B Delinchant

D Duret

L Estrabaut

L Gerbaud

H Nguyen Huu

B Du Peloux

H L Rakotoarison

F Verdiere

F Wurtz

An Optimizer using the Software Component Paradigm for the Optimization of Engineering Systems

Keywords: Software Environments for Optimization, Optimizer, Software Components, Optimization Framework

This paper presents CADES, a framework dedicated to system design, based on optimization needs. The framework is defined with a standard implementing the software component paradigm and a pattern to use it. Indeed, this pattern details how to create and use a component (the model of the device to design). This paper highlights the component standard, a generator based on analytical expressions of the system, and an optimization service.

I. Introduction: the exploration of the software components for the sizing and optimization of systems

This paper is an investigation of a new kind of software architecture, using the software component paradigm for sizing, by using optimization approaches, engineering components and systems. In the field of software architecture, one of the previous important paradigms, widely used in the area of optimization software, is object oriented languages. If object oriented approach is indeed very powerful, one of its limit is the reusability of objects from one development project to another. The new software component paradigm is a new approach of computer science [START_REF] Szypersky | Component Software -Beyond Object-Oriented Programming[END_REF] that wants to be a solution to this problem of reusability. Based on this approach, it will be shown in the following how it is possible to build global software architecture for optimization of components and systems with the goal of reaching the following needs:  Need 1°: Need of an architecture offering solutions for the availability of models for components and systems. This means especially reaching two sub-needs:

o Need 1° a: Need of a software architecture that allows managing methods and tools for designers to create the adapted model for each component and system, with multi physical modeling constraints (magnetic, thermal, mechanical…), and with many different levels of modeling (analytic, semi-analytic, numerical).

o Need 1° b: Need of software architecture that allows capitalization and the reuse models from one project to another.

 Need 2°: Need of an architecture offering solutions for managing as much as possible optimization strategies and algorithms (stochastic, deterministic, hybrid, Pareto…).

 Need 3°: Need of an architecture offering solutions for connecting software components containing models with software components containing optimization algorithms by managing constraints of reactivity, modularity and interoperability.

II. Pattern Framework/Standard

A. Initial definitions 1. Pattern

A pattern is a model describing how to implement a general concept in different cases. It is largely used in software development with "design patterns" which defines "good ways" to program recurrent architectural concepts. The definition of a pattern is the result of an archetype appearing (a primitive model) from different applications, then it will be used in next applications.

In the following, two kind of pattern will be detailed. The first will define the way of using the CADES Framework, with its 3 distinct entities (Generators, Components, and Services). The second will define the way of extending CADES functionalities through the ICAr software component standard.

Black box / White box

Models are generally contained into tools (like FEM simulation software, algorithms and equations implemented in a programming language), or standardized language (like Modelica1 or VHDL-AMS2 ). First kind can be considered as black box and second as white box. Both produce outputs thanks to input values, corresponding to the direct problem solving. Black boxes and white boxes may then be described as follows:

 Black box [START_REF] Delinchant | Mixing of FEM and Analytical Modeling for the Preliminary Design of a Transformer[END_REF]: brought by cybernetic, this concept enables manipulation of object without knowing its internal constitution. Only relations between inputs and outputs can be observed corresponding to its behavior.  White box [START_REF] Allain | Capitalisation and treatment of models for the optimisation of electric drives[END_REF]: It is characterized by a set of structured and ordered data. It defines outputs values depending on inputs ones, but also how to obtain these outputs, corresponding to its nature. As defined in Fig. 1 (b), component computes outputs from its inputs and computes also sensitivities (partial derivatives of outputs depending on inputs). This last capacity is important especially for optimization requirements as it will be shown in the following.

CADES framework owns several generation tools which create components from the following designer information:

 Multiphysical algebraic equations [START_REF] Duret | A Generator of Software Components dedicated to the Optimization of Engineering Systems[END_REF].

 Electromagnetic reluctance networks [START_REF] Du Peloux | Automatic Generation of Sizing Static Models Based on Reluctance Networks for the Optimization of Electromagnetic Devices[END_REF].

 Electromagnetic geometrical representations [START_REF] Rakotoarison | Methodology and tool for generating semi-analytical models used to pre-design electromagnetic MEMS (Mag-MEMS)[END_REF].

After creation, components can be used into CADES services like computation or optimization [START_REF] Wurtz | An Optimizer using the Software Component Paradigm for the Optimization of Engineering Systems[END_REF].

Optimization service offers different algorithms like SQP (sequential quadratic programming) [START_REF] Powell | On the quadratic programming algorithm of Goldfarb and Idnani[END_REF], genetic, hybrid, global, or mixed discrete-continuous.

This pattern is completed by specific tools:

 Composer [START_REF] Delinchant | A component-based framework for the composition of simulation software modeling electrical systems[END_REF] (use and generate components): allowing black box component composition to create system model.

 Projectors: translate components to other standards or project them into commercial environments (Matlab, etc.).

This pattern, which answers a part of needs 1°, 2° and 3°, and has to be applied to extend the CADES framework with other generators and other services based on the actual standard of components. Moreover, as it will be shown in the next paragraph, the extension of the component's standard induces far more extensibility of the framework.

C. Pattern for the design of software components

A pattern has also been defined for the standardization of components. ICAr (Interface for Component ARchitecture) represents the bases of our black box component and is issued from Vincent Fisher thesis works [START_REF] Fischer | Using Software Components for Multi-disciplinary Optimization[END_REF]. ICAr offers a unique interface to interact with all component capacities which depends on design requirements or needs. Such a component is called "multi-facets", where each facet may represents behavioral model, sensitivity computation, temporal simulation, accurate model, model visualization… Facets may also represent a part of the system to be simulated (electromagnetism + mechanic + thermic + economic + …). This pattern, which answers part of needs 3° and 1°b, specifies two major objects which have to be used for the definition of ICAr components. Component interface is the anchor point of an ICAr component. It defines input parameters of the system as well as all available facets which are defined and useful for the device design.

public

III. CADES v1.0: Component Architecture for the design through optimization of engineering systems

A. Application in the area of sizing and feasibility studies CADES v1.0 has been defined to be used as a design framework of engineering systems but especially for the preliminary design stage. Indeed, the first available generator was an algebraic equation generator, since it answers multiphysic system modeling needs, and also offers a quick reactivity to test many solutions.

During early stage of design, the problem is generally not fully defined (ill defined problem). A solution to converge to a good problem definition is to start the design by finding potential solutions and applying constraints to see how these solutions react (see Fig. 4). Then the problem formulation converges during solution convergence that requires many tries and then a specific modeling and optimization framework.

Problem formulation

Potiential solution definition

Test of constraints satisfaction As previously said, Component Generator creates a software component from a description (a model) of the device. Then, the creation of the model file do not requires any computer science skills, because the designer gives its equations and functions modeling the device as if he writes it plainly in a sheet of paper.

Moreover, as generator produces computation code, it also derivates symbolically the equation and produces also the sensitivity computation code. This full automated process avoids human errors and lets designers concentrate on complex task like finding modelling equations.

Component Generator provides an advanced support for special behavior. When mathematical expressions are inefficient for an accurate modeling, the designer can describe this behavior with various executable codes (for example C, java).

Based on previous tools [START_REF] Fischer | CoreLab: a component-based integrated sizing environment[END_REF], our generator architecture allows modularity and adaptability. Firstly, thanks to XML technology, just one XML file gathers all instructions supported by the generator. So, a modification of the parsing step simply consists in modifying this XML file. Secondly, information built after the analyzing step does not depend on a specific programming language. As a consequence, projection of this model to several executable codes (Java, C, C++) shall be possible more easily. Lastly, some disconnection between generation and packaging step allows various packaging for the software component. Nowadays, it is compliant with the open standard developed in our lab and creation of other component such as S-function 3 is possible.

CADES component

ModelSolver

ICAr pattern interfaces Generated code for a specific device Component 1.0 interfaces

Component Optimizer: an optimizer using software components

The optimizer service answers need 3°: connecting as easily and quickly as possible software components containing models to software components containing optimization algorithms [START_REF] Magot | Application aux composants magnétiques et aux filtres passif[END_REF].

That is why, the optimizer we have developed, has itself a software component architecture shown on fig. 7 . It allows:

 to plug the software component containing the model, and its associated sensitivity, in the optimizer.

 to plug two other kinds of software components as defined in the following.

The first kind of software component contains the optimization strategy. This allows immediately connecting the model to an optimization strategy. Until now we have developed optimizer components that contain algorithms like: gradient Sequential Quadratic Programming optimization (SQP), genetic optimization, global deterministic optimization, Pareto approaches, branch and bound algorithms [START_REF] Magot | Application aux composants magnétiques et aux filtres passif[END_REF]… It should also be noticed that those optimization components can be recursively plugged to each other in order to quickly create hybrid algorithms like a genetic combined with a SQP algorithm and this again without any programming skills for the final user of the optimizer.

The second kind of component is a graphical component, for the drawing of the parameterized picture of the structure.

It is not the aim of this paper to gives details about those components. What is interesting to notice is that again they are implemented by using patterns, concepts and technologies of software components previously described.

This architecture allows generically connecting new kinds of optimization algorithms to existing models without any need of programming, and so we reach need 3°.  definition of the objective function (minimize / maximize).

 starting of the optimization.

 plots of the evolution of the parameters during the optimization.

 drawing of the evolution of the device structure during the optimization.

C. Illustration of the CADES framework on an example

The use of the framework is described here on the example of the sizing of a three phase power transformer. As shown on figure 8, the physical model of this transformer is described in the SML language in the Component Generator tool. This generator is then able to automatically generate the CADES component  to draw plots of outputs values as functions of input values.

 to make sensitivity computation. You can especially see, on fig. 9, the computation of a jacobian matrix.

This service has for goal to do as many as possible computations on the component, especially in order to verify its right behavior.  to plug the software component containing the model, and its associated sensitivity.

Component Calculator

 to plug also the two other kinds of software components that this optimizer is able to manage.

First kind of component is the software component containing the optimization strategy. As previously said, much kind of strategies is available. Here, on the example, a classical deterministic algorithm is used:

Sequential Quadratic Programming approach (SQP). The software component encapsulates the SQP algorithm VF13, available in the Harwell Subroutine Library (See [START_REF] Hsl | VF13 subroutine[END_REF]), described in [START_REF] Powell | On the quadratic programming algorithm of Goldfarb and Idnani[END_REF]. This algorithm is written in FORTRAN, but there were no technical difficulties to compile it and to integrate it in the optimization software component.

The second kind of component is the graphical component, for the drawing of the parameterized picture of the structure of the transformer.

Component Optimizer

IV. Perspectives of the framework

A. Community of users creating standardized software components of engineering components and systems

Since the software component can be standardized, especially those containing models of components and systems, they can be stored and reused in order to reach need 1° b. So our aim is this capitalization, but also the creation of community of users, that have in common to exchange the same standardized components. This should allow creating communities of users sharing knowledge contained in the software components, and their skills for creating those models. With these communities, models of the engineering components and systems for optimization can be found and used more quickly.

B. Implementation and extension of the software pattern

As defined in this paper, two kinds of pattern have been proposed. First was the CADES framework pattern which describes how generators, component and services interact. Second is ICAr component pattern which describes how Component and Facets interact.

Both are a way to extend the CADES framework for new needs. Thus, scientific or industrial people who want to integrate modeling techniques can do it in a generator to produce components. Then they can take advantage of all available services (simulation, optimization, composition …). Moreover, services may also be developed like optimization services or tolerancing services to take advantage of existing generators but also to bring new design tools to designers with all existing and future community models.

If component v1.0 does not answer all requirements to develop specific pairs of generators/services, ICAr pattern can be used to extend the standard with new functionalities as it was already done with component visualization facet or temporal simulation facet. These extensions can be optional or required facets of component depending on which services are used. These extensions can be on facets functionalities (computing, sensitivity, visualization, documentation, light computing / heavy computing…) or on data types which are real number in Component v1.0. But a component may have for example two computation facets, one with input/output real number format and another with interval format to be used with specific global optimizer based on interval propagation [START_REF] Delinchant | Interval Arithmetic for the Design of Electrical Engineering Systems[END_REF].

V. Conclusions

Software component architecture is really a new paradigm in the implementation of optimization software.

It offers new opportunities to quickly connect models to algorithms, through standardized interface. It allows easy combination and offers a way to capitalize and reuse pre-existing models as well as pre-existing algorithms. The CADES framework which has been presented in this paper is an opportunity for designers and scientists to improve design methodologies by developing generators, capitalizing model and modeling skills around a modeler community, and by developing new services.

VI. References

Fig. 1 -

 1 Fig. 1 -(a) Generators create (b) components / (c) Services use (b) components.

Fig. 2 -

 2 Fig. 2 -Multi-facets component: (a) multi-information, (b) multi-physics.

FacetFig. 3 -

 3 Fig. 3 -ICAr interfaces: (a) Component, (b) Facet.

Fig. 4 -Fig. 5 -

 45 Fig. 4 -Early stages of a design process.

Fig. 6 -

 6 Fig. 6 -CADES Component v1.0 implementation example.

Fig. 7 -

 7 Fig. 7 -Structure of the optimizer: software components containing models are plugged together with software components containing optimization algorithms.

v1. 0 ,Fig. 8 -

 08 Fig. 8 -Use of the generator of the CADES framework: example of the optimization of a three phase power transformer.

Fig. 9 -

 9 Fig. 9 -Use of the software component of the three phase transformer in the component Calculator.

  . O.A. G.S. G.S.

Fig. 10 -

 10 Fig.10 -Use of the software component of the three phase transformer in the component Optimizer.

Table 1 :

 1 Black Box versus White Box.

		Black Box	White Box
	capitalisation /	Parameterized computable capacities	Customizable modelling
	diffusion	with knowledge protection	with knowledge diffusion
	composition	Weak coupling, multi-method allowed	Strong coupling only if same language
		3. Software Component as Black box implementation
	Software component are based on component paradigm which is largely used in electronic where electronic
	systems are realized by composition of elementary components. A software component is defined as an
	autonomous deployment entity, which encapsulate computer codes and describes by interfaces its interactions
	which are allowed with other components [1]. Software components are implementations of black box
	concept and are well suited to our engineering model encapsulation.
	In the following, a black box component is specified which is based on Sun Microsystems's JavaBeans
	software component standard, especially for portability requirements.
	B. Pattern of framework for environment using software component
	CADES (Component Architecture for Design of Engineering Systems) environment has been developed
	around a software component dedicated to optimization. The following design pattern formalizes the creation
	and use of component during design process. First of all, components (models) are created by "generators"
	and then can be used into "services".

http://www.modelica.org/

http://www.vhdl.org/


L Mokhtari
email: lounes.mokhtari@g2elab.grenoble-inp.fr

B Delinchant

T Chevalier

J-L Coulomb

Comparing Weak and Strong PEEC-MoM Coupling

Keywords: PEEC method, MoM method, weak/strong coupling, iterative methods

The aim of this paper is to compare strong and weak coupling of the PEEC and MoM methods applied to the modeling of magnetoharmonic problem, and to generalize the weak coupling approach. MoM and PEEC are dedicated to the modeling of specific parts of the overall device. As an example of validation, we have chosen the modeling of a simple transformer. Resolutions of several weak couplings are compared to the strong coupling approach. A new strategy of weak coupling is proposed to improve the time of resolution.

I. INTRODUCTION

T HE MODELING of multiphysic systems needs powerful modeling tools and simulation methods. In this context, a specialized electric and magnetic method will be coupled in order to model an electromagnetic device.

The PEEC (Partial Elements Equivalent Circuit) approach has been proved to be excellent for the modeling of many ranges of cabling and interconnections in electric structures. The ferromagnetic materials can be modeled by FEM or integral method like Method of Moments [START_REF] Chadebec | A Review of Magnetostatic Moment Method[END_REF] [START_REF] Antonini | PEEC modeling of linear magnetic materials[END_REF]. Couplings of these methods have been done successfully in [START_REF] Le Duc | Couplage de la méthode moment magnétique avec la mthode PEEC pour la rsolution des problmes magnètodynamique[END_REF][4] [START_REF] Tran | FEM-PEEC coupled method for modeling solid conductors in the presence of ferromagnetic material[END_REF].

In the following sections, different coupling strategies of PEEC and MoM method will be compared. The figure below (Fig. 1) shows the strategies commonly used for coupling programs.

PEEC MoM

PEEC MoM

PEEC MoM

MoM PEEC Strong Coupling

Gauss-LU Weak Coupling

Optimal Weak Coupling

Gauss-Seidel Weak Coupling

Tuning Tuning Direct Methods Direct Methods We have proposed an Optimal W eak Coupling strategy based on tuning models to improve the simulation performances. Both iterative and direct methods will be to solve each of strong and weak coupling.

II. PEEC AND MOM METHOD

A. The PEEC method

The principle of the PEEC method is to decompose electric circuits into several parts, then calculate the mutual between them also the contribution of each of these elementary circuits to the total inductance [START_REF] Tran | FEM-PEEC coupled method for modeling solid conductors in the presence of ferromagnetic material[END_REF]. The partial mutual between two electric conductors i and j is calculated as follows (1)

M p ij = µ 0 4.π • 1 S i S j Vj Vi 1 r i • dV i • dV j (1)
From ( 1), we build the impedances matrix ( 2)

Z m = R 1 + jωL 1 jωM 12 jωM 21 R 2 + jωL 2 (2)
The impedances matrix can be obtained directly with using software as INCA3D to the modeling of the conductors and a specific solver to extract the impedance matrix [6].

The source voltage formulation is defined as follows (3)

U = Z m • I + f em (3)
where f em is the well known electromotive force induced by the electric current I in the conductors.

B. The Method of Moments

The MoM is an integral approach well adapted to the modeling of magnetic and radiating regions with simple geometries [START_REF] Chadebec | A Review of Magnetostatic Moment Method[END_REF]. The global elements magnetic field is given by ( 4)

H = H ext + H red (4)
where H ext and H red are respectively the external magnetic field generated by the source field and the reaction of the material to the source field. In case of linear and unsaturable material, the discretization gives us the magnetic field in any point P of the material [START_REF] Tran | FEM-PEEC coupled method for modeling solid conductors in the presence of ferromagnetic material[END_REF].

H(P ) = H ext (P ) + 1 4π Vmag (3 M . r r 5 r - M r 3
).dV mag [START_REF] Tran | FEM-PEEC coupled method for modeling solid conductors in the presence of ferromagnetic material[END_REF] With making M (Magnetization of the element P ) out of the second term of the equation ( 5), we write the formulation of the magnetic field for all mesh elements (6)

[H] = [H ext ] + [g] • [M ] (6)
The matrix [g] is called the magnetostatic matrix, it represents the interactions between the material elements. It can be obtained directly with using some softwares specialized in the modeling of magnetic devices like LOCAPI [6].

C. Coupling terms

Lenz-Faradays law

The impact of the material magnetization [M ] on the conductor is the induced electromotive force [f em]. The vector potential created by [M ] is given by ( 7)

A = µ 0 4π Vmag M × r ||r|| 3 • dV mag (7)
The link between the potential vector and the electromotive force is given below (8)

f em = jω lc A • dl c (8)
The matrix formulation of ( 8) is

[f em] = [LF ] • [M ] (9)
where [LF ] is calculated with the geometric parameters of the system r and V mag .

Biot-Savarts law

The impact of the conductor current I on the material is the excitation magnetic field H ext . The magnetic field created by the electric current I is given by (9)

H ext (P ) = 1 4π V cond J s × r ||r|| 3 • dV cond (10)
where J s is the uniform density of the electric current I in the conductor. The expression of H ext in (10) becomes

[H ext ] = [BS] • [I] (11)
where [BS] is calculated with the geometric parameters of the system.

III. THE PEEC-MOM COUPLING

The modeling of electromagnetic devices requires the modeling of the electrical and magnetic part, but also the interactions between them. This interaction is usually called "Coupling".

A. Weak Coupling

In this coupling, each method is available separately (Fig. 2). The electric and magnetic systems are solved successively, the resolution of one is injected into the other and so that until achievement of the accuracy requirements.

B. Strong Coupling

The strong PEEC-MoM coupling developed in [START_REF] Antonini | PEEC modeling of linear magnetic materials[END_REF] aimed at grouping in one global matrix the overall modeling as detailled below.

We replace H ext in (6) by its formulation in (11), we obtain

[H] = [BS][I] + [g][M ] (12)
The material law says that (µ r -1)H = M so if we multiply the equation ( 12) by (µ r -1), we obtain

([I d ] -(µ r -1)[g])[M ] -(µ r -1)[BS][I] = 0 (13)
Lets consider :

([I d ] -(µ r -1)[g]) = [M M ] -(µ r -1)[BS] = [BS m ]
then (13) becomes

[M M ][M ] + [BS m ][I] = 0 (14)
The electric model ( 15) is built by replacing the formulation of f em in (3) by that of (9)

[LF ][M ] + [Z m ][I] = U (15)
Now, it is possible to group the magnetic (14) and electric (15) model in one global matrix system, which represents the overall model of the system (16)

M M BS m LF Z m • M I = 0 U (16)
The resolution of the system (16) gives as M I vector of unknowns composed by magnetizations of the material elements and the electric currents in the conductors, according to the sources vector 0 U .

IV. APPLICATION A. Device description

The application example is the modeling of a simple transformer (Fig. 2) with 2 current unknowns and a variable number of magnetization unknowns from 500 to 2000. Two square conductors are placed on two parallel planes and a magnetic core is placed through their center. The first conductor is powered by an alternative source voltage (1V, 1kHz). The second conductor is short-circuited.

V = 1volt f = 1kHz

Fig. 3. Geometry of the test device.

Note that the coupling models have been validated by a finit element approach. For the same results of currents and magnetizations, the time resolution and convergence between the strong and weak coupling are compared.

B. First Results

The curves of the first results are plotted in the figure below (Fig. 4).

Conclusion1

For a little mesh elements, the strong coupling using direct methods is a good solution but for a fine mesh, the iterative method like BiConjugate Gradient Stabilized Method is better adapted.

The Gauss Seidel method combined with the weak coupling shows very good performances comparing to the bicgstab method because the subsystems conditioning in the weak coupling is better than the conditioning of the overall system in the strong coupling.

C. Convergence Study

Here, the convergence and the stability of the different methods are studied and discussed. About the stability of the methods, oscillations appear when using the LU-Gauss method (Fig. 4) and the convergence of the model is delayed until the 10 th coupling iteration, unlike the Jacobi and Gauss-Seidel method. The oscillations persist even with using initial values very close to the truth values of magnetization and electric currents.

Conclusion 2

The weak coupling using iterative methods like Jacobi and Gauss-Seidel are more stable and converge faster than WC LU-Gauss method. The convergence of the WC LU-Gauss can be improved with using a parametric relaxation on the magnetization and current.

D. Optimal parameters

In the weak coupling built before (Fig. 1), for each coupling iteration, we compute only once the electric and magnetic model. We add two tuned loops to solve more than once the electric and magnetic model (Fig. 6). Now, for one coupling iteration, we make iterM and iterI iterations on respectively the magnetic and electric model. A test for 2000 mesh elements was performed to identify if there is an optimal number of iteration can be made locally on the electric and magnetic model to reduce the time resolution. From (Fig. 7), the optimal number of iteration that reduces the most the time resolution is 2 for Gauss-Seidel method. An algorithm was implemented to find these parameters so the overall model is automatically tuned with the optimal parameters.

E. Improvement of time resolution

The overall model of the system is tuned with the optimal parameters and solved. In the figure below (Fig. 8), we have plotted the new curves of time resolution regarding the number of mesh elements and given the number of coupling iterations. In the same figure we have let appear the curves obtained before in order to compare between them.

Conclusion 3

By using the optimal parameters, we have reduced the time resolution of the model comparing to the previous weak coupling.

We have also minimized the number of iterations on the coupling loop which is very important when coupling softwares on network and where an iteration on coupling takes a long time.

V. CONCLUSION Different strategies of weak coupling of the PEEC and MoM methods have been compared with the strong coupling. A new strategy of weak coupling (Fig. 6) is proposed, the results obtained prove that our weak coupling is: 1) better in term of time resolution than the strong one.

We also improved the gain of the first weak coupling (Fig. 2) by 27% and 17% for respectively Gauss-seidel and jacobi methods;

2) more stable than the weak coupling using direct methods; it speeds and relaxes the convergence of the model and

3) minimizes the iteration on coupling loop, which is very important in practice where we couple programs on network.

Adding to these gains, this strategy presents other advantages like, easy understanding because it repeats faithfully the evolution of physical phenomena in the system, more flexible, parallelizable tasks, and gives us the possibility to reuse the codes developed.

Fig. 1 .

 1 Fig. 1. Coupling strategies.

Fig. 2 .

 2 Fig.2. Algorithm of weak coupling iterative approach.

Fig. 4 .

 4 Fig. 4. Performances of the methods.

Fig. 5 .

 5 Fig. 5. Convergence of LU-Gauss, Jacobi and Gauss-Seidel method.

Fig. 6 .

 6 Fig. 6. Adjustabe Weak coupling loop.

  WC Jac [25 25 25 25] WC GS [16 16 16 16] WC G-LU [20 20 20 20]

Fig. 7 .

 7 Fig. 7. Time resolution of WC G-S regarding the local iteration number.

Fig. 8 .

 8 Fig. 8. Performances of the methods tuned with optimal parameters.


Analytical calculation of interaction forces between two cylindrical magnets is a difficult problem. Solutions can be obtained by elliptic integrals or by numerical computation. Thanks to the development of the analytical calculation for parallelepiped magnets (interaction energy, force and torque), the calculation of the force component between two cylinders can be obtained in analytical form by decomposition in a finite number of elements.

This type of calculation allows the calculation of many systems that uses the forces of interaction between cylinder magnets. For example it can be used to directly calculate the force exerted in permanent magnet bearings in non-centred position.

I. INTRODUCTION

Recent works on the analytical calculation of forces between permanent magnets allow calculating the interactions between magnets of parallelepiped shapes. The forces, interaction energy and couples can be expressed as analytical expressions, parameterized by the geometrical dimensions and magnetization of the magnets [1 to 4].

However, the problem of rod-shaped magnets has never been analytically solved. The simpler problem of the 3D calculation of the magnetic field is made with elliptic integrals. This field calculation needs two integrations. For the calculation of interaction energy and forces, four successive integrations are necessary.

We will present an innovative analytical method, to calculate the interaction forces between two magnets of cylindrical shapes. The final analytical expression is a series with a limited term number. The accuracy depends on the number of elements.

This method allows calculating the interaction forces in three directions, as well as energy interaction, through a relatively simple analytical expression.

II. CALCULATION PRINCIPLE

Npm=4

Npm=10 Npm=20

Figure 1 :

 1 Figure 1: Principle of cutting of the cylindrical magnet in parallelepiped elements.

Figure 1

 1 Figure1shows the principle of the cutting of the cylinder in parallelepiped elements. The first case corresponds to a cut in 4 parts, the second 10 and third 20. The number of elements is given by the parameter Npm (Number of elementary permanent magnets).III. INTERACTION ENERGYWe give in the page 2 of this summary only the expression of the energy of interaction. The forces are then obtained by derivation of this energy.IV. EXAMPLE OF CALCULATION

Figure 2 :

 2 Figure 2: Calculation of the forces components for a magnet moving above a fixed one.

Figure 2

 2 Figure2shows a direct example of use of these analytical results. It is the calculation of forces between two cylinder-shaped magnets. The two magnets have a diameter of 10 mm and a height of 10 mm. The second moves horizontally above the first and the distance between the magnets is 5 mm. Polarization J is

The parameters are given by: ( ) ( ) ------------------------------------------------normalized to 1 tesla. The results show the values of the three force components, Fx, Fy and Fz.

The accuracy of analytical results is directly related to the number of elementary parallelepiped components used for each of the magnets. The complete study of the accuracy shows that one can easily obtain a relative error less than 1/1000, when the number of elements is sufficient. For this example of two cylindrical magnets, the precision of 1/1000 is achieved when the number of elements is about 20, even for a gap down to 2 mm.

V. APPLICATION TO MAGNETIC BEARING

CALCULATION

For example, the calculation of the force exerted in a radial magnetic bearing is a relatively complicated. The permanent magnet magnetic bearings are composed of two rings in magnetic repulsion. Each magnet may be likened to two nested cylinders, representing the external volume, the other representing the internal volume. The problem of calculating the interaction forces between the two rings can be treated as one of four cylinders in interactions. The radial force can be expressed as an analytical expression involving a limited set of sums .

SUMMARY AND CONCLUSION

The calculation of interaction forces between two cylindrical magnets had never been resolved so far as analytical. Following the development of the analytical calculation of parallelepiped magnets, the calculation of these forces between two cylinders can be obtained in analytical form by decomposition to a finite number of elements.

This type of calculation allows the calculation of many systems that uses the forces of interaction between the cylinder magnets. For example it can be used to directly calculate permanent magnet magnetic bearings.