Simulation of electromagnetic devices using coupled models

Transcription

1 Manuscript received Aug. 2, 2007; revised Nov. 30, 2007 Simulation o electromagnetic devices using coupled models ION CÂRSEA Department o Computer Engineering and Communication University o Craiova Str. Doljului nr. 4, bl. C8c, sc., apt.7, Craiova ROMANIA DANIELA CÂRSEA High-School Group o Railways, Craiova ROMANIA ALEXANDRU ADRIAN CÂRSEA University o Craiova ROMANIA incrst@yahoo.com Abstract: his work presents numerical algorithms or simulation o distributed-parameter systems with direct applications in electrical engineering. he algorithms are developed in the context o the inite element method. Many works in the proessional literature present coupled models or the electromagnetic devices and this work is toward this direction with emphasis on the development o eicient algorithms in numerical computation o the coupled models. Our work describes the solution o coupled electromagnetic and heat dissipation problems in two dimensions and cylindrical-coordinates system or devices with cylindrical symmetry. he purpose o the work is to deine both conventional algorithms and parallel algorithms or coupled problems in context o the inite element method. he mathematical models or electromagnetic ield are based on potential ormulations. Some numerical results are presented. Key Words - Coupled ields; Finite element method; Domain decomposition. Introduction he reality orces us to deal with complex coupled systems where two or more physical systems interact. wo or more ields coexist in the same geometry, in the same electromagnetic device. hese ields interact. For example, induction heating is used or surace treatment o materials. In this practical application, the eddy currents generated by an electromagnetic inductor are used as the thermal heat sources through the Joule eect. More, any change in the physical or geometric parameters o an electromagnetic device will aect both magnetic and thermal ields. In our target examples the physical phenomena are electromagnetic and thermal. he physical properties o the materials are strongly dependent on the temperature, especially the ollowing characteristics: electric conductivity, magnetic permeability, and speciic heat and thermal conductivity. In this work we limit our discussion to coupled electromagnetic and thermal ields. Mathematical models or the problems in which the electromagnetic ield equations are coupled to other partial dierential equations, such as those describing thermal ield, luid low or stress behaviour, are described by equations that are coupled []. he coupling between the ields is a natural phenomenon and only in a simpliied approach the ield analysis can be treated as independent problem. In several cases, it is possible a decoupling and a cascade solution o the coupled equations. Another attractive and eicient approach o solving coupled dierential equations is to consider the set as a single system. In this way a single linear algebraic system or the whole set o dierential equations is obtained ater discretization, and is solved to a single step. I one or more equations are non-linear, non-linear iterations o the whole system are required. he equations o the electromagnetic ields and heat dissipation in electrical engineering are coupled because the most o the material properties are temperature dependent and the heat sources represent the eects o the electromagnetic ield []. ISSN: Issue, Volume 2, November 2007

2 he thermal eects o the electromagnetic ield are both desirable and undesirable phenomenon. hus, in conducting parts o some electromagnetic devices (coils o the large-power transormers, current bars, cables conductors, conductors o the electric machines etc) the heating is an undesirable phenomenon. he heat is generated by ohmic losses o the driving currents and eddy currents induced in conducting materials. But in induction heating devices or welding the heating is a desirable phenomenon. he thermal eect o the electromagnetic ield is the treatment base or many electric materials in industry [2]. With the terminology o the system theory, we identiy two kinds o the heat sources (and commands in an inverse problem): Distributed sources (electrical currents) Boundary sources (Dirichlet's condition, Neumann's condition, convection and radiation) In the heating o the electromagnetic devices, the internal heat sources (position, amplitude) are represented by: Ohmic losses rom driving (source) currents Ohmic losses rom eddy currents induced in conducting materials o the time variable magnetic ield Dielectric losses due to riction in the molecular polarisation process in solid dielectrics that orm the insulation o the high-voltage apparatus Hysteresis loss in magnetic problems. It is due to magnetic domain riction in erromagnetic materials. he boundary sources (commands) can be [3]: Dirichlet command, that is, an imposed temperature on the boundary o the spatial domain Neumann command that involves an imposed lux temperature on the boundary o the spatial domain Convective command (the temperature o the ambient medium or a cooling luid, a parameter o the cooling luid as the speed etc) Radiation commands (the temperature o the ambient medium or other parameters that are outside the spatial domain o the ield problem and inluences the temperature o a device by radiation phenomenon). 2 Mathematical modelling o the electromagnetic ield For numerical simulation o the coupled systems we must have in mind some practical aspects: Mathematical models o electromagnetic ield and thermal ield Mathematical tools or ield problems Mathematical methods or coupled problems A complete mathematical model or coupled electromagnetic-thermal ields involves Maxwell s equations and the heat conduction equation. Combining these equations yields a coupled system o non-linear equations. A complete physical description o electromagnetic ield is given by Maxwell s equations in terms o ive ield vectors: the magnetic ield H, the magnetic lux density B, the electric ield E, the electric ield density D, and the current density J. In low-requency ormulations, the quantities satisy Maxwell s equations [3]: H = J () B E = (2) t div B = 0 (3) div D = ρc (4) with ρ c the charge density, σ the electric conductivity, and µ the magnetic permeability. For simplicity we give up to the bold notations or vectors. he second set o relationships, called the constitutive relations, is or linear materials: B = µ H; D = εe; J = σe he B-H relationship is oten required to represent non-linear materials. he current density J in Eq. () must represent both currents impressed rom external sources and the internally generated eddy currents. he ormulation with vector and scalar potentials has the mathematical advantage that boundary conditions are more oten easily ormed in potentials than in the ields themselves. he magnetic vector potential is a vector A such that the lux density B is derivable rom it by the operator curl or ( ). he mathematical models or the electromagnetic ield problems may be included in two ormulations: Integral equation ormulations (Fredholm integral equations) Dierential equation ormulations (partial dierential equations o elliptic or parabolic type) Hybrid ormulations he complexity o the mathematical model or electromagnetic ield was one o the main reasons to ind and develop new computational methods. All methods can be included in one o the ollowing classes [3]: Manipulation o the equations so that some unknowns are eliminated Deinition o some potential unctions rom where the ield unknowns can be obtained by simple processing ISSN: Issue, Volume 2, November 2007

3 Finding o some assumptions that simpliies the computation or practical problems he potential ormulations seem attractive because o their computational advantages. One o these consists in the act the boundary conditions are easily ramed in the potentials than in the ield themselves. 2. he eddy-currents problem he time-varying magnetic ield within a conducting material causes circulating currents to low within the material. hese currents called eddy-currents can be unwanted or desirable phenomena. hus, the eddycurrents in electrical machines give rise to unwanted power dissipation. On the other hand the induction heating is a wanted phenomenon in industry o the metal treatment. Industrial equipment in which the eddy currents are essentially can be included in one o the ollowing classes: long structures, in which the electric ield and the current density posses only one component complex structures in which we use models 3D In the long structures, the currents are generated by an electric ield applied at the terminals o the conductor, or by a time-varying magnetic ield linking the loop ormed by the conductors. hese structures belong to electric transmission network or the distribution networks (bus bars, large-power cables etc). In these problems the applied voltage o the bar or cable is known and we seek to compute the current density distribution within the conductor in order to determine some electromagnetic quantities o interest (the electrodynamic orces, mutual inductances, local heating etc). he complex structures create diiculties in simulation and computation o their characteristics although these structures possess construction simplicity. One o these structures is the device or electric heating by electromagnetic induction. In this type the applications it is necessary to compute accurately the eddy currents. I the eddy-currents distribution is non-uniorm, the resulting hightemperature gradients may crack the workpiece. he problems are dierent in the two dierent types o applications but or any given application the presence o the saturable iron sheets introduces saturation phenomena and the problem becomes nonlinear. For each class we can apply general mathematical methods but it is more eicient to develop a particular algorithm or each kind o classes. he eects o the eddy currents are: he time-varying magnetic lux density is nonuniorm within the conductor. he alternating magnetic lux is concentrated toward the outside surace o the material (phenomenon known as the skin eect). Power losses are increased in the material Eddy current computation appears in two types o problems: Stationary problems where the structures are ixed and source currents are time varying Motion problems where the ield source is a coil in moving Many practical engineering problems involve geometric shape and size invariant in one direction. Let z denote the Cartesian co-ordinate direction in which the structure is invariant in size and shape. his is the case o a plane-parallel ield or translational ield problem, where A has one component, namely A z. his component is independent o the z coordinate and the Coulomb gauge is automatically imposed and V is independent o x and y. In such a case both the magnetic vector potential and the source current J S reduce to a single component oriented entirely in the axial direction and vary only with the co-ordinates x and y. Consequently, the component A z (or simplicity we give up the subscript z) satisies the diusion equation in ixed bodies []: A ( ν A) σ = J s (5) t or, in Cartesian co-ordinates: A A A (υ )+ (υ ) σ = -J s (6) x x y y t he boundary conditions are set-up or the single component A and can be Dirichlet and/or Neumann s condition. he interace conditions between two materials with dierent properties are: A A A = A = 2 2 ; υ υ N 2 N where n is the normal at the common surace o the two regions with dierent material properties. 2.2 Modelling o time-dependent ields he time dependent electromagnetic ield problems are usually solved using dierential models o diusion type. Many practical problems o great interest in electromagnetics involve time-harmonic ields and this case will be considered in this work. In general, computer sotware or time-varying problem can be classiied into two classes [3]:. time-domain programs 2. requency-domain programs ISSN: Issue, Volume 2, November 2007

4 ime-domain programs generate a solution or a speciied time interval at dierent time moments. Frequency-domain programs solve a problem at one or more ixed requencies. he irst class has some disadvantages. One o these consists in the large amount o data that must be stored to recover the ield behaviour. Although the second class has an essential advantage (a compact and a cheap program in terms o the computer resources), the area o problems that can be solved is limited. It is applicable only to linear problems (all phenomena are sinusoidal). he usual mathematical model or time dependent electromagnetic ield problems is with Maxwell s equations in their normal dierential orm. For low requency the displacement current term in Maxwell s equations can be neglected. At a surace o a conducting material the normal component o current density J n can be assumed to be zero. 3 Mathematical modelling o the thermal ield he thermal ield is described by the heat conduction equation [2]: [(cγ )( ) ] + [ k( ) ] = q (7) t where: (x, t) is the temperature in the spatial point x at the time t; point k is the tensor o thermal conductivity; γ is mass density; c is the speciic heat that depends on ; q is the density o the heat sources that depends on. In the coupled problems we use the ormula: 2 q = ρ ( ) J (8) with ρ the electrical resistivity o the material. Equation (7) is solved with boundary and initial conditions. he boundary conditions can be o dierent types: Dirichlet's condition or a prescribed temperature on the boundary; Neumann's condition; convection condition; radiation condition, and mixed condition [2]. hese boundary conditions have the ollowing orm on dierent parts o the boundary surace S: Dirichlet's condition: ( x, y, z, t) = ( x, y, z, t) S D Neumann's condition: [k + q ] S = 0 n n 2 Convection: [k + h( - )] S = 0 n 3 Radiation: [k + εσ ( 4-4 ) ] S = 0 n B 4 where the boundary surace S is: S = S S 2 S 3 S 4 he signiicances o the quantities that appear in the boundary conditions are: D is a known unction deined on the boundary S ; h is the convection coeicient; is the ambient temperature; q n is the normal heat lux; σ B is Boltzman's constant in radiation and ε is the emissivity coeicient. he coeicients o the heat transer as h and ε depend on the temperature and the surace quality. For these we use empirical ormulas based on the experiments. For many eddy-current problems the magnetic lux penetration into a conductor without internal sources o the magnetic ield is conined mainly to surace layer. his is the skin eect. he skin depth δ depends on the material properties µ, ω and σ so that or the small depths all eects o the magnetic ield are conined to a surace layer. In steady-state low-requency eddy current problems in magnetic materials, the mathematical model is the diusion equation deined by Eq. (6). he skin eect can be exploited in two directions: o reduce the space domain in analysis with a ine mesh close to conductor suraces o reduce the material volume since a signiicant proportion o the conductor is virtually unused he penetration depth is given by the ormula: 2 δ = (9) ωσµ For example, in a semi-ininite slab o conductor with an externally applied uniorm alternating ield, parallel to the slab, the amplitude o lux decays exponentially. In other words or problems with the skin depth very small all the eect o the ield is conined to a surace layer. In a numerical model based on inite element method (FEM) this eect can be exploited by the use o domain decomposition at the level o the problem. In this way we reduce the run-time o a program based on FEM. Designer engineers use the ormula (9) considering the permeability and the conductivity as numbers. In reality the two physical parameters change during heating. he changes in the value o δ aect the loss in the material and depend on the process (conduction or induction). For example, i the conductivity decreases by x, the depth increases by x, that is the current penetrates deeper into the metal. I the magnetic material heats, its resistivity (the inverse o the ISSN: Issue, Volume 2, November 2007

5 conductivity) rises but its relative permeability remains substantially constant up to the Curie point. In this point it drops suddenly to unit. Another simpliying assumption or the designer engineers is based on that all heat enters at the surace o the conductor. In reality, this is only true i the requency o the magnetic ield source is very high and the depth o heating is small compared with the geometrical dimensions o the conductor. his act can be exploited in numerical simulation o these devices by reduction o the analysis domain. For an accurate computation o the penetration depth o the magnetic ield we must consider two practical conditions: he heat is distributed in the conducting part here is an important heat lost by radiation at the conductor surace Radiation can be regarded as a simple surace loss subtracting rom the surace power input. he Stean- Boltzmann's law gives the radiation loss. I the body is radiating to a surace at absolute temperature Kelvin, the radiation loss is deined by: P ( 4 4 r = εr c ) 0 where ε r is the emissivity coeicient o the surace (dimensionless), and is the absolute surace temperature in grades Kelvin (K). he constant c 0 is W/m 2 K 4. For low temperatures, the radiation loss is negligible but in the induction-heating device it must be considered. Consequently, it is convenient to use coupled models and accurate methods or computation o the heat penetration in the conductors, especially in the induction heating devices. 3. ransient problems Many engineering applications are described by parabolic partial derivatives equations. When applying the FEM to time dependent problems, the time variable is usually treated in one o two ways: ime is considered as an extra dimension and shape unctions in space and time are used he nodal variables are considered as unctions o time and the shape unctions in space are used. A common approach or transient problems is to solve time dependent dierential equations by inite dierences approximation o time derivative terms, combined with some weighted residual method in space. A widely used inite dierence scheme or the irstorder equations is the so-called θ. Certain values o θ correspond to known methods or time stepping: θ =0 the orward dierence method; θ =/2 Crank-Nicholson s method; θ =2/3 central dierence method; θ = the backward dierence method. 3.2 θ-rule combined with Galerkin s method We illustrate the method by applying the θ-rule in time and Galerkin s method in space to the ollowing heat conduction equation: = ( k ) + q x Ω, t 0 (0) t ( x,0) = ( x) x Ω () k = g( x, t) x Γ, t 0 (2) n We shall present the numerical models obtained by two strategies. Applying the θ-rule to the heat equation (0) results in the ollowing spatial problem: (0) = ( x) ( m ) = θ[ ( k ) + q ] δ t (3) ( m ) ( m ) + ( θ )[ ( k ) + q ] x Ω k = g( x, t m ) x Γ, t 0 n where the superscript m denotes the iteration number, that is (m) =(x,t m ). Discretizing the Eq. (3) by the method o weighted residuals, with (n) approximated by = r i N i ( x) i = gives an algebraic equations system: ( m ) ([ M ] + [ K]) {} = {} b where the matrices [M] and [K] have the entries: M ij = N i N j dω Ω K ij = θ δt k N i N j dω Ω Instead o irst discretizing in time by a inite dierence method, irst we can apply the discretizing in space by the weighted residual method, with approximated by: ( x, t) = r t N x i = i ( ). i ( ) By this procedure a irst order dierential equations system is obtained in the orm: d{} + [ A]. {} = {} dt ISSN: Issue, Volume 2, November 2007

6 he θ-method or the time integration leads to [3]: ([ I ] + θ δt [ A]){ } = ([ I ] δt( θ )[ A]){ } ( m ) + { g} ( m ) { g} = δt ( θ{ } + ( θ ){ } ) For θ=0 the orward Euler's scheme is obtained and we can get (m) explicitly; otherwise, a linear system must be solved at each time step. When θ>/2 there is no stability restriction on time step δt, which can be convenient or the simulation algorithm. he choice o θ=/2 leads to an optimal combination o stability and accuracy. 4 Coupled models With a correct ormulation o the mathematical models and a good selection o the mathematical tools or a speciied ield problem, we must select the method or the numerical solution o the ield problem. Ones o these methods or ield problems are moment s method, inite element method (FEM), boundary element method (BEM), hybrid method BEM-FEM, inite volume method (FVM), and edges element method (EEM). In our works we considered the FEM [6]. his method can be viewed as a particular case o the general method o moments, or a case o the Rayleigh- Ritz method. When applying the FEM to time dependent problems, the time variable is usually treated in one o two ways: ime is considered as an extra dimension and shape unctions in space and time are used he nodal variables are considered as unctions o time and the shape unctions in space are used. 4. Coupled magnetic and thermal ields For magnetic ield we consider the A-ormulation, that is we deine the magnetic vector potential A by B = curl A. More, the domain is the same or temperature and the electromagnetic ield although in practice the interest is or dierent ield domains. In order to solve the transient coupled set o equations a numerical model can be developed using the inite element method [4]. he inite element discretization in space is used, leading to a system o irst-order dierential equations: A [ S A ] + [ K A ]{ A} + { J } = 0 (4) t [ S ] + [ K ]{} + [ K ]{} A = 0 (5) t A where the matrices have the entries deined in accordance the FEM. he subscripts A and reer to the magnetic and thermal ield respectively. he vector { J } is generated by the heat source. 2 q = ρ J = ρ H H he two equations are coupled and non-linear. Finally, the two models can be considered as a coupled system deined in matrix orm [2]: [] 0 A [] 0 { } S K A t + A A + J = 0 [] 0 { } {} 0 S K K A t In a discrete orm the unknowns are the nodal values o the temperature and the magnetic vector potential A. he non-linear equations or and A are straightorwardly obtained by a Galerkin's inite element method. For the case o 2D steady-state problems we do the ollowing approximations at the element level []: ( x, y) = r N j ( x, y) j j = A( x, y) = r N j ( x, y) A j j = where the interpolation unctions N j are basis unctions in the mesh over Ω, and r is the number o nodes o an element. he usual procedure or the FEM applications leads to a system o 2p equations where p is the total number o the unknowns in each ield problem. Finally, the coupled problem is described by a system o algebraic systems in the orm: A ( A,..., A p,,..., p ) = 0 (6) ( A,..., A p,,..., p ) = 0 (7) where the subscript denotes the original problem (A or the magnetic ield in the magnetic vector potential ormulation; or the thermal ield). 5 Iterative algorithms or coupled problem he inite element method has three distinct logical stages: pre-processing, processing (solution) and postprocessing. Each stage has an inherent parallelism that can be exploited or parallel computing. New algorithms or the parallel computers were developed and presented in the proessional literature. We shall limit discussion to one o them: domain decomposition [8]. his algorithm uses the subdomain-to-subdomain ISSN: Issue, Volume 2, November 2007

7 iteration. Although the procedure is well known, we must modiy it or coupled problems. 5. Conventional algorithms he numerical model or coupled problem deined by Eq. (6) and Eq. (7), can be solved by two dierent basic strategies [7]: Solving the equations or i and A i simultaneously Solving the equations or the two ields in sequence with an outer iteration, technique known as operator-splitting technique (or example Newton-Raphson procedure) In the area o the irst strategy, Gauss-Seidel and Jacobi methods are well known. We present these methods in brie. he Gauss-Seidel algorithm or coupled ields has the ollowing pseudo-code [7]: For m:=, 2, until convergence DO Solve ( m ) ( m ) A ( A,..., A p ;,..., p ) = 0 with respect to A (m) (m), A p Solve ( A,..., A p ;,..., p ) = 0 with respect to (m) (m), p In other words, the system is solved irstly with respect to A, using the values o rom the previous iteration. Aterwards, the equation derived rom the thermal ield model is solved using the computed values o A rom the current iteration. he equations A =0 or/and =0 are non-linear and must be solved by an iterative procedure (or example Newton-Raphson's method). he algorithm Jacobi-type is similar to Gauss- Seidel method, except that at the iteration m when we must solve the model or, the values or A are rom the previous iteration, that is A (m-). he algorithm has the ollowing pseudo-code: For m:=, 2, until convergence DO Solve (m) (m) (m ) (m ) A (A,...,A p ;,..., p ) = 0 with respect to A (m) (m), A p Solve ( m ) ( m ) ( A,..., A p ;,..., p ) = 0 with respect to (m) (m), p his algorithm has an inherent parallelism so that can be implemented in a parallel program. Practically, we decomposed the coupled problem in two subproblems: one or the magnetic ield, another or the thermal ield. At a time step o the algorithm, the numerical models or the two ields can be solved independently. 5.2 Advanced algorithms he domain decomposition method [8] is the best among three possible decomposition strategies or the parallel solution o PDEs, namely, operator decomposition, unction-space decomposition and domain decomposition. his is one o the motivations to present the principles o the domain decomposition methods in this section. he domain decomposition could be determined rom mathematical properties o the problem (real boundaries or interaces between subdomains), or rom the geometry o the problem (pseudoboundaries). For elliptic partial dierential equations, there exists a mathematical approach based on the ideas given earlier in 890 by Schwarz [8]. In Schwarz procedure there is an inherent parallelism with a data communication time or the passage o pseudoboundary data between the subproblems. here is no general rule or the domain or/and operator decomposition. It is deined in a somewhat random ashion. he problems and solutions that appear in the decomposition techniques depend on the ollowing aspects []: I it is used domain decomposition or the operator decomposition I the partition has disjoint or overlapping subdomains he type o boundary conditions that are set up on the pseudo-boundaries o the sub-domains I the decomposition is static or dynamic A general criterion or the decomposition does not exist so that the experience o the engineer can be a useul reerence or many algorithms and sotware products. 5.3 Decomposition techniques he desire o the scientiic community or aster processing on lager amounts o data has driven the computing ield to a number o new approaches in this area. he main trend in the last decades has been toward advanced computers that can execute operations simultaneously, called parallel computers. For these new architectures, new algorithms must be developed and the domain decomposition techniques are powerul iterative methods that are promising or parallel computation. Ideal numerical models are those that can be divided into independent tasks, each o ISSN: Issue, Volume 2, November 2007

8 which can be executed independently on a processor. Obviously, it is impossible to deine totally independent tasks because the tasks are so intercoupled that it is not known how to break them apart. However, algorithmic skeletons were developed in this direction that enables the problem to be decomposed among dierent processors. he mathematical relationship between the computed sub-domain solutions and the global solution is diicult to be deined in a general approach. In the area o the coupled ields we deine two levels o decomposition that is we deine a hierarchy o the decompositions: One at the level o the problem he other at the level o the ield In other words, we decompose the coupled problem in two sub-problems: a magnetic problem and a thermal problem, each o them with disjoint or overlapping spatial domains. his is the irst level o decomposition. At the next level, we decompose each ield domain in two or more subdomains. he decomposition is guided both by the dierent physical properties o the materials, and the dierence o the mathematical models. At this level o decomposition the Steklov-Poincaré's operator can be associated with ield problem [8]. his operator reduces the solution o the coupled subdomains to the solution o an equation involving only the interace values. One eicient and practical solution o elliptical partial dierential equations is the dual Schur complement method [3]. o the magnetic phenomenon is small compared to the diusion time o the heat transer. A cascade solution may be more eicient than a ully coupled model. In some applications there is a strict coupling between magnetic and thermal equation at each time instant, but in many situations we can do separate analyses o the magnetic ield and the thermal ield. It can be used a predeined temperature proile o a material or updating the magnetic ield at speciied temperatures. For example, at Curie temperature the material properties change dramatically. Ater this critical point the magnetic ield equation must be updated. he material characteristics are shown in the Fig. 2 he analysis domain can be divided in more subdomains with dierent solvers or each subdomain. In other words we can divide the analysis domain in accordance with the mathematical model o the problem. he numerical model can be obtained by θ-rule combined with the Galerkin s method. 6 Sotware products A inite element (FE) program may be developed in a modular orm (see the block diagram rom the Fig. ). FEM involves three stages:. Pre-processing 2. Solution (or processing) 3. Post-processing Each stage involves more steps that are not shown in the block-diagram. he details o the inite element programs are presented in a large proessional literature so that it is not the purpose to present them in this work. he inluence o the temperature on the material properties can be used in development o eicient programs in terms o the computing resources: memory and the execution time. Some relevant aspects in the design o the CAD sotware or coupled magneto-thermal problems are: he thermal source in the heat equation can be deined by the time-mean o the ohmic power loss. he motivation is simple: the time constant Fig. Block diagram or sotware CAD Fig.2 - Characteristics vs. temperature We must have a measure o conidence in the numerical solution. An approach or this requirement ISSN: Issue, Volume 2, November 2007

9 is a control o the solution accuracy. In adaptive mesh generation, the mesh is reined iteratively on the basis o error estimates. Advantages o this approach are: he solution is accompanied by an error estimate that is a measure o the conidence he solution is cheap because the nodes are added only where the accuracy is necessary he uninitiated can use complex programs without any ore-knowledge o the reinement strategies he disadvantages are: he matrix size is increased he sotware complexity is increased he CPU time increases with the estimating errors here are basically the ollowing methods o reinement [3]: h - reinement (subdividing elements) p - reinement (increasing the polynomial order p) r reinement (nodal positions are moved) hp reinement (a combination o h- and p- reinements) he domain or the magnetic ield can be reduced to a quarter o the device bounded by a boundary at a inite distance rom the device. For the thermal ield we consider the workpiece as the analysis domain. he penetration depth o the magnetic ield in the workpiece imposes the overlapping domains or the two ields [5]. he numerical model is considered in a cylindrical co-ordinates with the vertical axis Or and the horizontal axis Oz. 7 Some industrial applications In any electromagnetic device there are power losses that are transormed in heating so that the modelling o device involves coupled mathematical models. In electrical engineering the coupled electromagnetic and thermal ields represent both desirable phenomena and undesirable phenomena. wo examples illustrate this assertion: induction heating and the high-voltage (HV) electrical cables. Induction heating describes the thermal conductivity problem in which the heat is generated by eddy currents induced in conducting materials, by a varying magnetic ield. Induction heating is an eicient procedure or bulk-heating metals to a set temperature. he heating is generated by the eddycurrents induced rom a separate source o alternating current. Figure 3 shows a long cylindrical workpiece excited by a close-coupled axial coil [3]. he device has a cylindrical symmetry so that the problem can be reduced to a 2D-problem in the plane Orz. An axial section is presented in Fig. 4 with: - the workpiece, 2 the air and 3 the coil. he coil is assimilated with a massive conductor. In this case we cannot ignore the eddy currents in the coil. We consider a low-requency current in the coil so that the penetration depth is large. In this case we can decompose the whole domain o the ield problem into overlapped subdomains or the two coupled-ields. Fig.3 - Device or induction heating Fig. 4 Axial section o the device he mathematical model or the electromagnetic ield using A-ormulation is a 2D-scalar model in (r-z) plane: σ ( ra) υ ra = J s r t ( ) r (8) For the harmonic-time case, mathematical model or electromagnetic ield is: ISSN: Issue, Volume 2, November 2007

10 υ ( ra) υ ( ra) + r r r z r z (9) υ jσω ( ra) = J s r Another example that we present is a high-voltage cable with three phase conductors and a neutral conductor [3]. he HV cables are important components o the energetic system or distribution o the electric energy. Fig. 5 shows the cross-section o the system. he loads o the conductors are currents o amplitude equal to 250 A at the requency o 50 Hz. In post-processing stage o the FEM program, a lot o physical quantities can be obtained [3]. hey are o great importance or the electrical engineers in the evaluation o the device perormance. hese derived quantities are presented in user s manual o any sotware CAD [3]. he voltage amplitude is 7000 V. Fig. 6 he current density map Fig. 5 Cross section o the cable his high-voltage tetra-core cable has three triangle sectors with phase conductors and round neutral conductor in the lesser area o the cross-section above. All the conductors are made o copper. Each conductor is insulated and the cable as a whole has a three-layered insulation. he cable insulation consists o inner and outer insulators and a protective braiding (steel tape). he sharp corners o the phase conductors are chamered to reduce the ield crown. he corners o the conductors are rounded. Empty space between conductors is illed with some insulator (air, oil etc.) 8 Numerical results We shall present the results o the numerical simulation or the electrical cable described in the previous section. his system can be analysed or dierent operating regimes. When the cables are in load, the conductor currents can generate local heating that destroys the insulation and inally, the whole system. Consequently, the temperature distribution is o great importance or the designer. Each cable-core has its own insulation but there are two layers o insulation: inner cable insulation and outer cable insulation more thick than the internal insulation. Also, there is a protective steel braiding. Fig. 7 he temperature map he non-uniormity o the temperature is due to the non-uniormity o the current density in system. In Fig. 6 the map o the total current density is shown. In computation o the total current in the cable, the skin eect and proximity eect o the cable cores were considered. Fig. 7 shows the temperature map o the system. he orces that appear between dierent components and the displacements o the cable ISSN: Issue, Volume 2, November 2007

11 components generate another important ield in the cable. Stress analysis plays an important role in design o many dierent electrical components. Generally the quantities o interest in stress analysis are: displacements strains dierent components o stresses. In the electric cables the loading sources are: hermal strains Electric orces rom electromagnetic analysis Mathematical models are Navier's equations o elasticity [3]. Fig. 8 - he displacement vectors and map In Fig. 8 the map o the displacement and the vectors are shown [3]. he electric currents are the source o the electrodynamic orces. hese orces in short-circuit regime can be very strong and destroy the system coniguration. Also, the heat can lead to a dilatation o the conductors and important displacements can appear. Consequently, the orces ield is very important or the engineer, especially in the dangerous operating regimes. 9 Conclusions he problem o coupled ields in electrical engineering is a complex problem in terms o computing resources. In practice the coupled ields are treated independently in some simpliied assumptions. he accuracy o the numerical computation is poor. With the new architectures, a multidisciplinary research is possible. Some iterative procedures were presented with emphasis on the coupled problems. Domain decomposition oers an eicient approach or large-scale problems or complex geometrical conigurations. his method in the context o the inite element programs leads to a substantial reduction o the computing resources as the time o the processor. In coupled problems a hierarchy o decomposition can be deined with a substantial reduction o the computation complexity. he inite element method was used or the numerical result. he program Quickield was used in our target examples [3]. In our uture research we shall extend our results to coupled magnetic, thermal and stress analysis or important devices rom the energy distribution systems as the electrical transormers and reactors. More, we have in our projects some important objectives: optimal design o the electromagnetic devices using coupled models and gradient techniques [0]. Also, we published some works in the area o CAD or optimal control o the heat transer in large-power cables [3]. We shall extend the results o this research to electrical transormers using parallel algorithms. In some previous works we presented the computation o the electrodynamic orces in the large power transormers using uncoupled models. his approach was a practical approach or many designers motivated by the computation complexity and a limited computing power o the conventional computers. With the new computers we can use coupled models or transormers and other devices o high voltage and large power. In this work we limited our presentation to conventional algorithms. But FEM has an inherent parallelism in any stage: pre-processing, processing and post-processing. hese eatures will be exploited in our next sotware. We developed our own sotware or mesh generation using multiblock method with good results or parallel computing []. Reerences: []. Cârstea, I., Advanced Algorithms or Coupled Problems in Electrical Engineering. In: Mathematical Methods and Computational echniques in Research and Education. Published by WSEAS Press, ISSN: ; ISBN: Pg [2]. Cârstea, I., "Domain decomposition or coupled ields in electrical engineering". In: FINIE ELEMENS. heory and Advanced Applications. Published by WSEAS Press, ISBN: Pg [3]. Cârstea, D., Cârstea, I., CAD o the electromagnetic devices. he inite element method. Editor: Sitech, Romania. ISSN: Issue, Volume 2, November 2007

12 [4]. Marchand, Ch., Foggia, A., 2D inite element program or magnetic induction heating. In: IEEE ransactions on Magnetics, vol. 9, no.6, November 983. [5]. Cârstea, D., Cârstea, I., "A domain decomposition approach or coupled ields in induction heating devices". In: Proceedings o the 7-th International Conerence on Applied Electromagnetics (PES 2005), pg May 23-25, Nis, Serbia-Montenegro. [6]. Segerlind.L.J., Applied Element Analysis, John Wiley and Sons, 984, USA. [7]. Zenisek, A. Nonlinear elliptic and evolution problems and their inite element approximations. Academic Press Limited, 990. [8]. Quarteroni, A., Valli, A., Domain Decomposition Methods or Partial Dierential Equations. Oxord Science Publication. 999 [9]. Cârstea, I., "Advanced Algorithms or Coupled Problems in Electrical Engineering". In Volume: Mathematical Methods and Computational echniques in Research and Education. Published by WSEAS Press, ISSN: ; ISBN: Pg [0]. Cârstea, I., "Advanced Algorithms or Inverse Problems in Electrical Engineering". In Volume: Mathematical Methods and Computational echniques in Research and Education. Published by WSEAS Press, ISSN: ; ISBN: Pg []. Cârstea, D., "Multiblock Method or Database Generation in Finite Element Programs". In Volume: Mathematical Methods and Computational echniques in Research and Education. Published by WSEAS Press, ISSN: ; ISBN: [2]. Cârstea, I., Cârstea, D., Cârstea, A.A., "Simulation o electromagnetic devices using coupled models". In Volume: SYSEM SCIENCE and SIMULAION in ENGINEERING. Published by WSEAS Press, ISSN: ; ISBN: Pg [3]. *** QuickField program, version 5.4. Page web: Company: era Analysis. ISSN: Issue, Volume 2, November 2007