Solving Large-Scale Inverse Magnetostatic Problems using the Adjoint Method

Florian Bruckner,1, ∗ Claas Abert,1 Gregor Wautischer,1 Christian

Huber,1 Christoph Vogler,2 Michael Hinze,3 and Dieter Suess1

1Christian Doppler Laboratory of Advanced Magnetic Sensing and Materials,Institute of Solid State Physics, Vienna University of Technology, Austria2Institute of Solid State Physics, Vienna University of Technology, Austria

3Department of Mathematics, University of Hamburg, Germany

An efficient algorithm for the reconstruction of the magnetization state within magnetic compo-nents is presented. The occurring inverse magnetostatic problem is solved by means of an adjointapproach, based on the Fredkin-Koehler method for the solution of the forward problem. Due to theuse of hybrid FEM-BEM coupling combined with matrix compression techniques the resulting algo-rithm is well suited for large-scale problems. Furthermore the reconstruction of the magnetizationstate within a permanent magnet is demonstrated.

I. INTRODUCTION

Magnetic materials are used in a wide range of ap-plications, ranging from permanent magnets, magneticmachines, up to magnetic sensors and magnetic record-ing devices. Solving the Inverse Magnetostatic Problemallows to reconstruct the internal magnetization state ofa magnetic component, by means of magnetic field mea-surements outside of the magnetic part, which is of im-portance for quality control. Compared with the for-ward problem, where the magnetic state is known andthe strayfield is calculated, inverse problems are muchharder to solve, since they typically are much worseconditioned and often not uniquely solveable. Inverseproblem solvers are based on stable and reliable solversfor the forward problem. In the case of magnetostaticMaxwell equations, Finite Element (FEM) formulations,combined with methods to handle the open-boundary,have proven to be the methods of choice for many effi-cient and accurate methods [1–3].

The applications of inverse problems can be coarselydivided into optimal design and source identification.Optimal design problems define a desired strayfield andtry to calculate optimal material distributions or geome-tries to reach these requirements as accurate as possible[4, 5]. In contrast to this, source identification problemstry to reconstruct the state of existing magnetic compo-nents. The identification of (permanent) magnetic mate-rials has e.g. been successfully applied for reconstructingthe state of magnetic rollers used in copy machines [6],magnetically biased chokes [7], or even for the magneticanomaly created by ferromagnetic ships [8].

The presented solver for the inverse 3D magneto-static Maxwell equations, is based on the well-establishedFredkin-Koehler-Method [3], which uses a hybrid FEM-BEM coupling for efficient handling of the open-boundary conditions. Combined with a hierarchical ma-trix compression technique for the dense boundary inte-gral matrices, the algorithm is able to handle large-scale

∗ Correspondence to: florian.bruckner@tuwien.ac.at

problems. Additionally, the use of a general Tikhonovregularization (see e.g. [9]), provides a very flexiblemeans to define application specific regularizations.

II. ADJOINT METHOD

A. Forward Problem

The demagnetization field of a magnetic body is de-fined as hd = −∇u, where the magnetic scalar potentialu is given by

∆u = ∇ ·m in Ω (1)

∆u = 0 in R3 \ Ω (2)

with jump and boundary conditions

[u]∂Ω = 0 (3)[∂u

∂n

]∂Ω

= −m · n (4)

u(r)→ O(1/|r|) for |r| → ∞ (5)

where m is the magnetization and Ω is the magneticregion.

The forward problem requires the solution of the po-tential u on the region Ωh generated by the magnetizationin a magnetic region Ωm (see Fig. 1). This problem issolved by considering a single region Ω = Ωm ∪ Ωh withm(x) = 0 for x ∈ Ωh. The hybrid FEM/BEM methodintroduced by Fredkin and Koehler [3] is one of the mostaccurate methods for the solution of this problem and willbe used in the following. Consider the following splittingof the solution u:

u = u1 + u2 (6)

Here u1 is defined by

∆u1 = ∇ ·m in Ω (7)

∂u1

∂n= −m · n on ∂Ω (8)

u1 = 0 in R3 \ Ω. (9)

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Aug

201

6

2

Ωh

Ωm

FIG. 1. Discretized magnetization region Ωm (blue) andmeasurement region Ωh (brown). Since the strayfield prob-lem is scale-invariant, length units are omitted. The mag-netization is defined on a unit cube. Measurement boxes ofthickness 0.04 are located next to each side of the cube, usingan airgap of 0.1 (the 2 boxes in front are not shown in thefigure).

This Neumann problem is solved with the finite-elementmethod. While u1 solves for the right-hand side m andfulfills the jump condition of the normal derivative −m ·n, it is not continuous across ∂Ω as required. This jumpis compensated by u2 which is defined as

∆u2 = 0 in Ω (10)

[u2] = −[u1] on ∂Ω (11)[∂u2

∂n

]= 0 on ∂Ω (12)

u2(x) = O(1/|x|) if |x| → ∞ (13)

This system is solved by the following double-layer po-tential

u2 =

∫∂Ω

u1∂

∂n

1

|x− x′|dx (14)

For efficiency reasons the double-layer potential isonly computed on the boundary ∂Ω using a Galerkinboundary-element method. Subsequently these valuesare used as Dirichlet boundary conditions to solve u2

within Ω using the finite-element method.All potentials are calculated using piecewise linear ba-

sis function (P1) and the derived strayfield would be con-stant within each element. Thus, a mass lumping proce-dure needs to be used to project the field onto piecewiselinear basis functions which are defined on each vertex ofthe mesh.

B. Inverse Problem

The inverse problem can be understood as a PDE con-strained optimization problem. Due to the ill-posednessof the inverse problem, some additional informationneeds to be provided to allow reasonable results. This

can be achieved by using Tikhonov regularization, whichuses an additional penalty term which should be consid-ered for the optimization. A possible candidate for theobjective function is

J =1

2

∫Ωh

‖ −∇u− hm‖2 dx + αT (m) (15)

where hm is the prescribed (measured) field in Ωh andα is the Tikhonov constant corresponding to the regu-larization functional T (m). This functional should beminimized, constrained by

F = ∆u−∇ ·m = 0 on Ω (16)

with boundary conditions as given above. We aim tosolve this problem using a gradient based iterative min-imizer. The constraint F gives an implicit expressionfor u(m) which allows to directly calculate the desiredgradient by

dJ

dm=∂J

∂u

∂u

∂m+

∂J

∂m(17)

The inefficient calculation of the term ∂u∂m can be avoided

using the adjoint approach, which makes use of thederivative of the constraint equation to eliminate theproblematic term:

dJ

dm= λT

∂F

∂m+

∂J

∂m(18)

where λ is given by the adjoint equation(∂F

∂u

)Tλ = −

(∂J

∂u

)T. (19)

Since the Poisson problem is self-adjoint, the adjoint sys-tem (19) can be solved along the lines of the forwardproblem. Computing the variational derivative on theRHS yields

∆λ = ∇ · (∇u+ hm) on Ω. (20)

where the sources (RHS) live on Ωh and the solution isonly computed on Ωm. The same boundary conditionsas for the forward problem hold. Thus, the above de-scribed hybrid method is applied. The gradient of J isthen finally given by

dJ

dm= ∇λ+ α

∂T

∂m(21)

Note that ∇u and ∇λ are projected onto P1 before com-puting (20) and (21) respectively. The algorithm is im-plemented using Magnum.Fe [10], which is based on thefinite element library FEniCS [11]. This allows a verycomfortable definition of the regularization functional.Furthermore, automatic differentiation can be used forthe calculation of the corresponding partial derivatives.The algorithm was verified by comparison with a FEM-only implementation, using the dolfin-adjoint library [12],which allows to automatically derive the adjoint equationfor a given forward problem.

3

III. NUMERICAL EXPERIMENTS

The presented algorithm is validated by the reconstruc-tion of the flower-state within a magnetic unit cube. Thestrayfield is calculated within measurement planes nextto each side of the cube (see Fig. 1). The magnetic stateof the cube is parametrized by

m =

sin(ctiltθ) cos(φ)sin(ctiltθ) sin(φ)

cos(ctiltθ)

θ = z√x2 + y2

φ = tan−1(y/x)(22)

where ctilt is an open parameter that allows to change thestrength of the flower state. For the proper reconstruc-tion of the magnetic state additional knowledge about thesolution needs to be provided. For all presented resultsa smooth reconstructed magnetization is desired whichsuggests using the following default regularization func-tional

T (m) =

∫Ωm

(∇m)2 dx (23)

For this specific flower-state the absolute value of themagnetization is known to be constant. Thus, the solu-tion of the inverse problem could be simplified by usingthe following penaltization functional

T ∗(m) =

∫Ωm

(m2 − 1)2 dx (24)

The assumption of a constant magnetization may be agood approximation for (isotropic) permanent magneticmaterials. Due to the large magnetic remanence and therelatively small susceptibility the induced magnetizationmay be negligible (see [13] for a simple model of isotropicpermanent magnetic materials).

A Gaussian noise with zero mean and a standard de-viation σ = 10−4 has been added to the field, calculatedby the forward problem, which should simulate unavoid-able measurement errors. The minimization problem issolved by a gradient descent method combined with aline-search strategy. As expected, reconstruction withoutusing a proper regularization leads to large magnetizationvectors near to the edges of the unit cube. Increasing theregularization parameter α, first avoids the over-fitting ofthe noisy measurement data, but finally leads to blurringof the reconstruction results if α gets too large. Deter-mining the optimal alpha is a crucial step for the solutionof an inverse problem. The results for the reconstructionof a flower state with ctilt = 2 using α = 10−3 is visual-ized in Fig. 2. Although there are some deviations of thereconstructed magnetization from the reference state. Itcan be seen that the created strayfield is nearly identi-cal. As stated above this is a clear indication of the badcondition of the inverse problem.

Using the so-called L-curve method [14], allows to vi-sualize the trade-off between the reconstruction of thestrayfield and the fulfillment of the regularization con-straint. Plotting the regularization norm (also called so-lution norm) ‖T (m)‖ over the residual norm ‖−∇u−hm‖

Ωh

Ωm

FIG. 2. Reconstruction of a flower-state within a unit cubeaccording to Eqn. (22) using ctilt = 2. A cut throughthe y = 0 plane is visualized. Starting from the initialparametrized flower state in Ωm the magnetic strayfield iscalculated within the fieldboxes Ωh (green arrows). In or-der to simulate measurement errors a Gaussian noise withσ = 10−4 has been added to the forward strayfield. The re-constructed magnetization as well as strayfield are computedusing α = 10−3 (red arrows). The relative differences of ini-tial and reconstructed states are indicated by the gray-scale.Maximal relative errors of the x-components amount to 0.25for the magnetization, and 5 · 10−3 for the induced magneticfield, respectively.

for different regularization parameters α, shows an L-shaped curve. The optimal αopt can be selected at thecorner of the L-curve which means that α is large enoughto reduce the regularization norm significantly, but itdoes not change the residual norm too much. The re-sulting L-curves for the reconstruction of the flower statefor different noise levels are summarized in Fig. 3. Anoptimal value αopt ≈ 5 · 10−5 can be found.

The application of the presented method to optimaldesign problems should now be demonstrated by meansof a Halbach cylinder configuration. The goal is to finda magnetization configuration within a cylindrical do-main, which creates a homogeneous strayfield inside ofthe cylinder. The magnetization domain Ωm has an outerradius ro = 1.0, an inner radius ri = 0.6, and a height ofh = 2.0, while a cylindrical measurement domain Ωh withradius rm = 0.5 and the same height is used. The magne-tization vectors should have constant norm, which sug-gests using the constant-norm penaltization functional(24). The analytic solution for a cylindical Halbach ar-ray [15] can be expressed in cylindrical coordinates as

m(ρ, φ) = cos(φ) eρ − sin(φ) eφ (25)

where ρ, φ are the cylindrical coordinates, with the cor-responding unit vectors eρ, eφ.

As visualized in Fig. 4 there is a nearly perfect matchof the reconstructed and the analytical Halbach configu-ration.

4

10−4 2 · 10−410−3

10−2

10−1

100

101

102

α = 10−9 (over-fitting)

αopt = 5 · 10−5 (optimal)

α = 5 · 10−3

(over-smoothing)

Residual Norm ‖ −∇u− hm‖2

Reg

ula

riza

tion

Norm‖∇

m‖2

no noise

σ = 10−5

σ = 10−4

σ = 10−3

FIG. 3. L-curves for the reconstruction of the flower statefor different noise levels σ.

Ωh

Ωm

FIG. 4. Optimal design problem of a Halbach cylinder creat-ing a homogeneous strayfield inside of the cylinder. Startingfrom a homogeneous strayfield the presented algorithm repro-duces a Halbach like magnetization configuration within Ωm

(red arrows). A constant-norm regularization with α = 104

is used and shows a nearly perfect match with the analyti-cal solution (green arrows). The resulting strayfield is calcu-lated inside Ωh and shows a nearly homogeneous distribution(red arrows). The relative errors of the magnetization mag-nitude and the reconstruced strayfield are indicated by thegray-scale, and their maximum amount to 2% and 6%, re-spectively.

IV. CONCLUSION

An efficient algorithm for the solution of inverse prob-lems has been introduced. The use of the Finite El-ement library FEniCS allows to easily define applica-tion specific regularization functionals in a very flexi-ble way. Thus, the implemented algorithm is suitablefor a wide range of applications including reverse engi-neering of magnetic components, design and optimizationof magnetic circuits and topology optimization, respec-tively. Using the hybrid FEM-BEM method proposedby Fredkin-Koehler allows to handle the open-boundaryproblem accurately and without the need for global meshincluding a large airbox. Source identification has beenvalidated by the successful reconstruction of the magneticflower-state within a unit cube by means of Tikhonovregularization. The selection of a suitable regulariza-tion parameter has been demonstrated using the L-curvemethod. Finally, the application of the method to an op-timal design problem has been demonstrated by meansof an Halbach cylinder, which is nearly perfectly repro-duced.

V. ACKNOWLEDGMENTS

The authors acknowledge the CD-Laboratory AMSEN(financed by the Austrian Federal Ministry of Economy,Family and Youth, the National Foundation for Research,Technology and Development), the Vienna Science andTechnology Fund (WWTF) under Grant No. MA14-044and the Austrian Science Fund (FWF) under gant No.I2214-N20 for financial support.

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