3D FEM-BEM-Coupling Method to solve Magnetostatic Maxwell Equations

Florian Brucknera,∗, Christoph Voglera, Michael Feischlb, Dirk Praetoriusb, Bernhard Bergmaira, ThomasHubera, Markus Fugera, Dieter Suessa

aVienna University of Technology, Inst. Solid State Physics, AustriabVienna University of Technology, Inst. Analysis and Scientific Computing, Austria

Abstract

3D magnetostatic Maxwell equations are solved using the direct Johnson-Nédélec FEM-BEM coupling methodand a reduced scalar potential approach. The occurring BEM matrices are calculated analytically and approx-imated by H-matrices using the ACA+ algorithm. In addition a proper preconditioning method is suggestedthat allows to solve large scale problems using iterative solvers.

Keywords: FEM-BEM coupling, Johnson-Nédélec method, ACA+, preconditioner, magnetostatic Maxwellequations, scalar potential

1. Introduction

For the design of various applications ranging fromelectric motors to write heads in hard disks the so-lution of 3D magnetostatic Maxwell equations is re-quired. A straight forward method concerning imple-mentation is the Magnetic Moment Method (MMM)[1]. It does not require a mesh in free space and caneasily treat a nonlinear material law. A drawback ofthe method is that it leads to fully populated matricesof the size 3M×3M , whereM is the number of volumeelements. Recently, compression techniques such asH-matrices were applied in order to reduce complexity[2]. A well established tool for the solution of partialdifferential equations is the Finite Element Method(FEM) which can be applied to the magnetostaticproblem in its differential form. Various methods wereproposed to solve this open boundary problem, suchas infinite elements [3], ballooning elements [4], par-allelepipedic shell transformation [5], and asymptoticboundary conditions [6]. Drawbacks of these meth-ods are either a certain number of elements in theouter region or intrinsic systematic errors. For linearproblems the Boundary Element Method (BEM) al-lows to transform the volume integrals arising fromthe open boundary problem into corresponding sur-face integrals, which reduces the computational com-plexity of the problem. In order to be able to takenonlinear material laws into account, the BEM canbe merged with the FEM combining the advantagesof both methods. The FEM efficiently takes into ac-count the non-linearity of the problem, and the BEMtransforms the boundary condition from infinity to the

∗Corresponding authorEmail address: florian.bruckner@tuwien.ac.at (Florian

Bruckner )

surface of the magnet. Thus no finite elements arerequired outside of the magnet in order to solve thesystem accurately. For the strayfield calculation of agiven magnetization Fredkin-Koehler proposed an el-egant way, where FEM and BEM equations are calcu-lated sequentially [7]. For the solution of the completemagnetostatic Maxwell problem direct FEM-BEM cou-pling schemes, as the Johnson-Nédélec coupling pre-sented in this paper, provide robust algorithms. Inthis paper we combine a direct FEM-BEM couplingscheme with a H-matrix compression technique for thesolution of the magnetostatic Maxwell problem. Fur-thermore, we present a proper preconditioner whichleads to a fast and reliable solver.

The structure of the paper is as follows. Section 2summarizes the open boundary problem and its for-mulation with a reduced scalar potential. The cou-pling of FEM and BEM is discussed in section 3 fol-lowed by the construction of the BEM matrix in sec-tion 4, where the H-matrix method is used in orderto compress the involved boundary matrices. For thearising system of equations a proper preconditionerfor iterative solvers is proposed in section 5. Finallysection 6 shows numerical results and benchmarks fordifferent mesh sizes.

2. Reduced Scalar Potential Formulation

We start from magnetostatic Maxwell’s equationswhich are defined in the entire space. The full spaceis then divided into two partitions, since we use dif-ferent representations in these regions. The regionΩ+ contains all the magnetic parts of the problemwhich are described by FEM, whereas Ω− contains thesurrounding non-magnetic volume described by BEM.The superscripts + and − are therefore used for phys-ical quantities within Ω+ or Ω−, respectively (see Fig.

Preprint submitted to Computational Physics May 23, 2012

1). Maxwell’s equations, the material laws in both re-gions, and the jump conditions at the boundary of themagnetic parts are given by

rotH = j B+ = µ+H+ (1)

divB = 0 B− = µ−H− (2)

n×(H+ −H−

)= 0 (3)

n ·(B+ −B−

)= 0 (4)

where the current density j is the source of themagnetic field strength H which is related to the mag-netic fluxB via the permeability µ. The normal vectorat the boundary n allows to distinguish jump condi-tions for the normal as well as for the parallel compo-nent of the magnetic field.

Figure 1: Example geometry for a yoke containing a mag-netic region Ω+ and the surrounding area Ω−. In thisexample the external field is created by a coil located inΩ−.

In order to reduce the number of degrees of free-dom one introduces a scalar potential by splitting thetotal magnetic field into an external part Hext (whichis created by currents outside of Ω+) and the curl-free induced magnetic field which can be expressedas the gradient of a scalar potential u. Inserting thisdefinition of the magnetic field into the Eqn. (1) -(4) leads to the following system of equations for thescalar potential u. We introduce the normal derivativeof the potential at the surface of the magnetic partsas φ = ∂u

∂n :

H = Hext −∇u (5)

∇ ·(µ+∇u+

)= ∇ ·

(µ+Hext

)(6)

∇2u− = 0 (7)

u+ − u− = 0 (8)

µ+φ+ − µ−φ− =(µ+ − µ−

)n ·Hext (9)

If there are no currents within Ω+, an alternativeformulation of the scalar potential can be used. In thiscase, the internal field H+ is curl-free and can there-fore be directly expressed as the gradient of a totalscalar potential, which leads to some slightly modified

equations [8]. Since the total magnetic field can be cal-culated directly, numerical errors due to the subtrac-tion of the external field are avoided. The drawback ofthis method is that due to the different descriptions inΩ+ and Ω− the potential is discontinuous at the sur-face of the magnet and additional complications occurif the region Ω+ is not simply connected.

3. FEM-BEM coupling

For sake of simplicity, we restrict ourselves to lin-ear systems. Since µ is constant the divergence oper-ator on the right-hand side of Eqn. (6) only acts onHext which gives 0.

Starting with the FEM equations for the internalproblem, we use a Galerkin approach with polynomialtest and shape function Λi(x) for each node i of theproblem. Transforming Eqn. (6) into the weak for-mulations, performing an integration by parts, andconsidering that for linear systems the source termon the right-hand side vanishes, leads to the followingsystem of equations:∫

Ω

µ+∇Λi · ∇u+ dΩ−∫Γ

µ+ Λi∇u+ · n dΓ = 0 (10)

Using a FEM-only approach would require to defineeither Dirichlet or Neumann boundary conditions inorder to get a solvable system. Since we do not knowthese boundary conditions, we introduce the normalderivatives of the potential at the boundary as new in-dependent variables φ and add additional BEM equa-tions to the system, which then has a unique solution.

For these additional equations, one recognizes thatthe outer potential u− fulfills the Laplace equation.With the help of the fundamental solutionG = 1

4π1

|x−y| ,u− can be expressed by the third Green’s identity interms of its Cauchy data (the potential and its normalderivative) on the boundary Γ:

1

2u− =

∫Γ

(u−∇G−∇u−G

)ndΓ (11)

Mathematically, the coupling method resulting fromthis approach, was first analyzed by Johnson and Nédélec[9, 10] (the 1

2 factor is only valid for smooth surfacesand has to be modified at edges [11]). The terms onthe right-hand side are called double-layer and single-layer potential. For a general Galerkin-BEM, formula(11) is multiplied by a test function Ψm(x) and inte-grated over the boundary. In our implementation, weuse delta distribution as test functions for the BEMequations Ψm(x) = δ(x−xm), which leads to a point-matching approach that only evaluates the equationsat the centers of each boundary triangle. The mainadvantage of this approach is that the computationof the occurring boundary integrals is easier and lesstime-consuming.

2

Finally, we discretize our formulas by using linearshape functions Λj for the potential u and piecewise-constant shape functions 1n for its normal derivativeφ. Furthermore, u− and φ+ can be eliminated by useof the jump conditions (8) and (9). Our discretizedvariables as well as the test function used within theGalerkin formalism look like the following:

u+(x) =∑j

uj Λj(x) φ−(x) =∑n

φn 1n(x) (12)

Combining FEM and BEM equations to one to-tal system of equations we end up with a quadraticsystem M which fully defines our problem:

(M11ij M12

in

M21mj M22

mn

)(ujφn

)=

(RHS1

i

0

)(13)

The individual elements of the matrices can be cal-culated as follows. Remember that for the FEM partthese matrices are sparse whereas the BEM formalismleads to dense matrices:

M11ij =

∫Ω

µ+∇Λi · ∇Λj dΩ (14)

M12in = −

∫Γ

µ+ Λi 1n dΓ (15)

M21mj =

1

2Λj(xm)− 1

4π

∫Γ

Λj(y)xm − y

|xm − y|3n dΓy

(16)

M22mn =

1

4π

∫Γ

1n(y)

|xm − y|dΓy (17)

RHS1i =

∫Γ

(µ+ − µ−)n ·HextΛi dΓ (18)

The indices i, j run from 1 to the number of nodes ofthe geometry, whereas n,m run from 1 to the num-ber of boundary elements. Since only boundary nodescontribute to the boundary integrals in M12 and M21

all matrix elements which correspond to nodes withinthe volume are equal to 0.

4. Calculation of BEM matrices

The occurring BEM matrices are calculated ana-lytically as given in Ref. [12]. (An alternative butmore special solution was given by Lindholm [11]).The single and double-layer potential (E and D) atthe point x created by a triangle ∆ using the Greens’function of the 3 dimensional Laplacian are given by

E(x) =1

4π

∫∆

f(y)

|x− y|dΓy (19)

D(x) =1

4π

∫∆

f(y)x− y

|x− y|3n dΓy (20)

where f(y) stands for the shape function of the usedelements. For our implementation we confine our-selves to piecewise constant shape function for thesingle-layer potential and linear shape functions forthe double-layer potential. Since these integrals aretranslation- and rotation-invariant, we use a transfor-mation Φ to rotate the triangle into the x1-x2-planeand to move the reference point x onto the x3 axis(see Fig. 2). Doing this, we can write a general poly-nomial shape function as f(y) =

∑jk fjk y

j1 y

k2 and

we can break the problem down to the calculation ofthe integrals for the individual monomials that occurwithin the sum:

Ejk(x) =1

4π

∫∆

yj1yk2√

y21 + y2

2 + x23

dy1dy2 (21)

Djk(x) =x3

4π

∫∆

yj1yk2√

y21 + y2

2 + x23

3 dy1dy2 (22)

Figure 2: Diagram schematizing the triangle area used forthe integration process.

The divergence theorem∫

∆divG dΓ =

∫γGn dγ

can be used to further simplify the calculation bytransforming the surface integrals to line integrals [13].One can finally derive a recursive formula for the higherorder surface integrals that only depends on lowest or-der line integrals which can be solved analytically aswell as on the lowest order double-layer term D00.Furthermore D00 can be expressed as the surface an-gle corresponding to x and ∆ which allows a simplegeometric interpretation of these terms. For the con-struction of the surface angle one connects x with thethree corners of the triangle. The intersection pointsof these connection lines with the unit sphere definethe corners of a spherical triangle on the unit sphere.The area of this triangle finally gives us the value ofD00 (see Fig. 3).

Using the recursive formula [13] allows to calculatethe occurring integrals for shape functions of any or-der. For sake of clarity we now demonstrate how thisrecursion is derived for the lowest order single-layerpotential that is used in our algorithm. If we set G =(y1r ,

y2r

)with divG = 1

r +x23

r3 and r =√y2

1 + y22 + x2

3

3

Figure 3: D00 can be interpreted as the surface of the tri-angle on the unit sphere.

the divergence theorem gives:∫∆

1

r︸︷︷︸4πE00

+ x23

∫∆

1

r3︸ ︷︷ ︸4πx3D00

=

∫γ

(y1

r,y2

r

)n dγ (23)

We can now calculate the interaction matrix elementsof the piecewise constant single-layer shape functionsE00, by means of the line integral on the right-handside of equation (23) as well as of the surface angleD00.

The main numerical problem when dealing withBEM matrices is that they are dense, and thus exten-sive computation time and storage capacity is neededto handle them. This problem can be overcome by useof H-matrices [14], where some blocks of the dense ma-trices (M21, M22) are replaced by low rank approxi-mations. If a block can be simplified or not depends onthe admissibility criterion which tells whether the sizeof the corresponding pointsets is smaller than the dis-tance between source and destination pointset. If theadmissibility criterion is fulfilled, the correspondingblock describes far-field interaction and thus can beapproximated without introducing a significant error.On the other hand if source and destination pointsetsare very close to each other, then the correspondingmatrix is stored as a full matrix in order to obtain asufficient accuracy.

We use the ACA+ algorithm which allows to ob-tain low-rank approximations without having to eval-uate all elements of the original block [15]. Thereforthe approximated system matrix of rank k is expressedas Mn×m = Un×k Vk×m. U and V are created bysequentially extracting lines and columns of the orig-inal matrix Mm×n. Lines and columns of the matrixare chosen in a way that the biggest elements of theerror matrix R = M− M are eliminated.

5. Preconditioning

In order to solve the system of equations, we usean iterative solver (KINSOL [16]) which is an efficientmethod for large non-linear systems. It performs anInexact Newton iteration for general non-linear prob-lems. Each step of the non-linear iteration requires the

solution of a linear system of equations. For this pur-pose KINSOL uses the Generalized Minimal RESidual(GMRES) Krylov subspace method.

For linear system one can define a condition num-ber of the system matrix M which is directly relatedto the convergence speed of the iterative method andcan be expressed as cond(M) = ||M|| ||M−1|| (with aproper matrix norm). We empirically found that thecondition number is directly proportional to the sus-ceptibility of the simulated material for various mod-els (see Fig. 6). Therefore, especially for systems withhigh permeability, computation time can be decreasedsignificantly by use of a proper preconditioner.

In our current implementation, we use the pre-conditioning feature of the solver suite, that does aright-preconditioning, where one rewrites the originalmatrix equation as (MP−1)(Px) = RHS, with a pre-conditioner matrix P and the unknowns x = (uj , φn).This leads to a new system matrix MP−1 for the it-erative solver with a smaller condition number.

The preconditioner matrix P is a good approxima-tion of the system matrix M, but it is easier to solve.The most time consuming part of solving the systemof equations is due to the dense boundary matrices.For BEM matrices it can be proven that simple diag-onal scaling reasonably improves conditioning [17].

For the coupled system numerical studies showedthat using only the diagonal elements of the single-layer matrix as well as only the elements contributingto the 1

2 factor within the double-layer matrix leads toa sparse approximation of the system matrix M whichcan be used as an efficient preconditioner:

P =

(M11ij M12

in12Λj(xm) diag

(M22mn

)) (24)

For all our test geometries, computation times couldbe decreased dramatically, and additionally it does nolonger depend on the used susceptibility. Results arepresented in the following section.

6. Results

In this section, the algorithm is applied to calcu-late the induced magnetization of a magnetic spherewith susceptibility χ exposed to a homogeneous ex-ternal field and a yoke magnetized by an electric coil.Since the solution for the magnetic sphere in an ex-ternal field can be calculated analytically, it is wellsuited to check the accuracy of our algorithm. Thesecond example shows a more advanced applicationof our algorithm.

6.1. SphereThe analytical calculation of the homogeneously

magnetized sphere (with a magnetization M) showsthat the induced field Hd is also homogeneous withinthe sphere and its value is − 1

3M . Adding the mate-rial law M = χH, where H is the total field H =

4

Hd +Hext and χ is the magnetic susceptibility of thematerial, directly leads to the analytical result:

M =3χ

3 + χHext (25)

The comparison of the numerical results with Eqn.(25) shows that the mean magnetization (averagedover the volume of the sphere) differs by around 0.1%using 10535 elements. In order to demonstrate theperformance of the presented algorithm we measuredruntime and maximal memory consumption for dif-ferent number of elements (see Table 1). For highsusceptibilities the problem becomes more difficult tosolve due to the large condition number. For prac-tical applications the largest relevant susceptibility isaround χ = 105, which we therefor use for our studies.The calculated magnetization for a sphere with 1074elements is shown in Fig. 4.

Figure 4: Simulation results for the magnetization of themagnetic sphere in a homogeneous external field. TheMagnetization is nearly perfectly aligned to the externalfield which is applied in z direction.

No. elements 1074 10535 337505 1546384runtime [s] 1.470 26.45 1590.3 25324.1

memory [MB] 194 248 1499 5503

Table 1: Performance values of the sphere example for dif-ferent grid sizes.

6.2. Coil and yokeThis more complex example shows a cylindrical

coil around a magnetic yoke. The external field Hext

is created by a coil and is directly calculated by theBiot-Savart law. We again measured solver runtime as

No. elements 1077 10205 109384 1180305runtime [s] 5.027 28.33 734.3 14035.4

memory [MB] 218 374 1025 5782

Table 2: Performance values of the yoke example for dif-ferent grid sizes.

well as maximum memory consumption for differentdiscretizations (see Table 2). The calculated magne-tization for χ = 106 can be seen in Fig. 5.

Figure 5: Simulation results for a magnetic yoke magne-tized by an electric coil. 10205 elements were used for thediscretization of the yoke. The color of the arrows showsthe x component of the magnetization.

The calculated condition numbers as well as thecomputation time of the equation solver with and with-out preconditioning is illustrated for the yoke examplein Fig. 6.

7. Conclusion

We have introduced a FEM / BEM coupling methodwhich combines the advantages of both methods. Thealgorithm is applied to a magnetostatic problem withopen boundary. The arising dense boundary matricesare approximated by H-matrices in order to speed upcomputation and to decrease memory consumption.Furthermore an effective preconditioner has been pro-posed in order to improve the convergence of the iter-ative solver. Finally the performance of the algorithmis demonstrated by two different example calculations.

Acknowledgment

The authors would like to thank theWWTF ProjectMA09-029 and FWF Project SFB-ViCoM F4112-N13for the financial support.[1] Y. Takahashi, S. Wakao, A. Kameari, Large-Scale and

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100000

Condit

ion n

um

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Runti

me

[s]

Susceptibility

cond (no precond)runtime (no precond)

cond (precond)runtime (precond)

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