Simulation of electromagnetic devices using coupled models · Simulation of electromagnetic devices using coupled models ... the cooling fluid as the speed etc) ... "Simulation of

Simulation of electromagnetic devices using coupled models

ION CÂRSTEADepartment of Computer Engineering and Communication

University of CraiovaStr. Doljului nr. 14, bl. C8c, sc.1, apt.7, Craiova

ROMANIADANIELA CÂRSTEA

High-School Group of Railways, CraiovaROMANIA

ALEXANDRU ADRIAN CÂRSTEAUniversity of Craiova

[email protected] http://ioncarstea.webdrisign.net

Abstract: This work presents numerical algorithms for simulation of distributed-parameter systems with directapplications in electrical engineering. The algorithms are developed in the context of the finite element method.Many works in the professional literature present coupled models for the electromagnetic devices and this work istoward this direction with emphasis on the development of efficient algorithms in numerical computation of thecoupled models.

Our work describes the solution of coupled electromagnetic and heat dissipation problems in two dimensionsand cylindrical-coordinates system for devices with cylindrical symmetry.

The purpose of the work is to define both conventional algorithms and parallel algorithms for coupled problemsin context of the finite element method. The mathematical models for electromagnetic field are based on potentialformulations. Some numerical results are presented.

Key Words - Coupled fields; Finite element method; Domain decomposition.

1 IntroductionThe reality forces us to deal with complex coupledsystems where two or more physical systems interact.Two or more fields coexist in the same geometry, inthe same electromagnetic device. These fields interact.For example, induction heating is used for surfacetreatment of materials. In this practical application, theeddy currents generated by an electromagneticinductor are used as the thermal heat sources throughthe Joule effect. More, any change in the physical orgeometric parameters of an electromagnetic devicewill affect both magnetic and thermal fields. In ourtarget examples the physical phenomena areelectromagnetic and thermal. The physical propertiesof the materials are strongly dependent on thetemperature, especially the following characteristics:electric conductivity, magnetic permeability, andspecific heat and thermal conductivity.

In this work we limit our discussion to coupledelectromagnetic and thermal fields. Mathematicalmodels for the problems in which the electromagnetic

field equations are coupled to other partial differentialequations, such as those describing thermal field, fluidflow or stress behaviour, are described by equationsthat are coupled [1]. The coupling between the fields isa natural phenomenon and only in a simplifiedapproach the field analysis can be treated asindependent problem.

In several cases, it is possible a decoupling and acascade solution of the coupled equations. Anotherattractive and efficient approach of solving coupleddifferential equations is to consider the set as a singlesystem. In this way a single linear algebraic system forthe whole set of differential equations is obtained afterdiscretization, and is solved to a single step. If one ormore equations are non-linear, non-linear iterations ofthe whole system are required.

The equations of the electromagnetic fields andheat dissipation in electrical engineering are coupledbecause the most of the material properties aretemperature dependent and the heat sources representthe effects of the electromagnetic field [1].

WSEAS TRANSACTIONS on SYSTEMS and CONTROLManuscript received Aug. 2, 2007; revised Nov. 30, 2007

Ion Cârstea, Daniela Cârstea, Alexandru Adrian Cârstea

ISSN: 1991-8763 515 Issue 11, Volume 2, November 2007

The thermal effects of the electromagnetic field areboth desirable and undesirable phenomenon. Thus, inconducting parts of some electromagnetic devices(coils of the large-power transformers, current bars,cables conductors, conductors of the electric machinesetc) the heating is an undesirable phenomenon. Theheat is generated by ohmic losses of the drivingcurrents and eddy currents induced in conductingmaterials. But in induction heating devices for weldingthe heating is a desirable phenomenon. The thermaleffect of the electromagnetic field is the treatment basefor many electric materials in industry [12].

With the terminology of the system theory, weidentify two kinds of the heat sources (and commandsin an inverse problem):• Distributed sources (electrical currents)• Boundary sources (Dirichlet's condition,

Neumann's condition, convection and radiation)In the heating of the electromagnetic devices, the

internal heat sources (position, amplitude) arerepresented by:• Ohmic losses from driving (source) currents• Ohmic losses from eddy currents induced in

conducting materials of the time variable magneticfield

• Dielectric losses due to friction in the molecularpolarisation process in solid dielectrics that formthe insulation of the high-voltage apparatus

• Hysteresis loss in magnetic problems. It is due tomagnetic domain friction in ferromagneticmaterials.

The boundary sources (commands) can be [3]:• Dirichlet command, that is, an imposed

temperature on the boundary of the spatial domain• Neumann command that involves an imposed

flux temperature on the boundary of the spatialdomain

• Convective command (the temperature of theambient medium or a cooling fluid, a parameter ofthe cooling fluid as the speed etc)

• Radiation commands (the temperature of theambient medium or other parameters that areoutside the spatial domain of the field problem andinfluences the temperature of a device by radiationphenomenon).

2 Mathematical modelling of theelectromagnetic fieldFor numerical simulation of the coupled systems wemust have in mind some practical aspects:

♦ Mathematical models of electromagnetic fieldand thermal field

♦ Mathematical tools for field problems♦ Mathematical methods for coupled problemsA complete mathematical model for coupled

electromagnetic-thermal fields involves Maxwellsequations and the heat conduction equation.Combining these equations yields a coupled system ofnon-linear equations.

A complete physical description of electromagneticfield is given by Maxwells equations in terms of fivefield vectors: the magnetic field H, the magnetic fluxdensity B, the electric field E, the electric field densityD, and the current density J. In low-frequencyformulations, the quantities satisfy Maxwellsequations [3]:

JH =×∇ (1)

tB

E∂∂

−=×∇ (2)

0=Bdiv (3)

cDdiv ρ= (4)with ρc the charge density, σ the electric

conductivity, and µ the magnetic permeability. Forsimplicity we give up to the bold notations for vectors.

The second set of relationships, called theconstitutive relations, is for linear materials:

EJEDHB σεµ === ;;The B-H relationship is often required to represent

non-linear materials. The current density J in Eq. (1)must represent both currents impressed from externalsources and the internally generated eddy currents.

The formulation with vector and scalar potentialshas the mathematical advantage that boundaryconditions are more often easily formed in potentialsthan in the fields themselves. The magnetic vectorpotential is a vector A such that the flux density B isderivable from it by the operator curl or ( ×∇ ).

The mathematical models for the electromagneticfield problems may be included in two formulations:♦ Integral equation formulations (Fredholm integral

equations)♦ Differential equation formulations (partial

differential equations of elliptic or parabolic type)♦ Hybrid formulations

The complexity of the mathematical model forelectromagnetic field was one of the main reasons tofind and develop new computational methods. Allmethods can be included in one of the followingclasses [3]:• Manipulation of the equations so that some

unknowns are eliminated• Definition of some potential functions from where

the field unknowns can be obtained by simpleprocessing

WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ion Cârstea, Daniela Cârstea, Alexandru Adrian Cârstea


• Finding of some assumptions that simplifies thecomputation for practical problems

The potential formulations seem attractive becauseof their computational advantages. One of theseconsists in the fact the boundary conditions are easilyframed in the potentials than in the field themselves.

2.1 The eddy-currents problemThe time-varying magnetic field within a conductingmaterial causes circulating currents to flow within thematerial. These currents called eddy-currents can beunwanted or desirable phenomena. Thus, the eddy-currents in electrical machines give rise to unwantedpower dissipation. On the other hand the inductionheating is a wanted phenomenon in industry of themetal treatment.

Industrial equipment in which the eddy currents areessentially can be included in one of the followingclasses:♦ long structures, in which the electric field and the

current density posses only one component♦ complex structures in which we use models 3D

In the long structures, the currents are generated byan electric field applied at the terminals of theconductor, or by a time-varying magnetic field linkingthe loop formed by the conductors. These structuresbelong to electric transmission network or thedistribution networks (bus bars, large-power cablesetc). In these problems the applied voltage of the baror cable is known and we seek to compute the currentdensity distribution within the conductor in order todetermine some electromagnetic quantities of interest(the electrodynamic forces, mutual inductances, localheating etc).

The complex structures create difficulties insimulation and computation of their characteristicsalthough these structures possess constructionsimplicity. One of these structures is the device forelectric heating by electromagnetic induction. In thistype the applications it is necessary to computeaccurately the eddy currents. If the eddy-currentsdistribution is non-uniform, the resulting high-temperature gradients may crack the workpiece.

The problems are different in the two differenttypes of applications but for any given application thepresence of the saturable iron sheets introducessaturation phenomena and the problem becomes non-linear.

For each class we can apply general mathematicalmethods but it is more efficient to develop a particularalgorithm for each kind of classes.

The effects of the eddy currents are:

♦ The time-varying magnetic flux density is non-uniform within the conductor. The alternatingmagnetic flux is concentrated toward the outsidesurface of the material (phenomenon known as theskin effect).

♦ Power losses are increased in the materialEddy current computation appears in two types of

problems:♦ Stationary problems where the structures are

fixed and source currents are time varying♦ Motion problems where the field source is a coil

in movingMany practical engineering problems involve

geometric shape and size invariant in one direction.Let z denote the Cartesian co-ordinate direction inwhich the structure is invariant in size and shape. Thisis the case of a plane-parallel field or translationalfield problem, where A has one component, namelyAz. This component is independent of the z co-ordinate and the Coulomb gauge is automaticallyimposed and V is independent of x and y. In such acase both the magnetic vector potential and the sourcecurrent JS reduce to a single component orientedentirely in the axial direction and vary only with theco-ordinates x and y.

Consequently, the component Az (for simplicity wegive up the subscript z) satisfies the diffusion equationin fixed bodies [1]:

sJtA

A −=∂∂

−∇∇ σν )( (5)

or, in Cartesian co-ordinates:

s-J=tA

σ)yA

(υy

+)xA

(υx ∂

∂−

∂∂

∂∂

∂∂

∂∂

(6)

The boundary conditions are set-up for the singlecomponent A and can be Dirichlet and/or Neumannscondition. The interface conditions between twomaterials with different properties are:

N

A

N

AAA

∂

∂=

∂

∂= 2

21

1;21 υυ

where n is the normal at the common surface of thetwo regions with different material properties.

2.2 Modelling of time-dependent fieldsThe time dependent electromagnetic field problemsare usually solved using differential models ofdiffusion type. Many practical problems of greatinterest in electromagnetics involve time-harmonicfields and this case will be considered in this work.

In general, computer software for time-varyingproblem can be classified into two classes [3]:

1. time-domain programs2. frequency-domain programs



Time-domain programs generate a solution for aspecified time interval at different time moments.Frequency-domain programs solve a problem at one ormore fixed frequencies.

The first class has some disadvantages. One ofthese consists in the large amount of data that must bestored to recover the field behaviour. Although thesecond class has an essential advantage (a compactand a cheap program in terms of the computerresources), the area of problems that can be solved islimited. It is applicable only to linear problems (allphenomena are sinusoidal).

The usual mathematical model for time dependentelectromagnetic field problems is with Maxwellsequations in their normal differential form. For lowfrequency the displacement current term in Maxwellsequations can be neglected. At a surface of aconducting material the normal component of currentdensity Jn can be assumed to be zero.

3 Mathematical modelling of thethermal fieldThe thermal field is described by the heat conductionequation [2]:

qTTkTT(ct

=∇⋅−∇+⋅∂∂

])([]))([ γ (7)

where: T (x, t) is the temperature in the spatialpoint x at the time t; point k is the tensor of thermalconductivity; γ is mass density; c is the specific heatthat depends on T; q is the density of the heat sourcesthat depends on T. In the coupled problems we use theformula:

2)( JTq ⋅= ρ (8)with ρ the electrical resistivity of the material.

Equation (7) is solved with boundary and initialconditions. The boundary conditions can be ofdifferent types: Dirichlet's condition for a prescribedtemperature on the boundary; Neumann's condition;convection condition; radiation condition, and mixedcondition [2]. These boundary conditions have thefollowing form on different parts of the boundarysurface S:• Dirichlet's condition:

),,,(1

),,,( tzyxDT=StzyxT

• Neumann's condition:

0=|S 2]nq+nT

[k∂∂

• Convection:

0=|S 3)]T-h(T+nT

[k ∞∂∂

• Radiation:

0=|S 4]T-T 4B+

nT

[k )4( ∞∂∂

εσ

where the boundary surface S is:

4321 SSSSS ∪∪∪=

The significances of the quantities that appear inthe boundary conditions are: TD is a known functiondefined on the boundary S1; h is the convectioncoefficient; T∞ is the ambient temperature; qn is thenormal heat flux; σB is Boltzman's constant inradiation and ε is the emissivity coefficient. Thecoefficients of the heat transfer as h and ε depend onthe temperature and the surface quality. For these weuse empirical formulas based on the experiments.

For many eddy-current problems the magnetic fluxpenetration into a conductor without internal sourcesof the magnetic field is confined mainly to surfacelayer. This is the skin effect. The skin depth δ dependson the material properties µ, ω and σ so that for thesmall depths all effects of the magnetic field areconfined to a surface layer.

In steady-state low-frequency eddy currentproblems in magnetic materials, the mathematicalmodel is the diffusion equation defined by Eq. (6).

The skin effect can be exploited in two directions:• To reduce the space domain in analysis with a

fine mesh close to conductor surfaces• To reduce the material volume since a

significant proportion of the conductor isvirtually unused

The penetration depth is given by the formula:

ωσµδ 2= (9)

For example, in a semi-infinite slab of conductorwith an externally applied uniform alternating field,parallel to the slab, the amplitude of flux decaysexponentially. In other words for problems with theskin depth very small all the effect of the field isconfined to a surface layer. In a numerical modelbased on finite element method (FEM) this effect canbe exploited by the use of domain decomposition atthe level of the problem. In this way we reduce therun-time of a program based on FEM.

Designer engineers use the formula (9) consideringthe permeability and the conductivity as numbers. Inreality the two physical parameters change duringheating. The changes in the value of δ affect the loss inthe material and depend on the process (conduction orinduction). For example, if the conductivity decreasesby x, the depth increases by √x, that is the currentpenetrates deeper into the metal. If the magneticmaterial heats, its resistivity (the inverse of the



conductivity) rises but its relative permeability remainssubstantially constant up to the Curie point. In thispoint it drops suddenly to unit.

Another simplifying assumption for the designerengineers is based on that all heat enters at the surfaceof the conductor. In reality, this is only true if thefrequency of the magnetic field source is very high andthe depth of heating is small compared with thegeometrical dimensions of the conductor. This fact canbe exploited in numerical simulation of these devicesby reduction of the analysis domain.

For an accurate computation of the penetrationdepth of the magnetic field we must consider twopractical conditions:

• The heat is distributed in the conducting part• There is an important heat lost by radiation at

the conductor surfaceRadiation can be regarded as a simple surface loss

subtracting from the surface power input. The Stefan-Boltzmann's law gives the radiation loss. If the body isradiating to a surface at absolute temperature T∞Kelvin, the radiation loss is defined by:

)44(0 ∞−= TTcrrP ε

where εr is the emissivity coefficient of the surface(dimensionless), and T is the absolute surfacetemperature in grades Kelvin (K). The constant c0 is5.67.10-8 W/m2K4. For low temperatures, the radiationloss is negligible but in the induction-heating device itmust be considered.

Consequently, it is convenient to use coupledmodels and accurate methods for computation of theheat penetration in the conductors, especially in theinduction heating devices.

3.1 Transient problemsMany engineering applications are described byparabolic partial derivatives equations. When applyingthe FEM to time dependent problems, the timevariable is usually treated in one of two ways:• Time is considered as an extra dimension and

shape functions in space and time are used• The nodal variables are considered as functions of

time and the shape functions in space are used.A common approach for transient problems is to

solve time dependent differential equations by finitedifferences approximation of time derivative terms,combined with some weighted residual method inspace.

A widely used finite difference scheme for the first-order equations is the so-called θ. Certain values of θcorrespond to known methods for time stepping:• θ =0 the forward difference method;

• θ =1/2 Crank-Nicholsons method;• θ =2/3 central difference method;• θ =1 the backward difference method.

3.2 θ-rule combined with Galerkins methodWe illustrate the method by applying the θ-rule in timeand Galerkins method in space to the following heatconduction equation:

0,)( ftxqTktT

Ω∈+∇∇=∂∂

(10)

Ω∈= xxfxT )()0,( (11)

0,),( ftxtxgnT

k Γ∈=∂∂

− (12)

We shall present the numerical models obtained bytwo strategies.

Applying the θ-rule to the heat equation (10) resultsin the following spatial problem:

)()0( xfT =

Ω∈−+−∇∇−+

+∇∇=−−

xmqmTk

mqmTkt

mTmT

])1())1(()[1(

])())(([)1()(

θ

θδ (13)

0,),()(

ftxmtxgn

mTk Γ∈=

∂∂

−

where the superscript m denotes the iterationnumber, that is T(m)=T(x,tm).

Discretizing the Eq. (13) by the method ofweighted residuals, with T(n) approximated by

∑=

=r

ixiNm

iTmT1

)()()(

gives an algebraic equations system:

)1()(])[]([ −=+ mbmTKMwhere the matrices [M] and [K] have the entries:

∫Ω

Ω= djNiNijM

∫Ω

Ω∇⋅∇⋅⋅= djNiNktijK δθ

Instead of first discretizing in time by a finitedifference method, first we can apply the discretizingin space by the weighted residual method, with Tapproximated by:

∑=

=r

ixiNtiTtxT

1)().(),(

By this procedure a first order differentialequations system is obtained in the form:

fTAdtTd

=+ ].[



The θ-method for the time integration leads to [3]:

)()1(

)(

]))[1(]([])[]([

mm

m

gTAtITAtI

+−−

=⋅⋅+−θδ

δθ

))1(( )1()()( −−+⋅= mmm fftg θθδFor θ=0 the forward Euler's scheme is obtained and

we can get T(m) explicitly; otherwise, a linear systemmust be solved at each time step. When θ>1/2 there isno stability restriction on time step δt, which can beconvenient for the simulation algorithm. The choice ofθ=1/2 leads to an optimal combination of stability andaccuracy.

4 Coupled modelsWith a correct formulation of the mathematical modelsand a good selection of the mathematical tools for aspecified field problem, we must select the method forthe numerical solution of the field problem. Ones ofthese methods for field problems are momentsmethod, finite element method (FEM), boundaryelement method (BEM), hybrid method BEM-FEM,finite volume method (FVM), and edges elementmethod (EEM).

In our works we considered the FEM [6]. Thismethod can be viewed as a particular case of thegeneral method of moments, or a case of the Rayleigh-Ritz method.

When applying the FEM to time dependentproblems, the time variable is usually treated in one oftwo ways:

• Time is considered as an extra dimension andshape functions in space and time are used

• The nodal variables are considered asfunctions of time and the shape functions inspace are used.

4.1 Coupled magnetic and thermal fieldsFor magnetic field we consider the A-formulation, thatis we define the magnetic vector potential A by B =curl A. More, the domain is the same for temperatureand the electromagnetic field although in practice theinterest is for different field domains.

In order to solve the transient coupled set ofequations a numerical model can be developed usingthe finite element method [4]. The finite elementdiscretization in space is used, leading to a system offirst-order differential equations:

0][][ =++∂

∂

JfAAKt

AAS (14)

0][][][ =++∂∂

AATKTTK

tT

TS (15)

where the matrices have the entries defined inaccordance the FEM. The subscripts A and T refer tothe magnetic and thermal field respectively. Thevector fJ is generated by the heat source.

HHJq ×∇⋅×∇== ρρ 2

The two equations are coupled and non-linear.Finally, the two models can be considered as a coupledsystem defined in matrix form [12]:

[ ][ ]

[ ]

00

0

0

0=++

∂∂∂∂

JfTA

TKATKAK

tTtA

TSAS

In a discrete form the unknowns are the nodalvalues of the temperature T and the magnetic vectorpotential A. The non-linear equations for T and A arestraightforwardly obtained by a Galerkin's finiteelement method. For the case of 2D steady-stateproblems we do the following approximations at theelement level [1]:

∑=

=r

j jTyxjNyxT1

),(),(

∑=

=r

j jAyxjNyxA1

),(),(

where the interpolation functions Nj are basisfunctions in the mesh over Ω, and r is the number ofnodes of an element.

The usual procedure for the FEM applicationsleads to a system of 2p equations where p is the totalnumber of the unknowns in each field problem.Finally, the coupled problem is described by a systemof algebraic systems in the form:

0),...,1,,...,1( =pTTpAAAf (16)

0),...,1,,...,1( =pTTpAATf (17)where the subscript denotes the original problem (A for the magnetic field in the magnetic vector potentialformulation; T for the thermal field).

5 Iterative algorithms for coupledproblemThe finite element method has three distinct logicalstages: pre-processing, processing (solution) and post-processing. Each stage has an inherent parallelism thatcan be exploited for parallel computing. Newalgorithms for the parallel computers were developedand presented in the professional literature. We shalllimit discussion to one of them: domain decomposition[8]. This algorithm uses the subdomain-to-subdomain



iteration. Although the procedure is well known, wemust modify it for coupled problems.

5.1 Conventional algorithmsThe numerical model for coupled problem defined byEq. (16) and Eq. (17), can be solved by two differentbasic strategies [7]:♦ Solving the equations for Ti and Ai simultaneously♦ Solving the equations for the two fields in

sequence with an outer iteration, technique knownas operator-splitting technique (for exampleNewton-Raphson procedure)

In the area of the first strategy, Gauss-Seidel andJacobi methods are well known. We present thesemethods in brief.

The Gauss-Seidel algorithm for coupled fields hasthe following pseudo-code [7]:

For m:=1 , 2, until convergence DO• Solve

0))1(,...,)1(1;)(,...,)(

1( =−− mpTmTm

pAmAAf with respect to A1(m), Ap

(m)

• Solve

0))(,...,)(1;)(,...,)(

1( =mpTmTm

pAmATfwith respect to T1(m) , Tp

(m)

In other words, the system is solved firstly withrespect to A, using the values of T from the previousiteration. Afterwards, the equation derived from thethermal field model is solved using the computedvalues of A from the current iteration. The equationsfA=0 or/and fT=0 are non-linear and must be solved byan iterative procedure (for example Newton-Raphson'smethod).

The algorithm Jacobi-type is similar to Gauss-Seidel method, except that at the iteration m when wemust solve the model for T, the values for A are fromthe previous iteration, that is A(m-1). The algorithm hasthe following pseudo-code:

For m:=1 , 2, until convergence DO• Solve

01111 =−− ))(m

p,...,T)(m;T(m)p,...,A(m)(AAf

with respect to A1(m), Ap(m)

• Solve

0))(,...,)(1;)1(,...,)1(

1( =−− mpTmTm

pAmATfwith respect to T1(m) , Tp

(m)

This algorithm has an inherent parallelism so thatcan be implemented in a parallel program. Practically,

we decomposed the coupled problem in twosubproblems: one for the magnetic field, another forthe thermal field. At a time step of the algorithm, thenumerical models for the two fields can be solvedindependently.

5.2 Advanced algorithmsThe domain decomposition method [8] is the bestamong three possible decomposition strategies for theparallel solution of PDEs, namely, operatordecomposition, function-space decomposition anddomain decomposition. This is one of the motivationsto present the principles of the domain decompositionmethods in this section.

The domain decomposition could be determinedfrom mathematical properties of the problem (realboundaries or interfaces between subdomains), orfrom the geometry of the problem (pseudo-boundaries). For elliptic partial differential equations,there exists a mathematical approach based on theideas given earlier in 1890 by Schwarz [8]. In Schwarzprocedure there is an inherent parallelism with a datacommunication time for the passage of pseudo-boundary data between the subproblems.

There is no general rule for the domain or/andoperator decomposition. It is defined in a somewhatrandom fashion. The problems and solutions thatappear in the decomposition techniques depend on thefollowing aspects [1]:

• If it is used domain decomposition or theoperator decomposition

• If the partition has disjoint or overlapping sub-domains

• The type of boundary conditions that are set upon the pseudo-boundaries of the sub-domains

• If the decomposition is static or dynamicA general criterion for the decomposition does not

exist so that the experience of the engineer can be auseful reference for many algorithms and softwareproducts.

5.3 Decomposition techniquesThe desire of the scientific community for fasterprocessing on lager amounts of data has driven thecomputing field to a number of new approaches in thisarea. The main trend in the last decades has beentoward advanced computers that can executeoperations simultaneously, called parallel computers.For these new architectures, new algorithms must bedeveloped and the domain decomposition techniquesare powerful iterative methods that are promising forparallel computation. Ideal numerical models are thosethat can be divided into independent tasks, each of



which can be executed independently on a processor.Obviously, it is impossible to define totallyindependent tasks because the tasks are so inter-coupled that it is not known how to break them apart.However, algorithmic skeletons were developed in thisdirection that enables the problem to be decomposedamong different processors. The mathematicalrelationship between the computed sub-domainsolutions and the global solution is difficult to bedefined in a general approach.

In the area of the coupled fields we define twolevels of decomposition that is we define a hierarchyof the decompositions:

• One at the level of the problem• The other at the level of the field

In other words, we decompose the coupledproblem in two sub-problems: a magnetic problem anda thermal problem, each of them with disjoint oroverlapping spatial domains. This is the first level ofdecomposition. At the next level, we decompose eachfield domain in two or more subdomains. Thedecomposition is guided both by the different physicalproperties of the materials, and the difference of themathematical models. At this level of decompositionthe Steklov-Poincaré's operator can be associated withfield problem [8]. This operator reduces the solution ofthe coupled subdomains to the solution of an equationinvolving only the interface values. One efficient andpractical solution of elliptical partial differentialequations is the dual Schur complement method [3].

6 Software productsA finite element (FE) program may be developed in amodular form (see the block diagram from the Fig. 1).FEM involves three stages:

1. Pre-processing2. Solution (or processing)3. Post-processing

Each stage involves more steps that are not shownin the block-diagram. The details of the finite elementprograms are presented in a large professionalliterature so that it is not the purpose to present them inthis work.

The influence of the temperature on the materialproperties can be used in development of efficientprograms in terms of the computing resources:memory and the execution time. Some relevant aspectsin the design of the CAD software for coupledmagneto-thermal problems are:

• The thermal source in the heat equation can bedefined by the time-mean of the ohmic powerloss. The motivation is simple: the time constant

of the magnetic phenomenon is small comparedto the diffusion time of the heat transfer.

• A cascade solution may be more efficient than afully coupled model. In some applications thereis a strict coupling between magnetic andthermal equation at each time instant, but inmany situations we can do separate analyses ofthe magnetic field and the thermal field.

• It can be used a predefined temperature profileof a material for updating the magnetic field atspecified temperatures. For example, at Curietemperature the material properties changedramatically. After this critical point themagnetic field equation must be updated. Thematerial characteristics are shown in the Fig. 2

• The analysis domain can be divided in moresubdomains with different solvers for eachsubdomain. In other words we can divide theanalysis domain in accordance with themathematical model of the problem.

The numerical model can be obtained by θ-rulecombined with the Galerkins method.

n


ISSN: 1991-8763 522

Fig. 1 Block diagram for software CAD

We must have a measure of confidence in theumerical solution. An approach for this requirement

Fig.2 - Characteristics vs. temperature

Issue 11, Volume 2, November 2007

is a control of the solution accuracy. In adaptive meshgeneration, the mesh is refined iteratively on the basisof error estimates. Advantages of this approach are:

• The solution is accompanied by an errorestimate that is a measure of the confidence

• The solution is cheap because the nodes areadded only where the accuracy is necessary

• The uninitiated can use complex programswithout any fore-knowledge of the refinementstrategies

The disadvantages are:• The matrix size is increased• The software complexity is increased• The CPU time increases with the estimating

errorsThere are basically the following methods of

refinement [3]:• h - refinement (subdividing elements)• p - refinement (increasing the polynomial

order p)• r refinement (nodal positions are moved)• hp refinement (a combination of h- and p-

refinements)

7 Some industrial applicationsIn any electromagnetic device there are power lossesthat are transformed in heating so that the modelling ofdevice involves coupled mathematical models. Inelectrical engineering the coupled electromagnetic andthermal fields represent both desirable phenomena andundesirable phenomena. Two examples illustrate thisassertion: induction heating and the high-voltage (HV)electrical cables.

Induction heating describes the thermalconductivity problem in which the heat is generated byeddy currents induced in conducting materials, by avarying magnetic field. Induction heating is anefficient procedure for bulk-heating metals to a settemperature. The heating is generated by the eddy-currents induced from a separate source of alternatingcurrent.

Figure 3 shows a long cylindrical workpieceexcited by a close-coupled axial coil [13]. The devicehas a cylindrical symmetry so that the problem can bereduced to a 2D-problem in the plane Orz. An axialsection is presented in Fig. 4 with: 1- the workpiece, 2 the air and 3 the coil. The coil is assimilated with amassive conductor. In this case we cannot ignore theeddy currents in the coil. We consider a low-frequencycurrent in the coil so that the penetration depth is large.In this case we can decompose the whole domain ofthe field problem into overlapped subdomains for thetwo coupled-fields.

The domain for the magnetic field can be reducedto a quarter of the device bounded by a boundary at afinite distance from the device. For the thermal fieldwe consider the workpiece as the analysis domain. Thepenetration depth of the magnetic field in theworkpiece imposes the overlapping domains for thetwo fields [5]. The numerical model is considered in acylindrical co-ordinates with the vertical axis Or andthe horizontal axis Oz.

fp

f


ISSN: 1991-8763 523

Fig.3 - Device for induction heating

Thield ulane:

Foor ele

Fig. 4 Axial section of the device

e mathematical model for the electromagneticsing A-formulation is a 2D-scalar model in (r-z)

sJrArt

rAr

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∂

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)( υσ(18)

r the harmonic-time case, mathematical modelctromagnetic field is:

Issue 11, Volume 2, November 2007

sJrAr

j

zrA

rzrrA

rr

−=

−+

)(

)()(

υσω

∂∂υ

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Another example that we present is a high-voltagecable with three phase conductors and a neutralconductor [13]. The HV cables are importantcomponents of the energetic system for distribution ofthe electric energy. Fig. 5 shows the cross-section ofthe system.

secconAllconthrof (steareof con

8 NWesimpredifloathasysof

twooutins

The loads of the conductors are currents ofamplitude equal to 250 A at the frequency of 50 Hz.

In post-processing stage of the FEM program, a lotof physical quantities can be obtained [3]. They are ofgreat importance for the electrical engineers in theevaluation of the device performance. These derivedquantities are presented in users manual of anysoftware CAD [13]. The voltage amplitude is 7000 V.

Fig. 6 The current density map


Fig. 5 Cross section of the cable

This high-voltage tetra-core cable has three triangletors with phase conductors and round neutralductor in the lesser area of the cross-section above. the conductors are made of copper. Eachductor is insulated and the cable as a whole has a

ee-layered insulation. The cable insulation consistsinner and outer insulators and a protective braidingel tape). The sharp corners of the phase conductors chamfered to reduce the field crown. The cornersthe conductors are rounded. Empty space betweenductors is filled with some insulator (air, oil etc.)

umerical results shall present the results of the numericalulation for the electrical cable described in thevious section. This system can be analysed forferent operating regimes. When the cables are ind, the conductor currents can generate local heatingt destroys the insulation and finally, the wholetem. Consequently, the temperature distribution isgreat importance for the designer.Each cable-core has its own insulation but there are layers of insulation: inner cable insulation ander cable insulation more thick than the internalulation. Also, there is a protective steel braiding.

Fig. 7 The temperature map

The non-uniformity of the temperature is due to thenon-uniformity of the current density in system. InFig. 6 the map of the total current density is shown. Incomputation of the total current in the cable, the skineffect and proximity effect of the cable cores wereconsidered. Fig. 7 shows the temperature map of thesystem.

The forces that appear between differentcomponents and the displacements of the cable


components generate another important field in thecable. Stress analysis plays an important role in designof many different electrical components. Generally thequantities of interest in stress analysis are:

• displacements• strains• different components of stresses.In the electric cables the loading sources are:• Thermal strains• Electric forces from electromagnetic analysisMathematical models are Navier's equations of

elasticity [13].

Fig. 8 - The displacement vectors and map

In Fig. 8 the map of the displacement and thevectors are shown [13]. The electric currents are thesource of the electrodynamic forces. These forces inshort-circuit regime can be very strong and destroy thesystem configuration. Also, the heat can lead to adilatation of the conductors and importantdisplacements can appear. Consequently, the forcesfield is very important for the engineer, especially inthe dangerous operating regimes.

9 ConclusionsThe problem of coupled fields in electrical engineeringis a complex problem in terms of computing resources.In practice the coupled fields are treated independentlyin some simplified assumptions. The accuracy of thenumerical computation is poor. With the newarchitectures, a multidisciplinary research is possible.Some iterative procedures were presented withemphasis on the coupled problems.

Domain decomposition offers an efficient approachfor large-scale problems or complex geometricalconfigurations. This method in the context of the finiteelement programs leads to a substantial reduction ofthe computing resources as the time of the processor.

In coupled problems a hierarchy of decompositioncan be defined with a substantial reduction of thecomputation complexity. The finite element methodwas used for the numerical result. The programQuickfield was used in our target examples [13].

In our future research we shall extend our results tocoupled magnetic, thermal and stress analysis forimportant devices from the energy distribution systemsas the electrical transformers and reactors. More, wehave in our projects some important objectives:optimal design of the electromagnetic devices usingcoupled models and gradient techniques [10]. Also, wepublished some works in the area of CAD for optimalcontrol of the heat transfer in large-power cables [3].

We shall extend the results of this research toelectrical transformers using parallel algorithms. Insome previous works we presented the computation ofthe electrodynamic forces in the large powertransformers using uncoupled models. This approachwas a practical approach for many designers motivatedby the computation complexity and a limitedcomputing power of the conventional computers. Withthe new computers we can use coupled models fortransformers and other devices of high voltage andlarge power.

In this work we limited our presentation toconventional algorithms. But FEM has an inherentparallelism in any stage: pre-processing, processingand post-processing. These features will be exploitedin our next software. We developed our own softwarefor mesh generation using multiblock method withgood results for parallel computing [11].

References:[1]. Cârstea, I., Advanced Algorithms for

Coupled Problems in Electrical Engineering.In: Mathematical Methods andComputational Techniques in Research andEducation. Published by WSEAS Press,2007. ISSN: 1790-5117; ISBN: 978-960-6766-08-4. Pg. 31-38

[2]. Cârstea, I., "Domain decomposition forcoupled fields in electrical engineering". In:FINITE ELEMENTS. Theory and AdvancedApplications. Published by WSEAS Press,2007. ISBN: 975-960-8457-84-3. Pg. 54-61.

[3]. Cârstea, D., Cârstea, I., CAD of theelectromagnetic devices. The finite elementmethod. Editor: Sitech, 2000. Romania.



[4]. Marchand, Ch., Foggia, A., 2D finiteelement program for magnetic inductionheating. In: IEEE Transactions onMagnetics, vol. 19, no.6, November 1983.

[5]. Cârstea, D., Cârstea, I., "A domaindecomposition approach for coupled fields ininduction heating devices". In: Proceedings ofthe 7-th International Conference on AppliedElectromagnetics (PES 2005), pg. 95-96. May23-25, 2005. Nis, Serbia-Montenegro.

[6]. Segerlind.L.J., Applied Element Analysis,John Wiley and Sons, 1984, USA.

[7]. Zenisek, A. Nonlinear elliptic and evolutionproblems and their finite elementapproximations. Academic Press Limited,1990.

[8]. Quarteroni, A., Valli, A., DomainDecomposition Methods for PartialDifferential Equations. Oxford SciencePublication. 1999

[9]. Cârstea, I., "Advanced Algorithms forCoupled Problems in Electrical Engineering".In Volume: Mathematical Methods andComputational Techniques in Research andEducation. Published by WSEAS Press, 2007.

ISSN: 1790-5117; ISBN: 978-960-6766-08-4.Pg. 31-38.

[10]. Cârstea, I., "Advanced Algorithms for InverseProblems in Electrical Engineering". InVolume: Mathematical Methods andComputational Techniques in Research andEducation. Published by WSEAS Press, 2007.ISSN: 1790-5117; ISBN: 978-960-6766-08-4.Pg. 39-44.

[11]. Cârstea, D., "Multiblock Method for DatabaseGeneration in Finite Element Programs". InVolume: Mathematical Methods andComputational Techniques in Research andEducation. Published by WSEAS Press, 2007.ISSN: 1790-5117; ISBN: 978-960-6766-08-4.

[12]. Cârstea, I., Cârstea, D., Cârstea, A.A.,"Simulation of electromagnetic devices usingcoupled models". In Volume: SYSTEMSCIENCE and SIMULATION inENGINEERING. Published by WSEAS Press,2007. ISSN: 1790-5117; ISBN: 978-960-6766-14-5. Pg. 71-79.

[13]. *** QuickField program, version 5.4. Pageweb: www.tera-analysis.com. Company:Tera Analysis.



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