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MODELLING OF INDUSTRIAL CONVEYORIZED APPLICATORS USING HIGHER ORDER VECTOR FINITE ELEMENTS A.Hallac1 and A. C. Metaxas2 Engineering Dept, University of Cambridge, UK 2 St Johnʼs College Cambridge, UK 1 This paper presents a finite element time domain method for the solution of Maxwellʼs equations in microwave heating applicators using first and second order vector finite elements. Results are compared with experimental data and it has been shown that second order vector finite elements have many advantages over first order elements. Capitalising on the high accuracy and low computational cost attainable by higher order elements, an industrial conveyor belt system is numerically analyzed. Submission Date: February 21, 2006 Acceptance Date: October 24, 2006 INTRODUCTION Numerical techniques have been used widely in the past to model microwave heating applicators loaded with a variety of loads. The methods include finite differences, finite elements, transmission line methods and other techniques [Hussain M.Al-Rizzo et al, 2005; Dominguez-Tortajada et al, 2005; Yakovlev, 2006] while some earlier attempts of modelling using these techniques can be found in Dibben and Metaxas [1994]. This paper discusses the issue of using higher order finite elements for modelling large conveyorised microwave applicators as opposed to the usual formulation using first order finite elements. The most computationally intensive part of a finite element (FE) approximation is Keywords: industrial applications, FE approximation, modeling International Microwave Power Institute the solution of a large system of linear equations produced by the discretization of the domain being analyzed. In the frequency domain a single solution is required at the frequency of interest whereas in the time domain the system has to be solved at each time step over a band of frequencies. However, for practical microwave heating problems, the time domain solution is considerably easier to obtain than its frequency domain counterpart. This is due to highly ill conditioned matrices arising from the frequency domain method [Dibben and Metaxas, 1994 and 1997]. It has been proven that for first order tangentially continuous vector finite elements (TVFE) the condition number of matrices produced by the frequency domain is strictly proportional to the size of the solution domain [Dibben and Metaxas, 1996]. It is therefore difficult to achieve convergence when using existing iterative methods on high density meshes. For higher order elements this problem is even more severe [Hallac, 2003; 101 Hallac and Metaxas, 2003 and 2003a]. In the time domain, however, the second order basis functions produce real, symmetric and positive definite matrices allowing one to use a simple iterative method, such as the conjugate gradient method, while still achieving adequate convergence [Hallac and Metaxas, 2001]. Also, the time domain method is particularly well suited to the calculation of field distribution within the applicator and hence power density which is a key design parameter. In this paper we present the use of second order 3D time domain TVFE for the study of microwave heating problems. The formulation is described in Hallac and Metaxas [2003a] and uses interpolatory [Yioultsis and Tsiboukis, 1998] non-interpolatary [Savage and Peterson, 1996] second order vector finite elements for the electric field interpolation. These elements have degrees of freedom associated with their edges and faces and are well suited to model Maxwellʼs equations because they only enforce the tangential continuity of vector fields and do not generate spurious contributions or nonphysical resonances that are characteristic of nodal-based expansions [Carpes and Razek, 2000]. The efficiency and accuracy of the higher order bases are demonstrated through the analysis of batch and continuous multimode cavities with different dielectric loads including the use of a conveyor belt. The results are compared with those from a network analysis and other numerical solutions where applicable. THEORY This paper uses the well documented theoretical treatise starting from the time dependent vector equation describing the electric field and defining the weak form of that equation which includes the surface integral which surrounds the domain to be investigated [Hallac and Metaxas, 2003a]. This equation is 102 solved numerically using tetrahedral elements. The usual format of the vector basis functions is used to expand the electric field associated with the edges and facets of each element. Such a formulation produces a set of ordinary differential equations: [ S ]e(t ) + [Tσ + TABC ] d d2 e(t ) + [Tε ] 2 e(t ) = b(t ) dt dt (1) with the term b(t) representing the excitation currents and the surface integral term. The components of the FEntal matrices are: [ S ]i , j = ∫ (∇ Ωe Wi ) ⋅ (∇ W ) dΩe [Tσ ]i , j = σ eµo ∫ (Wi ⋅ W j )dΩe Ωe [Tε ]i , j = ε oε ' µo ∫ (Wi ⋅ W j )dΩe Ωe (2) (3) (4) and TABC is the elemental matrix resulting from the application of absorbing boundary conditions (ABC). A number of methods can be used to descretise the time derivatives, however, as we have reported previously, we have used the Newmark method [Dibben and Metaxas, 1994; Ehlers and Metaxas, 2003]. TVFE Basis It is well known that for practical microwave heating problems it is difficult to produce high quality unstructured meshes. For a simulation using first order basis elements, to increase the quality of the mesh the number of degrees of freedom must be substantially increased leading to excessive memory requirement. It is therefore necessary to increase the order of TVFE bases so as to reduce errors caused by the distorted meshes [Hallac, 2003], and to minimize computational costs. We consider a tetrahedral element with nodes Journal of Microwave Power & Electromagnetic Energy Vol. 40, No.2, 2006 numerical scheme but removes the necessity of specialized mesh generation software [Hallac, 2003]. Boundary Conditions Figure 1. Illustration of a tetrahedral element and the numbering of nodes. i, j, k and l as shown in Figure 1. The volume of a tetrahedron is denoted by V. Simplex (or volume) coordinates æi, æj, æk and æl at a point Pi are defined by Savage and Peterson [1996]. Using simplexes the second order interpolatory Yioultsis and Tsiboukis [1998] TVFEʼs constitute 12 linearly independent edgebased vector bases and 8 face-based functions. For example, on edge {i,j} the two bases ( {i,j} and {j,i} ) can be written as: r Wij = 8ζ i2 − 4ζ j ∇ζ j + −8ζ iζ j + 2ζ j ∇ζ i ( ) ( ) (5) and for face {i,j,k}, two basis functions ( {i,j,k} and {i,k,j} ) can be written as: r Wijk1 = 16ζ iζ j ∇ζ k − 8ζ jζ k ∇ζ i − 8ζ kζ i ∇ζ j r Wikj2 = 16ζ iζ k ∇ζ j − 8ζ jζ k ∇ζ i − 8ζ kζ i ∇ζ j (6) The conformity of degrees of freedom between adjacent elements is guaranteed by using a consistent global numbering. In contrast to the well known classic finite element method where all tetrahedrons have the same parent element, second order TVFE bases are non symmetric and each tetrahedron has its own parent element, where element parents have dissimilar local numbering. This adds a complication to the International Microwave Power Institute As it is usual the walls of the cavity and the feed waveguide are assumed to be perfect conductors. In the feed waveguide it is assumed that at the operating frequency only the dominant mode TE10 is supported while a first order ABC is used to model open boundaries [Dibben and Metaxas, 1994] [Hallac, 2003]. The boundary integral terms for the ABC and the source can be written as: [ S ]i , j = µo ∫ ν TE Γ 10 (Wi ⋅ W j ) dSo ) bi , j = µ0 ∫ Wi ⋅ Einc × n dS1 Γ ( )( ) (7) (8) where So and S1 are the ABC and excitation planes respectively in the feed waveguide, n) is a unit vector, W’s are the basis functions, _TE_10 is the waveguide phase velocity for the TE10 mode, and the incident field can be written as: πx ) Einc = ny sin( )sin(ωt − ν TE10 ) a (9) where ς is a unit vector in the y direction which defines the direction of the incident field. In practical microwave heating applications the source modelling described in equation (8) is adequate to represent the magnetron source [Hallac, 2003]. RESULTS Frequency domain matrices are known to suffer from ill conditioning when multimode applicators are analyzed [Dibben and Metaxas, 1996; Boffi et al, 1999]. The time domain solution overcomes the problem of ill conditioning and 103 provides a symmetric, positive definite system of equations to be solved at each time step. In this section for the solution of matrices the Conjugate Gradient method with zero fill-in incomplete LU preconditioner is used [Hallac, 2003]. shows the consistent performance of the second order edge elements with respect to the mesh quality in that any choice of the mesh parameters that increases the mesh quality would reduce computational errors. Mesh Quality Consideration Experimental Verification First order edge elements are very sensitive to the quality of the unstructured mesh. Mesh quality affects both the efficiency and accuracy of the FE solutions, since meshes with distorted elements make accurate results more difficult to achieve. For complex geometries it is difficult to mesh the domain using well-shaped elements with neither very small nor very large angles at corners. Therefore to obtain a high quality mesh it is necessary to refine the mesh, which increases substantially the number of degrees of freedom. Higher order TVFEʼs, on the other hand, are less sensitive to highly distorted meshes [Hallac and Metaxas, 2001]. The most common approach in estimating errors within a mesh is to view the mesh quality as being independent of the solution. The mesh quality for each element in a 3D mesh can be defined as [Berzins, 1998]: A complete characterization of microwave heating involves coupling the electromagnetic and the thermal fields through the temperature dependence of the material dielectric properties. Such a coupled problem, particularly when solved using finite elements, presents a severe CPU requirement and is not considered in this paper. Instead, by solving only Maxwell’s equations, the electromagnetic field distribution within a loaded applicator is obtained which, given a knowledge of the loss factor, yields the power density established in the material to be heated. In the past such power density data from simulations were compared with thermal imaging from experimental tests during the initial stages of heating but otherwise under similar conditions of load, dielectric properties, cavity size and so on and gave good agreement in multimode heating applicators. This suggests that such an approach gives sufficient insight and adequate accuracy for assisting in the design of microwave heating applicators [Ehlers and Metaxas, 2003]. To validate the numerical method, a different method is adopted whereby the reflection coefficient is measured using a network analyzer and compared with that derived from the simulations. The cavity tested was of rectangular shape of 40 cm x 30 cm, and height 35 cm. The position of the load measuring 12 cm by 12 cm and height 4.1 cm, is offset as shown in Figure 3, while the feed waveguide type WG9A is placed 15.8 cm and 13.4 cm from one side of the cavity box applicator. The load chosen for verification purposes is rubber whose dielectric properties were 8.3-j0.4. Reasonable correlation was obtained between the return loss measured M Q = 8.45528Ve 6 hi3 (10) where Ve and hi are the tetrahedron volume and average edge length respectively. For each element, MQ ranges from 1 for an equilateral tetrahedron down to very small values for highly irregular tetrahedrons. Figure 2 shows the average error in the first 10 resonant frequencies versus the mesh quality for a 1 m x 0.5 m x 0.75 m rectangular cavity. In all simulations the number of unknowns were kept around 3500. To obtain a different mesh quality whilst maintaining the number of degrees of freedom the nodes in the mesh are moved in arbitrary directions until the desired mesh quality is obtained. Figure 2 104 Journal of Microwave Power & Electromagnetic Energy Vol. 40, No.2, 2006 memory requirement of 120 MB. It took 4 hours and 9 hours to reach 4% and 0.5 % convergence respectively. A. Figure 2. Comparison of error norms of the first 10 resonant modes with respect to the mesh quality. The number of degrees of freedom are kept at 3500. using a network analyser and that computed from the finite element method [Hallac, 2003; Hallac and Metaxas, 2003a]. The discrepancies were attributed to the modelling of the ABC at a single frequency of 2.45 GHz and the fact that the experimental setup has some additional losses caused by geometrical imperfections that are not accounted for numerically. Using a personal computer with 1000 MHz Athlon processor the model is descretised using 30000 tetrahedral elements producing 186,118 unknowns with a Conveyor Belt Systems As conveyor belt systems are widely used in industry, it is necessary to develop an ability to model such a system including passive components and a conveyor belt. The conveyor belt carries the load through the input and output ports of an industrial applicator [Metaxas and Meredith, 1988]. First order ABCʼs are used to truncate the ports. A continuous industrial microwave applicator with a conveyor belt, similar to a microwave tempering unit for processing blocks of butter, is shown in Figure 4. Due to the complex nature of such a unit, experimental verification has not been carried out. The system is initially simulated in the absence of the load and conveyor belt so as to find out how the power propagates through the input and output ports. For comparison the results for a loaded system are also presented. Figures 5 and 6 show respectively the electric field patterns and the return loss characteristic for the industrial conveyor applicator. It is evident from Figure 5 that the energy dissipation within Figure 3. Plan diagram showing test model for numerical verification and the position of the load. International Microwave Power Institute 105 b) a) Figure 4. Continuous microwave conveyor-belt system. (a) Side plan and 3D view (b) Front plan and top plan. Dimensions are in centimetres. a) b) Figure 5. Normalised electric field distribution at the centre of the conveyor belt (y=20 cm). The frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence. (a) Unloaded (b) Loaded Figure 6. Return loss characteristic for a loaded and unloaded conveyor belt system. The frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence. 106 Journal of Microwave Power & Electromagnetic Energy Vol. 40, No.2, 2006 a) b) Figure 7. Computed field distributions at two different cross sections in the workload. The frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence. (a) Slice at the top of the loads (b) Slice at the conveyer belt a) the unloaded oven (that is, empty applicator) is more pronounced than in the loaded cavity when it contains a lossy material. Furthermore, as expected and clearly depicted in Figure 6, the bandwidth of an individual response in the loaded case is wider as the effective Q-factor of the system decreases due to the increased loss. Sharper resonances are characteristic of unloaded systems. For the simulation with the load and conveyor belt in situ, the conveyor belt needs to be transparent to the microwave energy, so polypropylene is used, having a relative permittivity of 2.55-j0.0008 at 2.45 GHz. A material with such a low loss factor allows the energy to also enter the workload from below. For the load, a butter-like material is used which has a relative permittivity of 4.05-j0.39 at 2.45 GHz in the temperature range -20 to –2oC, a value which remains nearly constant throughout this temperature range. Figures 7 and 8 show the field and power distributions at different cross sections respectively. The numerical simulations are carried out using 170,944 tetrahedral elements, with an average edge length of 3.5 cm, resulting in 1,066180 degrees of freedom and a memory requirement of 600 MB. If the model were to be discretised using first order tetrahedral elements with an average edge length of 1 cm, then up to 2 million unknowns would have to be solved for. CONCLUSIONS b) Figure 8. Computed power distributions at two different cross sections in the workload. The frequency of operation is 2.45 GHz and the simulation is terminated at 1 percent convergence. (a) Slice on upper surface of butter loads (b) Slice at y=12.5 cm International Microwave Power Institute In this paper, we have presented a three dimensional time domain TVFE method using second order vector finite element bases for the numerical modelling of microwave heating applicators. The comparison between first and second order TVFE show that higher order elements require less computer resources, especially as the problem size increases. One reason for the superiority of higher order elements is that errors are kept low even for 107 highly distorted meshes that have to be used for some realistic system geometries. The method then presents another promising tool for modelling large industrial systems. ACKNOWLEDGEMENTS The authors wish to acknowledge the many helpful discussions they held with Drs Wai Fu, Richard Ehlers and other members of the Electricity Utilization Group, Engineering Department at the University of Cambridge, UK. REFERENCES Al-Rizzo, H.M., Tranquilla, J.M. and Feng,.M., (2005), “A finite Difference Thermal Model of a Cylindrical Microwave Heating Applicator, Using Locally conformal Overlapping Grids: Part I- Theoretical Formulation,” Journal of Microwave Power and Electromagnetic Theory, 40(1), pp. 17-29. Berzins, M. (1998). “A Solution-Based Triangular and Tetrahedral Mesh Quality Indicator,” SIAM Journal on Numerical Analysis, 19(6), pp. 2051-2060. 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