iterative hybrid finite element boundary element method by Ryan Lian

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 4, JULY 1996

Element ng Syste K . S . Ismail, and R. A . Marzouk

Abstract-The Iterative Hybrid Finite Element-Boundary Element Method is applied for the analysis of induction heating system with rotational symmetry in unbounded free space. A mathematical boundary is defined arbitrarily in free space to enclose the system. Finite elements formulation is applied for the interior region and boundary integral formulation is applied on the boundary using the response function excited by a circular line current. A quasistatic magnetic field problem is solved in terms of magnetic vector potential. A ferromagnetic charge of nonlinear magnetic properties is considered. Iterative technique is used to correct the permeability values and the effect of nonlinearity is calculated. Accuracy of the method is verified with measurements of practical induction heating system .

I _ _

, , '

' 7

i I

Index Terms-Induction heating, iterative hybrid FEM-BEM. I

I. INTRODUCTION N THE analysis of induction heating systems, different numerical methods have been developed in 2-D and 3-D. Among these methods the finite element (FEM) and the boundary element (BEM) methods are the most popular [1]-[3]. A hybrid method based on the coupling of the FEM and BEM has been found advantageous for unbounded field problems [4]-[6]. The hybrid method provides the mean to deal with nonlinear regions by FEM formulation and infinite boundaries are approximated by boundary integral formulation on arbitrarily chosen mathematical boundary. Miyoshi and others have applied the general principle of the FEM-BEM to analysis of induction heating system with rotational symmetry taking into consideration linear materials [ 5 ] .This paper provides an extension of the hybrid method to deal with nonlinear magnetic properties of ferromagnetic charges. A magnetization curve giving the permeability as a function of the flux density is used to evaluate the permeability for each element of the charge. Starting from initial values corresponding to the rms value of the field intensity at the charge surface, an iterative technique is used to calculate the electromagnetic field problem. Relative changes of permeability values between successive iterations are used as a convergence criterion for the iteration process. The

r Fig. I . Basic induction heating system: I . the conducting charge; 2. the exciting coil.

accuracy of the method is verified by comparison with published literature. Results show very good agreement with measurements of a practical induction heating system. 11. GOVERNING EQUATION FOR

THE

VECTORPOTENTIAL

The basic induction heating system to be analyzed consists of a cylindrical charge surrounded by an exciting coil. The surface S of a cylinder is taken as a mathematical boundary to enclose the system as shown in Fig. 1. The FEM is applied for the interior region I'and the BEM is applied on the boundary S . Due to rotational symmetry, cylindrical coordinates are used and the only nonzero components of the current density and the magnetic vector potential are the 0-directed components which are denoted by J and A , respectively. The general differential equation for the vector potential in a linear isotropic medium assuming sinusoidal exciting current and negligible displacement current is given by [ 11:

Manuscript received May 4, 1994; revised December 1, 1995. The authors are with the Department of Electrical Engineering, College of Engineering, Baghdad University, Jadeira, Baghdad, Iraq. Publisher Item Identifier S 0018-9464(96)03488-7. 001 8-9464/96$05.00 0 1996 IEEE

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ISMAIL AND MARZOUK: ITERATIVE H Y B K I D FINITE ELEMENT-BOUNDARY ELEMENT METHOD FOK ANALYSIS

where p and o are the permeability and conductivity, respectively. From energy balance criterion of the system it is found that the instantaneous energy supplied by the source is equal to the instantaneous energy stored in the electromagnetic field in the interior and exterior regions. For the interior region the instantaneous energy is obtained by performing integration over the volume V. For the unbounded exterior region the instantaneous energy is obtained by application of the Poynting theorem over the surface S. Using the formulas governing the energy and the power flow in the system 171; the energy functional I of the system which is the variational representation of equation (1) is, given by

aA -

:>2I3

and

27r

Jrti dr dz

(7)

The integrals in (6) and (7) should be carried over the area of the element, it is also shown from (7) that T, take values only for the elements inside the coil where J is not zero. D,,is evaluated only for boundary nodes so it does not contribute anything to the equations of internal nodes. The total functional I is approximated by the sum of IL for all elements. Approximated numerical solution can be obtained by differentiating the approximate functional with respect to A,* for all nodes and equating the derivatives to zero. By collection of corresponding terms a set of simultaneous equations in terms of nodal vector potential and source current density is obtained given in matrix form as are the normal derivatives of A on the boundary.

151:

and the asterisk is the complex conjugate notation. 111. FEM DISCRETIZATION For the discretization of the interior region V the plane r-z is considered due to cylindrical symmetry and subdivided into a number of axisymmetric first order triangular elements. The vector potential in each element is defined in temis of thc vector potential at the vertices of the triangle by the following relation:

where {A}!“’ and { A } F ’ denote the vector potential of internal nodes and boundary nodes, respectively. IV. BEM FORMULATION

In free space the system differential (1) tends to:

c (;Ai

A =

(3)

i= I

where t j is called Ithe shape function and given by:

+ bir + ciz

(4) a i , bi, ci are constants to be evaluated for each element =

1x1. For the elernent k the functional Ik is approximated by using (3) as [ 5 ] : 3

3 1

X ,

i ~ 1= j = I

AiA,,*S,

i= I

(ATT,

+ A;TT)

c (A+DikqiF’+ AiDikqLF’”}

-- i = I

where q

a2 1 a a2 7+ - - + 7 ar r ar dz To find approximate solution to ( I O ) an adequate trial function G has been chosen such that, where V2

V2G -

-6( r - r , , z

z,)

( 1 1)

where 6 is the dirac delta, and G is the vector potential at ( r , z) excited by a circular line current at (ro, zo). Multiplying (10) by G and (1 1) by A , subtracting the equations, integrating over the volume I/ and applying Green’s theorem the following integral equation has been obtained [ 5 ] :

(5)

: --

a,,

The discretization of (12) results in a matrix equatiion

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IEEE TRANSACTIONS ON MAGNETICS, VOL 32, NO. 4, JULY 1996

of the form:

VI. EXTENSION TO NONLINEAR MEDIA

[F]{A}j-B’- [H]{A}p = (0) (13) where the superscript ( B ) stands for the BEM formulation.

The derivatives - in (12) are either an

g, r

(g)

according to the observation point

The response function G is given by [9]:

where h?

4rr1 (r

+ r(,12 + ( z

(15)

z,12

K ( k ) and E ( k ) are the complete elliptic integrals of the first and second kinds, respectively. Derivatives of G can either be obtained by numerical differentiation of (14) or more easily obtained using the following analytical expressions [9]:

ac -

-(z

- -

2 ~ [ ( +r r,)2

2,)

+ (Z

~,)~1~/~

In the previous analysis the permeability of the charge has been assumed constant and independent of field strength. For ferromagnetic materials the permeability is a function of the magnetic flux density and hence a single value is not sufficient to represent the magnetic properties. The charge in this case should be characterized by a magnetization curve attached to the programme. In an induction heating system there are two types of field variation which are: a. Time variation due to the exciting frequency. This type of variation can be allowed for by using the value of permeability corresponding to the rms value of the flux density [ l o ] ; and b. Space variation due to attenuation of the electromagnetic fields inside the charge. To take account of such variation the magnetic flux density should be calculated for each element of the charge and the corresponding permeability is then obtained from the associated magnetization curve. To correct the values of permeability an iterative procedure is used with the following steps: 1) Set the permeability to a value corresponding to the rms value of the exciting field at the charge surface. 2) Compute all submatrices of the system equation (20) 3) Solve the system equation (20) to obtain nodal vector potentials. 4) Compute the flux density for each element of the charge using the values of vector potential obtained in step ( 3 ) . The flux density for each element is given by [ 1 I]: ’

B==[lZl-

(17)

V . COUPLING OF FEM A N D BEM The boundary is discretized into straight line segments connecting the boundary nodes used in the FEM discretization. Since first order elements are used in the FEM aA formulation, therefore, - are guaranteed to be constant an across the boundary, hence: (9)P

-{&?

(18)

and since constant boundary elements are used, then: {A)?’

[C] {A)‘F’ C

(19)

where [C] is a transformation matrix. The coupling of (13) with (9) with substitution of (18) and (19) result in the final form of the system equation given by [SI: [&/I

Lscl

[&,I

[SCCl

where A,, 7,. are the vector potential and radius at the element centroid, respectively. Equation (21) enables the flux density to be computed from the gradients of the vector potential. The corresponding permeability is then obtained using the associated magnetization curve. The magnetization curve is normally stored as discrete data points, therefore, Lagrange interpolation formula is used to give the permeability values. To improve the iteration process a relaxation factor has been found necessary to correct the calculated value of permeability using the following formula [12]: pnew = pold

for0

T(pca1.

- &>Id)

(22)

<y< 1

where poldis the permeability of the previous itera-

+ [Dl [FI

[Cl

is the calculated permeability using the associated magnetization curve

peal

(20)

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ISMAIL A N D M A R Z O U K : ITERATIVE HYBRID FINITE ELEMENT-BOUNDARY hLEMENT METHOD FOR ANALYSIS

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y is the relaxation factor. knewis the new permeability value to be used in the next iteration. The relaxation factory is usually set at a value of 0.1 [ I l l . 5) Calculafe relative change of permeability value between successive iterations as follows: C

The convergence criterion for the iteration process is based on the m s value of relative changes taken over the total number of nonlinear elements (NLE) defined as:

A value of E = 0.01 is found to be adequate for such calculations. 6) Steps 2 to 5 are repeated until the specified convergence criterion is attained. It is necessary to mention here that BEM formulation does not depend on the charge permeability, therefore, the term ( [ D ][ F ] - ' [ H [] C ] )in (20) is computed only during the first iteration and kept in memory for next iterations to avoid unnecessary computations. A program based on the mentioned steps is written to deal with nonlinear charges. It can also be used for linear charges and i n this case only one iteration is required.

VII. NUMERICAL RESULTS The solution procedure that has been described in the previous sections is used to analyze two examples to check validity and accuracy. One example has analytical solution and has been analyzed by [ 5 ] . The other example is a practical induction heating system with nonlinear charge which has been previously considered by Stansel [ 131. The importance of the last example is that measurements are available in addition to calculations based on empirical formulas. The integrals over the triangular elements of (6) and (7) are performed numerically using the seven integrating points defined by the centroid, vertices and mid-side points of the element [14]. The line integrals of (8) are given by: D,L

= - (2r,

3Po

+ Y,)

JPO

where i and j are the boundary nodes of element k . To solve equation (20) it is found that diagonal dominance is not assured due to the effect of BEM [ 151, hence it is not suitable for iterative solution. Therefore, a direct

A Fig. 2. Single coil with squarc cross section; a and c = 0.1 m.

0.02 m, b

0.07 m ,

solution method based on Gauss elimination technique is used. To represent the problem in a suitable form for the FEM formulation, the problem domain is subdivided into a number of subregions; each subregion is defined by its coordinates, permeability and conductivity. For the coil subregion it should be noted that skin effect will operate on the coil, just as it does on the charge, tending to force the current on the inner surface of the copper [IO]. The coil is therefore, defined as a cylinder with uniform current density of internal radius and length equal to the actual coil values and a radial thickness equals to the skin depth corresponding to the operating frequency.

A . Sirigle Coil with Square Cross Section The magnetostatic field of a single coil with square cross section as shown in Fig. 2 is computed. This example has analytical solution given by equation (14). Values of the magnetic vector potential on the mathematical boundary are computed for a current density of 2500 A/m2. The interior region is subdivided into 200 triangular elements and the boundary is subdivided into 30 elements. Fig. 3 shows the computed values in comparison with the analytical solution, and the numerical results obtained by [ 5 ] . A computational error of about 1.5 percent is obtained with respect to analytical solution, which is better than that given by [ 5 ] .

B. Practical Induction Heating System This example represents a coreless induction heating furnace operating in the low frequency range. Problem data are given in Table I . This problem has been considered by Stansel [ 131 where measurements of electrical ]pa-

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 4, J U L Y 1996

POSiT!3N

Fig. 3. Computed nodal vector potential values along the mathematical boundary compared with analytical solution for J = 2500 Aim', A , B, C , and D indicate the positions on the boundary as shown in Fig. 2.

TABLE I

H E A T I NSGY S T E M (Based on Reference 1131, Ex. 9)

PKOBLF,M D A T A I.OK PRACTICAL INDUCTION

Coil inner diameter, (m) Coil length, (m) Number of turns Exciting ficld intensity, (kAiin) Frequency, (Hz) Charge materia! Charge diameter, (m) Charge length, (m) Charge conductivity, (Sim) Relative permeability corresponding to r.m.s. value of surface field intensity

0.08256 0.254 29 95.493 (rim) 60 steel SAE1045 0.06032 0.254 5.0 x IO" 18

rameters are given. A nonlinear charge with magnetic properties as shown in Fig. 4 is considered. The interior region is subdivided into 1044 axisymmetric triangular elements and the boundary is divided into 65 elements. Numerical results are obtained for two cases:

1) Assuming linear magnetic properties with permeability corresponding to the rms value of field intensity at the charge surface. 2) Using the actual magnetic properties of the charge as shown in Fig. 4. The power dissipated in the charge is computed numerically by the following equation [ 161:

7i-ow2rlA(r,z)I2 dr dz

ihdrpe

where o is the angular frequency (radiadsec).

(26)

0.0

j . : ,

(

IC '

Magnetizing Force, Oersteds Fig. 4. Magnetization curve of steel SAE 1045 (After Stansel, No. 5 , Fig. 24).

And the input impedance of the system is given by [SI:

jOhi

2 ~ Ar( r , z ) dr dz

coli

where: N is the number of the coil turns. S is the cross sectional area of the coil. Results obtained for case (b) are in good agreement with measurements as shown in Table 11.

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ISMAIL AND MAFIZOUK: ITERATIVE HYBRID FINITE ELEMENT-BOUNDARY ELEMENT METHOD FOR ANALYSIS

( 21

250.0

32 17

Linear c!wracter!stics ( MU = 18 ) Nonlinear characteristics

200 0

Fig. 5 . Surface power density distribution on the charge boundary (Case B).

TABLE I1 COMPUTED PARAMETERS OF T H E PRACTICAL INDUCTION HEATING SYSTEM

Parameter ~

Measured value [I31

Value Calculated by empirical Formula (131

computed value case(a)

case@)

0.0135 0.025

0.0137 0.020

0.0081 0.0186

0.0114 0.0226

of a practical induction heating system with ferromagnetic charge, and the effect of nonlinearity has been calculated with accuracy better than 10 percent in comparison with experimental value.

Charge resi5tance

(a)

Reactance

(a)

The effect of nonlinearity is obtained by comparison of the power dissipated in the charge for cases (a) and (b) and given by:

NLF =

The power dissipated in the charge for case (b) The power dissipated in the charge for case (a)

(28) For this example the nonlinearity factor is found to be 1.4. This computed value is about 4.8 percent less than Boden’s experimental value of the nonlinearity factor that has been founld to be 1.47 [ 101. Fig. 5 shows the surface power density supplied to the charge for the two cases (a) and (b).

VIII. CONCLUSION Hybrid FEM-BEM has been used for the analysis of induction heating system with nonlinear charges. Iterative procedure is used to correct permeability values according to the value of magnetic flux density at each element. This method has been applied to compute electrical parameters

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method for nonlinear electromagnetic field calculations,” IEEE Truns. Mugn., vol. MAG-24, no. I , Jan. pp. 150-153, 1988. 1131 Stansel, S . R . , Induction Heating. New York: McGraw-Hill, First Edition, 1949. [ 141 Zienkiewicz, 0. C . , The Finite Element Method in Engineering Science. New York: McGraw-Hill. Second Edition, 1971,

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in electromagnetics,” IEEE Truns. Mugn., vol. MAG-21, no. 5 , pp. 1829-1834, Sept. 1985.

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