Iterative Schemes for Coupled Multiphysical Problems in ElectricalEngineering

Sebastian Schops1,2

1 Technische Universitat Darmstadt, Graduate School for Computational Enginering, Dolivostrasse 15, Darmstadt2 Technische Universitat Darmstadt, Institut fur Theorie Elektromagnetischer Felder, Schlossgartenstrasse 8, Darmstadtschoeps@gsc.tu-darmstadt.de

Summary. This contributions addresses iterative couplingschemes for coupled model descriptions in computationalelectromagnetics. Theoretical issues of accuracy, stabilityand numerical efficiency of the resulting formulations areaddressed along with advantages and disadvantages of thevarious approaches. Three application examples are given:field-circuit coupling, a mechanical-electromagnetic and ther-mal-electromagnetic problem.

1 Introduction

Today, due to increased accuracy of modeling andsimulation, multiphysical problems become more andmore important in many engineering applications. Of-ten a monolithic approach, i.e., the solution of all sub-problems at once, is cumbersome or even impossiblebecause incompatible algorithms or software pack-ages are involved. Thus simulation engineers need tocouple subproblems in an efficient and stable way,where subdomains are solved separately. This intro-duces a splitting error, which is mitigated by an itera-tive procedure.

In this contribution we like to advertise the in-creased accuracy and stability due to iterative pro-cedures by discussing three examples: field-circuitcoupling in Section 1, a mechanical-electromagneticproblem in Section 2 and finally a thermal-electro-magnetic problem in Section 3. In the full contribu-tion also implementation issues and the practical rel-evance of those iteration schemes will be discussed.

2 Field-Circuit Problem

For field-circuit coupled models of electrical energytransducers, two general approaches are well estab-lished. A first approach consists of extracting lumpedparameters or surrogate models from a field modeland inserting these as a netlist into a Spice-like circuitsimulator. This is circumvented by monolithic cou-pling, where field and circuit models are solved to-gether. We propose a particular synthesis: the param-eter extraction is applied iteratively on time intervals.The eddy-current field problem on Ω is

σ∂ta(n)+∇×(

ν(|∇×a(n)|) ∇×a(n))= χj(n)

(a) 2D transformer model

Rloadv(t)

1 2 3 4 5

06

RM,2RM,1LM

(b) rectifier circuit with embeded PDE model

Fig. 1. Simple field-circuit coupled problem.

where a(n) is the magnetic vector potential after the n-th iteration (with homogeneous Dirichlet conditions),σ and ν are conductivity and reluctivity, respectivelyand the winding functions χ = [χ1, . . . ,χk, . . . ,χK ]

>

are functions of space that distribute the lumped cur-rents j in the 3D domain. The circuit coupling is es-tablished via integration

∂t

∫Ω

χka(n) dx+Rk j(n)k = v(n−1)k k = 1, . . . ,K

to the circuit system of differential algebraic equa-tions

AC∂tqC(ATCu(n), t)+ARgR(AT

Ru, t)+ALi(n)L

+AMj(n)+AVi(n)V +AIis(t) = 0,

∂tΦL(i(n)L , t)−AT

Lu = 0,

ATVu−vs(t) = 0,

with incidence matrices A∗ where v∗ = AT∗u and con-

stitutive laws for conductances, inductances and ca-pacitances (functions with subscripts R, L and C), in-dependent sources is and vs, unknowns are the poten-tials u and currents iL and iV.

In the full paper the convergence, [1, 2], of this it-eration scheme and tailored time integration will bediscussed. It will be shown that the optimal time inte-gration order depends on the iteration counter n, [6].

3 Field-Mechanical Problem

The Lorentz detuning of an accelerating cavity, whichis the change of the resonant frequency due to the me-chanical deformation of the cavity wall induced by the

2

Fig. 2. One cavity cell with field lines and exaggerated de-formation, [3]

electromagnetic pressure is a coupled electromagnetic-mechanical problem. In a first step, Maxwell’s eigen-problem is solved

∇×(

1µ0

∇× e(n))= ω

20 ε0e(n) on Ω

(n−1)

where e is the phasor of the electric field with ad-equate boundary conditions; µ0 and ε0 are the per-meability and permittivity of vacuum. From this themagnetic field h can be obtained. Both fields create aradiation pressure at the boundary of Ω (n−1)

p(n) =−12

ε0e(n)⊥(

e(n)⊥)∗

+12

µ0h(n)‖

(h(n)‖

)∗which gives raise to the linear elasticity problem inthe wall of the cavity

∇ ·(

2η∇(S)u+λ I∇ ·u

)= 0

for the displacement u(n) where p(n) is a boundarycondition on the inner boundary. We denote by ∇(S)

the symmetric gradient, while η and λ are the Lameconstants. Finally a deformed domain

Ω(n) ≡

x+u(n) (x) , x ∈Ω

(0),

is derived from the initial domain Ω (0) and the it-eration can be restarted with the computation of aneigenvalue. In the full paper this scheme will be dis-cussed in more details. The focus will be on the spatialdiscretization with Isogeometric Analyses using Non-Uniform Rational B-Spline (NURBS) and De-Rham-conforming B-Splines [3].

4 Field-Thermal Problem

In the previous sections we have discussed the mutualcoupling of transient and frequency-domain to staticproblems. The third example revisits the well-knowniterative coupling of frequency to time domain prob-lems. Again, the electromagnetic field is given by thecurl-curl equation, however since we are in frequencydomain we can regard displacement currents

εω2a(n)+σ(T (n−1)) jωa(n)+∇× (ν∇×a(n)) = χj.

where a is now a complex phasor. This is coupled tothe heat equation

ρ c∂tT (n) = ∇ · (k∇T (n))+Q(a(n), t)

by the Joule losses Q, where k is the heat conductivity,ρ the density and c the specific heat capacity. Besidesthe electric conductivity σ , all material parameters areconstant. The important modelling step is to relax thecoupling of both problems by introducing a averagedheat source

Q(n) :=1

t1− t0

∫ t1

t0Q(a(n)(t), t)ds.

obtained by converting the vector potential a backto the time domain. Convergence will be discussedin view of the works [4, 5] and the fractional stepmethod, [7].

Acknowledgement. Many people have contributed to thiswork. The author likes to thank A. Bartel, M. Brunk, M.Clemens, J. Corno, H. De Gersem, C. de Falco, M. Gunther,C. Kaufmann and E.J.W. ter Maten. This work is supportedby the German BMBF in the context of the SIMUROMproject (grant 05M2013), by the ‘Excellence Initiative’ ofthe German Federal and State Governments and the Gradu-ate School of Computational Engineering at TU Darmstadt.

References

1. A. Bartel, M. Brunk, M. Gunther, and S. Schops. Dy-namic iteration for coupled problems of electric cir-cuits and distributed devices. SIAM J. Sci. Comput.,35(2):B315––B335, 2013.

2. A. Bartel, M. Brunk, and S. Schops. On the convergencerate of dynamic iteration for coupled problems with mul-tiple subsystems. J. Comput. Appl. Math., 262:14–24,2014.

3. J. Corno, C. de Falco, H. De Gersem, and S. Schops. Iso-geometric simulation of lorentz detuning in supercon-ducting accelerator cavities. Presented at CEFC 2014. Inprep. 2014.

4. C. Kaufmann, M. Gunther, D. Klagges, M. Knorren-schild, M. Richwin, S. Schops, and E.J.W. ter Maten.Efficient frequency-transient co-simulation of coupledheat-electromagnetic problems. Journal of Mathemat-ics in Industry, 2014. To be published.

5. C. Kaufmann, M. Gunther, D. Klagges, M. Richwin,S. Schops, and E. J. W. ter Maten. Coupled heat-electromagnetic simulation of inductive charging sta-tions for electric vehicles. In Progress in IndustrialMathematics at ECMI 2012, Mathematics in Industry,2014. Springer. To be published.

6. S. Schops, A. Bartel, and M. Gunther. An optimal p-refinement strategy for dynamic iteration of ordinary anddifferential-algebraic equations. In PAMM, volume 13,pages 549–552, 2013.

7. P. K. Vijalapura, J. Strain, and S. Govindjee. Fractionalstep methods for index-1 differential-algebraic equa-tions. J. Comput. Phys., 203(1):305–320, 2005.