M Slazhniev

Kyung Hyun Kim

Se Hyun Suk Sim

Won Kim

Kyung Hyun

Research on MHD process for controlling the segregation of Al-Sn alloy

Keywords: electromagnetic stirring, processing, segregation, bearing alloy, microstructure

Introduction

The Al-Sn alloys there are widely using at manufacture of the sliding bearings in inserts of the internal combustion engines in machinery. In order to such bearing product must be have a high strength and tribological properties at it using. Necessary properties to possible get at providing of the special optimal bearing structure with interdendritic fine redistribution of Sn-inclusion between Al-grains [START_REF] Bushe | Transport[END_REF]. The main industry problem of manufacturing of such and another immiscible in liquid state metals in large difference of the density and melting point is consisted [START_REF] Abraham | [END_REF]. Due to that the aluminum as a base melt of such bearing alloys to have a melting point 660C and density at 700C 2380kg/m 3 , because Sn, as a sliding component have a 231,9C and density at 700C -6640kg/m 3 , that in result the strong tendency to segregation in the liquid state in a wide range of temperatures and concentrations is causing. For solving problem of segregation, the immiscible melts must be under continuous stirring in all volume with providing the high homogeneity in dispersed emulsion state [START_REF] Abraham | [END_REF]. In industrial case by means of the magnetic hydrodynamics (MHD) [3] and applying of the complex electromagnetic and hydrodynamic processing of prone to segregation immiscible bearing alloys, such as Al-25 and 40%Sn can be solved.

Developing methods and device for electromagnetic and MHD-processing of the immiscible metals

On the first stage of researches the original method and device, for electromagnetic processing bearing alloy, such as Al-Sn, has been developed and tested. The general means of electromagnetic processing (EMP) on applying of the one or two-dimensional controlled pulsing electromagnetic stirring of melt by alternating and/or traveling magnetic fields was based. Additionally for improving of the pulsing processing effect at the developing, the EMP there is combining with the simultaneously mechanical vibration of batch reactor during melt processing. Taking into account that for using in developing method device is presented as an intermediate vessel for Al-Sn processing before pouring to casting mold (or into twin roll machines for example) the batch reactor as a standard steel hand ladle with 180mm height and 170 mm has been used. The batch reactor at on one time is allowed to process the useful capacity from 3 to 5 liters or 5 to 15kg by Al-25 and 40%Sn alloy. Such decision for small or medium simple manufacture of the bearing product can be use, and easy especially in the small foundry or casting companies can be implemented, and not required the developing of the additionally expensive equipment and that is important can be easy operated by workshop workers. For activation of electromagnetic one or two-dimensional processing of the Al-Sn alloy into batch reactor, the special two-pole electromagnetic system, which generate the two mutually perpendicular variable magnetic fields (B1, B2) was designed (Fig. 1). For convenient practical application in real industrial conditions the electromagnetic processing sys-tem was executed as a stand, with bottom and lateral electromagnets, that is located under 90 angle to axes of it poles with distance between each designed to size of batch reactor (Fig. 1). Each from electromagnets had an own magnetic cores with 70*90mm cross section and coils (by 17 turns). The each coils was power supplied separately from own power low voltage sources (20~80V) standard industrial frequency 60Hz, that to allowed generate of the magnetic fields intensity (B1, B2) in the range 0.05~0.15T in the gap between poles. Due to that, each electromagnet powering individually, this is allowed to creating of the 1D, bottom/lateral, or 2D controllable electromagnetic stirring and melt processing with possibilities of separately controlling the magnitudes of AC voltages and initial shift phases  phases = 0, π/2, 2π/3 on each electromagnets. However, for improving of the force pulsing processing impact on the immiscible metals into batch reactor the additionally mechanical pulsating vibration by original way is applied. For this between the poles of electromagnets, the elastic refractory gasket (thickness up to 5mm) was placed (Fig. 1). At this the each one magnetic fields (B1, B2) has provided the attracting vibration force actions (Fattr) at its interactions with ferromagnetic material of batch reactor. The attracting pulsation force is directed to the poles of electromagnet side and had value Fattr = 12~27kg, (117~267N), that the oscillatory actions on the bath reactor with 120Hz makes. In the case of both electromagnets using the mechanical vibratory is characterized by magnitudes A vert = 1.5~2mm, 2A horiz = 0.5~1mm. Taking into account, that direction of electromagnetic pulsation force (Fem1, Fem2), which are activated in liquid alloy by imposed from electromagnets AC magnetic fields (B1, B2) are directed from electromagnet poles, but the attracting force had directed an opposite way. However, the doubled pulsating effect is provide the increasing of hydrodynamic pulse action, that appearing at the time of the transition amplitude of the magnetic field through zero. Indicated effect are presented on Fig. 2, where are observed the periodical running waves, that moving with melt at it stirring from electromagnet poles.

In result, by combination of bottom and/or lateral electromagnetic AC fields, the 1D (vertical or horizontal toroidal) multiple melt stirring into batch reactor with velocity in jet melt stream v = 0.1~1.1m/s (Fig. 2) is provided. For case of two-dimensional (2D), left-hand or right-turbulent electromagnetic in vertical plane melt stirring, that is actioned by two (bottom and lateral) pulsating electromagnetic forces -Fem1 and Fem2, the processing alloy is performed at multiplicity of volumetric circulation with radial velocity (f rad.circ. = 0.4~0.8Hz). In this time into submerged jet of melt flow cores (Fig. 2) the hydrodynamics pulsation with several harmonics and frequencies in range 0.4~5Hz are consisted (Fig. 3), that by the Pitot tube methods was researched [START_REF] Dubodelov | Joint 15 th Riga and 6 th PAMIR International Conference on Fundamental and Applied MHD[END_REF] . At processing on the melt surface the intensive intermixing not observed, that was allowing to prevent the melt oxidation and inputting inside the nonmetallic inclusions and etc.

In depending from the electrical modes of electromagnets coils switching (for case of 2D-stirring), at the initial phase angle  phases = 0, π/2, 2π/3 between the bottom and lateral magnetic fields (B1, B2) on batch reactor are impacted by means of the cyclic or parametric mechanical oscillation in the vertical/horizontal or simultaneously in two directions. However, this mechanical impact had an opposite direction in relation to pulsating electromagnetic forces PEMF (f pemf = 0.8~4.0kg, 7.85~39.4N, frequency f = 120Hz), which is generating in the melt by reacting electric currents excited by magnetic field (B) with this the magnetic field. The two basic alloys (Al-Sn) with content of Sn of 25/40% has selected for experimental approbation. For stabilize the characteristics of strength, elongation and "castability" to the composition of Al-Sn bearing alloy in small quantities the copper and silicon (within 1%) has been added. After melting Al and Sn in a melting resistance furnace and it heating to a temperature 660~800C by the using of impeller technology (GBF) the initial mixing by rotary activator (impeller) was produces with simultaneously degassing by argon gas blowing. After completion of the mixing (usually within 10 or 20 minutes), the melt has poured into preheated batch reactor, located on the electromagnetic stirring system. In next, by one of chosen schemes (with one or two electromagnets) the poured in the batch reactor melt was processed within 30 seconds or up to 2 minutes in depending from melt dose. The emulsified alloy after processing was pouring into the preheated to a temperature 180~200C chill mold for obtaining the specimens as a plate with 6 mm thickness and 450×320 mm size. Application the mechanical stirring by impeller (rotary activator) has provided the insufficient emulsification of Sn and microstructure, that there are no more than 60% of Sn with interdendritic redistribution (Fig. 4), but often the large size of Sn particles 50~85m was present. The result of 1D-electromagnetic processing with mechanical vibrational of Al-25%Sn, at using just one alternative magnetic field (0.1T), bottom or lateral, in comparisons with only mechanical stirring (Fig. 4) is showed the more deeper mixing and emulsification of Sn particles (5 ~ 40m). Sometimes the microstructure had of the interdendritic advantageous distribution, more dense and nonporous (Fig. 6). In case of EM-stirring without vibration the microstructure has a remaining gas porosity and minimal size of Sn particles makes no less 20~35m (Fig. 5).

Along with this, application of the 2D-electromagnetic melt stirring with additional parametric mechanical oscillation of batch reactor was shown deeper emulsification Sn in Al matrix (on example Al-25%Sn). Formation up to 90% zones of the homogeneous preferably interdendritic (optimal for bearing) distribution of Sn, 5÷25m (Fig. 6) without gas or shrinkage porosity in the casting state was obtained.

From the practical casting viewpoint the optimal temperature of processing the bearing alloy Al-25%Sn -700~720C and for Al-40%Sn the 700~760C in the performed experiments was set. Because in this case the processed melt is ready to pouring into casting mold and have a low overheating over the melting point of Al (as a base metal for such alloy) and that the device for processing are not require the using of the high-temperature expensive fireproof (refractory) material. In addition, the small overheating processed bearing alloy has a best condition for next solidification with forming the high homogeneity dispersed "bearing" microstructure. At the second stage the comprehensive experimental work about preparation, refining, emulsification of binary alloys Al-25 and 40%Sn by using of the industrial magnetodynamic installation MDN-6A-0.63 was performed. MDI, it is a multifunctional electro-technological device, which is the induction channel furnace with twin W-shaped channel and additional electromagnetic pump, that to provide controlled mixing of alloy in the them crucible and the induction channel by pulsing electromagnetic forces, with possibilities of melt heating with continuous controllable mixing.

The main idea of application of such unit is considered in the development of industrial MHD-technology for manufacturing bearing alloys, based on Al-25 and 40%Sn, that include it preparation (melting), refining, processing and further electromagnetic pouring to the casting mold or continuously casting machine. During research, the special mode of the lateral stirring with simultaneous degassing and intensive melt emulsification, by using suction melt in the central and pumping through the lateral channels, was developed and applied (Fig. 8). After preparing and processing alloy has been poured in the preheated to 180~200C chill-mold with 6 mm thick by using of three types of casting methods: manual gravity (with the mass flow rate of 0.1~0.2kg/sec), electromagnetic pouring by open stream through metal duct and by low electromagnetic pressure methods [3,[START_REF] Dubodelov | Joint 15 th Riga and 6 th PAMIR International Conference on Fundamental and Applied MHD[END_REF]. By practical viewpoint for industrial application the "lateral stirring" and "central suction" molten melt under the influence of pulsing electromagnetic forces (PEMF) in working area in MDI was checked, that was justified for "preparing" binary alloys (Al-Sn) as optimal. The processing of Al-Sn alloy in quantity of 100~150kg has been performed by continuously stirring through induction W-shaped channels, MDI working area and to the crucible at electromagnetic pump-ing in the circulation mode with flow rate 4~5kg/sec, temperature 740~760C during 10~15 minutes (Fig. 8). The analysis of microstructures (Fig. 9) there are show the high efficiency MHD-emulsification Al-Sn alloy in MDI, which the guaranteed uniform redistribution of Sn in the interdendritic space of Al in the prepared bearing alloy. By the experience of experimental researches, especially for alloy from immiscible in liquid state metals (Al-Sn or other), best result the scheme of central stirring, with suction through the central channel and the pumping through the side, has been obtained (Fig. 9). Stabile result is achieved at providing of multiple (20~30 times) stirring the volume of molten Al-Sn alloy in the MDI, at modes with a minimal harmful action of vortexes in the working area MDI, which at the exceeding of rotation speeds above 3 m/sec there are provoking an increase of Sn segregation. However, phenomenon of theta pinch-effect [3,[START_REF] Dubodelov | Joint 15 th Riga and 6 th PAMIR International Conference on Fundamental and Applied MHD[END_REF], that is appearing in the lateral channels MDI at passing through them the AC currents (I) more than 10~15kA, does contribute to increasing of segregation of tin inclusions. It can be explained by electromagnetic separation effect due to the difference electrical conductivity Sn and Al (Al/Sn 0.028/0.12 *mm 2 /m at 20C), that is reducing the melt emulsification process efficiency. Along with this, the high efficiency of emulsification Sn in the Al-matrix (is not more than 2~5m, which by frozen drops in rapid-cooled probe defined) can be obtained at the melt MHD-processing in the WA MDI by the pulsating MHD-action of volumetric electromagnetic forces (120Hz) and at creation of electromagnetic pressure not exceeding 20kPa (I = 8~10kA, B = 0.1~0.2T).

Conclusion

In this study the original method and device, for electromagnetic processing bearing alloy from immiscible in liquid state metals, such as Al-Sn, based on applying of the one, two-dimensional controlled pulsing stirring of melt by alternating and/or traveling magnetic fields combined with mechanical vibration of batch reactor is developed and described. In result of 1D-EMS applying has been achieved the deeper mixing particle of Sn, and obtained the dense (nonporous) structures in casting state with mainly interdendritic advantageous distribution of Sn in Al-matrix, makes sizes 5~40m, was stabile obtained. 2D-electromagnetic stirring with an additional parametric mechanical oscillation has provided to a deeper emulsification Sn in Al-matrix, with formation up to 90% homogeneous distribution zones of Sn (5~25m) and preferably interdendritic distribution without gas or shrinkage porosity. Best application MHDfacilities for emulsification immiscible metals (Al-Sn) by the pulsating volumetric electromagnetic forces (120Hz) has been found at generating in working are of MDI pressure not exceeding 20kPa, at intensive reversed stirring melt into induction channel and crucible (rate 4~7kg/sec), during no less than 10~15 minutes at 740-760C for 100~150kg of bearing alloy. Studied method and the electromagnetic device has been allow obtained a new technical result, concluded in the high level homogenization and emulsification Al-Sn alloy and can be implemented in mixers, melting industries furnaces, tundishes for continuous casting machines at manufacturing and contactless processing of bearing alloy.

Fig. 2 :Fig. 1 :Fig. 3 :

 213 Fig. 2: Forming the core of a submerged jet flow in the melt and the appearing the running waves on the alloy surface

Fig. 4 :

 4 Fig. 4: Mechanical stirrings by GBF, 730C, Al-25Sn

Fig. 8 :Fig. 9 :

 89 Fig. 8: Scheme of Al-25%Sn alloy MHD-processing in MDI at suction circulation mode (100~150kg)


Yueming Zhou

Zuosheng Lei

Bin Li

Rujun Wei

Yunbo Zhong

Zhongming Ren

Flow field in slab continuous casting mold driven by electromagnetic stirring under submerged entry nozzle clogging

Keywords: Slab Continuous Casting, Flow Control, SEN Clogging, Electromagnetic Stirring

In order to clarify the characteristics of a molten steel flow with or without the electromagnetic stirring under SEN clogging condition, a physical modeling system, which uses liquid mercury as the analog to simulate the flow of liquid steel in the continuous casting mold, has been developed. The liquid mercury velocity is measured by Ultrasonic Doppler velocity meter. The experiment results show that the SEN clogging will destroy the asymmetric of the flow in the mold and the uniform of the velocity in front of the initial solidifying shell. They will result in some products quality questions in the following process. The electromagnetic stirring is helpful to modify the flow field in the mold because the symmetry of the flow field can be optimized. If EMS current increases, the flow field will become more asymmetric and uniform.

Introduction

Submerged entry nozzle (SEN) Clogging is a serious problem to the slab continuous casting of steel because it has great influence on the fluid flow and product quality. Electromagnetic Stirring (EMS), an efficient way to modify the liquid steel flow field in the mold, has attracted many researcher's attitudes [1][2][3] . How and what style EMS can achieve the best liquid flow field and the best product quality when SEN clogging occurs is still an open problem.

In this research, a physical modeling system, which uses liquid mercury as the analog of liquid steel in the slab continuous casting mold has been developed in order to clarify the characteristics of a molten steel flow with and without EMS under SEN clogging condition. The liquid mercury velocity is measured by ultrasound Doppler velocimetry. A physical model using mercury is set up, whose architecture and dimension of main parts are showed in Fig. 1. In this system, mercury can be circulated by a electromagnetic pump from the mold bottom and flow up to the tundish. Mercury flow back to the upper zone of the mold via the Submerged Entry Nozzle (SEN). The SEN is designed as single-side clogged and the clogging ration is defined as the clogging area to the whole exit port area, see in the Fig. 1, such as non-clogging, 10% clogging, 25% clogging and 50% clogging. Experiments are operated under 0.4 m/min casting speed equal to the prototype mold of 0.89 m/min, and the up-points of the nozzle exit ports is located 34 mm below the liquid metal free surface in the mold. The vertical and horizontal velocity in the mold is measured by means of the ultrasound Doppler velocimetry (UDV) [4,5] in the measurement area showed in Fig. 1.

Experimental

Experimental Results and Discussion

The measured flow field of the front 1/4 plane without electromagnetic stirring is showed in Fig. 2. When the SEN is not blocked, the flow field in mold is an standard double roll structure and the maximum jet velocity is about 0.1m/s (Fig. 2 During slab continuous casting the free surface is very important to control the velocity of liquid metal flow under the mold flux. If the velocity of this area is too high, the possibility of mold flux entrapment will increase. Otherwise if the velocity of this area is too slow, the heat carried by hot liquid metal may be not enough to melt the mold flux which may lead to worse lubrication and worse surface quality. Based on lots of industrial practices,, it is estimated that the reasonable upper and lower limit of molten steel velocity are about 0.326 m/s [6] and 0.1 m/s [START_REF] Thomas | Fluid flow in the mold[END_REF][START_REF] Kubota | [END_REF] , respectively. According to similarity principle, the reasonable velocity zone of the liquid mercury near the free surface ranges from 0.044 to 0.15 m/s in the model mold. Fig. 3 shows the velocity of the flow fluid 2mm below the surface calculated from the measurement data by mathematical interpolation. It can be seen that the velocity near the surface is still reasonable when the clogging ration are 10% and 25%. But when the clogging ratio is 50%, the velocity on the non-clogging side increases above the upper limit while the velocity on the clogging side decreases below the lower limit. Fig. 4 gives the measured flow field when the EMS currents are 40A and 60A without SEN clogging. Compared with the flow structure showed in Fig. 2, it can be seen that EMS destroyed the typical double-roll flow structure totally. In the right part of the mold the jet flow throughout the nozzle is accelerated while in the left part the jet is braked partially and changes its flow direction. We can imagine that the flow can realize a horizontal circulation under the horizontal electromagnetic stirring force.

Fig. 5 gives the measurement flow field both the front and the back 1/4 plane when the EMS current and SEN clogging ratio are 40A and 50%, respectively. Compared with Fig. 4, it can be found that the electromagnetic stirring structure keeps the same, but the SEN clogging change the discharges quantity of liquid metal between clogging side and nonclogging side, which will lead to a different flow velocity.

In order to take both clogging effect and EMS effect into account and evaluate the flow field modification quantitatively, a new index S, called symmetric index, is introduced. And it is defined as:

   1 min( , ) max( , ) S n n i V U V U (1)
Where V is the velocity of one measurement point in the back 1/4 plane (for example, point A showed in Fig. 5 (c)) and U is the velocity of the corresponding point in the front 1/4 plane (for example, point A showed in Fig. 5 (c)), the two points are Centro symmetrical about the nozzle. Velocity of all the 45 points in every measurement area showd in Fig. 5 x It is obvious that when the SEN is non-clogging, the S index is equal to 1 with EMS or without EMS. Fig. 6 show the relationship between "S index" and SEN clogging rate. It is comprehensible that the S index decreases when the clogging ration increases. EMS can improve the S index in most of the clogging situations, so it is helpful to maintain the good slab quality to a certain degree. But how to obtain the best optimized flow field by EMS when SEN is clogged is still a open question, which needs a further and systematic study.

Conclusions

Physical model experiments are performed to simulate the flow field in slab continuous casting mold with EMS under submerged entry nozzle clogging condition. The influence of the degree of clogging and the current intensity of EMS on the symmetric degree of flow field are considered.

The experiment results show that under SEN clogging condition the flow in the mold is asymmetric and the velocity is not uniform in front of the initial solidifying shell, which are harmful to the products quality. The electromagnetic stirring is helpful to optimize the flow field in the mold because the symmetry of the flow field can be improved along with the increasing EMS current.

Fig. 1

 1 Fig.1 Schematic of experimental system

  Submerged Entry Nozzle with inner diameter 16mm and outer diameter 25mm and two sides rectangle exit ports with the size of 16mm (Height) × 12mm(Width). SEN immersion depth 34mm and Nozzle exit angle is 15 °. (a) Nonclogging. (b) 10% clogging. (c) 25% clogging. (d) 50% clogging.

  (a)). But if the SEN is clogged, see in Fig.2 (b), jet velocity of the left clogging decreases and the up circulation disappears while the velocity of the non-clogging right side increases, but the neutral symmetry in the mold decreases.

Fig. 2 .Fig. 3 .

 23 Fig. 2. The influence of SEN clogging ratio on flow field in the mold without EMS (a) Non-clogging, (b) 50% clogging.

Fig. 4 .Fig. 5 .

 45 Fig.4. Flow field of front 1/4 plane under different stirring current with SEN non-clogging (a) 40A Current, (b) 60A Current.

Fig. 6 .

 6 Fig.6. The relationship between S index and SEN clogging rate

Acknowledgment

This project financially supported by National Science Foundation of China (NO.51274137 and NO.50874133).


Huixing Wu

Yunbo Zhong

Yingbin Wang

Huai Wang

Zuosheng Lei

Zhongming Ren

Effect of the Electromagnetic Compound Field on Microstructure and properties of continuously cast Al-20wt%Sn alloy

Keywords: Al-Sn bearing alloy, electromagnetic vibration, grain refinement, tensile strength

Cylindrical ingots of Al-20wt%Sn bearing alloy were prepared by continuous casting under superimposing electromagnetic compound field (the static magnetic field and the alternating electric field were simultaneously imposed in the melt), of which the current intensity ranges from 0 to 2.2 × 10 4 A/m 2 and the magnetic flux density of the static magnetic field is 0.09 T. The microstructures and mechanical properties of the ingots were investigated. The results indicated that the grain size decreased with the increase of the current intensity. At the same time, both microhardness and tensile strength of the ingots increased. However, when the current intensity exceeded 1.8 × 10 4 A/m 2 , the grain size tended to be larger. At the same time, both micro-hardness and tensile strength of the ingots decreased. Based on the theory of electromagnetic vibration, a possible mechanism was suggested out.

Introduction

Aluminum-tin alloy has been widely used in automobile industry with excellent properties as one kind of bearing alloy. Traditional preparation methods of aluminum-tin alloy existed kinds of problems such as gravity segregation and coarse grain et al, which could be solved by electromagnetic vibration method. Electromagnetic vibration was firstly studied by Vives and Radjai [1][2][3][4] by simultaneously imposing of a static magnetic field and an alternating current which is perpendicular to each other into the solidification of aluminum alloys. The result showed that the microstructure was refined efficiently. Then, Mizutani [5] studied the effect of the intensity and the frequency on microstructural refinement of aluminum alloys. The results showed that there existed optimal value of both the intensity and the frequency, and refinement of primary α-dendrite particles was likely to be caused by the collapse of dendrite arms due to microexplosion and stirring of the melt. In this paper, the molten Al-20wt%Sn alloy was imposed simultaneously by a static magnetic field and an alternating current which is perpendicular to each other. The effect of operating parameters on the refining structure of Al-20wt%Sn alloy has been investigated experimentally. The equipment in this paper includes electrode, dummy bar, alternating current power et al. The size of the mold is 200mm in height, and the radius is 40mm. The average magnetic flux density remains 0.09 Tesla and the current intensity can be adjusted from 0 to 2.2 × 10 4 A/m 2 . The continuous casting sets were illustrated in figure 1.

The material used in this experiment was aluminum-20wt%tin alloy. It was heated to 730 ℃ by a heater and held for 20 minutes to make the molten metal more homogeneous. Turned on the switch of the cooling water and let it flow at a setting rate. Then poured the molten metal into the mold, and turned on the switch of the electromagnet and the alternating current power when liquid level was higher than the bottom of electrode. Adjust the value to let dummy bar go down at a setting velocity to ensure the steady of the liquid level of the molten metal. After the solidification, split the sample along the direction of cross section and polished it to observe the microstructure. Besides, the hardness and the tensile strength of the sample were also measured.

Results and discussion

Figure 2 shows the change in microstructure of cross section for the sample of various AC intensities at 0.09T static magnetic field. Obviously, the grain size decreased with the increase of the AC intensity, which is mainly attributed to the effect of the electromagnetic vibrations. However, when the current intensity exceeded 1.8 × 10 4 A/m 2 , the grain size tended to be larger.

F = J  B (1
) Solidification temperature of aluminum-20wt%tin alloy ranged from 228 ℃ to 630 ℃, which cost long time to finish solidification and resulted in large dendritic crystals. Formula (1) indicated that the intensity of electromagnetic vibration was enhanced with the increase of the current intensity, so the temperature gradient in molten metal tended to be gentler. As a result, more nucleation particles appeared in the molten metal and the final microstructure of the alloy was refined gradually. Meanwhile, molten metal in the mold obtained more Joule heat with the increase of the current intensity depended on the formula (2) below. Much more nucleation particles would be melted into the molten metal, so the final microstructure of the alloy tended to be larger when the current intensity increased to higher value.

Q = I 2 R t (2)
Conclusions 1) Simultaneous imposition of a static magnetic field and an alternating current which is perpendicular to each other could refine the microstructure of aluminum-20wt%tin alloy.

2) With the same magnetic flux density, the degree of grain refinement of the aluminum-20wt%tin alloy rose to a certain value with the increase of the current intensity and then decreased. The hardness and the tensile strength also showed the similar change.

3) When the current intensity was 1.8 × 10 4 A/m 2 and the magnetic flux density was 0.09 Tesla, grain refinement reached to a greatest degree which resulted in a peak value in the change of the hardness and the tensile strength, which was probably due to the combined action of change of Lorenz force and the temperature field. If controlling the current intensity and the magnetic flux density well, we could obtain refined microstructure with high mechanical properties.

  Experimental details (a) Envisage map (b) look down map Fig. 1: Sketch of experimental device of continuous casting in electromagnetic field 1-Electrode, 2-Inner lining, 3-Outer lining, 4-Molten metal, 5-Solidified metal, 6-Cooling water in, 7-Dummy bar, 8-Cooling water out, 9-alternating current power, 10-Electromagnet

2 Fig. 2 :Figure 3

 223 Fig. 2: The effects of various current intensity on microstructure in the cross section of Al-20wt%Sn at 0.09T Figure 3 shows the change of the hardness of the aluminum-20wt%tin alloy sample. The hardness rose from 36.03 to 40.6 HV when the current intensity increased from 0 to 1.8 × 10 4 A/m 2 , and decreased to 39.57 HV when the current intensity reached 2.2 × 10 4 A/m 2 . Similarly, though remaining steady from 0 to 1.4 × 10 4 A/m 2 , the tensile strength reached to a highest value of 131.51 MPa when the current intensity increased to 1.8 × 10 4 A/m 2 and then decreased, which was shown in figure 4.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Hardness change of Al-20wt%Sn at 0.09T

Acknowledgment

The authors gratefully acknowledged the financial support of Nature Science Foundation of China (Key Project No. 51034010), Science and Technology Commission of Shanghai Municipality (Key Project No. 13JC1402500), Natural Science Foundation of Jiangsu Province (BK2011501) and Innovation Fund for Technology Based Firms (12C26213202452).


Kang Deng

Zhenqiang Zhang

Zuosheng Lei

Zhongming Ren

Mercury Model on Flow in Continuous Casting of Slab with EMBr

A mercury model has been developed to investigate the influences of Electromagnetic Mold Brake Ruler (EMBr-Ruler) and Flow Control Mold (FC-Mold) on metal flow in continuous casting of slab with the practical casting velocity 1.3 m/min. The experiment shows that the FC-Mold can efficiently repress the surface flow and its fluctuation. The expanding space of the pouring flow is compressed by EMBr-Ruler and FC-Mold respectively, then the 'flow passage' can be developed near the narrow wall of the mold, where the vertical velocity of flow will increase sharply. It is bad for the development of the plug like flow in the mold.

Introduction

Electromagnetic Braking (EMBr) is widely applied as an useful method to control metal flow in the mold of continuous casting of slab [1][2] . Several years ago, Electromagnetic Mold Brake Ruler (EMBr-Ruler) and Flow Control Mold (FC-Mold) can gradually be used in industry to replace the local electromagnetic brake (EMBr). In this case, Yu et al. built a three-dimensional finite-volume mathematical model to study the influences of magnetic field intensity on EMBr effect [3] . Cukierski et al. applied a computational fluid flow model to investigate the effects of both the submergence depth of submerged entry nozzle (SEN) and magnetic field intensity of EMBr varying on metal flow in the mold [4] . Harada et al. built a mercury model to study the brake effect of a local magnetic field and a level magnetic field [5] . In this paper, to use mercury instead of molten steel, the flow is simulated in the model of continuous casting of slab with EMBr-Ruler and FC-Mold. In this experiment the velocity is measured by Ultrasound Doppler Velocimetry (UDV), and the braking effects of magnetic field on the metal flow can be presented.

Experimental Method

In the experiment the model is based on a scale of 1/6 of the size of a continuous caster and, the technique process is used in BAOSTEEL. In this paper, the similarities between the industrial prototype and modelling are maintained by the Froude number (Fr) and the Stuart number (St), and the geometric and technical details of above industrial prototype and modelling simulation can be presented on Tab. 1. On the other hand, the physical simulation apparatus and, the parameters of molten steel and mercury can be obtained from Ref. [6].

Experimental Result and Discussion

The flow patterns in the model can be shown in Fig. 2 for the cases of (a) no magnetic field, (b) EMBr-Ruler and (c) FC-Mold. It can be obtained that with EMBr-Ruler, the lower magnets repress the dispersion of the jet from the SEN outport, and make it impact to the narrow wall of the model with relatively big horizontal momentum. It enhances the horizontal impact strength to the narrow wall. With FC-Mold, the upper magnets also limit the space of the jet flow and enlarge the angle of the jet. It enlarges the distance that jet flow spreads to the narrow wall. Then with the coupled function of the two pairs of magnets, the maximum velocity impinging the narrow wall and the impinging position are both smaller than the values with EMBr-Ruler. In addition, from the flow regime perspective, when it is without magnetic field or with EMBr-Ruler, there are vortex centers in the upper recirculating flow region, and their velocities are relatively big, but with FC-Mold, the velocity of the upper recirculating flow is small, and the vortex center is not obvious. This indicates that FC-Mold gets the result of stabilizing the surface flow. From the measured results of downward flow, it can be shown that the 'flow passage' is developed near the narrow wall of the model when EMBr to be used, which makes the velocity of the downward flow enlarge, especially in FC-Mold, the channel effect is more obvious than that of EMBr-Ruler. This indicates that with EMBr, the inhibitory action of the magnetic field on the flow makes it redistribute. When the space distribution of the jet from the SEN outport is restrained, the major flow of the jet is released near the narrow wall, then the 'channel effect' is developed. It is not benefit for the formation of the plug like flow under the nozzle. In this experiment, the result shows that only the nozzle ports are placed in the region of magnets and the casting speed is suitable, the electromagnetic braking can be beneficial to the formation of the plug like flow.

Moreover, the experiment verifies that the horizontal velocity near the liquid surface decreases on the whole with the magnetic flux density increasing. the maximum horizontal velocities near the surface are 0.114,0.106 and 0.026 m/s corresponding respectively to the cases of without EMBr, EMBr-Ruler and FC-Mold. It shows a decreasing trend. In the cases of without EMBr and with EMBr-Ruler, their horizontal velocities on the surface are approximately parabola distribution, and the positions of the maximum horizontal velocities are all near 1/4 width of the model, but in FC-Mold the horizontal velocity on the surface is approximately linear distribution, and on the whole, the horizontal velocity and the turbulent intensity near the surface are low. Their maximum values are respectively 1/5 and 1/6 in comparison with that of without EMBr. This shows the braking effect of FC-Mold on the surface flow and its fluctuate behavior. It is the result of the braking effect of the upper magnets makes the upper recirculating flow turn to horizontal direction below it.

Conclusions

From the inhibition on the velocity and fluctuation of the surface flow perspective, the braking effect and the capacity of flow control of FC-Mold are better than EMBr-Ruler. It shows that the velocity of the surface flow is stable and the fluctuation of the liquid surface decreases, but it also shows that the vertical velocity near the narrow wall can be enhanced, which can develop a 'flow channel' near the narrow wall of mold, and its erosion to the initial solidification shell of slab may be intensified.

Tab. 1 Fig. 1

 11 Fig. 1 Distribution of magnetic flux density in the model

  without EMBr (b) EMBr-Ruler (c) FC-Mold Fig. 2 mercury velocity distribution in the model simulated the practical casting velocity 1.3 m/min


Thomas Bonis

Steve Oudot Datashape

A Fuzzy Clustering Algorithm for the Mode-Seeking Framework

In this paper, we propose a new fuzzy clustering algorithm based on the modeseeking framework. Given a dataset in R d , we define regions of high density that we call cluster cores. We then consider a random walk on a neighborhood graph built on top of our data points which is designed to be attracted by high density regions. The strength of this attraction is controlled by a temperature parameter β > 0. The membership of a point to a given cluster is then the probability for the random walk to hit the corresponding cluster core before any other. While many properties of random walks (such as hitting times, commute distances, etc. . . ) have been shown to enventually encode purely local information when the number of data points grows, we show that the regularization introduced by the use of cluster cores solves this issue. Empirically, we show how the choice of β influences the behavior of our algorithm: for small values of β the result is close to hard modeseeking whereas when β is close to 1 the result is similar to the output of a (fuzzy) spectral clustering. Finally, we demonstrate the scalability of our approach by providing the fuzzy clustering of a protein configuration dataset containing a million data points in 30 dimensions.

Introduction

The analysis of large and possibly high-dimensional datasets is becoming ubiquitous in the sciences. The long-term objective is to gain insight into the structure of measurement or simulation data, for a better understanding of the underlying physical phenomena at work. Clustering is one of the simplest ways of gaining such insight, by finding a suitable decomposition of the data into clusters such that data points within a same cluster share common (and, if possible, exclusive) properties.

In this work, we are interested in the mode seeking approach to clustering. This approach assumes the data points to be drawn from some unknown probability distribution and defines the clusters as the basins of attraction of the maxima of the density, requiring a preliminary density estimation phase (Chen et al., 2014b;[START_REF] Chazal | Persistence-based clustering in riemannian manifolds[END_REF][START_REF] Cheng | Mean shift, mode seeking, and clustering[END_REF][START_REF] Cho | Authority-shift clustering: Hierarchical clustering by authority seeking on graphs[END_REF][START_REF] Comaniciu | Mean shift: A robust approach toward feature space analysis[END_REF]Koontz et al., 1 1976). The theoretical analysis of this clustering framework has drawn increasing attention recently, see Chen et al. (2014a); [START_REF] Azizyan | Risk Bounds For Mode Clustering[END_REF]; Chen et al. (2015a,b); Arias-Castro et al. (2013). However, this (hard) clustering method provides a fairly limited knowledge on the structure of the data: while the partition into clusters is well understood, the interplay between clusters (respective locations, proximity relations, interactions) remains unknown. Identifying interfaces between clusters is the first step towards a higher-level understanding of the data, and it already plays a prominent role in some applications such as the study of the conformations space of a protein, where a fundamental question beyond the detection of metastable states is to understand when and how the protein can switch from one metastable state to another [START_REF] Chodera | Long-time protein folding dynamics from short-time molecular dynamics simulations[END_REF]. Hard clustering can be used in this context, for instance by defining the border between two clusters as the set of data points whose neighborhood (in the ambient space or in some neighborhood graph) intersects the two clusters, however this kind of information is by nature unstable with respect to perturbations of the data.

fuzzy clustering appears as the appropriate tool to deal with interfaces between clusters. Instead of assigning each data point to a single cluster, it computes a degree of membership to each cluster for each data point. The promise is that points close to the interface between two clusters will have similar degrees of membership to these clusters. Thus, fuzzy clustering uses a fuzzier notion of cluster membership in order to gain stability on the locations of the clusters boundaries.

Consider a smooth density f in R d . Under the mode seeking paradigm, clusters correspond to the modes of f . More precisely, considering the gradient flow induced by f:

y (t) = ∇f (u(t))
two points x and y are in the same cluster if the gradient flow started at x and the gradient flow started at y have the same limit which is a local maximum of f . A natural way to turn this approach into a fuzzy clustering algorithm is to follow a perturbed gradient flow instead, such as the diffusion process solution of

dY t = 1 β ∇(log f )dt + dB t , (1)
where B t is a d-dimensional Brownian motion and β is a temperature parameter controlling the amount of noise introduced in the gradient flow. We use the gradient of the logarithm of f here as this quantity arises naturally in practice. Indeed, since we only have access to a discretization of the space through the sampled data points, we mimic this perturbed gradient flow by a random walk on the data points. [START_REF] Ting | An analysis of the convergence of graph laplacians[END_REF] proved an isotropic random walk on a neighborhood graph approximates the previous diffusion process for β = 1 while other values of β are obtained by putting weights on the edges of the graph. At this point, one could perform fuzzy clustering by considering the first local maximum of the density encountered by the random walk, an approach wich has been proposed by Chen et al. (2014b). However, as emphasized by [START_REF] Luxburg | Getting lost in space: Large sample analysis of the resistance distance[END_REF], the hitting time to a single point for a random walk on the graph converges to irrelevant quantities when the number of data points goes to infinity. We can thus expect the clustering to fail in that case. Indeed, if we apply this method to the fuzzy clustering of two different Gaussian measures (see Figure 1). The obtained fuzzy memberships are unsatisfying. In order to circumvent this issue, we assign a zone of high density to each cluster, called cluster core and computed using the mode-seeking (hard) clustering algorithm ToMATo [START_REF] Chazal | Persistence-based clustering in riemannian manifolds[END_REF]. The fuzzy membership of a point to a given cluster is then given by the probability for the random walk started at this point to hit the corresponding cluster core first. After a presentation of the algorithm, we provide convergence guarantees for its output. We then study the empirical behaviour of our algorithm on synthetic data and provide a method to select relevant values for the temperature β. Finally, we perform some quantitative experiments on a few UCI datasets.

The Algorithm

Our algorithm is a fuzzy generalization of the ToMATo algorithm which relies on the concept of prominence. Let G be a graph and f be a real valued function on the vertices of this graph. For any α ∈ R, let The algorithm takes as input a finite set of points X = {X 1 , • • • , X n } together with pairwise distances d(X i , X j ). In practice only the distances are used, so there is no need for point coordinates. Additionally, the algorithm takes in the following set of parameters:

F α = f -1 ([α, +∞]) be the α-superlevel-set of f . A new connected component C is born in F α
• a density estimator f : X → R,

• a kernel k, for example the Gaussian kernel,

• a window size h > 0,

• a prominence threshold τ > 0,

• a temperature β > 0.

The first four parameters are in fact required by ToMATo for hard mode-seeking, upon which our algorithm relies. The last parameter is the one added in for fuzzy modeseeking, as per Equation (1). Given this input, our algorithm proceeds as follows:

1. It builds a weighted neighborhood graph G on top of the point cloud X , adding an edge with weight

w i,j = 1 + 1 -β β f (X j ) k d(X i , X j ) h (2)
between each pair of points (X i , X j ). Remark that it is possible to replace our kernel-based graph by a nearest neighbour graph.

2. It computes the cluster cores by running ToMATo with input X , d, log( f ), τ and the unweighted neighborhood graph Ḡ obtained from G by removing the edges with weights lower than 0.5 max(k). The output of ToMATo is a set of K clusters C 1 , • • • , C K . Each cluster C i corresponds to the basin of attraction of some peak of log( f ) of prominence at least τ within Ḡ. Up to a reordering of the data points, we can assume this peak to be X i . The i-th cluster core C i is then taken to be the highest and most stable part of C i , defined formally as the connected component containing X i within the subgraph of Ḡ spanned by those vertices

X j such that log( f )(X j ) > log( f )(X i ) -τ /2.
3. It computes the fuzzy-membership values µ 1 , . . . , µ K by solving the linear system A T µ = µ, where the matrix A is defined by:

A kl = δ kl if X k belongs to some cluster core K h (X k , X l ) otherwise,
where K h is the transition kernel of the random walk on the graph, i.e.

K h (X i , X j ) = w i,j z w i,z . (3)
The output of the algorithm is the set of fuzzy-membership values µ 1 , . . . , µ K computed at step 3.

3 Parameters selection 3.1 Density estimator, window size, kernel and prominence threshold

These 4 parameters are tied to the classical hard mode-seeking framework. The density estimator can be linked to the window size in practice, as is done e.g. in Mean-Shift [START_REF] Cheng | Mean shift, mode seeking, and clustering[END_REF] and its successors. For instance, one can consider the kernel density estimator associated to the kernel k. This not only reduces the number of parameters to tune in practice, but it also gives a way to select h using standard parameter selection techniques for density estimation, which is done for example in Chen et al. (2014b).

Finally, the prominence threshold τ is used to distinguish between relevant and irrelevant peaks in the discrete setting. It can be selected by running ToMATo twice: once to get the distribution of prominences of the peaks of f within the neighborhood graph Ḡ, from which τ can be inferred by looking for a gap in the distribution; then a second time, using the chosen value of τ , to get the final hard clustering. This procedure is detailed in [START_REF] Chazal | Persistence-based clustering in riemannian manifolds[END_REF].

Temperature parameter

This parameter is standard in fuzzy clustering. Outputs corresponding to large values of β will tend to have smooth interfaces between clusters, while small values of β will encourage quick transitions from one cluster to another. β can also be interpreted as a trade-off between the respective influence of the metric and of the density in the diffusion process: when β is small, the output of our algorithm is mostly guided by the density and therefore close to the output of mode seeking algorithms; by contrast, when β is large, the algorithm becomes oblivious to the density. In practice, one may get insights into the choice of β by looking at the evolution of a certain measure of fuzziness of the output clustering across a range of values of β. We elaborate on this in Section 5.

Convergence guarantees

In this section we provide guarantees to our fuzzy clustering scheme by exploiting the convergence of the random walk over the neighborhood graph to a continuous diffusion process.

As is usual in mode-seeking, we assume our input data points X 1 , ..., X n to be i.i.d random variables drawn from some unknown probability density f over R d . We also assume that the metric d that equips the data points is the Euclidean norm, and that f satisfies the following technical conditions:

• f is Lipschitz continuous over R d and C 1 -continuous over the domain Ω = {x ∈ R d | f (x) > 0},
• lim

x 2→∞ f (x) = 0,
• The SDE 1 is well-posed.

Standard sufficient conditions ensuring the well-posedness (particularly the non-explosion) of the SDE 1 can be found in [START_REF] Albeverio | Strong feller properties for distorted brownian motion and applications to finite particle systems with singular interactions[END_REF] or in [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF], for example one can assume ∇ log f to be Lipschitz continuous.

Our analysis connects random walks on graphs built on top of the input point cloud X using a density estimator f to the solution of Equation 1, for a fixed temperature parameter β > 0. Specifically, let M x,h denote the Markov Chain whose initial state is the closest neighbour of x in the point cloud X (break ties arbitrarily), and whose transition kernel K h is given by Equation 3. Following the approach of [START_REF] Ting | An analysis of the convergence of graph laplacians[END_REF], we show that, under suitable conditions on the estimator f , this graphbased random walk approximates the diffusion process in the continuous domain in the following sense: there exists s depending on h such that, as n tends to infinity, with high probability, M x,h t/s converges weakly to the solution of Equation (1). From there, under standard conditions for mode estimation on the window size h and on the density estimator f (see Chen et al. (2015b); Arias-Castro et al. ( 2013)), we obtain the convergence of the fuzzy-membership values µ i (x) computed by the algorithm to the membership defined from the underlying continuous diffusion process μi (x). Formally, letting v 1 , . . . , v K be the local maxima of f of prominence higher than τ , and C1 , • • • , CK , their associated cluster cores in the continuous domain (i.e. Ci is the connected component containing

v i in {x ∈ R d | log f (x) ≥ log f (v i ) -τ /2}
), we define μi (x) as the probability for the diffusion process solution of (1) to hit Ci before any other Cj .

x∈U |∇f (x) -∇ fn (x)| ≥ h(n) 2 ) = 0.
Then, for any compact set U ⊂ Ω, any > 0 and any i,

lim n→∞ P sup x∈U |µ i (x) -μi (x)| ≥ = 0.

Experiments

We first illustrate the effect of the temperature parameter β on the clustering output using synthetic data. We then apply our method on a couple UCI repository datasets and on simulated protein conformations data. In all our experiments we use a k-nearest neighbor graph along with a distance to measure density estimator [START_REF] Biau | A weighted k-nearest neighbor density estimate for geometric inference[END_REF] computed using the 2k nearest-neighbors.

Synthetic data

The first dataset is presented in Figure 2 As we can see from the output of the algorithm, for small values of β, the amount of noise injected in our trajectory is not large enough to compensate for the influence of the noise in the density estimation, so the result obtained is really close to hard clustering. Large values of β do not give enough weight to the density function which leads to a smooth transition between the two clusters on the top link. Intermediate values of β seem to give more satisfying results. In order to gain intuition regarding which value of β one should use, it is possible to look at the evolution of a fuzziness value for the clustering. For example, one can consider a notion of clustering entropy:

H = i j µ j (X i ) log(µ j (X i )), (4)
which gets lower when the fuzziness of the clustering increases. As we can see in Figure 2(f), the evolution of H with respect to β presents three distincts plateaus corresponding to the three behaviour highlighted earlier.

The second dataset we consider is composed of two interleaved spirals-see Fig- ure 3. An interesting property of this dataset is that the head of each spiral is close (in Euclidean distance) to the tail of the other spiral. Thus, the two clusters are wellseparated by a density gap but not by the Euclidean metric. We use our algorithm with two different values of β: β = 1 and β = 0.3. We also run the spectral fuzzy-C means on a subsampling of this dataset. The first thing we want to emphasize is that the result of spectral clustering and our algorithm using β = 1 are similar, this is to be expected as both algorithms rely on properties of the same diffusion operator, this also means that other fuzzy clustering techniques based on spectral clustering will fail on this dataset. Moreover, we can see that for β = 1, the density gap between the two spirals is not strong enough to compensate for the proximity of the two clusters in the Euclidean metric. On the other hand, for β 0.3 we recover the two clusters as we give more weight to the density structure.

UCI datasets

In order to obtain quantitative results regarding our fuzzy clustering scheme, we evaluate it in a classification scenario on a few datasets from the UCI repository: the Pendigits dataset (10, 000 points and 10 clusters), the Waveset dataset (5000 points and 3 clusters) and the Statlog dataset (6, 435 points for 7 clusters). We preprocess each dataset by renormalizing the various coordinates so they have unit variance. Then, for each dataset, we run our algorithm with various values of the parameter β between 0.3 and 5, but a single value of k and τ (given by a prominence gap), along with the fuzzy C-means algorithm for fuzziness parameters between 1.2 and 5. We also consider the fuzzy clustering algorithm proposed by Chen et al. (2014b), for which the cluster cores are reduced to a single point. Let X 1 , . . . , X n denote our sample points and Y 1 , . . . , Y n their respective labels taking values in {1, . . . , K }. In these datasets, there are only two plateaus, thus we choose β. in other words we take β inbetween the two plateaus by choosing the value of β maximizing the slope of H. In order to evaluate hard clustering algorithms, it is common to use the purity measure defined by

P = max π 1 n n i=1 K j=1 1 μj (Xi)=1 1 Yi=π(j) ,
where π is a map from the set of clusters {1, . . . , K} to the set of labels {1, . . . , K }. As this measure is not adapted to fuzzy clustering, we define the -entropic purity as

HP = max π 1 n i log   + j,π(j)=Yi μj (X i )   ,
for some > 0. The parameter is used to prevent the quantity from exploding due to possible outliers. This extension of the traditional purity can be useful to evaluate fuzzy clustering as it can be seen as an approximation of E[log( + j,π(j)=Y μj (X))] which enjoys the following property.

Proposition 2. Suppose that Y ∈ {1, . . . , K} and let > 0, then

arg max f ∈R d →R K , f 1=1 E[log( + f (X))] = (1 -) -1 (P(Y = j | X)) 1≤j≤K -.
Thus, for small values of , a fuzzy clustering minimizing the -entropic purity recovers the conditional probabilities of the labels with respect to the coordinates.

We provide the best 10 -3 -entropic purity obtained by each algorithm on all datasets in Table 5.2. As we can see, our algorithm outperforms the other fuzzy clustering algorithms on these datasets. In particular we can see that the simple fuzzy modeseeking algorithm of Chen et al. (2014b) fails on the Waveform dataset.

Alanine dipeptide conformations. We now turn to the problem of clustering protein conformations. We consider the case of the alanine-dipeptide molecule. Our dataset is composed of 1, 420, 738 protein conformations, each one represented as a 30-dimensional vector. The metric used on this type of data is the root-mean-squared deviation (RMSD). The goal of fuzzy clustering in this case is twofold: first, to find the right number of clusters corresponding to metastable states of the molecule; second, to find the conformations lying at the border between different clusters, as these represent the transition phases between metasable states. It is well-known that the conformations of alanine-dipeptide only have two relevant degrees of freedom, so it is possible to project the data down to two dimensions (called a Ramachadran plot) to have a comfortable view of the clustering output. See Figure 4 for an illustration, and note that the clustering is performed in the original space. In order to highlight interfaces between clusters, we only display the second highest membership function. As we can see there are 5 clusters and 6 to 7 interfaces.

d(f, g) = inf {∃λ ∈ Λ, λ ≤ , sup t |f (t) -g(λ(t))| ≤ },
where Λ denotes the space of strictly increasing automorphisms on the unit segment [0, 1], and where λ is the quantity:

λ = sup s =t log λ(t) -λ(s) t -s .
In diffusion approximation, standard results prove the weak convergence of a Markov chain to a difussion process in D([0, T ], R d ). A stochastic process M s converges weakly to a diffusion process Y in D([0, T ], R d ) as s tends to 0 if and only if

lim s→0 P(M s ∈ B) = P(Y ∈ B) (5)
for any Borel set B such that P(Y ∈ ∂B) = 0.

Let us state the convergence result when Y is the Solution of the Stochastic Differential Equation 1. For this case, b = 1 β ∇ log f and a = I d . Consider a family of Markov chains (M x0,s ) defined on discrete state spaces S s ⊂ Ω, transition kernels K s and initial states M x0,s 0 ∈ S s . For x ∈ S s and γ > 0, let

• a s (x) = 1 s y∈Ss K s (x, y)(y -x)(y -x) T ; • b s (x) = 1 s y∈Ss K s (x, y)(y -x); • ∆ γ s = 1 s K s (x, B(x, γ) c
), where B(x, γ) c is the complementary of the ball of radius γ centered at x. Proposition 3 (Adapted from Theorem 7.1 in [START_REF] Durrett | Stochastic calculus : a practical introduction[END_REF]). Let U be a compact subset of Ω. Let also B be a Borel set in D([0, T ], R d ) for some T > 0 such that P(Y x0 ∈ ∂B) = 0 for all x 0 ∈ U . For any > 0, there exist parameters ν and γ such that sup

x0∈U |P(M x0,s t/s ∈ B) -P(Y x0 t ∈ B)| ≤
whenever the following conditions are met:

(i) sup x∈Ss a s -a ∞ ≤ ν; (ii) sup x∈Ss b s -b ∞ ≤ ν; (iii) sup x∈Ss ∆ γ s ≤ ν; (iv) sup x0∈Ss M x0,s 0 -x 0 ∞ ≤ ν.

Weak-Convergence

In this section, we prove the following result. Then, for any T, > 0, for any compact set U ⊂ Ω, and for any Borel set B of D([0, T ], R d ) such that P(Y y ∈ ∂B) = 0 for all y ∈ U , there exists a constant C depending on d such that for s(n) = Ch 2 , we have

lim n→∞ P(sup x∈U |P(M x,h(n) t/s(n) ∈ B) -P(Y x t ∈ B)| ≥ ) = 0.
The proof relies on Theorem 3 of [START_REF] Ting | An analysis of the convergence of graph laplacians[END_REF] along with a proper control of boundary effects. Let T and be strictly positive real numbers, throughout the course of the proof, the notation M x,h(n) stands for the continuous time process M

x,h(n) t/s(n) . We denote by X n = (X 1 , ..., X n ) the i.i.d sampling which is also the state space of M x,h(n) . For α > 0, let

F α = {x ∈ R d | f (x) ≥ α} be the α superlevel-set of f and B α = {w ∈ D([0, T ], R d ) | ∀t, w(t) ∈ F α }
be trajectories staying in F α up to time T . Since Y does not explode in finite time, there exists α such that, for any x ∈ U , P(Y x ∈ B α ) ≥ 1 -/4. To obtain a good approximation of the trajectories of Y staying in F α using M x,h(n) , we only need to check assumptions (i)-(iv) of Proposition 3 on F α . F α is closed as f is continuous and it is also bounded as lim x 2→∞ f (x) = 0, it is therefore compact. Applying Theorem 3 from [START_REF] Ting | An analysis of the convergence of graph laplacians[END_REF] on the points of the compact set F α , we have, with probability 1,

(i) lim n→∞ P(sup y∈Xn∩F α a s -I d ∞ ≤ ν) = 0, (ii) lim n→∞ P(sup y∈Xn∩F α b s -∇f βf ∞ ≤ ν) = 0, (iii) sup y∈Xn∩F α ∆ h s = 0, (iv) lim n→∞ P( M x,h(n) 0 -x ∞ ≤ ν) = 0.
Thus, the assumptions (i)-(iv) of Proposition 3 are verified on F α . Since f is continuous, B α is an open set. Therefore, there exists n 0 > 0 such that for any n > n 0 ,

sup x∈U P(M x,h(n) ∈ B α ) ≥ sup x∈U P(Y x ∈ B α ) -/4 ≥ 1 -/2.
Therefore, for any Borel set B,

sup x∈U |P(M x,h(n) ∈ B) -P(M x,h(n) ∈ B ∩ B α )| ≤ /2.
Thus, we only need to approximate trajectories that do not leave F α to obtain a good approximation of P(M x,h(n) ∈ B). So we can apply Corollary 3 on these trajectories with an accuracy of /2 to obtain,

sup x∈U |P(M x,h(n) ∈ B) -P(Y x ∈ B)| ≤ .
Every step of the proof hold almost surely as n tends to infinity, thus the proof of Proposition 4 is complete.

Proof of Theorem 1

Let β and τ be strictly positive real numbers and let U ⊂ Ω be a compact set. Let C 1 , . . . , C K be the cluster cores used by the algorithm and computed with the density estimator f . These cluster cores are approximations of the sets C1 , . . . , CK obtained using the same computation with the true density f . Since, by assumptions, f is C 1continuous on Ω and ∇f is non-zero on the boundary of the C, we have (i) The Ci are compact sets of R d that are well-separated (i.e. Cj ∩ Cj = ∅ for all i = j).

(ii) For each i, the boundary of Ci is smooth.

By our assumptions on the convergence of f along with Theorem 10.1 of [START_REF] Chazal | Persistence-based clustering in riemannian manifolds[END_REF], ∀δ > 0, lim

n→∞ P( C-δ i ⊂ C i ⊂ Cδ i ) = 1, (6)
where Cδ

i = ∪ x∈ Ci B(x, δ) and C-δ i = Ci \ ∪ x / ∈ Ci B(x, δ
). Without loss of generality, we can assume that Ω has a single connected component. Let be a strictly positive real and consider x ∈ Ω, we let

• µ + i,δ (x)
be the probability that Y x hits Cδ i before any other C-δ j ,

• µ - i,δ (x) be the probability that Y x hits C-δ i before any other Cδ j .

Let us show that, for any i, a trajectory entering Cδ i has a high probability to enter Ci if δ is small enough. Since the Ci are closed and disjoint there exists δ 0 > 0 such that the Cδ0 i are disjoints. Moreover, since the Ci have smooth boundaries, there exists δ + i > 0 such that if d(x, Ci ) ≤ δ + i then, the probability for Y x to hit Ci before exiting Cδ0 i is at least 1 -/8.

Similarly, if a trajectory enters Ci , then it enters C-δ i with high probability. More precisely there exists δ - i such that if a trajectory hits Ci , then it hits C-δ i with probability at least 1 -/8.

Let δ = min(δ + j , δ - j ), by combining our results and using the strong Markov property of Y x we obtain that: Since C i ⊂ Ω and Ω has a single connected component, we have that P(τ (Y x t ) < ∞) = 1, in particular that means that there exists T 0 such that for any T ≥ T 0 , P(τ (Y x ) ≤ T ) ≥ 1 -/6. Using Proposition 4, we have that, almost surely P(P(τ (M x,h(n) ) ≤ T ) ≥ P(τ (Y x ) ≤ T )) -/6 ≥ 1 - Since P(Y x ∈ ∂B T ) = P(Y x ∈ ∂B) = 0, we can apply Proposition 4 on the set B T , and obtain

• µ + i,δ (x) -µ i (x) ≤ /4, • µ i (x) -µ - i,δ (
P(M x,h(n) ∈ B T ) -P(Y x ∈ B T ) ∞,U ≤ /4
Combined with our previous result, we obtain:

P(M x,h(n) ∈ B) -P(Y x ∈ B) ∞,U ≤ 3 /4
Using our assumption on Ĉi,n , we have P(M x,h(n) ∈ B) ≥ μi,h,n . Therefore, using our previous bound between µ and µ + : μi,h,n (x) -µ i (x) ≤ .

Similarly,

µ i (x) -μi,h,n (x) ≤ , concluding the proof.

Conclusion

We have provided a fuzzy clustering algorithm based on the mode-seeking framework relying on the approximation of a diffusion process through the use of a random walk. Despite the convergence issues of random-walk-based quantities for large data highlighted by [START_REF] Luxburg | Getting lost in space: Large sample analysis of the resistance distance[END_REF], we have shown that our algorithm does converge to meaningful values. Our thereotical result is backed up by encouraging experiments.

The main question still open regarding our algorithm is the choice of the temperature parameter β, while we have shown that the evolution of a quantification of the fuzziness of the clustering through the clustering entropy can give some hint about a correct choice for this parameter, it is not clear whether this can be done in all cases and for more complicated datasets.

Figure 1 :

 1 Figure 1: Fuzzy clustering output for an unbalanced mixture of gaussian. Red color corresponds to the right cluster, blue to the left one. Finally, green points have similar membership to both clusters.

  when α reaches a local maximum of f on G and we denote by α b,C the corresponding value of α. This component then dies at α = α d,C < α b,C when it gets connected, in F α , to another connected component C such that α b,C > α b,C . The prominence of C (and by extension, of the corresponding local maximum of f ) is then simply α b,C -α d,C .

Theorem 1 .

 1 Let β > 0 and assume ∇f is bounded from below on the boundary of the underlying cluster cores C. Let h : N → R + be a decreasing window size such that lim n→∞ h(n) = 0 while lim n→∞ h(n) d+2 n log n = ∞. Suppose the density estimator fn satisfies, for any compact set U ⊂ Ω and any > 0, lim n→∞ P(sup

  Output for β = 0.2. (d) Output for β = 1. (e) Output for β = 2. (f) Evolution of H with respect to β.

Figure 2 :

 2 Figure 2: Output of our algorithm on a simple dataset composed of two overlapping clusters. For fuzzy clustering green corresponds to an equal membership to both clusters.

  (a) and is composed of two high-density clusters connected by two links. The bottom link is sampled from a uniform density while the top link is sampled from a density that has a gap inbetween the two clusters. Standard mode seeking algorithms will have a hard time clustering the bottom link as a density estimation can create many "noisy" local maxima: for instance, ToMATo missclusters most of the bottom link (see Figure2(b)). We display the results of our algorithm for three values of β : β = 0.2 in Figure2(c), β = 1 in Figure 2(d) and β = 2 in Figure 2(e).

  (a) Output for β = 1. (b) Output for β = 0.28.(c) Fuzzy Spectral Clustering.

Figure 3 :

 3 Figure 3: Experiments on a cluttered spirals dataset.

Figure 4 :

 4 Figure 4: From left to right: (a) the dataset projected on the Ramachadran plot, (b) ToMATo output, (c) second highest membership obtained with our algorithm for β = 0.2

Proposition 4 .

 4 Let Y be the diffusion process solution of the SDE 1. Let h : N → R + be a decreasing function such that lim n→∞ h(n) = 0 and lim n→∞ h(n) d+2 n log n = ∞. Suppose our estimator fn satisfies, for any compact set U ⊂ Ω and any > 0, lim n→∞ P(sup x∈C |f (x) -fn (x)| ≥ h(n) 2 ) = 0.

  x) ≤ /4. The next step is to show that the approximation of µ + i,δ provided by the Markov chain is correct. For T > 0, letB = {w ∈ D([0, ∞], R d | ∃τ such that w(τ ) ∈ Cδ i and ∀t < τ we have w(t) ∈ Ω \ ∪ j C-δ j }, B T = {w ∈ D([0, T ], R d | ∃τ such that w(τ ) ∈ Cδ iand ∀t < τ we have w(t) ∈ Ω \ ∪ j

  x,h(n) ∈ B \ B T ) + P(Y x ∈ B \ B T ) ≤ P(τ (M x,h(n) ) > T ) + P(τ (Y x ) > T ) ≤ /2

Table 1 :

 1 Thus, we propose an automatic selection of β by computing the values of the clustering entropy H for multiple values of β and by selecting Entropic purity obtained by fuzzy clustering algorithms on UCI datasets.

	β = arg max	dH dβ	,

Acknowledgements. The authors wish to thank Cecilia Clementi and her student Wenwei Zheng for providing the alanine-dipeptide conformation data used in Figure 4. This work was supported by the French Délégation Générale de l'Armement (DGA), by ANR project TopData ANR-13-BS01-0008 and by ERC grant Gudhi (ERC-2013-ADG-339025).


H Harada

M Nagashima

T Toh

Electromagnetic sensor just below CC mold by using magnetic transformation of steel

Keywords: magnetic transformation, Curie point, continuous casting, unbalanced flow, solidification

Electromagnetic sensor to detect the unbalanced flow in CC strand by using the magnetic transformation just below CC mold has been proposed, noticing that the narrow face of slab has been cooled below Curie temperature at the beneath of mold. As preliminary experiments, laboratory experiments have been performed to clarify the relationship between the temperature and induced electric voltage, when AC magnetic field has been imposed on the slab surface. Moreover, plant tests have been conducted by installing the sensor just below the narrow face of mold to show that the proposed sensor could measure surface temperature under severe conditions and detect the unbalanced flow in CC strand.

Introduction

Molten steel flow in CC mold largely affects the cast slab quality and the productivity of CC. As shown in Fig. 1, unsymmetrical discharged flow from submerged entry nozzle causes unbalanced flow in CC strand and local remelting of solidified shell at the impinging points of the discharged flow. Additionally, the unbalanced flow leads to the entrapment of mold powder at meniscus and the inner defects due to deeply transported Ar bubbles. Therefore, suitable control of the discharged flow and the entire flow pattern in CC strand are necessary. At the same time, the detection of the unbalanced flow becomes important in order to escape from serious casting troubles and slab defects. Several detection methods of the unbalanced flow in CC mold have been reported [1] [2] . However, it is impossible to directly detect the unbalanced flow in CC strand. In this study, noticing that the unbalanced flow forms the large differences of surface temperature of slab just below the mold, and that strong cooling beneath the mold causes magnetic transformation of steel, the electromagnetically detecting methods of unbalanced flow in CC strand have been developed.

The principle of electromagnetic sensor and laboratory experiment for verification

In the continuous casting of slab, wide face of slab is supported with many rolls at CC strand. On the contrary, narrow face of slab is only supported with some rolls at the beneath of mold and is cooled with a large amount of water to suppress the bulging. Therefore, slab surface temperature of narrow face rapidly drops and then increases at the beneath of CC mold as shown in Fig. 2. At that time, slab surface temperature passes through Curie point (Tc), which is magnetic transformation temperature. Fig. 3 shows the principle of the proposed electromagnetic sensor. When AC magnetic field is applied normal to the slab surface, the distribution of magnetic field in the slab is dependent of magnetism of slab surface. When surface temperature (Ts) is higher than Tc, the distribution of magnetic field line is the same as the one in vacuum. On the contrary, when Ts is lower than Tc, the applied magnetic field is concentrated on the surface, this causes the big distortion of magnetic field. This change of magnetic field line is governed by the magnetism of slab surface, in other words, slab surface temperature. By measuring the induced electric voltage due to the change of imposed magnetic field line, surface temperature is presumed in this sensor. The sensor is composed of primary coil and secondary coil. A set of coils is mounted into the cylinder made of stainless steel. AC electric current of 1.5 Hz is supplied to primary coil by using a constant current power amplifier, AC magnetic field is imposed normal to the slab surface. The noise in secondary coils is minimized with low pass filter of 5Hz. Moreover, only the signal with 1.5Hz component is elucidated by using Lock in amplifier. First of all, laboratory experiment has been performed to clarify the relationship between sensor signals and surface temperature of slab. Slab samples are inserted into a heating furnace and heated into 1350 degree. A thermocouple is inserted into the slab, in which the position is 1mm from the surface of slab. After extraction of the heated slab, aforementioned sensor is set on the heated samples and the relationship between induced electric voltage of secondary coils and surface temperature of slab has been investigated. Fig. 4 shows the typical example of the obtained results. It can be found that the induced electric voltage is almost constant in high temperature and low temperature region. On the contrary, the electric voltage changes with temperature around Curie temperature. This relationship between the electric voltage and temperature can be expressed by using polynomial approximation. By using this relationship, surface temperature at the beneath of CC mold can be presumed.

Plant test for verification of the proposed sensor

A pair of sensor was installed under the respective narrow face of mold at bending type of continuous casting machine and plant tests have been performed to investigate the effects of casting conditions on the sensor signals. Table 1 shows the casting conditions. Fig. 5 shows the time-change of sensor signals, when flow rate of water spray changes. In Fig. 5, high electric voltage is equivalent to low surface temperature and low electric voltage is equivalent to high surface temperature. It can be seen that the electric voltage rapidly changes with correspondence of water flow rate. This results shows that the proposed sensor can be utilized to measure surface temperature of slab just below CC mold. Fig. 6 shows the successive change of surface temperature at both narrow faces in sequential casting. In Fig. 6 (a), which 1 st ladle and 2 nd ladle data are plotted, most plots are concentrating at the lower left corner of a chart. It can be confirmed that stable casting has proceeded. On the contrary, it can be seen that temperature fluctuation and unsymmetrical distribution becomes marked with progress of cast as shown in Fig. 6 (b). In this condition, nozzle clogging gradually proceeds and surface level fluctuation becomes marked in the latter part of cast. This means that nozzle clogging unstabilizes the discharged flow and forms the unbalanced flow in CC strand. Moreover, it is confirmed that the proposed sensor can directly measure surface temperature of cast slab to detect the unbalanced flow in CC strand.

T(Deg)

吐出反転流 TC 電 磁 セ ン サ ー 凝 固 シ ェ ル の 再 溶 解 (BO) 電 磁 セ ン サ ー 内欠 Electromagnetic

Conclusions

Electromagnetic sensor to detect the unbalanced flow in CC strand by using the magnetic transformation just below CC mold has been proposed. The obtained results are summarized as follows:

(1) Noticing that the narrow face of slab has been cooled below Curie temperature at the beneath of mold, the sensing method to measure the change of imposed AC magnetic field due to slab surface temperature, has been constructed.

Fig. 1 :Fig. 3 :

 13 Fig.1: Schematic view of unbalanced flow in CC strand. Fig.2: Schematic view of the surface temperature profile at the narrow face of slab.

Fig. 5 :

 5 Fig.5: Successive change of sensor signals in the condition which water flow rate of spray at narrow face changes.

( 2 )

 2 After obtaining the relationship between the temperature and sensor signals at laboratory experiments, plant tests have been performed by installing the sensor under the respective narrow face mold to show that the proposed sensor could measure surface temperature under severe conditions and detect the unbalanced flow in CC strand. face A (Degree)