E Rius

J Ph

Coupez

S Toutain

C Person

THEORETICAL AND EXPERIMENTAL STUDY OF COMPENSATED BRIDGES IN COPLANAR TECHNOLOGY: FLEXIBILITY OF THE COMPENSATION METHOD

Keywords: coplanar transmission lines, bandpass filters, millimeter-wa¨e circuits

This paper describes a systematic study of compensated bridges in coplanar technology. If bridges are necessary for filtering the parasitic coupled slot-line mode excited by coplanar discontinuities, they modify the electrical characteristics associated with the fundamental e¨en mode. Based on a low-pass filter approach, a ¨ery simple technique is used to compensate for this parasitic effect. Here, emphasis is placed on the flexibility of this compensation technique, which allows us to reduce the technological constraints. Se¨eral solutions based on this approach are presented in the case of transmission lines, discontinuities such as tee junctions, and classical distributed bandpass filters. This study is ¨alid for air bridges in classical coplanar technology, as well as for integrated dielectric bridges in multilayer coplanar technology. The concepts de¨eloped in this paper are illustrated theoretically and experimentally on multilayer de¨ices o¨er the 1᎐50 GHz frequency band.

I. INTRODUCTION

Many studies have shown that the coplanar waveguide can be considered as an alternative to microstrip lines in the milwx limeter frequency range 1 . Nevertheless, the problems associated with the present shortage of mature equivalent-circuit models and the problems related to the suppression of the parasitic coupled slot-line mode with bridges are the main drawbacks of this technology. For this last point, additional technological processes are needed to make bridges. Moreover, the bridge excess capacitance can be considered as the dominant performance-limiting factor at millimeter-wave frequencies. So, in order to avoid such a problem, several authors have proposed a very simple compensation technique based on the association of a high-impedance line with the wx low impedance presented by the bridge 2᎐5.

The present paper is focused on the flexibility of this compensation techniques. In fact, by taking a classical impedance match approach equivalent to a low-pass filter approach, it is pointed out that the length of the bridges as well as a precise positioning are not critical points.

We illustrate the concepts developed in this paper only in multilayer coplanar technology, but the validity of the compensation technique for air bridges in classical coplanar technology along with integrated dielectric bridges in multilayer coplanar technology has been already demonstrated. Flexibility will be illustrated first by matching bridges of different lengths on 50 ⍀ coplanar lines. Second, the compensation of tee-junction discontinuities will be considered. The case of cross-junction discontinuities, which we have also studied, although not reported here, give similar results. Finally, theoretical and experimental bandpass filter responses will be compared on considering, first, noncompensated bridges, and then compensated ones.

II. PARASITIC INFLUENCES OF THE DIELECTRIC BRIDGES

Both air and dielectric bridges introduce a shunt capacitance to ground due to the coupling between the CPW center conductor and the bridge span. Obviously, the characteristic impedance and the associated effective dielectric constant of the CPW transmission lines are significantly modified in the bridge region. Taking into account the low dispersive characteristic of the coplanar technology, in a first approximation, these variations can be calculated with a static method like, wx for instance, the finite-difference method 6 . In the case of wx an air bridge, Weller et al. 5 have shown that the excess capacitance is about 1.5 times the value expected from a simple parallel-plate approximation. From a numerical point of view, one should note that the presence of a bridge results in a sharp discontinuity for the coplanar waveguide. Fortunately, the overall length of a bridge is usually small. As a result, the influence of one single bridge on the transmitted RF signal remains negligible up to very high frequencies. This frequency depends both on the length of the bridge and on the transverse dimension of the line. However, practical coplanar designs require a large number of bridges, which results in an accumulation of parasitic effects on the circuit and, consequently, deteriorates its electrical performance Ž. Fig. 7 .

III. BRIDGE-COMPENSATED APPROACH

A. '' Low-Pass Filter'' Concept. In terms of technological feasibility, the best way to compensate for the excess capacitance is to introduce a length L of a high-impedance line on either Ž. side of the bridge Fig. 1 . Using ideal transmission-line theory for an impedance match, L can be easily calculated. In this way, the bridge capacitance is transformed into a Ž. low-pass filter structure L᎐C᎐L: T model with a very high cutoff frequency. The configuration based on the use of two bridges associated with a high-impedance line also can be a By comparison with the solution that consists of only reducing the width of the CPW line beneath the bridge in order to offset the decrease in impedance affected by the bridge, the main advantage of the compensation technique lies in the reduction of technological constraints: it requires neither very tiny strips nor a very precise positioning of the bridge. nique is its great flexibility. In fact, by using the low-pass approach, this method is valid whatever the bridge length. As the length of the compensation varies with respect to that of the bridge, only the cutoff frequency of the compensation structure is modified. To validate these concepts, we have built two dielectric bridge lengths on 50 ⍀ coplanar waveguides. The technological parameters for the technology used in this paper are s 9.6rh s 635 m for the alumina r substrate and s 4.3rh s 40 m for the upper dielectric r layer. The 50 ⍀ coplanar waveguide corresponds to a 80 m strip width and a 55 m slot width.

B. Flexibility of the Compensation Illustration on a 50 ⍀ Transmission Line. Another advantage of the compensation tech-

The bridge lengths are chosen in order to demonstrate the compensation techniques. The two bridges used have lengths Ž. of 135 and 300 m, respectively Fig. 2, curves 1 and 2 . It is obvious that such lengths will not be used in common prac-tice. The lengths L of the high-impedance compensations are calculated by using a circuit model coupled with a static analysis. For the two circuits, the experimental return-loss levels obtained at 40 GHz are better than y20 dB; these experimental results are in good agreement with theoretical ones.

IV. COMPENSATED BRIDGES AND JUNCTIONS

A. Compensated Bridges on Tee Junctions. The same approach as above is also applied to improve the millimeter-wave performance of tee-and cross-junction discontinuities. The use of bridges to suppress the excitation of the coupled slot-line mode in such geometries is a common practice. Bridges, nevertheless, modify the electrical parameters of the structure, which results in a shift of the resonator frequency response. A compensation technique thus allows one to modify the frequency in order to obtain the required value.

dielectric layer 1 F R E Q U E N C Y (G H
We will now deal with tee-junction discontinuities, but the conclusions of this study are also valid for cross-junction discontinuities and bends.

Ž. By using a very simple model model 0 , the structure 0 is designed to obtain a resonant frequency equal to 30.4 GHz Ž. Fig. 3 . As in the case of a transmission line, the model used is a circuit model coupled with a static analysis. Losses and eventual parasitic modes are not taken into account. On the other hand, according to the classical lumped model, in a first approach, the influence of the junctions is considered by properly defining the reference plane of the different connected lines at the junctions. In the model 0, the bridges are ignored.

A theoretical study and measurement are carried out to examine the behavior of such a structure when the strip width is modified at the junction level. It can be noted that the presence of the bridges is necessary, and provides a Ž. decrease of resonant frequency Fig. 4 . By modifying the junction geometry, i.e., by reducing the strip width, one can compensate for the bridges, and consequently increase the resonant frequency up to the desired value. Figure 4 also presents experimental results concerning different degrees of the compensation technique at the junction discontinuity. Without any compensation for the dielectric bridges, the Ž. measured resonant frequency is 23.54 GHz structure 1 . In structure 2, a first compensation is obtained by modifying the center of the junction with a reduction of the strip width. The frequency value measured is 26 GHz. In structure 3, the strip width is reduced at the junction center and under the bridges: a resonant frequency of 29.42 GHz is measured. The inductive character of the modified junction is noticeable. Finally, structure 4 is characterized by the use of reduced strip-width lines on a correctly chosen length, both on resonators and access lines. In this case, the measured resonant frequency is B. Flexibility of the Compensation Method. According to these results, along with the method flexibility, one can thus accurately design a resonator at a fixed frequency. As the compensation technique can be applied to many cases, e.g., for different bridge lengths as well as different numbers of bridges, various topologies of resonators are available for a given response. We have also tested different tee junctions; Figure 5 displays some experimental data. For example, four identical-length resonators with different bridge layouts and compensation techniques have been initially designed to give the same frequency response in both magnitude and phase.

The experimental results obtained with these devices show that the response is the desired one, and that accuracy is good whatever the layout.

In structures 1, the bridges are entirely compensated by Ž. the inductive junction center Fig. 5 . In structures 2 and 3, bridge lengths are longer than in the previous case: we thus add compensation. Finally, in structure 4, the single bridge is replaced by dual bridges.

Similar experiments performed on cross junctions and on open resonators gave identical results, and thus confirmed the validity and flexibility of the compensation technique. nificantly modifies their resonant frequency. In fact, by simply changing the resonator length, it is easy to obtain the desired resonant frequency. However, since the phase response is modified in this case, it thus differs from the theoretical one.

For instance, for a r4 short-circuited resonator, this g modification enabled us to adjust the first resonant frequency, but not the other ones that correspond to multiples of r4. By correctly using the compensation technique, the g response of the compensated resonator will exactly follow the ideal-model response. To illustrate this idea, some experiments are carried out on a 15.5 GHz resonator. Theoretical and experimental data are presented in Figure 6.

Graph A shows that bridges shift the first-and second-zero reflection frequency. By simply modifying the resonator length, only the first-zero reflection frequency is corrected, whereas the use of the compensation technique allows one to obtain the desired theoretical response over the whole frequency band. Experimental and theoretical data obtained Ž with a shortened resonator length are compared curve B, .Ž . Fig. 6 . Curve C Fig. 6 illustrates the comparison between experimental and theoretical data when a compensation technique is used. The good results observed validate the proposed technique.

V. APPLICATION TO MILLIMETER-WAVE BANDPASS FILTER

Finally, to validate both the above compensated-bridges concepts and flexibility, we designed a bandpass filter, which wx consisted of a classical short-circuited stub bandpass filter 7 ; it is implemented by using multilayer coplanar technology. This kind of wideband filter is chosen in order to avoid sensitivity problems all over the design. The electrical characteristics of this fifth-order filter are 31.5 GHz central frequency and 61% relative bandwidth. Figure 7 compares the ideal electrical response of this filter with the theoretical electrical responses of two identical filter configurations designed with 60-and 120-m-long uncompensated bridges. The responses display strong degradations in the bandwidth. Return losses are greater than y12 dB, and the central frequency is shifted. These simulations illustrate the need to limit the parasitic effects of the bridges by using an appropriate compensation.

In the final prototype, bridges are thus compensated and their lengths chosen in this case to exceed 100 m. Indeed, the length of the associated compensation is calculated in relation to the bridges one, which explains why the latter is not a critical point. Obviously, technological constraints are drastically reduced.

The experimental response of this filter presented in Figure 8 fits well with the desired response, and thus validates our approach.

VI. CONCLUSION

This work has dealt with the compensation of excess capacitance presented by bridges in coplanar technology. This compensation technique, based on high-impedance association, is a well-known method. Here, we only emphasized the highflexibility degree of this technique that, finally, allows one to significantly reduce the technological constraints associated with the bridge length and position.

For instance, flexibility enables one to design the same electrical basic function in magnitude and phase, with different bridge lengths, numbers, and configurations. For each configuration, the associated compensation line is correctly modified. Furthermore, such a flexibility allows one to perform broadband compensation.

The concepts exposed here are valid for both air bridges in classical coplanar technology and integrated dielectric bridges for multilayer coplanar technology. They are theoreti- cally and experimentally validated, in the multilayer case, on transmission lines, tee junctions, and bandpass filter at millimeter frequency. Concerning modeling, because of the low dispersive coplanar technology, the use of a circuit model approach coupled with a static analysis seems to be sufficient up to 50 GHz.

Figure 1

 1 Figure 1 High-impedance compensation in classical and multilayer coplanar technologies

Figure 2

 2 Figure 2 Compensated bridges in multilayer coplanar technology. Experimental results and layouts

Figure 3 1 F 3 Figure 4

 3134 Figure 3 Theoretical multilayer coplanar tee-junction response

Figure 5

 5 Figure[START_REF] Weller | Three-dimensional high-frequency distribution networksᎏPart I: Optimization of CPW discontinuities[END_REF] Flexibility of the compensated method. Experimental responses for compensated tee junctions

CFigure 6

 6 Figure 6Broadband compensation on tee junction

Figure 7 1 FREQUENCYFigure 8

 718 Figure 7 Parasitic influence of uncompensated bridges on a bandpass filter


Serge Toutain
email: serge.toutain@polytech.univ-nantes.fr

Christian Person
email: christian.person@enst-bretagne.fr

J Ph

Coupez
email: jp.coupez@enst-bretagne.fr

Electromagnetic theory, CAD and experimental approach, bases of an efficient education in microwave and RF techniques

This paper suggests some teaching approach and gives examples to make more efficient education in microwave and RF techniques and increase the interest of students in such courses.

Introduction

This paper is based on a twenty five year education experience in microwave and antenna theory and practice. This experience took place between 1970 and 2001 in three different french universities :

1. The « University of Lille » 2. The

The course in microwave and RF techniques and its necessary evolution

Several years ago and even now, the microwave engineer was specialized. One who developed active circuits did not study antennas or passive circuits. In all cases , electromagnetic theory constituted the basis of teaching and the mathematical approach was very important. Therefore, a lot of students, more interested in practical aspects did not choose to study microwaves as a specialization in electronic engineering.

Unfortunately, this physical reality was not easier to master than theory in the context of experimental benches which were difficult to mount and use. A long practice of the "slotted line" was necessary to be accepted in the restricted group of microwave experimenters.

Concerning circuit theory, formalism was not a problem : it was the natural continuation of low frequency circuit theory yet, this simple approach hid a lot of traps whereas the experimental approach a more precise vision of the reality.

Microwave power generation using semiconductor devices was just beginning with IMPATT and gunn diodes and one had to like solid-state physics to work in this area. As no reliable non-linear model existed, it was hazardous to develop microwave power amplifiers or oscillators.

As microwave teaching needs to associate mathematics and solid-state physics theory with high capacity in mechanics and experimentation, few candidates were interested in a course offering such techniques. However, because the narrowness of the job market, such situation was not actually a problem. So the teaching could retain a « French touch » with « closed » exercices following lectures and experiments being implemented just to verify the main theoretical results.

With the development of general public radio applications, the situation is very different. The need for Microwave and RF designers is growing and the course and teaching must be suited to this new situation. As a large number of students have choose a speciality in which microwave and RF techniques are present, the effort must be made towards the motivation of the students. Moreover, given the complexity of the new systems, the microwave and RF engineer must be prepared to deal with MMICs, passive and active circuits, antennas, bearing in mind that each element can be coupled to the others (EMC). Also, the designer must have a good practice of base-band circuits and signal processing. This situation coinstitutes a new challenge for the teacher and we do not claim to give an exhaustive response to this problem. In this paper we just want to suggest some examples of teaching approaches which have given good results in the motivation and training of our students.

Have a more physical and comprehensive approach of the propagation phenomena in order to appreciate the students'point of view better and lead them in to the necessary mathematical approach.

If electromagnetics remains the basis of any technical development at very high frequencies, we can now use 3D electromagnetic commercial software such as HFSS or Microwave Studio or home software allowing the observation of fields and current scaterring in complex devices. Thus, it is possible to show how a specific mode is propagated in a waveguide, how is diffracted by a discontinuity or how a guided wave is transformed into radiation. The materialization of these basic phenomena is a very strong lever in the understanding of propagation. It helps to convince the students of the necessity of a precise mathematical approach.

As many applications use planar or multilayered 3D technologies, we think that it is important for the student to understand the different influences of conducting planes and dielectric materials on the propagation conditions at specific fequencies. Roughly speaking, « low frequency conditions" as the existence of cut-off frequencies are related to the relative position of the conducting planes and dielectric materials imply wave propagation at higher frequencies. For a good understanding of such phenomena, very simple experiments can be mounted such as the study of non-homogeneous cross section rectangular waveguide. Figure 1 shows the basic idea. The slotted line,used for V.S.W.R or wavelength measurement, can also be used to set off the thinking of the students. For that, a dielectric slab is put along one of the vertical walls of the waveguide. Near, the cut-off frequency of the fundamental mode (point A), the scattering of the fields is slightly disturbed by the presence of the dielectric slab and it is easy to make a wavelength measurement to define the cut-off frequency for instance. When frequency is growing, the electromagnetic fields are concentrated in the dielectric material. It is more difficult to make measurements because of the weakness of the fields at the center of the guide and of the radiation of the slot (cf figure 1). By using home software it is possible to determine the field scattering and the ω-β diagram at each frequency at which the experiment is carried out and to explain all these odd phenomena. As the students must precisely explain their observations, they understand the concept of dispersion in waveguides more quickly. It is then easier to introduce dispersive models of planar waveguides as Getsinger model for the microstrip line or Cohn model for the microslot line, models which are used when millimetric hybrid integrated circuits are designed.

Do not to make of microwaves a subject separated from electronics but rather show their common points and the differences.

As the future microwave and RF engineers generally have a good knowledge of electronics and electrical engineering, we must use this basis to correctly understand the problems related to the concepts of lumped, quasi-lumped and distributed components. The study of microstrip low-pass filters gives us the possibility to illustrate these concepts simply and to show how it is possible to combine lumped elements and distributed elements in the same circuit in order to improve its performances. To implement this approach, we need a basic CAD software program such as PUFF (freeware by Caltech) and reasonably large circuits in order to modify them by using auto-adhesive copper sheets. The filter synthesis theory is generally well known by students and it is easy for them to determine all the elements (inductances and capacitances) of the filter when the template is fixed. A simulation of this lumped element filter is done by PUFF then, the inductances and capacitances are determined by using transmission line sections of respectively high and low characteristic impedance. are verified. This modus operandi generally araises the students curiosity and increases their motivation. So, it is very interesting to develop this experimental approach to autonomy. We develop this aspect in the last example that we propose.

4.The practical discovery of autonomy can ehance the interest in theoretical lectures.

To improve the efficiency of lectures it is necessary to araise the interest of the students. One possibility among many is to propose lab. projects with a possibility of a delayed supervision by the teacher. To be efficient, such an approach must use very simple concepts and very simple elements. The discovery of microstrip antennas can be used as an example of such an approach.

Microstrip patch antennas are open resonators easy to build using auto-adhesive copper sheets. If we make a substrate with a coaxial feed line as presented in figure 3, the first experiment consists in search the matching point on the patch antenna. Making the hypothesis of a field scattering under the patch, it is relatively easy to determine its place. In a first step, the coaxial feeder is put on one of the two radiating discontinuities of the open resonator and the equivalent impedance is determined by using a network analyzer. Then, a second measurement is taken putting the coaxial feeder in the median plane, between the center and the end of the patch. The realistic hypothesis made on the field scattering allows to find the perfect matching point precisely and to verify this fact experimentally. So the gain, the efficiency and the radiating pattern can be experimentally determined and a simple model of the patch (like for instance the transmission line model) can be verified. Starting from this basis, it is possible to imagine a lot of variations : develop a wideband antenna using parasitic resonators on each side of the patch, develop dual band antenna by cutting slots into the autoadhesive copper sheets etc… In these cases, the use of 2.5D electromagnetic CAD software can be introduced in order to compare the hypotheses made by students with the simulation. Such a comparison is very interesting to develop the physical awareness of the students. Another aspect of such an experience is in the necessity for the teacher to supervise the work of each student in order to synthesize the questions that crop up during this training in autonomy. In this respect, the practice of groupware can bring some interesting solutions. Then, in the following lecturess, the teacher can concentrate on the points presenting some difficulties for the students. These teaching approaches have been used successfully in several cases . However, they do not constitute a panacea and we are interested in discussing other teaching experiences to develop student motivation and improve the efficiency of education in microwave and RF techniques.

Auto-adhesive copper sheet

Substrate

Coaxial feeder background

Figure 1 :

 1 Figure 1 : Experimental study of non-homogeneous cross section rectangular waveguide

Figure 2 :

 2 Figure 2 : Design of a low pass microstrip line filter

Figure 4 :

 4 Figure 4 : The study of patch antennas


Eric Rius

Jean Philippe Coupez

Serge Toutain

Christian Person

Pierre Legaud

Theoretical and Experimental Study of Various Types of Compensated Dielectric Bridges for Millimeter-Wave Coplanar Applications

Keywords: Bandpass filters, coplanar transmission lines, millimeter-wave circuits

This paper describes a new approach for the integration of dielectric bridges in microwave-integrated-circuit uniplanar technology. Taking into account the electrical influence of these elements on a coplanar waveguide, a simple way to compensate these parasitic effects is presented. It consists of locally tuning the dimensions of both the bridges and associated compensation lines. Theoretical and experimental results on coplanar waveguides and T-junctions are given up to the millimeter frequency range. Finally, to validate the proposed technique, a bandpass filter including compensated dielectric bridges is implemented.

I. INTRODUCTION

Many studies have shown that the coplanar waveguide can be considered as a good alternative to microstrip lines in the millimeter frequency range [START_REF] Houdart | Coplanar lines: Application to broadband microwave integrated circuits[END_REF]. However, there is one major drawback in dealing with coplanar design. The parasitic and strongly dispersive coupled slotline mode may be excited by nonsymmetrical coplanar-waveguide discontinuities, such as bends or T-junctions for example. As even and odd modes are fundamental modes for the coplanar waveguide, it is necessary to eliminate one of them, i.e., the coupled slotline mode. This is achieved by the insertion of bridges that enforce identical potential values on each side of the ground plane. Generally, bridges in microwave integrated circuits (MIC's) are implemented by thin metallic wires ( 25 m), which are bonded to the ground planes. However, such an operation induces numerous problems in reproducibility and positioning, which deteriorate the circuit performances. Moreover, the mechanical fragility of such bridges does not guarantee circuit reliability.

In this paper, we present how these problems can be overcome by using a new multilayer uniplanar approach for bridges integration. Nevertheless, dielectric bridges appear as strong discontinuities on coplanar circuits, and must be compensated. Thus, we propose a way to reduce the parasitic influence of the bridges by locally tuning the dimensions of both the bridges and associated compensation lines. We then illustrate how this technique can be successfully applied to the millimeter functions design.

II. THICK-FILM MULTILAYER UNIPLANAR TECHNOLOGY FOR DIELECTRIC BRIDGES INTEGRATION

The multilayer uniplanar technology is built by dielectric layers deposition on a substrate using a thick-film technological process. The metallic layer is then directly screen-printed, or screen-printed and etched, on the upper side of the dielectric layers. In our case, the dielectric material used is a photosensitive Heraeus ink (KQ 110) with a low-permittivity value equal to 3.9 and dielectric losses of 5×1 0 -5 [START_REF] Barnwell | High performance material technologies for advanced circuit assemblies and MCMs[END_REF]. With this ink, low-loss propagation characteristics have been observed over a 1-50-GHz frequency band, with "on wafer" measurements, carried out on simple transmission lines. Finally, such a process enables us to obtain robust integrated bridges and eliminates possible problems in the positioning and reproducibility of these elements. In addition, it provides an easy control of bridge dimensions and shapes.

III. PARASITIC INFLUENCE OF THE DIELECTRIC BRIDGES

Due to the reduced height of the dielectric layers (typically between 10-20 µm for the MIC process used), an additional and significant capacitive parasitic coupling exists between the inner conductor of the coplanar waveguide and the upper short-circuited strip. Obviously, the characteristic impedance and effective permittivity of the basic coplanar waveguide are significantly modified in the bridge region. Taking into account the low dispersive characteristic of the coplanar technology, these variations can be calculated in a first approximation with a static finite-difference method [START_REF] Barnwell | High performance material technologies for advanced circuit assemblies and MCMs[END_REF]. In this study, bridges were modeled by an equivalent transmission line, taking into account the modified electrical parameters of this local multilayer structure. Fig. 1 shows the influence of the dielectric layer (B) and of the dielectric bridge (C) on the electrical parameters of a 50-coplanar waveguide (A).

IV. COMPENSATED BRIDGES APPROACH:" L OW-PASS FILTER" CONCEPTS

To minimize the distortion due to the low characteristic impedance of the bridge discontinuity, it becomes possible to match the electrical parameters with those of the transmission line by locally reducing the inner conductor width under the bridge. Unfortunately, this method leads to obtaining very tiny strips and, consequently, increases technological constraints [see Fig. 1 (D)].

Our approach is to associate high characteristic impedances on each side of the bridge to compensate its equivalent low characteristic impedance [START_REF] Stephan | Integration of various types of compensated dielectric bridges for mm coplanar applications[END_REF]. This is achieved by reducing in an appropriate way the width of both the center conductor under the bridge and the compensation). Based on a similar approach, the structure composed of two bridges separated by an inductive compensation element can be also realized. A low-pass filter with a model is obtained in this case (Fig. 2: B-type compensation). The main advantage of this type of structure lies in the reduction of technological constraints: indeed, it neither requires very tiny strips nor a very precise positioning of the bridge. To validate these concepts, many different dielectric bridges configurations (with and without compensations) on 50-coplanar waveguides have been built on an alumina substrate ("r =9 :6, thickness = 635 µm). Relatively high values of the dielectric bridges lengths have been chosen especially to increase their parasitic effect on the coplanar waveguides and then to illustrate the compensation technique efficiency. "On wafer" measurements have been carried out on these structures over the 1-50 GHz frequency band. In Fig. 3, curve 1 represents return losses of a simple 50-multilayer coplanar waveguide and curve 2 corresponds to the case of a similar transmission line with a 135-µm-long bridge. In this case, it should be noted that the performances strongly deteriorates. As illustrated in Fig. 4, the two types of compensation yield acceptable return losses. Moreover, the compensation technique does not produce significantly additional insertion losses.

V. C OMPENSATED BRIDGES AND JUNCTIONS

Preliminary measurements were made to verify the integrated bridges efficiency in suppressing the parasitic mode on different types of discontinuities and to verify the effect of the compensation technique. For instance, Fig. 5 shows experimental results of an asymmetric short-circuited stub (theoretically, a 30-GHz resonator). The dielectric bridges modify the electrical performances of the resonator, its response is shifted to a lower frequency (20 GHz). Thus, by using a compensation technique, the resonant frequency is corrected to the required value.

VI. APPLICATION TO MILLIMETER-WAV E BANDPASS FILTERS

Finally, to validate the above compensated bridges concepts, we designed basic microwave functions such as filters that use integrated bridges. For instance, Fig. 6 deals with a classical short-circuited stubs bandpass filter [START_REF] Matthaei | Microwave Filters, Impedance-Matching Networks and Coupling Structures[END_REF] integrated by using a multilayer technology. In this case, bridges are thus compensated. To illustrate the flexibility of the compensation technique, two different filter prototypes were tested. A first prototype (A) uses, for example, a 60-µm-long bridge on the central resonator. The compensation of the second prototype (B)was determined in relation with the 100-µm bridge located on this central resonator. Indeed, the bridges dimension, for instance, is not critical because it is always possible to modify their lengths by adjusting the associated compensation. For the two filters, the frequency responses are identical. We only present in Fig. 6 the measurements relative to the second prototype (B). Experimental results are in agreement in both magnitude and phase with the desired response. Some discrepancies appear at high frequencies (between 40-50 GHz), but they are due to calibration errors during the measurements.

VII. CONCLUSION

Compared to the classical uniplanar technology, one main advantage of the multilayer uniplanar technology is to provide robust and reproducible integrated bridges. Indeed, with this new technology it becomes possible to control both the shape and dimensions of bridges. Thus, it is easier to model their parasitic effects on the fundamental propagation mode. Furthermore, a broad-band compensation technique has been developed and successfully applied to optimize the bridges integration on a coplanar structure. The fair agreement between theoretical and experimental results on millimeter-wave filter structures, for example, shows that a static analysis coupled with a transmission-line approach is sufficient up to 50 GHz. Consequently, this method can be very easily included in commercial computer-aided design (CAD) softwares.

Fig. 1 .

 1 Fig. 1. Electrical parameters of the multilayer technology.

Fig. 2 .Fig. 3 .

 23 Fig. 2. "Low-pass filter" concepts.

Fig. 4 .

 4 Fig. 4. A-type compensation and B-type compensation.

Fig. 5 .

 5 Fig. 5. Compensation technique on discontinuities.

E. Rius, J. P. Coupez, and C. Person are with the Laboratoire d'Electronique et Systèmes de Télécommunications, 29285 Brest, France, and also with the Université de Bretagne Occidentale, 29285, Brest France, and the Ecole Nationale Superieure des Télécommunications de Bretagne, 29285 Bretagne, France. S. Toutain was with the Laboratoire d'Electronique et Systèmes de Télécommunications, 29285 Brest, France. He is now with the Laboratoire des Systèmes Electronique et Informatique, 44306 Nantes, France. P. Legaud is with the the Centre National d'Etudes des Télécommunication, 22302 Lannion, France.

ACKNOWLEDGMENT

The authors thank the CNET, Lannion, France, for providing "on wafer" measurements and Prof. M. Ney for his helpful suggestions.


Yann Toutain
email: yann.toutain@enst-bretagne.fr

Jean-Philippe Coupez

Christian Person

Microstrip miniaturized loop-filters with high out-of-band rejection for future 30 mobile terminals

This paper deals about the implementation of miniaturized microstrip band-pass filters operating on a very narrow bandwidth. We propose an original synthesis for achieving high out-of-band rejection by means of a Z'd order resonant structure, for which transmission zeros are introduced through mutual coupling effects. Quite good performances are obtained regarding losses, size and spurious resonances for the proposed topology, which can be considered as a good solution for future mobile on-board equipments.

The length of the resonator is used for defining the center operating frequency while the coupling between the feeding Jines extremities contributes to the transmission zero location. The main effort for the designer lies in the definition of an appropriate decomposition of the resonator shape, regarding the required filtering response.

Quite a fair agreement is obtained between the HP-Momentum simulation and the experimental results as presented on the figure 2.

"' "'

-i:::-... ;.,f

-VA " � L./ ' ..- e,__..-'--"" \ 1/ i...---- - •ii ulat. � ree tlte \ /./'" -exp4 rimei talrE !iults \ '/ \ l/ ,.

l.5GHz

Frequency 2.5 GHz Fig. 2.

Comparison between measured and simulation on a single resonator structure with multiple coupling

The complete structure, integrated on an appropriate substrate (a=l0.2, h=635 f.UU ), is quite small (7 mm* 5 mm, see figure 3), with performances in accordance with new mobile system requirements. Nevertheless, only one transmission zero can be introduced with a single resonator structure. Moreover, this zero is necessarily located at a frequency higher than the center operating one because of in-phase and out of phase recombination constraints. The insertion of a transmission zero at a lower frequency requires at least a second order topology with multiple non-adjacent couplings. Indeed, in relation with the non co-directive coupling effects existing between the two resonators, great difficulties are encountered for retrieving out-of-phase recombination between the different possible paths for the signal, and therefore the associated zero.

In the other hand, transmission zeros can be recovered when operating with symmetrically coupled configurations. The compartmental study reported on figure 5 clearly underlines the possibilities of generating a transmission zero, depending on the symmetry axis ensured with such co-directive coupled resonators.

  Fig. I.Single resonator with a transmission zero : Principles

Fig. 3 .

 3 Fig. 3. Photography of the single resonator structure.

Fig. 4 .

 4 Fig. 4.No-transmission zero topologies by means of two unsymmetrical coupled resonators

Fig. 5 .Fig. 7 .

 57 Fig. 5. Laterally coupled resonators -Symmetrical configurations for recovering transmission zero -With symmetrical coupling architectures, (fi gure 5), the out-of-phase destructive conditions can be easily recovered each apart with respect to the operating bandwidth. The resonator shape can be adjusted so as to control the effective electrical length of one path with respect to the other. The main difficulty lies in keeping the fundamental resonance conditions at Fo, which may basically be affected by these coupling phenomena.

FigFig

  Fig. 9.

Acknowledgement: FRANCE TELECOM (contract 98 1BCTI7 CNET FT) has supported this work.


Gaëtan Caillaut
email: g.caillaut@brgm.fr

Cécile Gracianne
email: c.gracianne@brgm.fr

Nathalie Abadie
email: nathalie-f.abadie@ign.fr

Guillaume Touya
email: guillaume.touya@ign.fr

Samuel Auclair
email: s.auclair@brgm.fr

Automated construction of a French Entity Linking dataset to geolocate social network posts in the context of natural disasters

Keywords: Automated geotagging, French Entity Linking, Wikipedia, Twitter, Crisis Management, Natural Disaster

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

When a natural disaster occurs, the main issue is to make sense of what is going on, with the necessity to be able to qualify the severity of the situation and to follow its evolution over time. Response organizations address this issue by collecting as much information as possible in order to increase their "situational awareness" and to build a realistic "common operational picture" on which to undertake effective actions. After having considered for a long time that the only exploitable data for disaster management should come from specialized services guaranteeing their reliability and their technical quality, the experience of recent years has shown that citizen data could allow -when well exploited -to bring additional knowledge and to build better targeted response [START_REF] Kaufhold | Retrospective Review and Future Directions for Crisis Informatics[END_REF]. Federating communities of users accustomed to daily comment events on live, social media channel the data of millions of connected "citizen sensors" [START_REF] Goodchild | Citizens as sensors: the world of volunteered geography[END_REF]) endowed with five senses, able to share testimonies spontaneously and quickly. In practice, the richest information usually comes from those citizens being closest to the area affected by the disaster. Because the natural disaster affects their immediate environment, these "local citizens" [START_REF] Grace | Social Triangulation: A new method to identify local citizens using social media and their local information curation behaviors[END_REF]) are indeed more inclined -both in the physical and digital spheres -to help or to exchange objective information on the field situation [START_REF] Akter | Big data and disaster management: a systematic review and agenda for future research[END_REF][START_REF] Starbird | Learning from the crowd: Collaborative filtering techniques for identifying on-the-ground Twitterers during mass disruptions[END_REF]. The latter tend to exchange information on the effects of disasters, victims or other types of useful information, while populations farther away from events relay this information, or express their empathy to the victims [START_REF] Olteanu | What to expect when the unexpected happens: Social media communications across crises[END_REF]. For these reasons, monitoring and analysis of social media has become at the heart of the concerns of crisis managers [START_REF] Rasmussen | Risk, crisis, and social media: A systematic review of seven years? research[END_REF], especially Twitter. Although it is not the most widely used social media, the Twitter platform present particularly useful features for reporting and monitoring natural disasters such as publication of short messages in real time, free streaming Application Programming Interface (API) making it possible to automate monitoring tasks [START_REF] Auclair | SURICATE-Nat: Innovative citizen centered platform for Twitter based natural disaster monitoring[END_REF], and ability to attach pictures. Its real-time monitoring and analysis may turns in delineating rapidly the extension of the area impacted by an earthquake [START_REF] Fayjaloun | Integrating strongmotion recordings and Twitter data for a rapid shakemap of macroseismic intensity[END_REF] or a flood [START_REF] Arthur | Social sensing of floods in the UK[END_REF], or identifying emergency situations of interest for the rescue or security forces [START_REF] Qadir | Crisis analytics: big data-driven crisis response[END_REF][START_REF] Wang | Social media analytics for natural disaster management[END_REF].

As such, several works considering information extraction for natural disaster crisis management tried to place their results on a map to make it directly usable [START_REF] Stanek | A hybrid decision support system for disaster/crisis management[END_REF][START_REF] Zhang | A visual analytics framework for microblog data analysis at multiple scales of aggregation[END_REF] or advocate for alternative way to build those maps such as crowdsourcing mapping [START_REF] Hunt | Crowdsourced mapping in crisis zones: collaboration, organisation and impact[END_REF]. A very low proportion of tweets have an intrinsic geolocation (less than 1 %), like GPS coordinates [START_REF] Cheng | You are where you tweet: a content-based approach to geo-locating twitter users[END_REF]. Consequently, geolocation of the information should be inferred from the information present in the text and/or the metadata of the tweet itself. This can be done thanks to the recent advances in Natural Language Processing (NLP) allowing to extract accurate word representations, in a vectorial form, from raw text. These vectors, or word embeddings, hold precious semantic and syntactic latent features, including geospatial features. While the first word embedding approaches resulted in a fixed set of vector for a given vocabulary [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF][START_REF] Pennington | GloVe: Global Vectors for Word Representation[END_REF], modern approaches [START_REF] Devlin | BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding[END_REF] propose to first train a generic model and then to fine-tune it on any downstream task. Among others, works have been carried out to fine-tune such systems to solve the well-known Named Entity Recognition (NER) and the Entity Linking (EL) tasks. The first one aims to detect and classify named entities, such as persons or organizations, in running text, while the second one aims to disambiguate these entities by linking them to an identifier, usually a node in a Knowledge Base (KB). This allows enriching the text with external information which can help either an automated system to provide a better prediction or a human agent to better understand, or to refute, the outputs of such systems.

In this work, we propose to apply EL on toponyms, or any geolocatable entity, mentionned in tweets written during crisis such as earthquakes or floods. We use gazeteers, such as OpenStreetMap, as target for our EL system. Such a system should enable end-users to collect fine to coarse grained information about the spatial entities mentioned in a tweet, such as, most importantly, its coordinates on Earth, but also its area or its population. Depending on these information, rescue teams may behave differently. For instance, an intervention in a densely populated city center does not require the same human and material resources than another one in a less populated residential neighborhood. However, training such a system requires quite a large annotated corpus, which makes solving this task a real challenge, especially when dealing with French tweets. Indeed, French resources tend to be less numerous than their English counterparts. This is why we first tackle the problem of collecting and annotating a French corpus dedicated to the Entity Linking (EL) task. Due to the scarcity of geotagged tweets, collection and annotation must be made manually, which is heavily resources consuming. Following a long line of works [START_REF] Logan | Barack's Wife Hillary: Using Knowledge Graphs for Fact-Aware Language Modeling[END_REF][START_REF] Merity | Pointer sentinel mixture models[END_REF][START_REF] Bunescu | Using Encyclopedic Knowledge for Named entity Disambiguation[END_REF], we propose an automated alternative using hyperlinks in Wikipedia pages as a supervision signal. Such a dataset would probably not fit Twitter data, therefore we plan, in a second time, to augment our dataset with Twitter data. Nevertheless, it has already been shown that deep neural networks can be effectively pre-trained on a generic task then fine-tuned to fit other, more specific, tasks, the most obvious example being BERT. [START_REF] Rajapaksha | BERT, XLNet or RoBERTa: The Best Transfer Learning Model to Detect Clickbaits[END_REF] successfully trained large Transformer models to detect clickbait on Twitter data even though the base pre-trained models were not necessarily trained on Twitter. Also, J. [START_REF] Wang | NeuroTPR: A neuro-net toponym recognition model for extracting locations from social media messages[END_REF] showed that augmenting a Twitter dataset with data from Wikipedia had a positive impact on the predictive performance of their system. We will first do a quick review of works proposing solution to extract spatial features from raw texts. Then, we will show the benefits of an EL system over the others and we will describe the process we used to automatically build our annotated dataset for EL. Finally, we will introduce our system architecture, which should fix some limitation of current state-of-the-art approaches.

RELATED WORKS

Since the recent advances in Natural Language Processing (NLP), Named Entity Recognition approaches have been mostly relying on deep learning architectures [START_REF] Li | A Survey on Deep Learning for Named Entity Recognition[END_REF] representations of words, that can be leveraged to classify token and identify those representing spatial named entities, yielding state-of-the-art results. The most famous approaches are BERT [START_REF] Devlin | BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding[END_REF] and ELMO [START_REF] Peters | Deep Contextualized Word Representations[END_REF], which rely, respectively, on the Transformer [START_REF] Vaswani | Attention is All you Need[END_REF]) and LSTM architectures [START_REF] Gers | Learning to forget: Continual prediction with LSTM[END_REF]. Spatial Named Entity Recognition is a popular subtask of NER that can be performed with the same neural architectures but requires dedicated annotated corpora for training and testing. J. [START_REF] Wang | NeuroTPR: A neuro-net toponym recognition model for extracting locations from social media messages[END_REF] relied on a subset of Wikipedia to generate automatically a dataset to train a toponym recognition model. They trained a model on this dataset, another on the WNUT2017 Twitter Dataset [START_REF] Derczynski | Results of the WNUT2017 shared task on novel and emerging entity recognition[END_REF], and a last one on both. They evaluated their models on a Twitter dataset extracted during Hurricane Harvey, in 2017, and showed that the model trained on Wikipedia performs worse than the model trained on WNUT2017. However, using data from Wikipedia is far from useless since the models trained on both corpora surpass them by a large margin.

These architectures can be further specialized and fine-tuned for geotagging purposes. This means associating the tokens representing a spatial named entity, no longer with a class label, but either with predicted geographical coordinates or with the identifier of a geographical entity of known location listed in a gazetteer. In both cases, leveraging the contextual distributed representations of toponyms produced by modern language models may prove especially useful. Indeed, toponyms often refer to multiple places in the world, for instance, five French municipalities are called Chaumont. Previous works in the field of spatial entity resolution, either based on supervised approaches or not, have shown the benefits of using contextual knowledge (nearest neighbors, spatial relations, popularity, importance, etc. . . ) for place names disambiguation. BERT or ELMO based language models have been pre-trained and made available for many languages. Geotagging systems can hence be derived from such models.

Approaches for geographic coordinates prediction or geographic region classification. The quality of the results of gazetteer-based approaches depends strongly on the quality of the input gazetteer, and in particular on its completeness. To overcome this difficulty, especially for areas where few location data are available, many text geotagging approaches propose to first discretize the surface of the Earth, or the studied area, into cells. In this context, the geotagging task is a classification task with 𝑛 classes, 𝑛 being the number of cells. Early approaches of that kind, are presented in the survey proposed by [START_REF] Melo | Automated geocoding of textual documents: A survey of current approaches[END_REF]. More recently, deep learning architectures have been proposed to solve this classification task, like in [START_REF] Gritta | Which melbourne? augmenting geocoding with maps[END_REF] or in [START_REF] Yan | The Integration of Linguistic and Geospatial Features Using Global Context Embedding for Automated Text Geocoding[END_REF]. In both approaches, some geographical knowledge is introduced by feeding the model with the same spatial grid where each cell is filled with knowledge regarding the candidate geographic entities: in the former, the candidates population count is added, in the later it is the frequency of co-occuring place mentions. None of these approaches can predict an exact location, since their theoretical precision is bound to the cell size, even though it is often the coordinates of the cell's barycenter that are given as results. This can be somehow mitigated by modulating the cell size according to some criteria: for instance, it seems reasonable to set a smaller cell size in an area densely populated. This kind of approach can also be used to predict a vague location that can then be leveraged by another, more precise, geotagging method. Such an approach is proposed by [START_REF] Cardoso | A Novel Deep Learning Approach Using Contextual Embeddings for Toponym Resolution[END_REF]. A first deep learning architecture is used to predict a probability distribution over geo-spatial regions in a hierarchical spatial grid and this result is then combined with the centroid coordinates of the grid cells and a second loss function to predict geographical coordinates.

Gazetteer-based approaches. The last kind of geotagging approaches takes advantage of gazetteers, such as Open Street Map1 (OSM), Geonames2 or BDTOPO ®3. Gazetteers store the names of any type of geographic entity, such as cities, streets, lakes or buildings, with at least a spatial reference alongside (i.e.: coordinates, a geometry, etc. . . ). Further information regarding geographic entity properties may be added, like their nature, their geometric properties, their population, etc. Such information, providing some contextual knowledge about geographic entities, may reveal useful for place name disambiguation. That is why many works have focused on gazetteer construction to improve spatial named entity linking results [START_REF] Overell | Using co-occurrence models for placename disambiguation[END_REF][START_REF] Brando | Disambiguation of named entities in cultural heritage texts using linked data sets[END_REF][START_REF] Spitz | So far away and yet so close: Augmenting toponym disambiguation and similarity with text-based networks[END_REF][START_REF] Kim | Similarity matching for integrating spatial information extracted from place descriptions[END_REF][START_REF] Ardanuy | Toponym disambiguation in historical documents using semantic and geographic features[END_REF]. Besides, gazetteers may provide very accurate location information for each place name, which can be crucial for emergency response applications. Predicting such geographic entities is the same as solving Entity Linking (EL), a task consisting in aligning text and entities from knowledge bases. Most of the previous works have focused on unsupervised approaches based on two steps :

• Candidates selection: For each spatial named entity mention, this step aims at selecting the most similar place names in the gazetteer. It is mostly performed with string similarity measures [START_REF] Recchia | A Comparison of String Similarity Measures for Toponym Matching[END_REF].

• Candidates ranking: This step aims at finding, among the selected gazetteer entries, the most likely to be represented the same place as the spatial named entity mentioned in the text. This is performed either by applying a set of heuristics based on geographic entity distances [START_REF] Habib | Improving Toponym Disambiguation by Iteratively Enhancing Certainty of Extraction[END_REF][START_REF] Derungs | From text to landscape: locating, identifying and mapping the use of landscape features in a Swiss Alpine corpus[END_REF][START_REF] Blank | Geocoding place names from historic route descriptions[END_REF], their spatial relations [START_REF] Kim | Similarity matching for integrating spatial information extracted from place descriptions[END_REF][START_REF] Paris | Linking spatial named entities to the Web of data for geographical analysis of historical texts[END_REF], their importance [START_REF] Amitay | Web-a-where: geotagging web content[END_REF][START_REF] Brando | Disambiguation of named entities in cultural heritage texts using linked data sets[END_REF] or with approaches implementing supervised learning algorithms, like SVM or LambdaMART, with hand-crafted features to describe spatial named entity mentions and geographic entities listed in the gazetteer [START_REF] Martins | A machine learning approach for resolving place references in text[END_REF][START_REF] Daiber | Improving efficiency and accuracy in multilingual entity extraction[END_REF][START_REF] Speriosu | Text-driven toponym resolution using indirect supervision[END_REF][START_REF] Santos | Using machine learning methods for disambiguating place references in textual documents[END_REF]. Many works use a corpus built from the English Wikipedia to train their learning algorithm.

More recently, some approaches have tried to solve this task with deep-learning-based classification methods. [START_REF] Xu | DLocRL: A deep learning pipeline for fine-grained location recognition and linking in tweets[END_REF] proposes DLocRL, a deep learning pipeline for locations recognition and linking in tweets, that computes a matching score between a given tweet and a location profile built from geographic data extracted from Foursquare4. They also add a post-processing step based on a Geographical Pair Linking algorithm that leverages the geographical coherence between co-occurring spatial entity mentions. To overcome the limitation due to the overwhelming number of classes, some works propose to train two classifiers -the first one receiving the text containing (spatial) entities and the second one the features of the (spatial) entities to predict -in parallel to predict the same spatial embedding [START_REF] Botha | Entity Linking in 100 Languages[END_REF].

Whatever the strategy adopted, there are two major challenges for geotagging applications based on deep learning: integrating geographical knowledge into the model and building an appropriate corpus to train the model.

USING WIKIPEDIA TO TRAIN A SPATIAL ENTITY LINKING PIPELINE

As suggested in the previous section, predicting entities from KB or gazeteers, such as OSM, would have several benefits. Amongst them, gazeteers store many precious information that could be extremely valuable in case of an emergency, such as, for instance, the height of a building or the presence of a gas pipe near in a damaged area.

Unfortunately, there are very few datasets mapping text spans to OSM entities, making it impossible to train an EL system to predict OSM entities. Actually, to the best of our knowledge, such mappings exist only between the couple Wikipedia/Wikidata. For that reason, Wikipedia and Wikidata are ideal resources to train EL systems, since all Wikipedia pages have a corresponding entity in the Wikidata graph. One can use the internal hyperlinks in Wikipedia pages to train a system to (1) detect entities, since they are identified by hyperlinks, and (2) align them to the Wikidata graph. Actually, many works have been relying on Wikipedia to extract supervisions signals from hyperlinks [START_REF] Botha | Entity Linking in 100 Languages[END_REF][START_REF] Merity | Pointer sentinel mixture models[END_REF][START_REF] Ghaddar | Winer: A wikipedia annotated corpus for named entity recognition[END_REF][START_REF] Nothman | Learning multilingual named entity recognition from Wikipedia[END_REF]), but only a few uses the French Wikipedia. Other works have been using Wikipedia to train models capable of resolving, specifically, toponyms and others geolocatable entities [START_REF] Martins | A machine learning approach for resolving place references in text[END_REF][START_REF] Geiß | The Wikipedia location network: overcoming borders and oceans[END_REF]. In the case of a spatial entity, the Wikidata graph often contains links to other geospatial databases, such as GeoNames or OSM, making it possible to retrieve fine-grained spatial features. Even if the Wikidata entity lack these links, spatial features can, most of the time, be retrieved directly from the Wikidata KB.

However, models trained on Wikipedia may not be well-suited to deal with social network data, since Wikipedia pages and social network posts generally do not share the same writing style. It is still possible to alter the Wikipedia pages, for instance by randomly removing or swapping characters, or changing cases, which should make the model more robust. But such alterations could conflict with the model's tokenization methods, which could results in a lot of <unk> token (J. [START_REF] Wang | NeuroTPR: A neuro-net toponym recognition model for extracting locations from social media messages[END_REF], which does not hold any useful information. Furthermore, such artificial data augmentation are not sufficient since they cannot generate data with the same vocabulary and slang used on social networks. That being said, the lack of annotated social network corpora, especially in French, will inevitably force us to rely on the Wikipedia dataset, since it is the only publicly available corpus with direct mappings between entities and text spans. Nevertheless, Wikipedia is insufficient on its own. As pointed by J. [START_REF] Wang | NeuroTPR: A neuro-net toponym recognition model for extracting locations from social media messages[END_REF], the best performances are obtained with a combination of the Wikipedia dataset and a small social network dataset, WNUT2017 [START_REF] Derczynski | Results of the WNUT2017 shared task on novel and emerging entity recognition[END_REF].

Building an Entity Linking dataset automaticaly

The Wikimedia foundation provides dumps of Wikipedia5 and Wikidata6. The Wikidata dump consists in a big JSON file whose size is approximately 70 GB as of November 2021. It is relatively easy to read since it is encoded as a single JSON array and entities can be processed incrementally by reading the file line by line. Hence, to process the file, one needs to read the dump line by line and parse the JSON string to get a representation of the current Wikidata entry. The sole difficulties lie in understanding the Wikidata architecture, which is not really user-friendly, and in processing efficiently and quickly such a large dataset.

The Wikipedia dumps are released as separate XML files: one per language. The French Wikipedia dump, as of November 2021, is around 5.1 GB. Being plain, standard, XML, the dump is quite easy to read too, although, as we will see in the following, it is not really usable. Indeed, Wikipedia pages are written in a specific language, the wiki markup language, also known as wikitext or wikicode. This language is known to have no clearly defined syntax and extensively rely on (nested) templates expansion, making it extremely difficult to write a reliable parser7. Indeed, previous work shows that standard parsing methods could not reproduce the expected Wikipedia output [START_REF] Dohrn | Design and implementation of the sweble wikitext parser: unlocking the structured data of wikipedia[END_REF]. As a consequence, researchers seeking to leverage the Wikipedia corpora only can use unreliable tools. Currently, WikiExtractor (Attardi 2015) seems to be the most used software to extract text from Wikipedia dumps, despite that a lot of texts were removed because the tool could not handle some templates from the French Wikipedia dump. Wikitextprocessor8 seems to be a very promising tool since it is capable of expanding templates much more reliably, at the cost of an non-negligible increase in processing time. However it does not yet handle non-English Wikipedia dumps, so it is not suitable for applications on other languages.

We also explored solutions using the Wikipedia API. Indeed, it is possible to retrieve the wikitext of a page using the API, then ask the API to do template expansion on the wikitext. However, the template expansion API is limiting the size of its input, and rate limits disallow making too much queries. These limitations could be bypassed by hosting our own Wikipedia instance. While running a Wiki instance is made easy by using the official Docker image, importing the whole Wikipedia dump is not feasible in a reasonable time frame since development of the import tool have been abandoned9.

In the end, we did not have much choices but to rely on the HTML output of the Wikipedia website. However, scrapping the whole website is excluded since it would take too much time, and it would be inappropriate regarding the number of queries made to the Wikipedia web server. Furthermore, we do not need the full Wikipedia corpus since we aim working with social network data. Hence, training a model on the entire Wikipedia corpus may introduce biases into the system, making it inefficient on social network data. So, we extracted a small subset of the French Wikipedia corpus following the same scheme as [START_REF] Merity | Pointer sentinel mixture models[END_REF]. Using Pywikibot, we extracted the list of featured and good articles, which are articles tagged as being well-written by real users from the Wikipedia community. This resulted in 6027 articles, as of January 2022. We then scanned these pages in order to extract links to other Wikipedia pages. These links are very important, since we will use them as gold reference to train our Entity Linking system. As we intend to use a system similar to the one proposed by [START_REF] Botha | Entity Linking in 100 Languages[END_REF], we also need a short description per entities. So, we also downloaded the pages referring to the entities we extracted from the 6027 featured and good articles, resulting in 336 743 articles, from which we extracted the first paragraph since it is supposed to be a short description of the page, according to the Wikipedia's guidelines10.

Not so much cleaning was necessary to make the HTML pages usable, since most of the work have been done by the Wikipedia backend. We removed everything that is not raw text, such as tables and pictures and replaced the mathematical formula by the textual representation, provided by Wikipedia, intended to text-based web browser. We also removed common sections, such as "references" or "see also" sections, which we think are irrelevant for most of the use-cases. Finally, we tagged the links with [E] tags. For instance, if a link to the page Paris with the text "Ville lumière" (City of light) is found, it is replaced by the string "[E=Paris]Ville lumière[/E]". This allow to preserve both the original text, as well as the title of the linked page.

For each page we extracted from Wikipedia, we also extracted the following properties from the linked Wikidata entity:

QID A string identifying a Wikidata entity. For instance, the QID of Paris is Q90.

Description A short description of the entity. Not always reliable.

Label

The name of the entity. For instance the label of the entity Paris is "Paris".

Aliases A list of aliases for the entity. For instance, both "Paname" or "Ville-Lumière" refer to the entity Paris.

Type A suggested type for the entity. One of "GEOLOC", "ORG", "PERSON", "DATE" or "OTHER". The Type property is not extracted directly from Wikidata. We built an ad hoc set of rules to predict the type of an entity. These rules check the presence of some properties in the entities. For instance, the entity Paris is an instance of (property P31) of the entity Capital (whose QID is Q5119). Then, it is probably a spatial entity, so we set the value "GEOLOC" to the Type property. The number of instances of each entity type is shown in Table 1, but the data extracted from our set of rules has not yet been audited, so we do not recommend relying on it for a production system. Our classification probably needs further refinement considering the high number of "OTHER" entities.

We released our dataset on the HuggingFace hub11. It contains 6023 documents and 304 826 distinct entities. Among the 37 051 326 words, 2 913 959 are part of an entity (an entity can be made of multiples words). A total of 1 619 961 entities have been annotated. The scripts we used to produce this dataset are available as a Git repository12. Currently, the scripts are specific to the French Wikipedia, since some links are hard-coded, as well as the name of the sections to remove during the cleaning phase. However, it should not be too difficult to edit the scripts in order to make them work with any language.

A PIPELINE FOR NAMED ENTITY RECOGNITION AND ENTITY LINKING

We build a system heavily inspired from the work of [START_REF] Botha | Entity Linking in 100 Languages[END_REF]. This system encompass two Transformerbase encoder trained conjointly to produce similar outputs. The first encoder is called the mention encoder and the second one the entity encoder. In the original work, the mention encoder takes as inputs the title of a Wikipedia page and a sentence from the same page, in which one entity is tagged. The entity encoder takes as input a textual description of the tagged entity. Their system achieve impressive results, especially considering that it was able to produce accurate representations for many languages, even on languages on which it has not seen during training. However, we argue that this system, as is, is not well-suited for real-world applications. First, the system can only process sentences from Wikipedia, since the Wikipedia title is a part of its inputs. As such, it may be difficult to use this system on Twitter data. Second, the system can compute a representation for a single entity that has to be tagged beforehand. Hence, this system has to be used in conjunction with an accurate Named Entity Recognition (NER) system. Yet, we think that doing both NER and EL in the same time may be preferable because it may reduce both the size of the model and processing time. Indeed, the dual encoder architecture is already heavy, since it encompass two distinct BERT models13. Adding a third system to do NER would increase significantly the size and complexity of an already complex architecture.

To alleviate the two mentioned limitations, we first propose to not rely the mention encoder on Wikipedia specific features, such as the title of a page. In this case, the mention encoder takes only text as input. Second, the mention encoder should not be limited to only one entity per input sentence and be designed to (i) detect any Named Entities and (ii) compute Entity embeddings for each of them. In this configuration, the system has to be trained to minimize the following dual objective:

𝑁 𝐸 𝑅(𝑜 𝑛𝑒𝑟 , 𝑦 𝑛𝑒𝑟 ) + 𝐸 𝐿(𝑚 𝑒𝑙 , 𝑒 𝑒𝑙 )

Where 𝑜 𝑛𝑒𝑟 are the predicted NER labels, 𝑦 𝑛𝑒𝑟 are the expected NER labels, 𝑚 𝑒𝑙 are the output entity representations from the mention encoder and 𝑒 𝑒𝑙 are the output entity representation from the entity encoder. The 𝑁 𝐸 𝑅 function is the classical cross-entropy loss and 𝐸 𝐿 is a loss function based on the cosine similarity.

CONCLUSION

Geotagging social network posts requires to be able to extract accurate and relevant spatial features from raw text. Several strategy exists, and we chose to train entity embeddings by leveraging links found in Wikipedia pages. In this work, we propose a set of open-source tools to automatically scrap data from Wikipedia and annotate them using links found in Wikipedia articles. We also release a French dataset from the featured and good articles with both NER and EL annotations.

While we do not have, yet, experimental results, we propose nevertheless further developments of the method proposed by [START_REF] Botha | Entity Linking in 100 Languages[END_REF] to make it more suitable in real-world applications since it does not rely specifically on Wikipedia metadata (such as the page's title) nor requires entity annotations. Experiments need to be carried out to validate our proposed pipeline.

Future works will be dedicated to extending our dataset. First, we will manually annotate a set of French tweets extracted during natural disaster events, such as the Alex storm that hits France in September 2020. Such efforts are critical, since, as stated previously, a good system trained only on Wikipedia (or any mainstream dataset) will have difficulties performing evenly on a Twitter dataset. Second, we will improve the NER annotations, since, currently, we provide only a few class labels for Named Entity tagging and these annotations are given by an ad-hoc set of rules.

Finally, we plan to integrate time related features into our modeling. Indeed, people are likely to post messages about a specific event, such as an earthquake, in the same time span as the event. This could enable clustering online posts around a shared event and help the disambiguation of some spatial entities by leveraging knowledge extracted from other posts in the same cluster.

  . Modern neural architectures produce contextual <CoRe/WiP/Practitioner> Paper -<Track Name> Proceedings of the 19th ISCRAM Conference -Tarbes, France 2022

Table 1 . Number of distinct entities' types as well as their appearance in the corpus.

 1

	Entity Type	Count Instances
	PERSON	85 670	301 714
	GEOLOC	98 749	552 583
	ORG	2180	13 849
	DATE	2024	38 816
	OTHER	116 118	712 683

1https://www.openstreetmap.org/ 2http://www.geonames.org/ 3https://geoservices.ign.fr/documentation/donnees/vecteur/bdtopo <CoRe/WiP/Practitioner> Paper -<Track Name> Proceedings of the 19th ISCRAM Conference -Tarbes, France 2022

4https://foursquare.com/ 5https://dumps.wikimedia.org/ 6https://www.wikidata.org/wiki/Wikidata:Database_download <CoRe/WiP/Practitioner> Paper -<Track Name> Proceedings of the 19th ISCRAM Conference -Tarbes, France 2022

7https://utcc.utoronto.ca/~cks/space/blog/programming/ParsingWikitext 8https://github.com/tatuylonen/wikitextprocessor 9https://www.mediawiki.org/wiki/Manual:MWDumper 10https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Lead_section <CoRe/WiP/Practitioner> Paper -<Track Name> Proceedings of the 19th ISCRAM Conference -Tarbes, France 2022

11https://huggingface.co/datasets/gcaillaut/frwiki_good_pages_el 12https://github.com/GaaH/frwiki_good_pages_el 13We suppose this is why the authors use only the four first BERT's layers <CoRe/WiP/Practitioner> Paper -<Track Name> Proceedings of the 19th ISCRAM Conference -Tarbes, France 2022

ACKNOWLEDGMENTS

The work presented in this article was carried out within the framework of the RéSoCIO project co-financed by the French National Research Agency (ANR) under the grant ANR-20-CE39-001.


Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

Allaa Eddine Boumesbah and Thomas Henneron Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Yncrea Hauts-deFrance, ULR 2697 -L2EP, F -59000 Lille, France.

A parametric geometric metamodel is built for a nonlinear magnetostatic problem, using proper orthogonal decomposition approach combined with radial basis functions interpolation. Furthermore, the geometrical variation of the problem is modeled using a RBF interpolation for smooth mesh deformation. The metamodel is applied for a single-phase EI inductance, the aim is to create precise flux cartographies based on few solutions of the original finite element (FE) model. The results show that the POD-RBF approach can reduce efficiently the evaluation time of a parametric nonlinear magnetostatic problem.

Index Terms-Model Order Reduction, Proper Orthogonal Decomposition, Radial Basis Function, mesh deformation.

I. INTRODUCTION

HE finite element (FE) method is widely used to model electromagnetic devices. This method ensures precise results at the expense of a consequent computation time, especially when dealing with parametric problems. Then, model order reduction (MOR) methods had proven their efficiency in reducing the computation time for parametric studies. The proper orthogonal decomposition (POD) is one of the most popular MOR approaches [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. I. Coherent structures[END_REF]. Based on the solutions of the FE model for different values of parameters (called snapshots), the POD approximates the solution of the FE model in a reduced basis. Then, the parametric FE model is projected onto a reduced basis, decreasing the order of the numerical model to be solved for new values of parameters. Another approach consists in constructing a metamodel by interpolating the parametric FE solution expressed in a reduced basis for new values of parameters without solving the matrix system and avoiding the nonlinear iterative scheme. Based on the principle, different approaches had been proposed in the literature, such as the POD with interpolation (PODI) [START_REF] Rama | Real-time modelling of the heart using the proper orthogonal decomposition with interpolation[END_REF], using an optimization process, or the orthogonal interpolation method (OIM) [START_REF] Farzam Far | Orthogonal interpolation method for order reduction of a synchronous machine model[END_REF], using polynomial interpolations, for example. Another approach has also been proposed combining the POD with the radial basis function (RBF) interpolation. Initially developed for mechanical and thermal problems [START_REF] Adamczyk | Development of a nondestructive technique for measuring thermal conductivity of material with small anisotropy based on application of the reduced order technique[END_REF] [5], it had been also applied to a nonlinear magnetostatic problem [START_REF] Henneron | Surrogate model based on the POD combined with the RBF interpolation of nonlinear magnetostatic FE model[END_REF]. Moreover, when the problem involves geometric parameters, an adaption of the FE mesh is required. Remeshing the geometry at each modification can deteriorate the precision of the FE model by introducing significant numerical noise. A solution to preserve the problem's consistency is to deform the initial mesh using the RBF interpolation [START_REF] De Boer | Mesh deformation based on radial basis function interpolation[END_REF] [START_REF] Henneron | Mesh Deformation Based on Radial Basis Function Interpolation Applied to Low-Frequency Electromagnetic Problem[END_REF]. The method consists of imposing a displacement on a set of nodes and interpolating the remaining nodes' position.

In this work, we propose to build a metamodel of a nonlinear magnetostatic parametric problem involving geometric and electric parameters using the POD-RBF approach to approximate the FE solution and the RBF interpolation for the mesh deformation. First, the study case and the FE model are represented in sections II and III. The RBF interpolation method is introduced in section IV. In section V, results for mesh deformation using the RBF interpolation are presented. The approach to build the POD-RBF metamodels is explained in section VI, and the results are presented in section VII.

II. STUDY CASE

The study case is a 2D single-phase EI inductance, with one electrical parameter and two geometrical parameters. An initial geometry is meshed with 36859 elements and 18553 nodes (Fig. 1.). The geometric parameters are the thickness of the air gap 𝑒 varying from 0.1 𝑚𝑚 to 0.9 𝑚𝑚, the width of the central column of the magnetic core 𝑑 varying from 20 𝑚𝑚 to 40 𝑚𝑚. The electric parameter is the phase current 𝑖 of the winding defined in the interval [0,50] 𝐴.

III. FINITE ELEMENT MODEL

The governing equations for the nonlinear magnetostatic FE model are derived from Maxwell's equations. The vector potential formulation used to solve the problem is given by: 𝒄𝒖𝒓𝒍(𝜈(𝑩) 𝒄𝒖𝒓𝒍(𝑨)) = 𝑱 𝒔 (1) where 𝜈 is the reluctivity depending on the magnetic flux density 𝑩, 𝑨 the magnetic vector potential and 𝑱 𝒔 the source current density depending on the current 𝑖. The discretized nonlinear 2D FE equation system can be expressed by: 𝑺(𝑿). 𝑿 = 𝑭 (2) where 𝑿 ∈ 𝑹 𝑛 is the vector of unknowns corresponding to the components of the magnetic vector potential, 𝑺(𝑿) 𝑛×𝑛 the curl-curl stiffness matrix and 𝑭 ∈ 𝑹 𝑛 the source vector. Then, the FE solution 𝑿 depends on the current 𝑖, the width of the central column 𝑑 and thickness of the air gap 𝑒.

T winding

Nonlinear ferromagnetic core

IV. RADIAL BASIS FUNCTION INTERPOLATION METHOD

The RBF interpolation consists in approximating a multivariate function 𝑓(𝒙), using a linear combination of basis functions 𝜙, pre-calculated for 𝑁 𝑡 training points. The approximation of 𝑓(𝒙) is given by: 𝑓(𝒙) ≈ ∑ 𝛼 𝑖 𝜙 𝑖 (𝒙)

𝑁 𝑡 𝑖=1

(3) Where 𝜙 𝑖 (𝒙) = 𝜙(‖𝒙 -𝒙 𝑖 ‖) is a radial function depending on the Euclidean distance between 𝒙 and 𝒙 𝑖 ; and 𝛼 𝑖 is its associated coefficient. Different kinds of functions can be used to define 𝜙 𝑖 (𝒙) such as Gaussian functions for example. The determination of the coefficients 𝛼 𝑖 can be done by solving the matrix system:

𝒀 = 𝑮𝜶 (4) with 𝒀 = [𝑓(𝒙 1 ), … , 𝑓(𝒙 𝑁 𝑡 )] 𝑡 ,𝜶 = [𝛼 1 , … , 𝛼 𝑁 𝑡 ] 𝑡 and 𝑮 = [ 𝜙 1 (𝒙 1 ) ⋯ 𝜙 1 (𝒙 𝑁 𝑡 ) ⋮ ⋱ ⋮ 𝜙 𝑁 𝑡 (𝒙 1 ) ⋯ 𝜙 𝑁 𝑡 (𝒙 𝑁 𝑡 ) ]
This approach can be generalized for vector function 𝒇(𝒙) ∈ 𝑹 𝑞 by interpolating each component.

For a new set of inputs 𝒙 𝑛𝑒𝑤 , 𝑓(𝒙 𝑛𝑒𝑤 ) can be approximated by:

𝑓 𝑗 (𝒙 𝑛𝑒𝑤 ) = ∑ 𝛼 𝑖 𝑗 𝜙 𝑖 (𝒙 𝑛𝑒𝑤 )

𝑁 𝑡 𝑖=1

(5) Where 𝑓 𝑗 is the 𝑗 𝑡ℎ component of the function 𝑓 and 𝛼 𝑗 is its corresponding coefficients.

V. MESH DEFORMATION BASED ON RBF INTERPOLATION

In order to model the geometric variation of the problem, mesh deformation is performed using the RBF interpolation [START_REF] De Boer | Mesh deformation based on radial basis function interpolation[END_REF] [START_REF] Henneron | Mesh Deformation Based on Radial Basis Function Interpolation Applied to Low-Frequency Electromagnetic Problem[END_REF] for two consecutive times to consider the two varying geometric parameters 𝑒 and 𝑑. For each mesh deformation, two sets of nodes are defined, 𝑁 𝑓 nodes where the displacements are imposed and 𝑁 𝑖 nodes where the displacements are approximated. In this case, the function to be interpolated is the displacement function denoted 𝒅(𝒙). By applying the RBF interpolation, the displacement 𝒅(𝒙 𝑘 ) for a node 𝑛 𝑘 ∈ 𝑁 𝑖 is approximated by: 𝒅(𝒙 𝑘 ) = ∑ 𝛼 𝑖 𝜙 𝑖 (𝒙 𝑘 )

𝑁 𝑓 𝑖=1 (6) An indicator for the conformity of the mesh 𝑑 𝑒 is defined by: 𝑑 𝑒 = 𝑑𝑒𝑡 𝑒_𝑑𝑒𝑓 𝑑𝑒𝑡 𝑒_𝑖𝑛𝑖𝑡 [START_REF] De Boer | Mesh deformation based on radial basis function interpolation[END_REF] with 𝑑𝑒𝑡 𝑒_𝑑𝑒𝑓 and 𝑑𝑒𝑡 𝑒_𝑖𝑛𝑖𝑡 are the determinants of the deformed and the initial element 𝑒, respectively. The indicator is positive in the case of a conformal mesh, a negative indicator indicates an overlapping between the elements, and thus the non-conformity of the deformed mesh. Figure 2 presents an example of mesh deformation. The RBF interpolation can also be used to model a more complex mesh deformation; another example is to impose a sinusoidal displacement 𝑑 𝑥 on the nodes on the border of the central column of the magnetic core, such as:

𝑑 𝑥 = ±10sin ((20.5 -𝑦)𝜋 60 ⁄ ) (8) The deformed mesh and the distribution of the deformation indicator are represented in Fig. 3.

VI. METAMODEL BASED ON POD-RBF

A. Principle of POD combined with RBF

The POD-RBF approach is applied to the parametric FE model [START_REF] Adamczyk | Development of a nondestructive technique for measuring thermal conductivity of material with small anisotropy based on application of the reduced order technique[END_REF][5] [START_REF] Henneron | Surrogate model based on the POD combined with the RBF interpolation of nonlinear magnetostatic FE model[END_REF]. The method consists of first gathering a set of data, using the method of snapshots, by solving the original FE model ( 2) for a set of parameter values:

𝑴 𝒔 = [𝑿 1 ⋯ 𝑿 𝑚 ] (9
) Where 𝑴 𝒔 is the snapshots matrix, 𝑚 is the number of snapshots and 𝑿 𝑖 is the FE solution at the 𝑖 𝑡ℎ set of parameter values. Then, the snapshots matrix is decomposed using the singular value decomposition (SVD).

𝑴 𝒔 = 𝒀𝚺𝑾 𝑡 (10) The left orthogonal matrix of the SVD decomposition 𝒀 corresponds to the reduced basis, and the right orthogonal matrix 𝑾 is composed of the FE solution projected into the reduced basis for each set of parameter values. The POD reduced basis 𝚿 of size 𝑛 × 𝑚, is given by: 𝚿 = 𝒀𝚺 (11) For any new set 𝒙 𝑛𝑒𝑤 of parameter values, the approximated solution 𝑿 𝑎𝑝 , can then be defined by: 𝑿 𝑎𝑝 = 𝚿𝑽 𝑡 (𝒙 𝑛𝑒𝑤 ) (12) Where 𝑽 𝑡 (𝒙 𝑛𝑒𝑤 ) is obtained by performing a RBF interpolation on the right orthogonal matrix 𝑾 𝑡 , each element 𝑣 𝑙 for 𝑙 = 1, … , 𝑚 is interpolated for the new set 𝒙 𝑛𝑒𝑤 by: 𝑣 𝒍 (𝒙 𝑛𝑒𝑤 ) = ∑ 𝛼 𝑙𝑖 𝜙 𝑖 (𝒙 𝑛𝑒𝑤 )

𝑚 𝑖=1

(13) The coefficients 𝛼 𝑙𝑖 are determined by solving an equation system as in (4).

B. Greedy algorithm

The choice of snapshots has an impact on the precision and the efficiency of the POD-RBF approach, the aim is to build a precise metamodel using a low number of snapshots. Two greedy algorithms are proposed, one based on a precomputed snapshots library [START_REF] Henneron | Surrogate model based on the POD combined with the RBF interpolation of nonlinear magnetostatic FE model[END_REF], where the exact error of the metamodel can be computed, and the other based on the evaluation of the residual of (2) with approximated solution from (12) to obtain an error approximation without having to solve the FE model for a large number of parameter values sets. We denote 𝒑 a coordinate in the parameter space such as 𝒑 = (𝑖, 𝑒, 𝑑).

The first algorithm consists of; first, calculating a sufficiently large snapshots library 𝑴 𝑋 , for a set of parameters chosen based on the knowledge of the engineer of each studied device . Then, an iterative process is initiated, where at each iteration a new snapshot corresponding to the maximum relative error of coordinate 𝒑 𝑚 is considered to update the POD-RBF metamodel. The metamodel and the error are evaluated in the grid of parameters space used to precompute the snapshots library. The process is repeated until the stop criterion is satisfied.

For the second algorithm, the metamodel and the norm of residual 𝒓 are evaluated on a sufficiently large grid of parameters space 𝛀, defined by the user. At each iteration a new snapshot is added for the coordinate 𝒑 𝑚 corresponding to the greatest norm of the residual. The process is repeated until the stop criterion set on the residual norm is met.

The stop criterion for the first algorithm 𝜖 1 is set to a maximum error between the FE solution and the metamodel solution lower than 5%. While for the second algorithm, the stop criterion 𝜖 2 is set to a maximum norm of the residual lower than 5. It is important that the snapshots library 𝑴 𝑋 for the first algorithm, and the grid 𝛀 for the second algorithm, are large enough. Because, in the first case, the snapshots are selected from the library, and in the second, the snapshots are chosen within the coordinates of the grid. Having a reduced library or a coarse grid will reduce the precision of the metamodel. The required time to solve the FE model for the 2574 calculation points is 16 min, using an i7 CPU with a 16 Go RAM. The results of the original FE model are taken as a reference. Fig. 5 presents the distribution of the magnitude of the magnetic flux density 𝑩.

VII. RESULTS

A. FE model

B. POD-RBF metamodel using algorithm 1

A POD-RBF-based metamodel is built using the first greedy algorithm. The snapshots library 𝑴 𝑿 is composed of the FE model solutions for the 2574 defined calculation points. To satisfy the stop criterion, the greedy algorithm selects 151 snapshots. Once the snapshots library is calculated, the total time required for the greedy algorithm convergence and the evaluation of the POD-RBF metamodel for the 2574 calculation points, is 4 𝑚𝑖𝑛, an additional 16 𝑚𝑖𝑛 is required to create the snapshots library. To assess the precision of the metamodel, the The error distributions as a function of 𝑑 and 𝑖 for fixed 𝑒 values are presented in Fig. 6. The global mean relative error between the flux cartographies obtained using the metamodel and the original FE model is 1.6%, while the maximum local error is 5%. The first algorithm chooses the snapshots for the coordinates corresponding to the maximum error, thus the maximal local error is relatively low and the error distribution is relatively even. Fig. 7 presents the difference of the magnitude distributions of 𝑩 obtained from the FE model and the POD-RBF approximation for the same parameter values as in Fig. 5. The magnitudes of the error are small compared with those of the magnetic flux density (Fig. 5). The maximal values of the error are located in the internal corners where the ferromagnetic core is saturated.

C. POD-RBF metamodel using algorithm 2

A second POD-RBF metamodel is built, using the residualbased greedy algorithm. The metamodel is evaluated on a grid composed of the same 2574 points where the FE model is solved. 206 snapshots are required to reach the stop criterion. The convergence of the residual-based algorithm requires 3 ℎ, it is significantly higher than the time required by the exact error-based algorithm, as in the second algorithm the snapshots are not precomputed and thus the FE model must be solved at each iteration to get a new snapshot. Furthermore, the evaluation of the residual generates an additional computational time, as it requires the assembly of the FE matrix for every point of the grid and at each iteration. The convergence time of the residual-based algorithm is also higher than the required time to solve the FE model on the defined grid. Indeed, solving equation ( 2) consumes more time than matrix assembly; however, in the iterative process of the residual-based algorithm, the matrix assembly step is performed over 500000 times, which explains the increase of calculation time. The distributions of the error on the flux cartographies obtained using this metamodel are represented in Fig. 8. The global mean error on the flux calculated using the residual-based metamodel is 4.1%, while the maximum local error can reach 51%. As shown in Fig. 8., the error is mainly concentrated around the low values of current. The residual used as the criterion of snapshots choice in this algorithm is an absolute value; thus, the algorithm often neglects the points where the solution is low, even if the relative error in these coordinates is high. The residual-based algorithm selected more snapshots than the exact error-based algorithm. However, the precision of the first algorithm is better. This is due to the fact the residual does not give a precise evaluation of the error; instead, it gives an approximation of the error without having to calculate the exact solution.

Fig. 9 presents the difference of the magnitude distribution of 𝑩 obtained from the FE model and the residual-based metamodel for the same parameter values as in Fig. 5. The magnitudes of the error are of the same order as those in Fig. 7.

D. Comparison

Two POD-RBF metamodels had been established, for a magnetostatic nonlinear parametric problem, allowing to approximate the FE solution as a function of three parameters. The two metamodels are built using two different algorithms, with different choice criteria for the snapshots. These criteria will impact the precision of the metamodels and the convergence of the algorithms. In order to evaluate the convergence rate of the two algorithms, two types of errors are defined, the maximum error in the sense of Euclidean norm 𝑒 𝑛2 , representing the maximum local error, and the mean error in the sense of Frobenius norm 𝑒 𝐹𝑟𝑜 , representing the global mean error.

Fig. 1 .

 1 Fig. 1. The single-phase EI inductance (a) and associated mesh (𝑑 = 30 𝑚𝑚 and 𝑒 = 0.5 𝑚𝑚) (b).

Fig. 2 .

 2 Fig. 2. Deformed geometry (a) and the distribution of the deformation indicator (b), for 𝑑 = 20 𝑚𝑚 and 𝑒 = 0.1 𝑚.

Fig. 3 .

 3 Fig. 3. Deformed geometry (a) and the distribution of the deformation indicator (b), for a sinusoidal displacement.

  First, the original FE model in (2) is solved for a regular grid of 2574 points, for 9 air gap thicknesses 𝑒 from 0.1 𝑚𝑚 to 0.9 𝑚𝑚, 11 𝑑 values from 20 𝑚𝑚 to 40 𝑚𝑚 and 26 current values 𝑖 from 0 to 50 𝐴. Magnetic flux associated with the winding is calculated as a function of the three varying parameters. Cartographies of linkage flux as a function of 𝑑 and current 𝑖, for fixed values of 𝑒, are represented in Fig. 4.

Fig. 4 .

 4 Fig. 4. Original FE model flux cartographies as a function of 𝑑 and current 𝑖 for 𝑒 = 0.1 𝑚𝑚 (a) and 𝑒 = 0.9 𝑚𝑚 (b).

Fig. 5 .

 5 Fig. 5. Magnitude distribution of 𝑩(𝑇) for 𝑒 = 0.1 𝑚𝑚, 𝑑 = 0.9 𝑚𝑚 and 𝑖 = 40𝐴.

Algorithm 1 2 ‖𝑿 2

 122 Greedy algorithm with exact error evaluation Input: 𝑴 𝑿 = [𝑿 1 ⋯ 𝑿 𝑄 ], 𝒑 1 Output: 𝚿, 𝛼 𝑙𝑖 -𝑚 = 0 while 𝑚𝑎𝑥(𝒆) > 𝜖 1 do -𝑚 = 𝑚 + 1 if 𝑚 > 1 then -select the coordinate corresponding to the maximum error such as 𝒑 𝑚 = 𝑎𝑟𝑔𝑚𝑎𝑥(𝒆) end if -𝒑 ← [𝒑, 𝒑 𝑚 ] -𝑴 𝒔 ← [𝑴 𝒔 , 𝑿 𝒑 𝑚 ] -perform a SVD on 𝑴 𝒔 , update 𝚿 and 𝛼 𝑙𝑖 -evaluate the error 𝒆 = [𝑒 𝑘 ] 𝑘=1 𝑄 with 𝑒 𝑘 = ‖𝑿 𝑘 -𝑿 𝑘 𝑎𝑝 ‖ Greedy algorithm with residual evaluation Input: 𝒑 1 , 𝛀 = [𝜔 1 , … , 𝜔 𝑄 ] Output: 𝚿, 𝛼 𝑙𝑖 -𝑚 = 0 while 𝑚𝑎𝑥(𝒓) > 𝜖 2 do -𝑚 = 𝑚 + 1 if 𝑚 > 1 then -select the coordinate corresponding to the greatest residual such as 𝒑 𝑚 = 𝑎𝑟𝑔𝑚𝑎𝑥(𝒓) end if -𝒑 ← [𝒑, 𝒑 𝑚 ] -𝑴 𝒔 ← [𝑴 𝒔 , 𝑿 𝒑 𝑚 ] -perform a SVD on 𝑴 𝒔 , update 𝚿 and 𝛼 𝑙𝑖 -evaluate the residual 𝒓 = [𝑟 𝑘 ] 𝑘=1 𝑄 with 𝑟 𝑘 = ‖𝑭 -𝑺(𝑿 𝑘 𝑎𝑝 )𝑿 𝑘 𝑎𝑝 ‖ 2 and 𝑿 𝑘 𝑎𝑝 computed by (12) end while error between the fluxes calculated by the metamodel 𝚽 𝑎𝑝 and the FE model 𝚽 𝐹𝐸 , is defined as:

Fig. 6 .

 6 Fig. 6. First algorithm POD-RBF metamodel error distribution for 𝑒 = 0.1 𝑚𝑚 (a) and 𝑒 = 0.9 𝑚𝑚 (b).

Fig. 7 .

 7 Fig. 7. Difference of the magnitude distribution of 𝑩(𝑇) for 𝑒 = 0.1 𝑚𝑚, 𝑑 = 0.9 𝑚𝑚 and 𝑖 = 40𝐴.

Fig. 8 .

 8 Fig. 8. Second algorithm POD-RBF metamodel error distribution for 𝑒 = 0.1 𝑚𝑚 (a) and 𝑒 = 0.9 𝑚𝑚 (b).

Fig. 9 .

 9 Fig. 9. Difference of the magnitude distribution of 𝑩(𝑇) for 𝑒 = 0.1 𝑚𝑚, 𝑑 = 0.9 𝑚𝑚 and 𝑖 = 40𝐴.

𝑒 𝐹𝑟𝑜 = ‖𝑿-𝑿 𝑎𝑝 ‖ 𝐹𝑟𝑜 ‖𝑿‖ 𝐹𝑟𝑜

(15)

The evolution of error represented by the two norms as a function of the number of iterations, for the two algorithms is plotted in Fig. 10. Although globally converging in the same way (Fig. 10.a), the maximum error converges best for the first algorithm. The first algorithm chooses at each iteration the snapshots corresponding to the higher maximum error, resulting in fast convergence of the maximum error. On the other hand, the residual-based algorithm chooses the snapshots corresponding to the higher residual norm; thus, if the maximum relative error does not represent an important residual norm, many iterations can pass without any improvement of the maximum relative error, which can be seen in the constant parts in Fig. 10.b. The impact of the choice criteria on the selected snapshots can be seen in Fig. 11. The distribution of snapshots coordinates in the parameters space selected by the two algorithms is plotted for the 151 first snapshots. The performances of the metamodels resulting from the two algorithms, in terms of precision and computational time for the same number of snapshots, are summarized in Table 1.

The POD-RBF is an interpolation-based method; it offers the advantage of approximating the solution without accessing the FE full order matrix system. Nevertheless, the residual-based algorithm requires the assembly of the FE matrix, for each calculation point at each iteration, in an additional timeconsuming step, in comparison with the first algorithm. Moreover, the residual only gives an approximation of the exact error. The snapshots selected by the residual-based algorithm do not necessarily correspond to the most important error; thus, more snapshots are required to reach the same precision as the first algorithm.

The evaluation time of a new set of parameter values for the POD-RBF-based metamodels is significantly lower than the solving time for the FE model. The necessary time to build a POD-RBF metamodel depends on the used algorithm. Nevertheless, omitting the construction time of the metamodel, the use of a POD-RBF-based metamodel can significantly reduce the computation time.

VIII. CONCLUSION

A parametric nonlinear magnetostatic problem has been studied. A consistency preserving mesh deformation is carried out using a RBF interpolation. Two POD-RBF metamodels of the FE solution are built, following two different algorithms. The type of algorithm impacts the choice of snapshots, the accuracy and the construction time of the metamodel. The established metamodels allow a significant reduction of the calculation time. POD-RBF metamodels of the parametric FE solution can be used in an optimization process for different applications or to be coupled with other numerical models to simulate a more complex system.