of interest forconductivity 0- are summarized in Table 2, see also [9].

We shall also assume that the current density is split into two parts, an impressed (im-posed) current Jimp, and actual (ohmic) current J related to the electric field E throughOhm's law stated above.

In regions where the field quantities are smooth, the four conservation laws (2.1)-(2.4)imply four differential Maxwell equations:

aBVxE+- =0

ataDVxH-- =J (2.11)at

VoD =p

VoB =0

Across an interface between two media with different material constants, say medium 1 andmedium 2, the conservation laws imply the following jump conditions:

n X (E2 - E1) = 0

nx(H2-H1)=0

no (D2 - DJ) = 0

no (B2 - B1) = 0

(2.12)

For piecewise smooth functions, the four differential equations (2.11), and the four jumpconditions (2.12) are equivalent to the original four Maxwell laws in the integral form.

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Boundary conditions

In the following discussion we will restrict ourselves to two kinds of boundary conditionsonly. For a boundary adjacent to a perfect conductor we have [34]:

nxE =0

nxH =K(2.13)

noD = ps

noB =0

where ps is a surface charge, and K denotes a possible surface current density. As K andps are in general a-priori unknown, only the first and last conditions of (2.13) provide usefulinformation:

nxE =0

noB =0

A dual boundary condition (a magnetic symmetry wal0 [14] reads as follows:

nxH =0

noD =0

Radiation conditions

(2.14)

(2.15)

For steady-state problems in unbounded domains, Maxwell's equations have to be comple-mented with radiation conditions eliminating waves coming from infinity.

3 Analysis of a Model Problem

In order to discuss the main mathematical issues we will restrict ourselves in this paper toa presentation of the following model problem.

The problem is set up in a bounded domain n consisting of two disjoint parts ni,i= 1,2,occupied by media with different material constants Ci, µi, ai, i = 1,2, and separated by aninterface r 12. Boundary r of the domain n consists of two disjoint parts r 1 and r 2. Wewish to find an electric field E = E( aJ), and magnetic field H = H (aJ), aJ E n, satisfyingthe following equations:

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• steady-state form of the Maxwell equations in ni, i = 1,2,

v X E+jwµiH = 0

v X H - jWciE - aiE = J'mp

v 0 (ciE) = p

Vo (µiH) = 0

• jump conditions on interface rI2:

n X [E] = 0

nx [H] =0

no [cE] = ps

no [µH] = 0

(3.1)

(3.2)

where [] denotes the jump of the corresponding quantity across the interface, e.g.,

(3.3)

• boundary conditions on r1:

nxE =0

no (µH) = 0

• boundary conditions on r2:

nxH =0

no (cE) = 0

The impressed current Jimp has to satisfy the usual compatibility conditions [14]:

(3.4)

(3.5)

Vo J'mp = 0

no J'mp = 0

in ni,i = 1,2

on r12 (3.6)

Volume charge density p and surface charge density ps are additional unknowns in theproblem. In order to eliminate p, we take the divergence of (3.1h:

-jwVo(cE)-Vo(aE)=VoJimp=O ,

replace (3.1h with the equation:

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(3.7)

(3.8)

and use the original Gauss law to calculate density p after the field E has been determined.Similarly, the jump condition (3.2h is used to calculate the surface charge density ps, and(3.8) implies a new jump condition in the form

no [(jwc + alE] = 0

and a new boundary condition on r2, replacing (3.5h

no ((jwc + a)E) = 0

(3.9)

(3.10)

Equivalently, the elimination of the charge densities can be done for the integral form of theMaxwell equations, or for the Maxwell equations understood in the distributional sense withthe assumption that the charge density p includes distributions corresponding to the surfacecharge densities.

The choice of the model problem reflects all the fundamental difficulties related to theradiation/ absorption problems of interest, except for open boundary conditions.

Reduction to the (reduced) wave equation

Let E, H be a sufficiently regular solution to the model problem. Eliminating H from theequations, interface and boundary conditions above, we arrive at the following boundary-value problem for the electric field E.

• Reduced wave equation and the Gauss law in ni,i = 1,2,

v X (~V X E) - (W2ci- jwai)E = _jwJ'mp

µiV 0 ((jwci + ai)E = 0

• jump conditions on r12,

n X [E] = 0

n X [IV X E] = 0µ

no [(jwc + a)E] = 0

• boundary condition on r1,

nxE=O

• boundary conditions on r2,

n X (IV X E) = 0µ

no ((jwc + a)E) = 0

8

(3.11)

(3.12)

(3.13)

(3.14)

Conversely, let E be a solution of (3.11)-(3.14). With (3.1h defining H,

1H = ---:--V X E

JWµi(3.15)

we automatically satisfy (3.1h and (3.1h- Taking the divergence of (3.1h, we get equation(3.1)4' Next formulas (3.15) and (3.1h imply jump condition (3.2h and boundary condition(3.5h. Finally, (3.15) implies that

1n 0 [µH] = - -;-n 0 [V x E] = 0

JW(3.16)

due to the continuity of the tangential component of E guaranteed by jump condition (3.2h.Similarly, boundary condition (3.14h implies boundary condition (3.4h-

Thus the reduced wave equation formulation is equivalent to the original problem.

REMARK 1

1. The redundancy of the Gauss law for the magnetic field can be derived from the integralform of the Maxwell equations. For the steady-state case, the Faraday law (2.1) reducesto:

rEo dT + jw r BodS = 0Jas Jsand, in particular, for a closed surface S, it implies (2.4)

(3.17)

2. We emphasize the formal character of the presented equivalence proof. All considera-tions on VoH have been done under the assumption that the solution E to the reducedwave equation is sufficiently regular to allow considering the divergence V 0 (I V x E).

µ

Such a regularity result may depend upon the regularity of the domain n.

I

Variational formulation

We introduce the space of admissible solutions:

(3.18)

and the space of Lagrange multipliers

(3.19)

9

The boundary condition in (3.18) is understood in the sense of the generalized Green formula[10], and the boundary condition in (3.19) in the sense of traces.

Following the usual procedure for the mixed methods, we arrive at the following varia-tional formulation:

E E W, P E V

r (V x E) 0 (V x F) daJIn- k (w2c - jwa)(E + Vp) 0 F daJ

= -jw k(Jimp 0 F) daJ, VF E W

= 0 Vq E V

(3.20)

Substituting in (3.20h F = V q, we obtain a variational equation for Lagrange multiplierp,

{

PEV(3.21 )in (w2c - jwa)Vp 0 Vij daJ = 0 Vq E V

which implies that (at the continuous level) p = O. For that reason p is sometimes called thedummy variable.

Integrating (3.20h by parts and using the standard Fourier lemma argument, we canwrite down the strong form of equation (3.20)1' The wave equation (3.11h gets replaced by

(3.22)

with the remaining equations, interface and boundary conditions unchanged. Variationalargument (3.21) can be reproduced by taking divergence of (3.22),

(3.23)

and by multiplying (3.22) with a unit vector n normal to r2, and extending (3.22) to r2.

As a result we get the Neumann boundary condition on p on r2,

(3.24)on r2ap = 0an

built into the variational formulation (3.21). Note that the boundary condition (3.14himplies that

1noV X (- V x E) = 0 on r2

µi(3.25)

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Stability, Existence and Uniqueness

In the following analysis we will restrict ourselves to the case of loss less media only, assuminga = O. We begin by rewriting (3.20) in an abstract form:

!E E W, P E V

a(E,F) -w2(E,F) -w2b(p,F) = wI (F) VF E W

b(q,E)=O VqEV

with

11 -a(E,F) = -(V x E) 0 (V x F) daJ

nµ

(E, F) = in cE 0 F daJ

b(p, F) = in cVp 0 F daJ = (Vp, F)

I(F) = -j in Jimp 0 F daJ

A more compact form reads as follows;

{(E,p) E W x V

B((E,p), (F, q)) = L((F, q)) V(F, q) E W x V

where

• W x V is a Hilbert space with the norm:

II(E,p)1I2 = IIEllw + Ilpll~= IIEI12 + IIV x EI12 + w411Vpl12

• B (', .) is a sesquilinear form on W xV:

(3.26)

(3.27)

(3.28)

(3.29)

B((E,p), (F, q)) = a(E, F) - w2(E, F) - w2b(p, F) + w2b(q, E) (3.30)

• L(·) is an antilinear form on W X V:

L((F, q)) = wl(F) (3.31 )

Note that placing the factor w in front of IIVpll in the definition of the norm in space W x Vis equivalent to replacing the Lagrange multiplier p in the formulation with the product w-2p.

11

It is well known that the eigenvalue problem

{E E W, A E C

a(E, F) = A(E, F) VF E W(3.32)

admits eigenpairs (0, W 0), (Ai, Wi) where the zero eigenvalue is of infinite multiplicity andthe remaining eigenvalues Ai are real, positive, of finite multiplicity, and form a sequenceconverging to infinity. The corresponding eigenspaces form an orthogonal decomposition ofthe L2-space with the weighted L2-product (3.272), Introducing a sequence of orthonormaleigenvectors eI, e2, ... corresponding to positive eigenvalues Ai, we can represent an arbitraryvector F from L2(n) in the form:

00

F = Fo + L:Fieii=l

(3.33)

where Fo E Wo, (Fo, ei) = 0, i = 1,2, ... , and Fi denote the spectral components withrespect to eigenvectors ei, Fi = (F, ei)'

Consequently, we have the following spectral representation:

00

a(E, F) - w2(E, F) = -w2(Eo, Fo) + L:(Ai - w2)EiFii=l

(3.34)

Finally, we shall assume that the domain n and boundary conditions are selected in such awayl that:

Eo E W 0 {=} ~<P E V : Eo = V <P (3.35)

Now, a straightforward calculation using Lagrange multipliers leads to the following result:

sup IB((E,p),(F,q))1 = max IB((E,p),(F,q))j =: µII(F,q)lI=l II(F,q)lI=l

whereµ2 = IIEol12 + w4

11Eo + Vpl12 +t IA: ~ ~2121Ei12i=l

The formula above allows for an easy calculation of the stability constant:

µ' 2: IIEolI'+w4l1Eo + Vpll' + "jin{ I"; ~ ~'I rECI + ",llE,I'

. { (IAi _W21)2 . }II 112 411 112

~ mIll 1, 1+ Ai ' Z = 1,2,... E W + w Eo + Vp

1We emphasize that n needs not to be simply connected!

12

(3.36)

(3.37)

(3.38)

and, therefore,

1+ Ai .~ max{l, IAi _ w21' z = 1,2, ... }µ= w211Vpli ~ w211Eo + Vpll = w211Eoil ~ (1 + w2)µ

(3.39)

This results in an estimate for the global LBB (inf sup) constant an the final stability result:

(3.40)

Note that, in particular, the solution is uniformly stable for w ---t O.

Combined with the standard theory (see e.g. [32]), the stability estimate above impliesthe following theorem.

THEOREM 1Let Ai denote the eigenvalues (3.32). Then, for any w2 i= Ai, problem (3.26) possesses aunique solution and the stability estimate (3.40) holds. I

REMARK 2

1. Enforcing the divergence condition (the Gauss law) explicitly by introducing the La-grange multiplier is essential in obtaining the formulation whose stability propertiesdo not deteriorate with diminishing frequency w. A majority of results reported in theliterature, see e.g. [15, 35], are based on the simplified formulation:

{EEW

a(E, F) - w2(E, F) = wl(F) VFEW(3.41 )

The corresponding stability result for the curl-free component reads then as follows:

(3.42)

which indicates the deteriorating stability of the formulation for small w.

2. Both non-zero conductivity a i= 0 and possible radiation conditions improve stabilityand, in particular, eliminate resonance.

I

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4 Discretization

Galerkin Approximations

Introducing finite-dimensional subspaces W heW and Vh C V, we obtain the approximateproblem by replacing all unknowns and test functions from W, V with their approximationsfrom W h, Vh.

Eh E Wh,Ph E l1h

in(V X Eh) 0 (V X Fh) daJ -in (w2c - jwa)(Eh + Vph) 0 Fh daJ

= -jw in(Jimp 0 Fh) daJ, VFh E Wh

k(w2c-jwa)EhoVqhdaJ =0 V%EVh

(4.1)

Note that, in general, the discrete pressure Ph is non-zero unless, again, V(Vh) C W h. Insuch a case, we can repeat the same procedure as for the continuous problem, using in (4.1hFh = Vqh,qh E Vh·

An hp Finite Element Approximation. A Triangular Element ofVariable Order

We will present the concept using a triangular element but exactly the same technique appliesto rectangles, and in 3D, to prisms, cubes and tetrahedra.

Let ]{ denote a triangle with straight sides Si, i = 1,2,3. We associate with the trianglea specific order of approximation P = PK which may vary with element I<. Additionally,with each of the sides we associate a possibly different order of approximation Pi, i = 1,2,3,with the assumption that

1 ::; Pi::; P i= 1,2,3 (4.2)

Due to approximability requirements, in practice we select P = max{pl, P2,P3}. We introducenow two spaces of element shape functions:

• the scalar space (to approximate the Lagrange multiplier p) identified as the space ofpolynomials of order P + 1 which, over each of the three sides Si, reduce to polynomialsof order Pi + 1, i = 1,2,3,

(4.3)

14

• the vector space (to approximate the E-field) identified as the space of vector-valuedpolynomials of order p for which the corresponding tangential components, over eachof the three sides Si, reduce to polynomials of order Pi, i= 1,2,3:

(4.4)

Here Ti denotes a tangent vector to side Si, i = 1,2,3. One can think of either space as asubspace of polynomials of order p with additional constraints imposed along the sides. Thecondition that a polynomial of order p (in one dimension) must reduce to a polynomial ofsmaller order Pi < p is equivalent to satisfying p - Pi scalar constraints2 and, therefore, thedimension of the scalar space V(K) is:

or equivalently,

(p + l)(p + 2) )dimV(K) = 2 - (p- PI) - (p- P2) - (p- P3 (4.5)

dimV(K) = 3 + (PI - 1) + (P2 - 1) + (P3 - 1) + (p - l)(p - 2) (4.6)2

Similarly, we have for the vector space:

dimW(K)(p + l)(p + 2)

= 2 - (p - pJ) - (p - P2) - (p - P3)2

= (PI + 1) + (P2 + 1) + (P3 + 1)

+(p - 1) + (p - 1) + (p - 1)

(p - 1)(p - 2)+2 2

(4.7)

The two spaces satisfy the same compatibility condition as their continuous counterparts.

THEOREM 2The following compatibility condition holds:

<P E V(K) ¢:::::? V <P E W(K)

I

(4.8)

Proof: The proof is trivial. I

In practice both spaces are constructed as spans of predefined element shape functions.For the scalar approximation we use the shape functions introduced in [5].

2These constraints can be explicitly written down using e.g. L2-product and Legendre polynomials.

15

~2

Figure 1: Triangular element with variable order of approximation

The element is illustrated in Fig. 1. It consists of seven nodes: three vertex nodesVI,V2,V3, three mid-side nodes al,a2,a3" and a middle node a4' Each of the three mid-side nodes and the middle node may have a different, corresponding order of approximationPI, P2, P3, P4, respectively. Introducing the area coordinates )1} = 1 - el - 6, A2 = 6, A3 = 6,we define the corresponding scalar-valued shape functions as follows:

• vertex nodes shape functions,

Xi = Ai, i = 1,2,3

• mid-side nodes shape functions,

pl-l

.II P2-i)AXl,i = J=O,J=I=i PI 1

pl-l . .

II (2 J' ., 2=1.. ~-~)(l-~ , ... ,p,-l

J=O,Jt;i PI PI PI )

with formulas for X2,i and X3,i obtained by permuting indices,

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(4.9)

(4.10)

• middle node shape functions,

X4,i,j,k =

i-I j-l k-lm n mII (AI - -) II(A2 - -) II(A3 - -)m=O P4 n=O P4 1=0 P4i-I i m j-l j n k-I k III(- - -) II(- - -) II(- --)m=O P4 P4 n=O P4 P4 [=0 P4 P4

(4.11)

wi th 1 ~ i, j, k ~ P4 - 1, i + j + k = P4.

The choice of the actual global degrees of freedom (dof) is to some extent arbitrary and itreflects the global continuity requirements. For the scalar case, for instance, we can use theLagrangian nodes introduced in the definition of the shape functions. More precisely, if ai,jdenote the Lagrangian nodes corresponding to mid-side nodes aI, a2, a3 and middle nodea4, the corresponding dof are defined simply as:

q ---t q(vd i = 1,2,3

q ---t q( ai,j) i = 1,2,3, j = 1, ... ,Pi

THEOREM 3The Lagrangian dof defined above are V(K)-unisolvent. I

(4.12)

Proof: As the number of dof equals exactly the dimension of space V(K), compo (4.6),it is sufficient to show that once all dof are zero, the corresponding function q E V(K) mustvanish. I

In order to enforce now the global continuity of the scalar approximation it is sufficientto assume that the values at the vertex nodes and the mid-side nodes are shared by adjacentelements.

Note that the Lagrangian dof do not coincide with the dual basis to the original shapefunctions. Consequently, the restrictions of the global basis functions (defined as the dualbasis corresponding to the global dof) to an element, do not coincide with the original elementshape functions and one has to use the generalized assembling procedure discussed later.

The choice of the dof for the vector-valued approximation is more elaborate. The conceptis illustrated in Fig. 2. Formally we introduce seven nodes: three tangential mid-side nodesaLa~,a~, three normal mid-side nodes a1,a2,a3, and again the middle node a4. Thetangential nodes are of order PI, P2, P3, resp., whereas all normal nodes and the middle nodeare of order P = P4. Next, similarly to the scalar approximation, for each of the nodes

17

~2

~I

Figure 2: Vector triangular element with va.riable order of approximation

we introduce the corresponding Lagrangian nodes a~,j' ai,j' a4,j' Note that the Lagrangiannodes for the tangential mid-side nodes include the vertices. Finally, the dof are defined asfollows:

• tangential dof:

• normal dof:

• middle dof

F ---t F (a t ") 0 'r i J" i = 1, 2, 3, j = 0, ... ,Pi1.,) ,

F ---t F(a7,J 0 ni,j i= 1,2,3,j = 1, ... ,Pi-1

F ---t F(a4,j) 0 ik k = 1,2,j = 1, ... , (p - 2)(p - 1)/2

(4.13)

(4.14)

(4.15)

Here 'ri,j denotes tangential unit vectors constructed at points aL and directed from thevertex node with a smaller (global) number to the vertex node with a greater number, ni,j

denotes outward normal unit vectors constructed at points ai,j' and ik, k = 1,2 stands forthe global cartesian unit vectors. Note that for elements with straight edges vectors 'ri,j andni,j are independent of index j. We record the more general definition for the sake of curvedelements to be discussed later.

18

THEOREM 4The dof described above are W(I<)-unisolvent. I

Proof: As for the scalar case, number of dof equals the dimension of the space and,therefore, it is sufficient to verify linear independence. Assume that E is a function fromW(I<) for which all dof vanish. It follows from vanishing of the tangential componentsat vertices that E is zero at vertices. Next for each side Si, from vanishing of the normalcomponent at additional P - 1 points and the tangential one at Pi - 1 points, it follows thatboth the normal and tangential components must vanish. Consequently, E is zero along eachside. Finally, from the standard argument on the Lagrange interpolation, E must vanisheverywhere. I

Finally, let us note that by enforcing common values of the tangential components foradjacent elements we construct the desired H( curl, n)- conforming approximation.

REMARK 3

1. Location of points a~,j' ai,j is arbitrary as long as their number remains the same. Alsovectors ni,j may be replaced with other non-tangential vectors.

2. Formally the presented definitions can be extended to accommodate curvilinear defor-mation of elements, using in particular parametric maps. The tangential componentalong a common interelement side is no longer a polynomial, and enforcing its continu-ity at Pi + 1 points does not, in general, imply the global continuity. The approximationis no longer conforming (internal), and the integrals in the variational formulation areinterpreted element-wise. The choice of points a~,j' ai,j becomes essential and the ap-proximation theory for the p-version of the FEM suggests using Gauss-Lobatto orGauss integration points instead of the uniformly distributed Lagrange points.

I

Convergence Analysis

Let us assume for simplicity that n is a polygonal domain. Denoting by 7ft a triangulationof the domain we define the FE spaces W heW and VhC V as follows.

(4.16)

and(4.17)

19

We have the following essential result.

THEOREM 5The following compatibility condition holds:

(4.18)

I

Proof: Let ¢>E Vh be an arbitrary function. Then by (3.35) V¢> E Wand we haveto show only that (V¢»IK E W(K) for every element K. But this is guaranteed by thecompatibility condition (4.8). Conversely, if E E W h then there exists ¢> E V such thatV¢>= E and it is sufficient to show that ¢>K E V(K). But this again follows from (4.8).I

Repeating precisely the same steps as on the continuous level, we end up with the fol-lowing estimate for the discrete LBB constant:

-1 { 2 1+ Aih. }Ih :::;max 1 +w , I' 21,2 = 1,2, .../\ih - W

(4.19)

where Aih denotes the discrete eigenvalues resulting for the Galerkin approximation of (3.32).We emphasize the essential role of condition (4.18) in deriving the result above.

Combining the discrete stability result with the classical theory on asymptotic conver-gence (see e.g. [7]), we arrive at the final convergence result.

THEOREM 6There exists a threshold value ho and a constant C, independent of h, such that

for every h :::;ho. Constant C depends upon frequency w but it is bounded away from zerofor w ---t O. I

Obviously p = 0 and, with condition (4.8) satisfied, Ph = 0 as well, and (4.20) reduces tothe estimate:

liE - Ehllw :::;C inf liE - EhllwFhEWh

In practice, of course, due to the round off error, Ph is never exactly zero.

20

(4.21 )

REMARK 4

1. Parametric elements. Once the elements are deformed, the compatibility condition(4.18) is no longer satisfied, the discrete Lagrange multiplier Ph i= 0, and the presentedproof of the stability result does not go through. A simple perturbation argument,however, suggests that the method should work as long as the distortion from theaffine elements is small [3]. Numerical experiments indicate that, for small w, theproposed formulation does remain stable for curvilinear elements whereas the classicalone discussed in Remark 2 does not.

2. hp -error estimates. Theorem 6, when combined with the interpolation error esti-mates for hp approximations, see [1, 24], results in standard hp error estimates, withexponential rates of convergence for analytic solutions.

I

Assembling procedure. Constrained approximation

Let ei E W, i = 1, ... ,N denote the global basis functions to approximate E, F. We nowdiscuss shortly how to calculate the global stiffnes matrix a (ei, e j ), and the global load vectorl( ej), corresponding to the presented hp discretization. We have,

(4.22)

The basis functions ei are vector-valued.

Over each element J{, basis function e j can be represented as a linear combination of theelement scalar-valued shape functions XI, I = 1, ... , M,

3 M~~ J. Je j = L...J L...J CjIXI'lJ, Cjl E 1RJ=ll=l

(4.23)

where iJ denotes the J-th unit vector in 1R2. The element contribution to the global loadvector can then be calculated as follows:

IK(ej) = JKfoej2 M

= IKe'£. 'E cflxliJ)J=l 1=1

2 M

= L L cfIIK(xliJ)J=ll=l

21

(4.24)

Similarly, we get for the stiffness matrix,

aI«ei, ej) = JI«V X ed 0 (V x ej) daJ2 M 2 M

= aI«L L C[kXli/, L L cflXliJ)/=lk=I J=II=l

2 M 2 M

= L L L L c[kcflbI«XkiI, XliJ)/=1 k=l J=I 1=1

(4.25)

Thus, in order to assemble the global matrices, we simply have to determine the coefficientscfl' Usually, we like to think backwards, from the point of view of the contributor, not thereceiver. In other words, if we have a shape function Xl, we want to determine all global basisfunctions ej, and the corresponding coefficients cll' The first step is accomplished throughthe information on connectivities and the constrained approximation. Once the set of theinvolved basis functions ej has been established, the corresponding constraint coefficients cllcan be evaluated by applying to ej the local, element dof cPI ( a dual basis to XI),

(4.26)

In practice, the original, element dof cPI may be cumbersome to calculate (see [5] for thedescription of various hp interpolation procedures). In such a case, functionals cPI maybe replaced with a different set of functionals 'l/Jk, resulting in a system of equations forcoefficients cll (with multiple right-hand sides),

M

L cfl < 'l/Jk,XI >=< 'l/Jk,(ej 0 iJ) >1=1

(4.27)

Systems (4.27) are formed and solved for groups of shape functions connected with a partic-ular side (edge, base, or side in 3D) of a constrained element. The most convenient choicefor functionals 'l/Jkare the usual Lagrange dof.

REMARK 5 Note that unless hierarchical mid-side node shape functions are used, when-ever any of the mid-side node orders of approximation Pi is different from p, different sets ofscalar-valued shape functions have to be used to represent basis functions corresponding totangent and normal dof. I

5 Numerical Experiments

Test cases. All presented results are two-dimensional. Fig. 3 displays three test cases.The corresponding exact solutions are polynomials of order 5, 7, and 2, respectively. In

22

Case 1 Case 2 Case 3

Rectangular Domain Trapezoidal Domain 2-Media Domain

N N D

£=2.0 I I C=LI. II I I ,-J <=2.0

µ=1.0IN D/

µ="::.V I D DIµ=2.01 µ=1.0 IDD 1

0=0.0 0=0.0 0=0.0 0=0.0

D

D = Dirichlet b.c.

N = Neumann b.c.

N

Figure 3: Test cases

N

the third case, due to the discontinuity of constants µ and c, the corresponding normal (i.e.horizontal) component of the solution is discontinuous across the vertical interface.

Geometry representation and initial mesh generation. The domain is viewed as aunion of disjoint figures. Topologically, each of the figures is a triangle or rectangle and it isspecified as an image of the reference triangle or rectangle by a particular parametrization.The concept is illustrated in Fig.4. In practice, the parametrizations may be explicit orimplicit, with an additional assumption on compatibility of parametrizations being enforced.For details concerning practical implementation, we refer to [6]. The initial mesh generationis based on the idea of an algebraic mesh generator and hp-interpolation. The conceptis illustrated in Fig.4. Given, for each reference figure, a number m of divisions in the"horizontal" and "vertical" directions (compatible for neighboring elements, the initial meshis always regular), the reference blocks are covered with uniform, regular grids consistingof elements i<. By constructing a composition of the standard affine map 17transformingmaster element k onto element i< and the (restriction of) the block parametrization aJb,we construct a map from master element k onto a curvilinear element K identified as theimage of element i< under the particular parametrization aJb:

K = T(k) = aJb(K), T = aJb 017 (5.1 )

In principle, this map could be used directly to define the curvilinear element, i.e. in theelement calculations. In practice, we chose instead to approximate it with polynomials using

23

'1,

1\

K

Figure 4: Geometry representation and mesh generation

the idea of parametric approximation. More precisely, given a particular order of approxima-tion for element J{, we replace transformation T with its hp-interpolation corresponding toorders (Pl,P2,P3,p). The idea of the hp-interpolation follows from convergence theory for hpapproximations [1] and has been introduced in [31]. Roughly speaking, the hp-interpolationcombines the classical interpolation for vertex nodes with local HJ-projections for higher-order nodes. Given a sufficiently regular function, the corresponding hp-interpolant exhibitsthe same orders of convergence (in terms of both hand p) as the corresponding global HJ-projection (solution to the Laplace equation with Dirichlet boundary conditions imposedusing the HJ projection on the boundary).

For the first and the third case, the parametrization maps are linear and the resultingmeshes consist of affine elements. In the case of trapezoidal domain, two geometry repre-sentations are used. The first one is based on dividing the domain onto two triangles andusing linear maps for each triangle. Consequently, the corresponding meshes consist againof affine elements. In the second case, the trapezoid is parametrized directly with a bilinearmap and the resulting triangular elements have curved sides.

Discontinuous solutions. We begin with a solution of Case 3 and a simple illustrationof the fact that we can handle discontinuous solutions. Fig. 5 displays the x-component ofthe E-field reproduced using a mesh of quadratic elements.

24

4 elements, order =2, x-component of E-field 00=2.5

o-0.571428

-1.14286

-1.11429

-2.28571

-2.85714

-3,42857..-4.51143

-5.14286

·5,71429

-8.28571

-8.85714

-H2857·8

2-Media Domain

N

e= 1.0 I e= 2.0

µ=2.0 µ = 1.0

0 I 0=0.0 0=0.0 I 0

N

0= Dirichlet BeN=Neumann Be

Exact solution is piecewise quadratic

Figure 5: Case3. Capturing discontinuities

h-convergence rates for affine elements. Figures 6 and 7 display h-convergenceplots for test cases 1 and 2, with the trapezoid in the second test parametrized with apiecewise linear map, and meshes of qudratic and cubic elements. The convergence rates arein agreement with the theoretical estimates.

h-convergence rates for curvilinear elements. Fig. 8 presents convergence ratesfor meshes of cubic elements and two values of frequency w. The method converges but,compared with the affine elements, the rates are suboptimal. Please note the stable behaviorof the method for the small frequency case. A sample solution, including components of theE-field, Lagrange multiplier p, and the divergence of the E-field is displayed in Figures 9and 10. Note that the Lagrange multiplier does not vanish.

Sensitivity with respect to a mesh distortion. As a final example we present asimple illustration of the sensitivity of the solution with respect to a curvilinear distortion ofthe mesh. Figure 11 presents the x-component of the E-field for the first test case obtainedusing just two quadratic elements. Depending upon the magnitude of the distortion, thesolution changes but not dramatically. In the corresponding experiment using the standardformulation the solution simply falls apart.

A simplistic explanation of the stability problems. We conclude this section witha simple argument explaining why our formulation can handle curved elements while theclassical one cannot. As discussed in the previous section, for a mesh consisting of affineelements and piecewice polynomial approximation, the eigenvalues of the V x (V x E) op-erator include a multiple zero eigenvalue with the corresponding curl-free eigenvectors, and

25

eWE ·2oZ>-C)

Q;CW"Oi.3

-6

Error Analysis, Rectangular Domain

-Iog(h)

(0 = 2.5

Exact solution is quintic

-s::J- Order=2

-0- Order=3

N

£=2.0olµ=1.0IN

0=0.0

oD=Oirichlet BeN=Neumann Be

Figure 6: Case 1. Experimental convergence rates for quadratic and cubic element.s

Error Analysis, Trapezoidal Domain by 2 Triangles (0 = 2.5

Exact solution is a polynomial of degree 7

'C'4eWE 3oZe;2Q)c

!:!:!.,C)o..J

.,0.0 0.5 1.0 1.5

-Iog(h)2.0

----v- Order=2

-0- Order=3

D=Dirichlel BeN=Neumann BC

Figure 7: Case 2 with linear parametrization. Experimental convergence rates for quadraticand cubic elements

26

Error Analysis, Trapezoidal Domain by Bilinear Map Order=3

Exact solution is a polynomial of degree 7

g 4WEoZ~2Q;cWc;o

...JO

-2

-Iog(h)

-SJ- 00 = 0.0 I

-{)_ 00= 2.5

N

D=Dirichlel BC

N=Neumann Be

Figure 8: Case 2 with bilinear parametrization. Experimental convergence rates for cubicelements

128 elements, order=3,exact solution is a polynomial of order 7

v-component of E-field

0) = 2.5trapezoidal domain(b'f bllInear map}

D=Dirichlel BC

N=Neumann Be

Curl of E-field

0.<199908

04300630.362017

0.293072

0.22"'260.155181

008623530.G172898

-4.0516557

-0.120601

-0.189547

-0 2584g2

..{}.327438

-0.3116383-0.485320

6.606725.6521

.,6Q747

3.74285

2.78823

"'3360878976

.{).0756482

-1.03027

-1.9649

-2.93952

-3.89415....."-5.8034

-6.75802

Figure 9: Case 2 with bilinear parametrization. E-field

27

128 elements, order=3,exact solution is 8 polynomial of order 7

Divergence of E-field

(J) = 2.5trapezoidal domain

(by bli06lll' map)

D;Dirichlel Be

N=Neumann Be

Lagrange multiplier p

0.105167

0.0924021

0.07963690.0668716

O.O~l064

0.0.413411

0.0285759

0,0158106

0.00304538

.0.00071987..~~:.=-0.0.480156-0.0607800

.(),073S461

0.0002Q6428

0.000256392

0.0002163560.00017632

0.000136284

9.624E·5

5621E-5

1.617E-5

-2.385E-5

-8.389E-5

-0.000103931

-0.00014396]

-0.00018-4003

-0000224039

-o,ClQ0264015

Figure 10: Case 2 with bilinear parametrization.

2 elements, order=2, x-component of E-field,

exact solution is quintic

00=2.5 rectangular domainN

Perturbation of the central midside node along the X-axis:

O.B0.4142860.0285716

-0.357143-0.742857·1.12857-1.51429-1.9-2.28571-2.67143·3.05714·3.44286·382657-4.21429....

£=2.0

olµ=1.0 IN0= 0.0 0""""''' Be

N""'llUmamBC-20%+40%original position

Figure 11: Case 1.Sensitivity of solution to a mesh distortion

28

a sequence of positive eigenvalues. We emphasize that more than 50 % of the eigenvectorscorrespond to the zero eigenvalue. When the mesh is distorted, the multiple zero eigenvaluespreads into a cloud of non-zero eigenvalues whose size depends upon the distorsion. Forinstance, for the experiment discussed above, a distortion of only 5 % is sufficient to spreadthat cloud beyond the first (physically meaningful) positive eigenvalue. These numericaleigenvalues interact with the forcing frequency wand we simply run into a numerical reso-nance, exciting one or more corresponding modes. It is these excited modes that drasticallychange the solution and prohibit the convergence in the L2-norm. We can proceed with ex-actly the same interpretation for the mixed formulation, except that, in that case, the entireanalysis takes place in the subspace of W h consisting of the approximate E-fields whichsatisfy the divergence condition. If the number of the imposed scalar constraints equals ex-actly the multiplicity of the zero eigenvalue (with no constraints imposed), the correspondingnull space is eliminated and the spectrum consists only of the positive eigenvalues. This isprecisely guaranteed by the compatibility condition (4.18). In such a case, when we disturbthe mesh, there is simply no zero eigenvalue to spread and we do not run into the instabilityproblems.

6 Conclusions

In the paper a new formulation for the steady-state form of Maxwell's equations has beenproposed. The formulation involves the electric field E and a Lagrange multiplier p eventhough, on the continuous level or for affine elements approximation, the multiplier p = O.

A novel, variable order of approximation based on the mixed formulation is proposed andproved to be stable with a stability constant bounded away from zero for w ---t O.

A perturbation argument and numerical experiments indicate that the method remainsstable for parametric elements.

The proposed methodology is independent of the dimension and lays down foundations fora general class of both two- and three-dimensional hp-discretizations of Maxwell's equations.

Acknowledgment

The authors would like to thank Professors I. Babuska, D. Neikirk, and J.T. Oden formany interesting discussions on the subject. The work of the first author has been partiallysupported by the NSF under Contract DMS 9414480, and the support for the second authorwas provided from DOD-Air Force Grant F49620-96-1-0032 (P.I. - D. Neikirk).

29

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