Mar 24, 2000 - This paper deals with the simulation of induction heating of conducting ... the conductor and, according to Faraday's law, triggers eddy currents.
Modelling and Simulation of Induction Heating R. Hiptmair
J. Ostrowski y
R. Quast z
March 24, 2000 Abstract This paper deals with the simulation of induction heating of conducting workpieces with a complex shape and topology. We assume that its conductivity is high, which, owing to the skin effect bars the fields from penetrating deep into the conductors. This justifies the use of a simple magnetostatic model that describes the magnetic field outside the workpiece and the inductor. Field concentrating plates can also be taken into account. After possible holes in the conductors have been patched with cutting surfaces, a scalar magnetic potential can be employed. The actual computation relies on a boundary integral equation of the second kind, discretized by means of piecewise constant boundary elements. Thus we get an approximation of the surface currents. Then we use the skin effect formula to determine the rate of heat generation inside the workpiece. Keywords. Induction heating, scalar magntic potential, skin effect, boundary elements.
1 Introduction Induction heating has become a standard procedure when it comes to handling metals as part of a manufacturing process [3]. Hardly any other non-intrusive technology can compete with induction heating in terms of speed, controllability and heating power. Controlling the process, however, entails a detailed quantitative insight into how electric energy is finally converted into heat. These insights can be gained by measurements, but numerical simulation promises to yield them faster, more comprehensively and more cheaply. During induction heating, a conducting item is exposed to a time-dependent electromagnetic field generated by an alternating current in an inductor, usually some coil. The field penetrates the conductor and, according to Faraday’s law, triggers eddy currents. It is the Ohmic losses due to the eddy currents that, eventually, heat the conductor. Much work has been spent on developing codes for the numerical simulation of induction heating. Some approaches resort to semianalytic methods [10, 12], but they are confined to very simple geometries. Some settings feature cylindrical symmetry and have been tackled by codes based on essentially two-dimensional models [15, 16]. A survey of techniques that can be applied to genuinely three-dimensional induction heating problems is given in [13]. There the Sonderforschungsbereich 382, Universit¨at T¨ ubingen. This work was supported by DFG as part of SFB 382. y Sonderforschungsbereich 382, Universit¨at T¨ ubingen and Daimler-Chrysler corporation. This work was sup-
ported by DaimlerChrysler corporation. z Research center Ulm of DaimlerChrysler corporation.
1
Figure 1: Typical setting for induction heating: Inductor I , workpiece C and two plates M authors stick to vector valued surface currents as principal unknowns in the boundary element method. In this paper, we consider the situation depicted in Figure 1. The conductor C may be some technical item like a bolt, an axle, or a screw and may feature a rather complex topology with a few holes drilled into it. The inductor I has the topolgy of a torus and may neither intersect nor touch the conductor. It might be a copper pipe bent into a coil carrying some coolant inside. We point out that the item to be heated might be turning slowly in order to achieve even heating. Chunks of highly permeable non-conducting materials occupying the region M are placed close to the inductor to deflect the magnetic fields. All the shapes are usually available in the form of CAD data, their surfaces composed of smooth facets. The inductor is fed with a sinusoidal alternating current of 10 to 40 kA at medium-range frequencies of 5 to 30 kHz. The size of the items is a few cm, and the whole process takes only a couple of seconds. It is important to realize that the two main physical effects, electromagnetic induction and heat conduction, occur on vastly different time-scales. This means that there is hardly any change in the temperatur of the conductor within one cycle of the electromagnetic fields. Thus a partial decoupling can be employed and the simulation can be done by carrying out the following two steps in turns [6, 14]: 1. Compute Ohmic losses based on conductivities that are determined from on a stationary temperature distribution. 2. Update the temperature distribution by taking into account the heat generation computed in the first step. This can be done by some nonlinear implicit timestepping. The focus of this paper is on the electromagnetic aspects of the problem, among which the skin effect commands attention: It denotes the rapid decay of external high-frequency electromagnetic fields away from the surface of a conductor. Roughly speaking the fields are present only 2
up to a certain skin depth Æ below the surface of the conductor. As a consequence, the bulk of the inductor and the conductor have little impact on the electromagnetic fields. This has led us to assume perfect conductors into which the fields cannot penetrate and where only surface currents are flowing. So we obtain a magnetostatic model of the interactions of the conductors with the electromagnetic fields. The magnetic field in the unbounded space outside I and C can be determined from traces on the surface that are connected via boundary integral equations. Ultimately, the entire electromagnetic problem is reduced to an integral equation on the two-dimensional surfaces of C ,
I , and M . The plan of the paper is as follows: In the next section we introduce and justify the magnetostatic model (7) on which our computations are based. We then derive the boundary integral equations with particular attention to the treatment of complex topologies. In section four we discuss how to get a unique solution by the additional minimization of the magnetic field energy. Finally, we present the results of computations conducted for an industrial workpiece.
2 Modelling of electromagnetic phenomena The electromagnetic phenomena under consideration are governed by Maxwell’s equations. However, at the relevant frequency range and in the presence of high conductivities we can resort to the simpler eddy current model. In the frequency domain for a constant angular frequency ! > 0 it reads [9].
urlE =
Here
i! B ;
urlH = E + jI
;
B = H :
(1)
B and H stand for the magnetic induction and the magnetic field, respectively, and 2
L1 (R 3 ) is the magnetic permeability, which can be assumed positive and constant inside C ,
I , and M . In the air region N := R 3 n ( C [ I [ M ) it is set to 0 . The conductivity 2 L1 ( L ) is nonzero and uniformly positive only within L := C [ I and vanishes
j
elsewhere. The exciting current I is confined to the inductor I and has to be divergence-free. Then the system (1) has a unique solution 2 ( ; R 3 ), 2 0 (div; R 3 ) [2].
urlH
H H url
B H
Outside L we have = 0 and we would like to use a magnetic scalar potential v there [1, 4]. Yet, its existence is guaranteed on simply connected domains only. To reach this we introduce cutting surfaces SI := S0 ; S1 ; : : : ; Sp , p 2 N . They have to meet the following requirements: 1. Each of the S has to be an open subset of a piecewise smooth two-dimensional manifold.
SI I and S C , = 1; : : : ; p. 3. S \ S = ; if 6= , ; 2 f0; : : : ; pg. 4. E := R 3 n ( L [ S0 [ S1 [ : : : [ Sp ) is simply connected. 2.
We take for granted the existence of such a set of cutting surfaces. Sloppily speaking, the number p corresponds to the number of holes in C . In the current setting I has exactly one hole, so that the cutting surface SI is always needed. The specification of the cutting surfaces can be done manually or automatically. For a more profound discussion of cutting surfaces we refer 3
to [5] and the references cited therein. For the sake of simplicity, we are setting p = 1 in the sequel, i.e. there is exactly one hole in the workpiece to which the cutting surface SC is to belong. Besides, SI and SC must not cut through the deflection plates.
H
grad
Then there is a v 2 H 1 ( e ) such that = v in E . Denoting by [v ℄S the jump of v across some externally oriented surface, Ampere’s law teaches that
[v ℄SI = I ; [v ℄SC = IC ;
(2)
where I 2 R is the fixed current in the inductor and IC 2 R , corresponds to the (unknown) total eddy current around the hole of C . Note that we can dispense with an exciting spatial current I in this case, as the total current flowing in the inductor is known in advance.
j
B
E
Following [4], eliminating and and taking into account (2), we cast (1) in weak form: Seek ( ; v ) 2 I such that for all ( ; q ) 2 0
H
Z
L
X
w
1
urlH; urlw dx + i!
X
Z
hH; wi dx + i!
L
Z
grad v; grad q dx = 0 : (3)
E
2 R we have introduced the affine space (related to the Hilbert space X 0 ) (w; q ) 2 H ( url; L ) H 1 ( E ); w n = grad q n on L ; X J := [q℄ = J; [q℄ = onst: for all = 1; : : : ; pg ; S S
Here, for J
with norm (on
X 0)
I
k(w; q)k2X := kwk2H( url; ) + kgrad qk2L2 ( ) : L
E
We follow the convention that H 1 ( E ) is the closure of smooth functions with bounded support in E with respect to the norm k kL2( E ) (Beppo-Levi space, [8, Vol. 4,XI,x1]). The bilinear form of (3) is elliptic and continuous with respect to this norm and, thus, by the Lax-Milgram lemma, there exists a unique weak solution in I .
grad
By using test functions enforces
q
X
2 H01( N ) in (3) it is immediate that the variational formulation
v n
S
= 0 ; S 2 fSI ; SC g :
(4)
B
H
:= . For exThis is essential for getting a divergence-conforming magnetic induction cellent conductors is very large in L and it might be feasible to pass to the limit = 1, altogether. From a physical point of view, this will make the electric field and magnetic induction vanish in L . Hence, (3) reduces to the magnetostatic model, whose variational formulation reads: Seek v 2 I such that
Y
Z
h grad v; grad qi dx = 0 8q 2 Y 0 ;
E
where
Y J , J 2 R , is the affine space (related to the Hilbert space Y 0 ) Y J := fq 2 H 1( E ); [q℄S = J; [q℄S = onst:g : I
4
C
(5)
Theorem 1 As (5) in H 1 ( E ).
! 1 uniformly the solution of (3) converges weakly towards the solution of
Proof. The proof will be carried out for the special case of 1 and constant First we introduce the Neumann vectorfield 2 H 1 ( N n SI ) defined by
div( grad ) = 0 in N n SI ;
[ ℄SI = I ;
hgrad ; ni = 0
n
SI
! 1 in C .
on N
=0:
We write ~ 2 H 1 ( C [ N n SI ) for the H 1 -extension of . Then the variational problem (3) can be recast into: Seek ( ~ ; v~ ) 2 0 such that for all ( ; q ) 2 0
H
Z
1
url H~ ; urlw dx + i!
L
w
X
= i!
Z D
ZL
L
H
E
X
~ ; w dx + i! H
h grad ~; wi dx
Z
E Z
grad v~ ; grad q dx =
grad ~; grad q dx : (6)
E
Testing (6) with its solution ( ~ ; v~ ) we conclude
1
2
url H~
L2( ) kgrad ~kL2( ) ; kgrad v~ kL2 ( ) kgrad ~kL2 ( ) ;
~ kgrad ~kL2 ( ) :
H 2 L ( ) N
C
N
N
C
C
Thus, there is a sequence of increasing conductivities tending to 1 such that 1 2
when
url H~ H~ grad v~
! 1. As for all 2 C 10 ( C )
lim !1
* E0 weakly in L2 ( C ) ; * HC0 weakly in L2 ( C ) ; * He0 weakly in L2 ( N ) ;
1 1 ~ ; 2 2 url H
url H~ ; L2( ) = lim !1 C ~ ; lim H = H ; 0 L2( ) ; !1 L2 ( ) 1
C
L2( C ) = 0 ;
C
C
and
we see
~ ; url H 1
L2 ( C )
~ ; + i! H
L2 ( C )
= ( grad ~; )L2 ( C ) ;
HC0 = grad ~ in L2( C ). As a consequence of ((He0 + ); grad q )L2 ( ) = i! ( grad ~; grad q~)L2 ( ) ; E
C
5
for q 2 H 1 ( E ) and q~ its H 1 -extension into the interior of C , we almost recover the magnetostatic model (5) in variational form, when testing with real-valued q . Finally
0 = lim (grad v~ ; url )L2 ( E ) = (He0 ; url )L2 ( E ) !1
for all
2 C 10 ( E ) establishes urlHe0 = 0 and He0 2 grad H 1( E ), hence.
2
Less rigorously,pthe magnetostatic limit also arises from a boundary layer analysis [18] for skin depths Æ := 1= 2! L , where L is the characteristic length of the workpiece. It shows that neglecting the electromagnetic fields in the interior of the conductors provides an O (Æ )approximation to the eddy current model. However, no quantitative bounds tell what ratios of L and Æ are required for a prescribed accuracy.
For CF45 steel at = 20Æ C, frequencies of 10kHz and B < 1T we get Æ 0; 1mm, which is reasonably small compared to L 50mm. Yet, the situation changes with increasing temperature or field strength. Then conductivity and permeability are decreasing: For CF45 steel at = 1000Æ C, frequencies of 10kHz and B > 2T we get Æ 5mm. Nevertheless, we will ignore the interior of C and I in the field calculation. We caution that the validity of the numerical results has to be checked carefully. Remark. The temperature dependence of the coefficients and the ferromagnetic behavior of the material can be taken into account when converting the calculated surface currents into spatial currents. The material parameters enter through the skin effect formula (cf. Sect. 5). It is useful to know that (5) reflects the minimization of the energy of the magnetic field, because it is equivalent to the minimization problem
inf f
Z
h grad q; grad qi ; q 2 Y I g :
E
Next, we examine the impact of the choice of the cutting surface. Lemma 1 If we choose two different sets fSI ; SC g and fSI0 ; SC0 g of cutting surfaces giving rise to E and 0E , related function spaces J and 0J , and different solutions v 2 I and v 0 2 0I of (5), then v v 0 is locally constant with respect to those subdomains of N that are bounded by the cutting surfaces and N .
Y
Y
Y
Y
Proof. We explain the idea of the proof for the simple situation sketched in Figure 2, where there is only one toroidal conductor L , whose hole is closed by two different disjoint cutting surfaces SC and SC0 . Then E := R 3 n ( C [ SC ) and 0E := R 3 n ( C [ SC0 ). The nonconduting region is split into 1 and 2 by the two cutting surfaces. The solutions v and v 0 of (5) have constant jumps across SC and SC0 , respectively. Assuming suitable crossing directions for the cutting surfaces, we set (
v (x) , if x 2 1 ; v~(x) := v (x) + [v ℄SC , if x 2 2
; v~0 (x) :=
(
, if x 2 1 ; v 0 (x) 0 0 v (x) + [v ℄SC0 , if x 2 2 :
Then
v~ 2 fq 2 H 1 ( 0E ); [q ℄SC0 = onst:g ; v~0 2 fq 2 H 1 ( E ); [q ℄SC = onst:g : 6
SC0
1
2
L
SC
Figure 2: Choice of two different cutting surfaces (cf. Lemma 1). In addition, the “magnetic energies” are not affected by switching to v~ and v~0 . As both v and v 0 are minimizers of the respective energies, we conclude Z
Z
h grad v; grad vi dx = h grad v0; grad v0i dx :
0E
E
The uniqueness of the solution of variational problems of the form (5) then establishes v and v 0 = v~. In terms of the magnetic field form
= v~0
2
H = grad v the variational problem (5) in E reads in strong I
urlH = 0 ; div(H) = 0 ; H ds~ = I ; hH; ni = 0 on L ; (7) H ~ is a line integral of the magnetic field along a closed path circling the inductor where H ds once. This, together with H = E = 0 inside the conductors and the assumption of surface currents, is our magnetostatic model. The previous lemma confirms that H is not affected by the choice of the cutting surfaces.
Remark. The boundary condition also results from the jump conditions at the interface two materials
M of
[n H℄M = k ; [n B℄M = 0 ;
k is a surface current. Assuming vanishing fields inside the conductor we end up with n H = k ; hH; ni = 0 ; which shows a way how to calculate the surface current k for a given solution for H in E . where
One might guess that
div( grad v ) = 0 in E v = 0 on L n [v ℄ = I ; [v ℄SC = onst vS n S = 0; S 2 fSI ; SC g : 7
(8)
is a more appropropriate strong form of (5), but beware: Whereas (5) has a unique solution, (8) has not, because we can impose arbitrary constant jumps [v ℄SC and still (8) remains solvable. This is a very important observation, since all formulations based on (8) need extra conditions to recover the unique solution of (5).
3 Boundary Integral Equations Boundary integral equations establish a relationship between the Dirichlet boundary values 1 1 vj E 2 H 2 ( E ) and Neumann boundary data := vn 2 H 2 ( E ) for the solution v of the Neumann problem (8) on the unbounded domain E . First let us consider the case M = ;. Among several variants of boundary integral equations (cf. [7]) we choose
(x)v (x) = V ( )(x) K (v )(x) ;
x 2 E ;
(9)
with the single layer potential
V ( )(x) :=
Z
N (x; y) (y) dS (y);
x 2 E ; 2 H
1 2 ( E )
E
and the double layer potential
K (q )(x) :=
Z E
N (x; y) q (y) dS (y); n(y)
x 2 E ; q 2 L2( E ) ;
where
N (x; y) :=
1 1 ; 4 jx yj
x; y 2 R3 ; x 6= y x
is the fundamental solution for the (scaled) Laplacian. The function ( ) is the solid angle which is equal to 12 where E is smooth and can have different values at edges and corners. The boundary of E includes the cutting surfaces, on which the potentials have to be evaluated for both sides. Let S denote a generic, piecewise smooth cutting surface endowed with a crossing direction given by a unit normal vectorfield ( ), 2 S . The normal vector points from side S to S + (see Figure 3).
ny y
We pick q
2 Y J , J 2 R , and first scrutinize the single layer potential restricted to S : q VjS ( )(x) = n =
Z
S+
q N (x; y) + (y) dS (y) + n
Z S
q N (x; y) n
S
Z
N (x; y)
S
dS (y) = 0 ;
q (y) dS (y) n
8x 2
R3
;
(10)
where we used the continuity of the single layer potential and the restrictions on q . Similar arguments show for the double layer potential
KjS (q )(x) = [q ℄S
Z S
N (x; y) dS (y) ; n(y) 8
8x 2 E :
n
n
E
E
n+
S+ S
Figure 3: Notations for cutting surfaces
x 2 E there is a double layer representation Z X v N (x; y) N (x; y) N (x; y) (y) v (y) dS (y) + [v ℄S dS (y) : n n(y) n(y) S 2fS ;S g S
Using this, we conclude from (9) that for all
v (x) =
Z L
I
C
(11) In addition, recalling that vn
( v )(x) =
Z
L
= 0 on L , we can rewrite the representation formula as
N (x; y) v (y) dS (y) + [v ℄SC n(y)
Z SC
N (x; y) dS (y) + I n(y)
Z SI
N (x; y) dS (y) ; n(y) (12)
where
x 2 E . If M 6= ;, we have to take care of the additional surface M Z N (x; y) (x)v (x) = v (y) dS (y) n(y)
Z N (x; y) v v (y) dS (y) + N (x; y) (y) n n(y)
Z Z N (x; y) N (x; y) dS (y) + I dS (y) + [v ℄S n(y) n(y) S S L
(13)
M
C
I
C
and boundary integral equations on M come into play ( Z
(x)vM (x) =
N (x; y)
M
vM (y) nM
= 1 ):
N (x; y) v (y) dS (y) : nM (y) M
The transmission conditions on M
v
vM = 0 ;
v n 9
r
vM =0 n
(14)
provide the necessary link between (14) and (13). First, they make it possible to convert (14) into
(x)v (x) =
Z
N (x; y)
M
1 v N (x; y) (y) + v (y) dS (y) ; r n n(y)
(15)
where r is the constant relative permeability inside M . Using this in (13) we finally get
N (x; y) v (y) dS (y) + (r n(y)
Z
( (x) + r (x)) v (x) =
L
+I
Z SI
N (x; y) dS (y) + IC n(y)
1)
Z
M
Z SC
N (x; y) v (y) dS (y) + n(y)
N (x; y) dS (y) ; n(y)
(16)
with
(x) =
81 > :
x 2 L ; x 2 o M ; , if x 2 E , if , if
1
and
(x)
8 > :
0
x 2 L ; x 2 o M ; , if x 2 E , if , if
for smooth boundarys L , M . If C is simply connected, i.e. SC = ;, (16) is a valid 1 boundary integral equation of the second kind for the unknown function v 2 H 2 ( L [ M ). It has a unique solution [8, Vol. 4,Ch. XI,x2,Thm. 5]. Yet, as already noted in Sect. 2, if we have a hole in C , (16) is underdetermined, because it does not allow us to fix the jump IC = [v ℄SC . Therefore, we have to incorporate the informaton that has been lost when passing from (5) to (8): We have to enforce the minimization of magnetic energy explicitely.
4 Minimization of Magnetic Energy In order to translate the minimization of magnetic energy into an equation for [v ℄SC note that
1 Emag = 2 1 = 2 =
1 2
Z
RZ3
A Z A
= IC , we
H(x) B(x)dx h grad v; grad vi dx + 21
v 1 v dS + n 2
Z M
Z
h grad vM ; grad vM i dx
M
vM v dS ; nM M
where A := E n M . As a consequence of the boundary and transmission conditions for v we end up with the expression
I 0 Emag = 2
Z SI
I v dS + C 0 n 2 10
Z SC
v dS ; n
(17)
Next we recall lemma 2 from [8, Vol. 4,Ch. XI,Part B,x2], which states that for a double layer potential on a piecewise smooth surface
u(x) = the gradient away from
grad u(x) =
Z
g (y)
N (x; y) dS (y) n(y)
(18)
is given by
Z
(n(y ) grad g (y)) grady N (x; y) dS (y)
8x 62 :
Let us assume that we have used slightly different cutting surfaces SI0 and SC0 for the computation of v . Then, according to lemma 1 the expression (17) for the magnetic energy remains the same. Moreover, (16) provides us with the following double layer representation for all o 2 := R 3 n ( L [ M [ SI0 [ SC0 ):
x
Z
v (x) =
L
Z
+
S0+ I
Z
S0I
N (x; y) v (y) dS (y) + (r n(y)
N (x; y) + v (y) dS (y) + n(y)
Z
S0+ C
N (x; y) v (y) dS (y) + n(y)
1)
S0C
M
N (x; y) v (y) dS (y) n(y)
N (x; y) + v (y) dS (y) n(y)
N (x; y) v (y) dS (y) : n(y)
x 2 o (n(y ) grad v (y)) grady N (x; y) dS (y)
Then (17) immediately yields for all
grad v(x) = (r
Z
1)
Z
Z
M
Z
(n(y ) grad v (y)) grady N (x; y) dS (y) ;
L
as the contributions of different sides of cutting surfaces cancel due to the equality of tangential gradients of v + and v . Without deflection plates M , the above formula is the familiar BiotSavart law. The magnetic field energy (17) can now be written as
1 Emag = 0 I 2
Z
n(x)
(r
1)
SI
1 + 0 IC 2
Z Z SC
n(x)
Z
(n(y ) grad v (y)) grady N (x; y) dS (y)
M
(n(y ) grad v (y)) grady N (x; y) dS (y) dS (x) +
L
(r Z
1)
Z
(n(y ) grad v (y)) grady N (x; y) dS (y)
M
(n(y ) grad v (y)) grady N (x; y) dS (y) dS (x) :
L
(19) 11
We have to find the minimum of Emag with respect to the independent variable IC . Please note that also v depends on IC in a linear affine fashion as is clear from (16). This means that we actually have to minimize a quadratic function in IC , which can easily be done analytically: First, we use (16) to calculate solutions v10 and v01 for the particular total currents I = 1; IC = 0 and I = 0; IC = 1, respectively. The general solution of (16) is then given by
v (x) = I v10 (x) + IC v01 (x) :
(20)
Insert (20) in (19), and the induced total eddy current IC results from the condition
0 =
Emag (v (IC )) : IC
From it we find
I IC = 2
(
Z
n(x)
(r
SI
n(x)
(r
SC
+ ,( Z SC
n(x)
(r
1)
Z
(n(y ) grad v10 (y)) grady N (x; y) dS (y)
M
Z
(n(y ) grad v10 (y)) grady N (x; y) dS (y) dS (x)
L
1) Z
(n(y ) grad v01 (y)) grady N (x; y) dS (y) dS (x)
L
(n(y ) grad v01 (y)) grady N (x; y) dS (y)
M
Z
+ Z
1)
Z
Z
)
(n(y ) grad v01 (y)) grady N (x; y) dS (y)
M
)
(n(y ) grad v01 (y)) grady N (x; y) dS (y) dS (x) :
L
(21) As soon as we know IC , (20) gives the desired unique solution of our surface eddy current problem.
5 Boundary Element Method The surfaces of I , M and C are equipped with a shape regular surface mesh h composed of flat rectangles. Discretization of the boundary integral eqautions (16) relies on a piecewise constant approximation vh of v and is based on midpoint collocation [11, Sect. 4.4]. The singular collocation integrals over the elements are evaluated exactly using stable analytic expression derived by O. Steinbach. Thus we get linear systems of equations for the unknown coefficients of the piecewise linear approximations of v10 and v01 . They are solved iteratively by means of the BiCGStab Krylov-method [17]. Since the discrete integral operator of the second kind is well conditioned, only a moderate number of iterations have to be carried out. 12
kx
grad x n x
What deserves attention is the evaluation of the surface currents ( ) := v ( ) ( ), 2 L , because there is no meaningful gradient of vh. A smoothing of vh is inevitable and can be carried out through an L2 -projection of vh onto the space S ( h ) of piecewise linear, continuous functions on L that feature a constant jump across SC \ L .
x
We end up with the discrete variational problem: Seek v~h Z
hv~h(x); ph(x)i dS (x) =
L
Z
2 S(
h ) such
that
hvh(x); ph(x)i dS (x) 8ph 2 S (
h)
:
L
This gives rise to a linear system of equations with a sparse, well-conditioned symmetric positive definite (mass)matrix. A few steps of the CG-method give us a solution of satisfactory accuracy. From v~h the current can be obtained as plain surface gradient.
k
6 Simulation Results We applied our method to the induction heating of the workpiece from Figure 1. The computed surface current distribution is shown in Figure 6. The ultimate goal is to compute the time-dependent spatial temperature distribution in the workpiece. To that end we have to determine the power density of Ohmic losses from the surface currents. The first step consists of employing the skin effect formula: For any 2 C that is fairly close to the surface, we denote by 2 C the nearest point on the surface. For almost all 2 C this point is uniquely defined. Then we set
x
x
x
j(x) = j0 e where z
:= jx
1+i z Æ
; Æ=
r
2 !
x j. j0 is fixed for all x belonging to x by the condition Z1
k = j(x(z)) dz k
x
0
where is the surface current density in . As the currents are assumed to be time harmonic the effective power density of Ohmic heating amounts to 12 j j= . For instance, it can be evaluated at the centres of spatial elements. The results provides a discrete source term for the non-linear heat equation. Thus the temperature dependence of the conductivity can be taken into account.
j
We first verified our computations by comparing it with measurements of the surface temperature of a cylinder. The configuration of the experiment is shown in Figure 4. The internal diameter of the inductor was 5 cm, it was 1 cm thick and made of copper. The cylinder with a radius of 4 cm was made of CF45 steel and it was 10 cm long. Figure 5 shows the results of measurements of the maximum surface temperature and computed values for different exciting alternating currents of about 10.5 kA, 7.9 kA and 5.3 kA (peak current), at a frequency of 10 kHz. The results of electromagnetic computations in the case of a real rotating workpiece are shown in Figure 6. A qualitative result for the temperature distribution on the surface in Figure 7.
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Figure 4: Cylinder with inductor
Figure 5: Maximum surface temperature of the cylinder. The curves tagged with bullets represent measured values, the plain solid lines arose from simulation.
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Figure 6: Surface currents of workpiece and inductor. Up to the left and down to the right one can see the impact of the field concentration plates.
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Figure 7: Temperature of the workpiece
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