application of asymptotic expansions by Ryan Lian

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Application of Asymptotic Expansions to Model Two-Dimensional Induction Heating Systems. Part I: Calculation of Electromagnetic Field Distribution F. Bioul and F. Dupret Center for Systems Engineering and Applied Mechanics (CESAME), Université Catholique de Louvain (UCL), B-1348 Louvain-La-Neuve, Belgium We analyze the effect of a radio-frequency alternating magnetic field generated in the vicinity of solid or liquid electrically conducting components, such as used in induction heating processes. The field can penetrate only into a thin magnetic skin located beneath the conductor surface, where the generated heat and stresses are concentrated. This most often leads to major numerical difficulties, especially for very thin magnetic skins. Therefore, we have developed a mathematical model of the electromagnetic field distribution inside the conductors for planar and axisymmetric configurations by using a matched asymptotic expansion technique. Among other features, our method takes the curvature of the conductor surfaces into account. A practical numerical implementation of our model is detailed here, and numerical calculations are carried out in order to extend the model to limiting cases such as curvature discontinuities and corners. These calculations compare successfully with complete numerical computations. Index Terms—Computer simulation, electromagnetic fields, floating zone process, induction heating.

I. INTRODUCTION

NDUCTION heating is widely used in today’s industry and in particular in operations such as metal melting and heat treatment, bulk crystal growth, semiconductor wafer processing, curing of organic coatings, and high-speed sealing and packaging [1]–[3]. The radio-frequency (RF) ac electric current generated inside the inductor(s) induces an alternating electromagnetic (EM) field both in the ambient gas and inside a very thin layer located beneath the surface of all surrounding electrically conducting components. The typical thickness of this layer is called the skin depth and is given by (1) where is the field angular frequency, and and are the electric conductivity and permeability of the conducting components, respectively. In turn, the generated alternating EM field induces ac electric currents inside the conductor skin depth. A threefold effect results from these currents: first, Joulean heating, which can be used to create a molten zone in some cases; second, free surface deformation, resulting from the normal component of the Lorentz force; and third, generation of a generally undesirable free surface shear flow due to the EM force. Since the design and investigation of an induction heating system usually relies on the execution of expensive and time-consuming experiments, numerical modeling today represents an indispensable tool, which combines with experimental methods to help design new apparatuses and determine optimal processing conditions [4]–[7]. Typically this technique can be applied to the crystal growth floating zone process [8]–[17].

Digital Object Identifier 10.1109/TMAG.2005.854325

However, in most cases, numerical simulation of induction heating will involve EM field calculations that must be accurate at low cost. Since the conductor skin depth is generally quite thin, discretization of this layer can be expensive, hence requiring a fine computational grid and most often leading to high numerical difficulties. Nonetheless, when this layer is thin with respect to the other system dimensions, the EM fields can accurately be approximated in the layer, thereby providing a semianalytical solution inside the conductors that can be matched to a numerical solution of the EM field in the surrounding gas. By this way, the effect of the EM field on heat and stress generation inside the skin depth can be accurately evaluated at low cost. The global EM field precision depends on the accuracy of the analytical approximation inside the conductors, which itself depends on the characteristic dimensionless skin depth . Previous treatments in the literature were generally based on using the analytical expression obtained for a flat surface [18], which is accurate to order near the surface. This solution, which is widely used in induction heating numerical simulations, is acceptable only if the skin depth is much smaller than the local radius of curvature of the conductor surface. Unfortunately, this is generally not the case in the entire domain, especially in the presence of corners. This issue will be addressed in the present paper. Let us here mention the related work of Mestel [19], [20] who first provides an approximation of higher order in for the planar induction problem, as based on an integral formulation, and second a numerical integral method to compute the EM field around an isolated corner. Also, in [21], Rytov presents a second-order approximation of the electric and magnetic fields in the three-dimensional (3-D) case, however without any practical implementation envisaged. The matching procedure between the analytical and the numerical solution plays a key role in obtaining an efficient and accurate numerical method. The classical matched asymptotic expansion technique is generally applied as follows: the external EM field is numerically computed by replacing all conducting components by perfect conductors (like in [10], [17]). In

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BIOUL AND DUPRET: APPLICATION OF ASYMPTOTIC EXPANSIONS TO MODEL 2-D INDUCTION HEATING SYSTEMS

the planar and axisymmetric cases, the numerical normal field derivative (which is directly linked to the surface current density) is then used to match the external expansion to a simplified analytical solution inside the conductor. However, applying this procedure to an infinitely flat surface does not provide the exact solution in the one-dimensional case, and moreover leads to difficulties in the vicinity of geometrical corners (where the normal derivative of the field is singular [20]). In [19] and [22], an improved matching procedure of first order in is presented on the basis of a Robin (or Leontovich) condition, which does not lead to singularities and gives the exact solution if the curvature of the conductor surface is zero. However, this method is of low order and does not apply in the presence of corners. Noting that Robin–Leontovich conditions are in fact impedance boundary conditions (IBCs), it should be mentioned that higher order IBCs are abundant in the EM wave propagation literature [23], which however is not directly the investigation domain of the present paper. Nevertheless, since the equations describing induction heating and harmonic wave propagation share the same structure, some results will be quite similar. This paper presents an induction heating model for planar and axisymmetric systems, which is based on matched asymptotic expansions and expressed in the form of a high-order Robin–Leontovich IBC. The model, which is developed in Sections II–V, provides an accurate value of the EM field along the conductor surfaces even in regions with relatively small curvature radius and can be extended to the treatment of corners. After discussion, an efficient numerical implementation of the matching procedure is developed in Section VI. Finally, in Section VII, the model is extended to limiting cases with curvature discontinuities and corners, and is successfully compared with results obtained by means of complete numerical computations carried out in the gas and the conductors. In a related paper [24], the associated equivalent heat flux and surface stress are calculated with a view to applying the present technique to the simulation of floating zone bulk crystal growth. Additional applications are presented in [25].

Fig. 1. (a) Sketch of the three domains. (b) Sketch of the three subregions, the coordinate system (n; s), and the angle 8.

Applying these hypotheses to Ampère’s law, which relates to the electric current density , and to Ohm’s laws, which reduces , it is easy to obtain the relation to (2) As the magnetic induction is divergence free, the magnetic can be introduced in such a way that vector potential with the additional “gauge condition” . Using Faraday’s law, one can also write the electric field in terms of a scalar electric potential and : (3) where the first term of the right-hand side corresponds to the electrostatic field, while the second term represents the induced electric field. Introducing the potential definitions into (2), the equation governing the vector potential in the entire space is written as (4)

II. PROBLEM FORMULATION Let us first present the modeling hypotheses generally assumed in induction heating processes [4], [5]. of 1) The induction vector is a linear function the magnetic field , with the magnetic permeability of vacuum, in order to approximate the behavior of nonferromagnetic materials (replacing by the material permeability , our theory also applies in the case of linear or weakly nonlinear ferromagnetic materials, provided is nearly constant inside each component, without any direct contact between these different components). induced by electric conductor 2) The electric field motion across the magnetic field (with the conductor velocity in the laboratory reference system) is neglected with respect to the electric field . is neglected with respect to the 3) The convection current conduction current , with the volume density of the electric charges. is supposed negligible 4) The displacement current with respect to the conduction current .

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The alternating current used in induction heating processes allows one to express all the fields in the form , where denotes real part and is a complex vector. This representation will be used in the sequel. A sketch of our particular problem is given in Fig. 1(a). We consider a conducting region , bounded by and submitted to a given transverse electric potential difference, together with a conducting region , bounded by and where no electrical potential difference is imposed, the whole surrounded by the external free-space region . We assume without loss of generality that is composed of perfect conductors, called inductors, that is composed of nonperfect conductors, called susceptors, and that the free-space in has the EM properties of vacuum. We focus on the transverse planar (TP) and azimuthal axisymmetric (AA) cases, with the following simplifications. • In the TP case, a purely transverse electric current is assumed in the inductors, and hence

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, and , implying that inside the components where there is no imposed potential difference. and â&#x20AC;˘ In the AA case, using cylindrical coordinates their standard basis , a purely azimuthal electric current is assumed in the inductors, , and and hence , implying that inside the components where there is no imposed potential difference. inside perfect conductors, the On the other hand, since following relations, coming from (3) for a perfectly symmetric inductor, are verified in : in TP case in AA case

(5) (6)

where is the imposed electric potential difference on the in the TP case, Ith inductor, for a reference thickness or between and in the AA case. For nonperfectly symmetric inductors including radial slits, these relations can be modified to take losses of symmetries into account [17], [26], allowing one to still consider the fields generated outside these inductors as two-dimensional (2-D). It should also be noted that letting some furnace components rotate around the axis contributes to increase the system axisymmetry [13]. Introducing complex vectors and definition (1) into (4), the vector potential equation reduces, in (and only in) the considered TP and AA cases, to

Fig. 1(b)] are the outer region, located deeply inside the suscepthicktors (beyond the skin depth), the inner layer with ness inside the susceptors, and the external region in the free space (outside the susceptors and the inductors). The vector potential in each subregion is written in the form of an appropriate asymptotic expansion for small , and the expansions in adja, or denote cent regions are matched. Superscripts , the region where the expansion is defined. In this section, we consider a general 2-D problem, with a reference plane defined by or in the TP or AA is case, respectively. The curvilinear coordinate system corresponds to the intersection defined such that the curve of the susceptor surface with the reference plane, where stands for the outward normal coordinate and for the tangential coordinate in the reference plane along curve [Fig. 1(b)]. are orthogonal to the curve. The straight iso-lines The skin depth is then written as a product of the constant , characteristic skin depth by a position-dependent term which is assumed to be constant on each different conducting component and to be in range: (11) On the other hand, the vector potential can be expressed as a function of the coordinates and . One can then formally assume the following expansions1 for the nonzero component of as : (12a) (12b)

in in

since since

continuous on with and condition (BC):

(7) (8)

(12c)

and the following boundary

(9)

where is a given vector depending on the potential difference imposed in the Ith inductor, and or in the TP or AA case, respectively. Equations (7) and (8) with BC (9) provide a unique solution for every . III. ASYMPTOTIC EXPANSION Dimensionless variables, indicated by a tilde, are defined as follows: (10) and are a characteristic length and transverse elecwhere tric potential gradient, respectively. The characteristic dimensionless skin depth is defined as , where is a characteristic electrical conductivity. Henceforth, for the sake of simplicity, all variables will be written in dimensionless form without a tilde. Parameter is the assumed small parameter in the asymptotic expansions of in certain subregions. The latter [as shown in

denotes a curvilinear system depending where on to be introduced at a later stage. These asymptotic expansions are not series since partial sums might not tend to the exact solution when increasing the number of terms, while for a given number of terms the expressions tend to the exact solution when . Noting that the alternating field vanishes deeply inside the electric conductors (and everywhere inside the perfect conductors) and that the electric potential gradient is zero inside the susceptor components, it can be shown by using (3) that must exactly equal zero, in the same way as (since if then all conductors are perfect). The asymptotic expansions (12) will lead to an analytical approximation of developed in the following section. On the is easy to find by means of an apother hand, the solution propriate numerical method. The expressions of and are then matched along . All developments are first carried out for the TP problem, and the theory is subsequently extended to the AA problem. 1For the sake of clarity, a function f (x; y ) is called o (x) or O (x) as x (where y is a n-tuple value) when the following definition is satisfied:

if 8C > 0 9 > 0 s:t: jxj < ) jf (x; y)j C jxj; f (x; y ) = O (x) if 9C > 0 and > 0 s:t: jxj < ) jf (x; y)j C jxj

f (x; y ) = o (x)

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BIOUL AND DUPRET: APPLICATION OF ASYMPTOTIC EXPANSIONS TO MODEL 2-D INDUCTION HEATING SYSTEMS

IV. PLANAR CASE The analytical approximation of inside the inner region is now developed in the TP case, while the matching procedure will be discussed afterwards.

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Introducing the expansion (12b) into this last equation, and extracting the powers of up to order 3, the following differential system is obtained: (18a)

A. Analytical Inner Solution An analytical approximation of is found by rewriting the equation in the normal-tangential coordinate system, by introducing appropriate expansions in this equation, and finally by using a similarity change of variables. is such that the square of an The coordinate system elementary length is given by

(18b) (18c) whose solutions writes as follows: (19a) (19b)

(13) and denote the metric coefficients. where stand for the meridional principal Letting at coordinate , where is the angle curvature of between the normal vector to the surface and the horizontal is analytical and that vector [Fig. 1(b)], we assume that , which means that the curvature radius is much larger than the skin depth. The following metrics apply for the new coordinate system in the vicinity of the susceptor surface: (14) In TP problems, where inside the inner domain as

, one can rewrite (7)

(15) In the

system, the Laplacian operator writes as

(19c) and are integration functions to be where determined by appropriate matching with the external solution, while a prime symbol denotes -derivation along the boundary. Let us mention that (19a) and (19b) remain valid when depends on , while (19c) requires corrective terms. Hence, without correction, expansions (12b) and (12c) will only be valid up to order when depends on . B. Matching Procedures At this stage, the analytical approximation of the inner sois known up to order , except that the integration lution must still be found. These functions depend on functions on since the potential vector and the external solution its normal derivative are continuous across the surface of a nonperfect conductor. This leads to the following conditions: and (20)

(16) Let

denote the value of at the surface (for ). Since will vary on a small length of characteristic size across the skin depth, the order of magnitude of its normal derivative is . If everywhere, we can assume , and the similarity variable that can be introduced while remains unchanged. This normal coordinate stretching allows us to compare the orders of magnitude of the different terms of the equation and to neglect the appropriate terms. After this change of variable, (15) multiplied by can be simplified by using (16) and expressions (14):

(17)

or, explicitly and using expansions (12b) and (12c), and relations (19a), (19b), and (19c): (21) and

(22) Observing that each external solution satisfies the , the problem remains to find Laplace equation in order to easily compute boundary conditions (BCs) for it with an appropriate numerical method. Next, we present two procedures to compute these BCs.

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Exactly in the same way, it is now possible to compute , which provides , while calculations provide the expression

(27) 2) Procedure II: Another way to solve the problem is to use a one-step procedure. A single BC can indeed be found by reorganizing (22) as a development of powers of :

(28) Fig. 2. Illustration of matching procedure I for a given s: first step (top) and second step (bottom).

1) Procedure I: A theoretical way to find BCs for , as suggested in [19], is to proceed step by step by successively solving the equations associated to each power of and then matching the inner and external solutions. Although this method is usual from a theoretical viewpoint, it will be observed that it is not optimal from a numerical viewpoint. The number of steps determines the method order. The matching procedure up to order is illustrated in Fig. 2. During the first step, is computed from its known BCs: on on

(23)

provides . LetThe computation of ting the first terms of the inner and external normal derivatives (22) be equal, quick calculations show that

and by introducing (21) into this last equation. This technique , its normal directly leads to the following BC linking on derivative and its second tangential derivative for :

(29) This BC is composed of a series of powers multiplied by , plus an additional term involving the second tangential derivative of (and derivatives of for higher order terms in ). This last term is in agreement with the generic form of a second-order absorbing boundary condition in two dimensions [27]. Since (29) will produce the inner approximation up to order , we call it the third-order BC. Observing that the fol: lowing relations hold along the boundary (30) (31)

(24) Once is found, the procedure is pursued in the same way, and it is possible to compute from its now known BCs by using (21): on on

(25)

provides . LetThe computation of ting the second terms of the inner and external normal derivatives (22) be equal, calculations show that

(26)

the third-order BC (29) is equivalent to a generalized IBC relating the tangential magnetic field to the tangential electric field and its derivatives. The BC provided by our model can be compared to the analytical solution found for the particular geometry described in Fig. 3 in the TP case. This geometry allows us to check the third-order BC found by means of Procedure II, in a nontrivial case. Using polar coordinates and imposing the transverse cominside the inductor, with an ponent of as equal to arbitrary integer, one can prove that the expression of in the susceptor is (32) where is an integration constant and is the modified Bessel function of the first kind of order . Since the normal

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V. AXISYMMETRIC CASE In AA problems, where , it is more convenient , which can be to define the magnetic flux function expressed as a function of the coordinates and . In this case, is defined such that the curvilinear coordinate system the square of an elementary length is given by (35) Fig. 3. Sketch of a planar problem providing a reference analytical solution.

derivative of is continuous at the interface (for following relation can directly be found from (32):

), the

and are the metric coefficients defined where by (14). Moreover, the section of the conductor surface by the , while the meridional plane is given by holds in the interface vicinity relation lines. along Applying twice the curl operator to the divergence-free vector field provides the desired vector Laplacian operator

(33) where sion of

. Using the following asymptotic expan, valid when is large [28]:

(36) Performing the same analysis as in the planar case, the following equation is finally obtained inside the inner region:

(34) and observing that and , one can directly check that BC (29) is valid up when . to order 3) Discussion: A numerical problem encountered with Procedure I is that the imposed BC at every matching step is based on the normal derivative of computed from the previous step. Since the normal derivative of a function is less regular than the function itself, the successive approximations of will become less and less regular. Therefore, this procedure is useless for high-order expansions in as leading to nonregular solutions. This implies that in general only a scheme up to order can be applied. On the other hand, the BC found by means of Procedure II leads to an approximation up to order for smooth surfaces and seems to be very stable, to exhibit good convergence properties and to produce regular solutions. Contrarily to Procedure I, the computational cost of Procedure II is approximately the same whatever the order of the method is, since there is a single BC to be applied. Moreover, in the case of an infinitely flat surface with zero curvature, this condition provides the exact solution. Let us note that the first-order condition (with only the term) is the same as the one found in [19], [22]. It is important to emphasize that Procedure II only gives the value of and its normal derivative at the surface. This was expected since we are interested only in these values. Nonetheless, when a more accurate solution is needed inside the conductor (and not only at its surface), Procedure I should be used to find all the integration . functions

(37) where and are the two prin. cipal surface curvatures of Applying exactly the same method as in the planar case, the third-order axisymmetric BC found from Procedure II is

(38) Since the following relations hold along the boundary

(39) (40) this BC is again equivalent to a generalized IBC. The above BC (38) can again be compared to the analytical solution found for the geometry described in Fig. 3 in the AA case. Using cylindrical coordinates and imposing the azimuthal

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component of expression of

as equal to inside the inductor, the in the susceptor is found to be

BC (29), the right-hand side of the weak formulation can be interm: tegrated by parts except for the

(41) . Since the normal where is a constant and , the folderivative of is continuous at the interface lowing relation can be found from (41): (42) where . Using the asymptotic expansion (34) of , and observing that while and , one can directly when . Other check that BC (38) is valid up to order analytical results, not presented here, can be obtained inside a spherical susceptor and also validate condition (38) when the two principal curvatures and are equal and differ from zero. VI. NUMERICAL IMPLEMENTATION The analytical inner solution has to be matched with the external solution, which verifies the Laplace equation inside with the BC on provided by the selected matching procedure. The numerical method used to solve the Laplace equation depends on the matching procedure used as described below. Procedure I is applied only up to first order in for the above explained reasons. The boundary-element method (BEM) can then be used since we only need to compute the normal derivative of along the boundary. The advantage of BEM is that this normal derivative is directly provided by the method, contrarily to the finite-element method (FEM) case. Let us emphasize that, if matching procedure I and BEM are applied to a geometry including corners, singular shape functions will be required to discretize the normal derivative, since this normal derivative is singular at the corners of a perfect conductor. On the other hand, Procedure II will be applied up to order . The FEM is used to compute the solution with this matching procedure since the curvature and the second tangential derivatives of , which are present in the BC, are computed by appropriate integration by parts while this is not allowed with BEM. FEM is based on the weak formulation of the problem, which is written as follows in the TP case. such that for all test functions , with on Find : (43) on

(44)

Unfortunately, real process geometries generally present curvature discontinuities and corners, implying that the curvature might be nonanalytical. Our model is not anymore valid for such cases but can be applied everywhere except at the critical points. and the third-order Using the previous definition of

(45) where stands for the conductor boundary without its denotes the curvature everywhere except at the corcorners, ners where it is set to zero, while a function subscript means that this function is evaluated at corner , and the operator applied to a function denotes the jump of this function across corner . The role of the parameter , which is theoretically equal to 1, will be explained later. The curvature square needed in the BC can be approximated at the discretization nodes by drawing a circle passing by the node of interest and its two neighbors and computing its radius. The disadvantage of BC (45) is that the real corners (in opposition to numerical corners) must be identified. To avoid this operation, the BC can be simplified. First, a simple dimensional analysis shows that the second-derivative term in (29) is one order of magnitude higher than the curvature in the vicinity of term can generally be neglected. corners, implying that the Second, since the model developed is no longer valid in the vicinity of corners, the parameter in (45) can be set to zero, whereas numerical experimentations should be carried out to validate this technique. VII. RESULTS AND DISCUSSION In this section, the extension of our model to limit cases with curvature discontinuities and corners is compared with numerical computations obtained by solving the complete equation in the entire domain. Only the TP results are shown since the results obtained in the AA case are very similar. The FEM is used to compute the â&#x20AC;&#x153;exact solution,â&#x20AC;? which however is not the analytical solution but a numerical solution of the . A highly refined complete relation (4) computed in mesh in the skin layer is needed to compute this exact solution without letting spurious oscillations appear. Moreover, the smaller the skin depth, the finer the computational grid is required to be. We compare results obtained for the four test problems described below and shown in Fig. 4. In each case, a real potential equal to unity is imposed inside the inductor, while the skin depth is equal to 5 10 and there are three symmetry planes represented by dashed-dotted lines. The geometry is voluntarily nonsymmetric with respect to the nonzero curvature region in order to avoid possible mutual compensations of some terms.

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Fig. 5. Test T1, discontinuous curvature, R = 0:25. order 1 with matching procedure I, order 1, order 2, order 3, order 3 with = 0.

Fig. 4. Sketches of the four test configurations. Dashed-dotted lines are symmetry planes. In test T1, @V is composed of a circle arc joined to straight lines. In tests T2 and T3, @V is composed of straight lines and a Bezier curve defined by the control points A; B; C; and D . The maximum curvature along the Bezier curve in T2 is linked to parameter p by the relation = 16=3 2p.

These test cases allow us to compute the solution either near a surface with a discontinuous curvature or near a corner. In test T1, the curvature is discontinuous at the junctions between the circle arc and the straight lines. In test T2, the curvature is continuous everywhere, but can be increased in order to tend to a , corner. In test T3, there is a concave corner with and finally in test T4, there is a convex corner with . Figs. 5â&#x20AC;&#x201C;8 show the normalized error modulus for in each test case, where moduli are normalized by means of the skin depth . Unless otherwise stated, calculations are performed with matching procedure II. Fig. 5 shows the approximation errors for test T1. The model hypotheses are not valid anymore since the curvature is discontinuous. Nevertheless, we can check that the higher the BC order is, the smaller the approximation error will be. The case of curvature discontinuities is not critical but accuracy can significantly be increased by applying the third-order BC. Indeed, the error modulus is approximately divided by 30 from the first- to the third-order method. Let us note that the numerical noise present when the error is lower than 1 10 results from the discretization of both the reference and the term, approximate solutions. One can also observe that the which leads to some implementation difficulties, only weakly influences the error associated with the third-order condition. Finally, let us emphasize that the approximation errors obtained with matching procedure II vanish in zero-curvature regions, contrarily to those obtained with procedure I. Before analyzing the effect of corners, let us focus on test is equal to 5. As , the hyT2, when the product

Fig. 6. Test T2, continuous curvature with = 100. order 1 with matching procedure I, order 1, order 2, order 3, order 3 with = 0.

potheses of our model are not valid anymore. Fig. 6 shows the associated approximation errors. The first-order BC with Procedure I produces a large error at the corner, as a consequence of the singularity of the normal derivative. On the other hand, we can observe that, close to this sharp curvature zone, the classical BCs of second- and third-order produce higher errors than the first-order method with Procedure II. However, if the third-order term, the error is strongly reduced BC is applied without the in the sharp curvature zone. This numerical experiment shows that the solution with the smallest error seems to be obtained with the third-order BC applied everywhere, without the term in sharp-curvature regions. This is exactly what is done when the function is used and the parameter is set to zero in (45) when real corners are present. Fig. 7 shows the approximation errors in case of a convex is used in the BC. We can corner with test T3. The function observe that effectively the error is the smallest in the vicinity of the corner when parameter is set to zero in the third-order BC, . whereas this error is very similar to the one obtained with Our model applied to convex corners gives good results since,

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the second-order BC, the method becomes divergent near the corners. VIII. CONCLUSION

Fig. 7. Test T3, concave corner with [ ] = =4. order 1,

= 0; order 3 with = 0; order 3 with = 1.

Fig. 8.

order 2 with

Test T4, convex corner with [ ] = =2. order 1 or order 2 with order 3 with = 0; order 3 with = 1.

= 1; order 2 with = 0;

in this case, there is only weak interaction between the inner solutions inside the conductors. Numerical experiments carried out with different mesh refinements show that the second-order diverges at the corner, while the method method with converges with . In the case of a concave corner with test T4, we observe in Fig. 8 that the error obtained with the third-order BC is the . Concave corners are critical smallest, especially with since the error is relatively large in their vicinity. Moreover, the more concave the corner is, the worst the approximation error computed by our model will be. This is why the use of the third-order BC is highly recommended in such regions. As in the case of a convex corner, the second-order BC with diverges at the corner, while this BC is exactly equal to the since the corner sides are straight first-order BC when . lines where As a conclusion, the accuracy of the approximate solution will be the best with the third-order BC provided the term be set to zero in sharp curvature regions, and provided the parameter be equal to zero. If this parameter is set to zero with

A planar or axisymmetric 2-D model has been developed to represent the radio-frequency alternating electromagnetic fields observed in the vicinity of electric conducting components. In addition, a practical numerical implementation has been detailed, leading to accurate solutions with low computational cost when the skin depth is small but not negligible, with an appropriate extension to the treatment of geometrical corners. Some of the conclusions of our paper, although found by a different approach, fit with those of [19]. The latter paper is especially dedicated to reduce the EM field error in the interior of the conductors, with a view to further reduce the error on the Lorentz force analytically computed inside the fluid. Our objective was better to obtain satisfactory results near the conductor surface, with a view to principally reduce the errors on the equivalent heat flux and magnetic surface forces, since the latter are of prime importance in induction heating processes (cf. the related paper [24]). Moreover, the single boundary condition found for the external potential problem leads to easy implementation, increasing result accuracy in the critical regions. However, since the BCs of the external 3-D problem are much more difficult to address, our model is limited for the moment to planar and axisymmetric cases. Moreover, we have only considered constant conductivity (and permeability) materials. Extension to nonconstant conductivities is nonetheless easy with use of the parameter introduced in (11), except when the material properties are discontinuous. In that case, the expansion cannot be rigorously carried out further than to order . Nevertheless, these critical regions can be removed from the analysis as it was done for corners. On the other hand, our method could easily be extended to problems where a thin fluid film with different solid and fluid material properties is formed between the gas and the solid conductor. ACKNOWLEDGMENT F. Bioul acknowledges a Belgian FNRS Research Fellowship. REFERENCES [1] E. J. Davis and P. G. Simpson, Induction Heating Handbook. New York: McGraw-Hill, 1979. [2] S. Zinn, S. L. Semiatin, I. L. Harry, and R. D. Jeffress, Elements of Induction Heating: Design, Control, and Applications. Palo Alto, CA: ASM, 1988. [3] J. Bohm, A. Lüdge, and W. Schröder, “Crystal growth by floating zone melting,” in Handbook of Crystal Growth, D. Hurle, Ed. Amsterdam, The Netherlands: North-Holland, 1994, vol. 2, ch. 4, pp. 213–257. [4] P. Gresho and J. J. Derby, “A finite element model for induction heating of a metal crucible,” J. Cryst. Growth, vol. 85, pp. 40–48, 1987. [5] L. Egan and E. Furlani, “A computer simulation of an induction heating system,” IEEE Trans. Magn., vol. 27, no. 5, pp. 4343–4354, Sep. 1991. [6] A. Mühlbauer, A. Muiznieks, and H.-J. Lessmann, “The calculation of 3D high-frequency electromagnetic fields during induction heating using BEM,” IEEE Trans. Magn., vol. 29, no. 2, pp. 1566–1569, Mar. 1993. [7] C. Chaboudez, S. Clain, R. Glardon, D. Mari, J. Rappaz, and M. Swierkos, “Numerical modeling in induction heating for axisymmetric geometries,” IEEE Trans. Magn., vol. 33, no. 1, pp. 739–745, Jan. 1997.

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BIOUL AND DUPRET: APPLICATION OF ASYMPTOTIC EXPANSIONS TO MODEL 2-D INDUCTION HEATING SYSTEMS

[8] A. Mühlbauer, W. Erdmann, and W. Keller, “Electrodynamic convection in silicon floating zones,” J. Cryst. Growth, vol. 64, pp. 529–545, 1983. [9] H. Riemann, A. Lüdge, K. Böttcher, H. J. Rost, B. Hallman, W. Schröder, W. Hensel, and B. Schleusener, “Silicon floating zone process: Numerical modeling of RF field, heat transfer, thermal stress, and experimental proof for 4 inch crystals,” J. Electrochem. Soc., vol. 142, pp. 1007–1014, 1995. [10] A. Mühlbauer, A. Muiznieks, J. Virbulis, A. Lüdge, and H. Riemann, “Interface shape, heat transfer and fluid flow in the floating zone growth of large silicon crystals with the needle eye technique,” J. Cryst. Growth, vol. 151, pp. 66–79, 1995. [11] T. Munakata and I. Tanasawa, “Study on silicon melt convection during the RF-FZ crystal growth process II. Numerical investigation,” J. Cryst. Growth, vol. 206, pp. 27–36, 1999. [12] G. Ratnieks, A. Muiznieks, L. Buligins, G. Raming, A. Mühlbauer, A. Lüdge, and H. Riemann, “Influence of the three dimensionality of the HF electromagnetic field on resistivity variations in Si single crystals during FZ growth,” J. Cryst. Growth, vol. 216, pp. 204–219, 2000. [13] G. Ratnieks, A. Muiznieks, A. Mühlbauer, and G. Raming, “Numerical 3D study of FZ growth: Dependence on growth parameters and melt instability,” J. Cryst. Growth, vol. 230, pp. 48–56, 2001. [14] R. Hermann, J. Priede, G. Behr, G. Gerbeth, and L. Schultz, “Influence of growth parameters and melt convection on the solid liquid interface during RF-floating zone crystal growth of intermetallic compounds,” J. Cryst. Growth, vol. 223, pp. 577–587, 2001. [15] G. Raming, A. Muiznieks, and A. Mühlbauer, “Numerical investigation of the influence of EM-fields on fluid motion and resistivity distribution during floating-zone growth of large silicon single crystals,” J. Cryst. Growth, vol. 230, pp. 108–117, 2001. [16] T. Munakata, S. Someya, and I. Tanasawa, “Suppression of Marangoni convection in the FZ melt by high frequency magnetic field,” J. Cryst. Growth, vol. 235, pp. 167–172, 2002. [17] G. Ratnieks, A. Muiznieks, and A. Mühlbauer, “Modeling of phase boundaries for large industrial FZ silicon crystal growth with the needle-eye technique,” J. Cryst. Growth, vol. 255, pp. 227–240, 2003.

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Manuscript received March 23, 2005; revised June 10, 2005.

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