Geometrical singularities effects on the calculation of the magnetic force, an equivalent behaviour approach
The Authors
Olivier Barre, L2EP – Ecole Centrale de Lille, Villeneuve d'Ascq, France
Pascal Brochet, L2EP – Ecole Centrale de Lille, Villeneuve d'Ascq, France
Abstract
Purpose – The purpose of this paper is to introduce a simplified method to calculate an estimation of local forces acting on a body submitted to electric or magnetic fields. With experimentations, the method is thereafter evaluated.
Design/methodology/approach – When an external strength exists on a body, its deformation is an effect always observed. With materials with low elasticity modulus, such a deformation becomes visible and its measurement can be used to validate numerical simulations. Using similarities between electric and magnetic behaviour laws, magnetic problems can be modelled with an electric field approach and studied with an experiment that also uses an electric field.
Findings – Geometrical singularities and their effects on calculations are not always well taken into account by a finite element resolution. An adaptive mesh refinement is often required. If such mesh refinement is refused, another solution can be explored. The goal is to know the external stress distribution induced by the field. The methods only focus on this stress distribution and assume that the magnetic or electric field distribution is imprecise when it is calculated near geometrical singularities. The stress distributions suggested are verified with experiments.
Originality/value – Using new materials with particular physical properties provides a new concept of experimental validation.
Article Type:
Research paper
Keyword(s):
Mechanical testing; Experimentation; Numerical methods.
Journal:
COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
Volume:
26
Number:
1
Year:
2007
pp:
48-61
Copyright ©
Emerald Group Publishing Limited
ISSN:
0332-1649
Introduction
The design of electric systems that transform magnetic energy into mechanical energy requires a good knowledge of the forces distribution induced by the magnetic field. The calculation of such forces has been underlined by several authors and recent publications show that this subject is not closed. Using the energy method, numerical simulations can provide global characteristics such as a mechanical torque delivered by an electric motor. Looking for formulations that take into account physical phenomena also implies the design of experiments that can check them step by step. The energy method can be seen as a universal method. Electric or magnetic behaviour laws are very similar. Consequently, it should be possible to study magnetic problems that are produced into magnetostatic with an electrostatic approach and electrostatic experimentations. More precisely, magnetic field components that are calculated near geometrical singularities are not reliable; the force distribution is also unreliable. Can solutions for this problem be easily confronted to the experiment by using an electrostatic point of view? Can the duality between electric and magnetic fields provide new experimental cases?
Stress formulation and energy methods
The calculation of the forces generated by fields is a current problem (Bossavit, 2003; Henrotte, 2004; Ren et al., 1995). To solve this problem, several formulations exist (de Medeiros et al., 1999; Muller, 1990; El Amraoui et al., 2004). In this paper, the energy method is privileged. It assumes that magnetic or electric energy variation of an insulated system is equivalent to the external force work. This concept provides equations of the forces induced by magnetic and electric fields (Hameyer and Belmans, 1999; Coulomb and Meunier, 1984; Hadjali et al., 2004; Gabsi et al., 1999; Reyne et al., 1988). Hence, a body, with relative magnetic permeability μ r or relative electric permitivity ɛ r will be submitted to external forces. Equation (1) is the local force associated to the magnetic field and equation (2) is the local force associated to the electric field: Equation 1 Equation 2 However, this concept introduces very restrictive conditions of use. The material must have a linear behaviour when the preceding local formulations are used. But such restrictions allow a resolution of many problems with a similarity principle (Penman and Fraser, 1983). The following equations are behaviour laws and highlight the duality between the magnetic field (equation (3)) and the electric field (equation (4)): Equation 3 Equation 4 B, the magnetic induction and E, the electric field, are material answers to the external source field H or D. ɛ r and μ r are characteristics of the medium. Following these two equations, it is natural that formulations for the magnetic energy (equation (5)) and the electric energy (equation (6)), by unit of volume, are also very similar (Li et al., 1994): Equation 5 Equation 6 To conclude, the behaviours laws induce such similarities. They provide a new approach to study force calculations. Therefore, if numerical methods are used to calculate the field distribution and local force distribution, the experiments that are selected in order to study magnetic problems by the electric fields effects have to present the same difficulties for the numerical resolution. That means that the problem of calculating magnetic field components near discontinuities must be present in the electric approach. Thus, it is initially necessary to identify the components associated with the field that are preserved during a change of medium (Figure 1). It becomes obvious that the other components are not preserved. They provide discontinuities that establish certain similarities in the problems of numerical calculation.
Figure 1 shows that the normal component of magnetic induction is kept between two different substances. With the electric field, it is the tangential component of the electric field that presents this characteristic. Hence, discontinuities exist and are only observed with the tangential component of the magnetic induction and with the normal component of electric field. Such properties can be presented as consequences of equations (7) and (8). Equation (7) implies continuity for tangential magnetic field components when there is no current. In static case, equation (8) shows that it is the same conclusion for the tangential electric field component: Equation 7 Equation 8
Geometrical singularities
To be economically viable, equipments, like electric motors, use manufacturing processes that generate parts the geometry of which has singularities that cannot be easily interpreted by finite elements resolutions. For example, the rotor of an electric engine has many teeth and is easily produced by stacking stamped plates. Sharp angles, which are met on the top of teeth, are singularities that can intervene on the reliability of simulation results (Allegre et al., 1996). Corners remain an open problem (Cividjian et al., 2004). Such unfavourable situations must be considered as case studies. A parallelepiped plunged in a magnetic field can be sufficient; it is shown with Figure 2.
To generate a magnetic field, a magnetic core with an air-gap is often used as soon as a high field intensity is required. In the case of the electric field generation, such a process is not necessary. The disappearance of the magnetic core and the air-gap allows greater possibilities of observation or measurement. Moreover, it is very easy to build materials with high relative permittivity and weak modulus of elasticity.
Using experimental measures as validations of formulations is a process that has been used with success (Barré and Brochet, 2003; Barré et al., 2005; Park and Park, 2001; Reyne et al., 1988). For example, a magnetic material, called test material is submitted to a high magnetic field (Figure 3). An heterogeneous field distribution appears around this body (Figure 3(a) and (b)). Low relative magnetic permeability material is used and singularity effects are negligible. Some magnetic forces appear on this body and they produce deformations that are visible if this material has a weak modulus of elasticity. A numerical simulation associated to a finite element model of this body can be used to find the magnetic field distribution and consequently, the local forces that are acting on it (Figure 3(c)). With this force distribution, it is possible to calculate the final geometry of the body (Figure 3(d)). The deformation measured during the experiment and the calculated deformation are the same, the equation associated to the local force distribution is validated (Barré and Brochet, 2003). Any experiment under well controlled environment gives interesting results (Ren and Razek, 1990, 1994).
Validation of this experimental method under electric field
This experiment, using a magnetic field, shows that this experimental concept of validation can provide a result with a sufficient accuracy to discriminate formulations associated to magnetic forces induced by magnetic fields (Barré and Brochet, 2003). But in this experiment, a low relative permeability material undervalues the effects of singularities. Hence, using material with high electric relative permittivity, singularities effects encountered in finite element simulations of electric field should be similar to these encountered in finite element simulations of the magnetic field. But the first step must be to verify the accuracy of the electric force distribution calculated with the energy method without any singularity effect. A volume of test material, the geometry of which is shown in Figure 4, is submitted to the electric field generated by two conducting plates, the geometry of which is shown in Figure 5. A potential of 11 kV is set to (A) and a ground potential, 0 V, is set to (B).
A Z independent behaviour is assumed and allows a 2D finite elements resolution. Such behaviour is easily verified with the observation of the deformation along Z-direction. In Figure 6(a), a test volume is submitted to a high electric field. A camera, with a macro photography lens, is used to take a photo of the geometry with and without the electric field. A deformation is noticed and it appears that this one is constant along Z. Figure 6(b) and (c) is upper views of the test volume and the accuracy of such photos is sufficient to discover a displacement up to 1/100 mm.
With an 11 kV direct current source, the electric field distribution is calculated with a 2D model of the test bench associated to Figure 5. With this field distribution (Figure 7(a)) the local force distribution can be calculated by using the equation (4). Figure 7(b) shows such a stress distribution and Figure 7(c), the body deformation. With this geometry, which does not have geometrical singularities, a low deformation is observed. Using equipments shown in Figure 6(a), such a low deformation can be measured. A deformation of 0.1425 mm is calculated (Figure 8(a)). Figure 8(b) shows the camera; it takes a photo of the test volume with and without the electric field. (D) is a fixed probe that is also visible on photos (Figure 8(b)). This probe is used to measure the body displacement and a deformation of 0.15 mm is observed.
To conclude, material characteristics are well known, low displacements are encountered and the method used to measure them is able to show such a low displacement. Local forces that appear, when an electric field is applied, are perfectly described by the energy methods (Barré et al., 2005).
The influence of the body geometry on the field and on the local force calculations
Material behaviour and local force formulation are verified on a simple test bench. It is now possible to change the body geometry and use such body deformations to study the calculation under geometrical singularities. The distribution of the local force is highly dependent on the distribution of the field. A result, which is not in conformity with reality, can thus be associated to an imperfect knowledge of the field distribution and not to the formulation.
The geometry associated to the test bench
A parallelepiped is built using the preceding material. It is placed in the electric field generated by the system (Figure 9(a)). An 11 kV voltage is applied and the deformation is deduced from two photos (Figure 9(b)-(e)). The first one is the initial position without an electric source, and the second one is the final position when the electric source is connected. A 0.55 mm deformation is noticed. With this experimental result, the distribution of force attached to any resolution method can be evaluated.
Field and stress distributions obtained with traditional method
As this system is Z independent, the 2D resolution can be used. The Vector Field Opera 2D software is used to calculate the field distribution. When this field distribution (Figure 10(a)) is used to calculate the stress distribution, it appears that near singular points, not very realistic stress intensities are obtained. Figure 10(b) shows the local force distribution associated to the left side of the body. It appears that a manifest discontinuity of the local force exists near the singular point. This discontinuity does not disappear even if an auto adapted mesh is used. If such a force distribution is used without caution, the calculated deformation is in the opposite direction of the one coming from the experiment. In addition, if stress values around singularities are not taken into account, the computed deformation is 0.3 mm, whereas for the experiment it is 0.55 mm. Mismatching between experiment and simulation can be associated to an unreliable field distribution. It is not unusual to see derivations in field or induction during the simulation (Hameyer et al., 1998).
Simplifying assumptions associated to stress calculation and their consequences on the numerical results
Close to singular points, numerical methods used to calculate the field distribution do not give reliable results. Unfortunately, electric or magnetic fields cannot be easily measured and the methods that propose solutions in such field calculation cannot be easily verified. Increasing the mesh density is not the desired solution. Even if equation (1) or equation (2) provide good stress distributions, the results depend on the field distribution reliability. An auto adapted mesh requires an error estimator (Marmin et al., 1998, 2000; Vanti et al., 1993; Raizer et al., 1989; Song-Yop et al., 1988). This one is applied on the whole geometry and in this case, only a small part of the system is concerned. A more physical approach is thereafter used and only three simplifying assumptions are presented (a, b, c). They can be used in post-processing during the force calculation. It is not the field distribution that is desired but its mechanical effect. As it had been noticed, the effect of the corner appears in the forces calculation and it is only attached to a small number of elements, practically 3 or 4:Ha. With high relative permittivity, the electric field can be assumed to be normal to the surface of the body and electric field intensities, given by simulation, are representative. Hence, electric field components that are used in the local force formulation are: Equation 9 Hb. The tangential component of the electric field is not reliable, it is set to 0 but the normal component is kept. Hc. A linear extrapolation of the calculated distribution of force can be done to estimate this distribution near singular points. Field values associated to the element attached to the corner are not used. To calculate local forces, a linear extrapolation is done using the two nearest other elements. Only the force distribution associated to the left side of the body is presented (Figure 11). Indexes a, b and c are, respectively, associated to Ha, Hb and Hc. Even if differences are visible, especially near singular points, they do not really appear on the integration of the force distribution (Table I). On the whole left boundary, the difference is less that 3.8 per cent.
Stress integration provides numerical values that are very similar. It is usual to conclude that the effect of the corner on the global force is generally diminished (Allegre et al., 1996). But when the stress distribution is applied on the body, the deformations associated to each distribution are not similar and differences have reached 10 per cent (Figure 12).
With these three examples, three stress distributions have been produced. By applying on the body the stress distribution coming from one of these hypotheses, the deformation of the body can be calculated. If the electric field modulus coming from simulation is representative (Ha), the calculated deformation shown on Figure 12(a) is very close to the experimental value. When the tangential component of the electric field is set to 0 (Hb), Figure 12(b) shows that the predicted deformation is lower than the real one. If linear extrapolation is used to build stress distributions near singular points (Hc), the produced deformation, shown on Figure 12(c), is representative and the conclusion is that Hc is the best approach. This linear extrapolation of the force density is easily done. It uses only the elements that are near the corner to calculate the force density associated to the element attached to the corner. Keeping in mind that the mesh density is very low, this evaluation of force density can be acceptable in problems that are expensive in calculation time.
The imperfect geometry associated to this realisation, and its effect on local force distributions
Deformations and especially deformations high enough to be observed are required within such experimental validation. Many studies propose to calculate the field or the force distribution associated to a body deformation (Bossavit, 1992; Ren and Razek, 1992; Ren et al., 1995). Here, even if the displacement is not negligible, it must be compared to the accuracy of the body geometry or the accuracy of the geometry associated to the conducting plates. Even if 0.5 mm can be considered as high displacement, accuracy in the position of each element also approx 0.5 mm. Another way to study the influence of displacement on the force calculation reliability consists in studying the influence of the body position on such force calculation. If a displacement of 0.5 mm of the body does not induce variation of the local force distribution, the deformation of the body is not significant enough to modify the local distribution of force used in calculation. Hence, using two different cases, shown in Figure 13, it is possible to verify the influence of the geometry variation on this calculation.
The force distribution on the body is the same in these two cases (Figure 14); the distribution of local electric force is not strongly modified when a small displacement is applied to the test body. That means that a displacement induced by the electric field will not affect the local force calculation.
Conclusion
The geometries that contain singular points require caution when finite element methods are used to solve associated problems, such as local forces calculation. Modifying the model or using auto-adapted mesh are possible solutions. They are generally applied to the whole structure and do not focus necessarily on the identified problem. But the unreliable values of local forces are only attached to a limited area of the system. It should be possible to estimate such values with another method. Even if the problem of unreliable values is identified, its effect is not so important because what matters in the design of electrical machines is generally the global forces. Within this experiment, this effect is enhanced and it appears in this example, that a simple post-processing can provide more realistic values near geometrical singularities.
Equation 1
Equation 2
Equation 3
Equation 4
Equation 5
Equation 6
Equation 7
Equation 8
Equation 9
Fixed graphic 1
Fixed graphic 2
Figure 1In the case of two different substances, some components of magnetic (1.a) or electric (1.b) fields are maintained when the border is crossed
Figure 2Example of a rotor coming from an electric motor and showing a singularity
Figure 3Validation of the force distribution using calculated and measured deformation on a test material
Figure 4Geometry of the test body in the electric field experimental validation
Figure 5Electric field generation system and the test body
Figure 6Method used to visualize body deformation
Figure 7Application of the validation method with an electric field
Figure 8Calculated deformation and its measurement on the body
Figure 9The test bench with an electric field point of view
Figure 10Electric field and force distribution on the body without any interpretation of the geometrical singularity effects
Figure 11Local force distribution associated to the simplifying hypothesis
Figure 12Calculated deformation using force distribution associated to each simplifying assumption (it is 0.55 mm in the experiment)
Figure 13In order to evaluate the impact of the geometrical variation on the local force distribution, two study cases are taken into account
Figure 14Stress distribution associated to the left border of the test body when it is set at 15 mm from the vertical electrode (a) and when it is set a 14.5 mm from the vertical electrode (b)
Table IResults of integration of these stress distributions
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About the authors
Fixed graphic 1Olivier Barre is A&M Engineer, Doctor in Science and Professor in Electrical Science. He works in a technical school (Lycée G.Eiffel) and is also a Researcher in the Laboratory L2EP (Laboratoire d'Electrotechnique et d'Electronique de Puissance). His researches relates to the modelling of physical phenomena and their introduction in the design and the optimisation of the electrical machines. Olivier Barre is the corresponding author and can be contacted at: Olivier.barre@ec-lille.fr
Fixed graphic 2Pascal Brochet is a full time Professor at Ecole Centrale de Lille (France) and a Researcher in the Laboratory L2EP (Laboratoire d'Electrotechnique et d'Electronique de Puissance). His researches are on the design of electrical machines, particularly methodology, specific tools like optimisation, design of experiments, numerical models. Pascal Brochet is the corresponding author and can be contacted at: pascal.brochet@ec-lille.fr
