Pierre-Antoine Versini

Auguste Gires
email: auguste.gires@leesu.enpc.fr

Ioulia Tchinguirinskaia
email: ioulia.tchinguirinskaia@leesu.enpc.fr

Daniel Schertzer
email: daniel.schertzer@leesu.enpc.fr

Performance of Green roof in stormwater management regarding highresolution precipitation fields

Keywords: Green roof, precipitation, modelling

At the basin scale, green roofs can represent an efficient tool to manage stormwater in urbanized areas. Widely implemented and/or completed with additional sustainable urban drainage systems, they show a positive impact on urban runoff: decrease and slow-down of the peak discharge and decrease of runoff volume. To assess green roof performances at this scale, a specific module dedicated to simulate their hydrological behaviour has been developed in the Multi-Hydro rainfallrunoff model. Its distributed structure gives the opportunity to test the susceptibility of green roof response regarding spatial distributions of precipitation. Based on radar rainfall fields, an ensemble of 50 realistic downscaled rainfall fields with a resolution of 10 m in space has been generated by using multifractals downscaling technique. Simulations have been conducted on a small urban catchment close to Paris (France) where most of the buildings roofs are assumed to easily accept the implementation of green roof. Although green roof confirm their ability to reduce urban runoff, these results illustrate that peak discharge reduction seems to be clearly dependant of spatial distribution of precipitation. Implementation of green roof can also produce concomitance situations and higher peak discharges than those produced by impervious roofs.

INTRODUCTION

Green roofs have become relatively commonplace over the last 20 years in urban areas for many reasons. Indeed, they can perform in: reduction of heat island through increasing evapotranspiration, improvement of the air quality, protection of biodiversity and stormwater management. Their use in urban runoff management is surely the most significant argument used to promote their implementation. At the building scale, the main performance of green roofs in quantitative management of storm water is known to be: (i) the reduction of runoff volume at the annual scale, and (ii) the peak attenuation and delay at the rainfall event depending on the green roof structure, the rainfall intensity and the antecedent soil moisture conditions [START_REF] Simmons | Green roofs are not created equal: the hydrologic and thermal performance of six different extensive green roofs and reflective and non-reflective roofs in a subtropical climate[END_REF][START_REF] Stovin | The hydrological performance of a green roof test bed under UK climatic conditions[END_REF][START_REF] Versini | Assessment of the hydrological impacts of green roof: from building scale to basin scale[END_REF].

As roof areas represent a significant part of the surfaces of city centres where no space is available for new infrastructures, green roof can also appear as a useful tool to solve operational issue at the basin scale by reducing the runoff volume and/or attenuating peak discharge. Most of the work conducted for now on the hydrological impact of green roof were focused at the building scale: [START_REF] Voyde | Hydrology of an extensive living roof under sub-tropical climate conditions in Auckland, New Zealand[END_REF][START_REF] Gregoire | Effect of a modular extensive green roof on stormwater runoff and water quality[END_REF][START_REF] Stovin | The hydrological performance of a green roof test bed under UK climatic conditions[END_REF]. These works usually present the results provided by an experimental monitored green roof, used to collect continuous runoff and precipitation data and also often to develop some specific (global) models [START_REF] Šimůnek | Development and applications of the HYDRUS and STANMOD software packages, and related codes[END_REF]2 Palla et al., 2009). To our knowledge, very few studies [START_REF] Carter | Vegetated roofs for stormwater management at multiple spatial scales[END_REF][START_REF] Palla | Modelling storm water control operated green roofs at the urban catchment scale[END_REF][START_REF] Versini | Use of green roofs to solve storm water issues at the basin scale -Study in the Hauts-de-Seine county (France)[END_REF] have been conducted at the basin scale and even less by using a distributed rainfall-runoff model able to take into account the spatial distribution of green roof and precipitation. It is the originality of this paper, which is focussed on the hydrological impact of green roof at the basin scale regarding precipitation fields characterized by high spatio-temporal resolution.

METHODOLGY AND CASE STUDY

To assess green roof impact at the basin scale distributed precipitation, the Multi-Hydro [START_REF] El Tabach | Multi-Hydro: a spatially distributed numerical model to assess and manage runoff processes in peri-urban watersheds[END_REF][START_REF] Giangola-Murzyn | Multi-Hydro: physically based modelling to assess flood resilience across scales, case studies in Paris region[END_REF][START_REF] Gires | Impacts of small scale rainfall variability in urban areas: a case study with 1D and 1D/2D hydrological models in a multifractal framework[END_REF] distributed rainfall-runoff model has been used. It is a numerical platform that makes interact several models representing a specific portion of the water cycle in an urban environment: surface runoff and infiltration depending on a land use classification, sub-surface processes and sewer network drainage. A specific module dedicated to simulate green roof behaviour has been added in Multi-Hydro (inspired from that presented in Versini et al., 2015a, anddetailed in Versini et al., 2015c). It has been calibrated to simulate the hydrological behaviour of a green roof structure comprising an extensive vegetation layer (sedum), a growing medium layer (thickness of 3 cm) and a drainage layer. Distributed structure of the Multi-Hydro model enables to analyse the impact of the spatial variability of precipitation and land use, and especially the precise location of green roof.

Multi-Hydro has been applied on the Loup basin (0.5 km 2 ) in a highly populated and urbanized city (Villepinte, France) located close to Paris. 20% of the basin area is occupied by buildings dedicated to industrial activities. It has been assumed that green roof can be implemented on these roofswhich are all flat. Note that Multi-Hydro was previously applied on the Loup basin with satisfactory results in [START_REF] Gires | Impacts of small scale rainfall variability in urban areas: a case study with 1D and 1D/2D hydrological models in a multifractal framework[END_REF]. In order to test the impact of spatial variability of precipitation, an ensemble of 50 realistic downscaled rainfall fields with a resolution of 10 m in space was generated from Radar data (1 km-resolution). The downscaling process relies on the framework of Universal Multifractals [START_REF] Schertzer | Physical modelling and analysis of rain and clouds by anisotropic scaling and multiplicative processes[END_REF]. It is assumed that rainfall is generated through a space-time multiplicative cascade process characterized by two parameters (the mean intermittency and the multifractality index). More details on the downscaling process can be found in [START_REF] Gires | Multifractal analysis of an urban hydrological model on a Seine-Saint-Denis study case[END_REF]. The process was applied on two complex events on the studied basin for which radar data was available: (i) a common event (9 February 2009) characterized by a total rainfall depth of 8.3 mm and a duration of approximately 6 h (return period estimated to 1 month), (ii) a severe event (15 August 2010) characterized by a total rainfall depth of 25 mm and a duration of 6h (return period estimated to 2 years).

RESULTS AND DISCUSSION

Multi-Hydro simulations have been carried out for both rainfall events. The results obtained with green roof implementation have been compared to those obtained for the current situation (without any green roof implemented). Related hydrographs are illustrated in Figure 2. Green roof performances on peak discharge and volume runoff have particularly been studied.

First of all, it has to be noticed that current basin response can significantly be modified depending on the spatial variability of rainfall distribution. Peak discharge for the impervious scenario varies between 0.10 and 0.15 m 3 /s for the first event and between 0.5 and 0.9 m 3 /s for the second one.

When green roofs are implemented, this variation is still significant: between 0.06 and 0.11 m 3 /s, and between 0.5 and 1.4 m 3 /s for both events respectively. Green roofs appear to impact urban runoff in terms of peak discharge and volume depending on precipitation -total amount and intensity-and initial substrate saturation: the higher the precipitation, the more saturated the substrate, and the lower the reduction in terms of stormwater. Results are less pronounced than those usually observed at the building scale because they depend on the green roof area, representing here only 20% of the basin area. At the roof scale, this low event would be completely retained by the substrate (in dry conditions).

The reduction of volume runoff can reach more than 36% as it can be noticed for the more common event. This reduction is only about 12% for the 2-year return period event characterized by a higher amount of precipitation. Note that the variation of volume regarding the 50 downscaled scenarios is not significant (standard deviation of 1%) as the amount of precipitation falling on the basin is usually always the same for the 50 members of the ensemble.

Peak discharge can also be significantly affected by green roof implementation depending on the rainfall event. Regarding the common event, average peak discharge decrease is about 35% and ranges between 10% and 56% depending on the downscaled scenario, illustrating the influence of spatial variability of precipitation. These satisfactory results are most of all concentrated on the first rainfall peak. Here, basin response to the main rainfall intensity is clearly attenuated but likewise dispersed. Conversely, the response to the second rainfall peak is systematically increased by green roofs (from 0 to 33% with an average value of 10%). The range of simulated values is also more dispersed, increasing the sensitivity of the basin regarding precipitation. First peak decreases and second peak increases are quite correlated, meaning attenuation of the first one seems to cause gain 4 of the second one. In this case, the substrate stores precipitation during the first part of the event, before releasing it simultaneously with the second rainfall peak. The same kind of situation is noticed and even more pronounced for the 2010 event. Although green roof can reduce the first peak discharge for most of the downscaled scenarios (between -9% to 65% with an average value of 32%), the response to the second large rainfall peak is systematically increased (average increase of 60%). In a general way, green roof response to successive rainfall peaks can alternatively increase or decrease peak discharge at the basin outlet depending on the spatio-temporal distribution of precipitation. These examples represent "concomitance" situations. For some events characterized by several rainfall peaks, the fast response of the saturated substrate generated by the "second peak" of the rainfall coincides with the slow response of the green roof produced by the initial portion of the rainfall event. These situations are related to both the spatio-temporal distribution of precipitation and the specific configuration of the studied basin (especially on the basin geometry and land cover, and the sewage network arrangement). The superposition in space and time of responses from impervious and a green roofed areas to a complex rainfall event (composed by several rainfall peaks for example) can generate a peak discharge higher than that produced by the current (impervious) situation.

CONCLUSION

The Multi-Hydro distributed rainfall-runoff model has been used to assess green roof performances in stormwater management regarding high-resolution precipitation fields.

Regarding the presented examples and additional works, green roof implementation appears to reduce urban runoff in terms of peak discharge and volume depending on rainfall event characteristics (in terms on intensity and duration) and the initial state of the substrate saturation. The wide implementation of green roof should be useful to avoid some flooding (or sewer combined overflows) issues in several cases depending on the initial conditions. They will particularly perform to mitigate the effects of common rainfall events and/or very short (but sometimes intense) rainfalls that are expected to be more often in Paris area under climate change.

Nevertheless an important point has to be carefully considered: concomitance situations that can occur for a particular basin configuration and/or a rainfall event. As illustrated with the Loup basin, depending on the considered downscaled rainfall scenario, the combination of basin configuration and precipitation can produce higher peak discharge than that produced in the current situation (impervious roofs).

Finally, spatial variability of rainfall fields seem to influence the basin response as peak reduction can increase/decrease of about +/-25% regarding the average response. This demonstrates the interest to use high-resolution precipitation data (as those provided by X-band radar for instance) in stormwater management.

Figure 1 .

 1 Figure 1. Loup basin: land use spatial distribution with pixel of size 10 m x 10 m, and sewer network inputted on Multi-Hydro

Figure 2 .

 2 Figure 2. Simulation results obtained by using the 50 downscaled rainfall fields for the 09/02/2009 (top) event and the 15/08/2010 (bottom): current situation (without any green roof) is depicted in black and green roofing scenario in blue.

ACKNOWLEDGEMENT

This work is part of the Climate-KIC funded Blue Green Dream project (http://bgd.org.uk/) that aims to move forward a new paradigm for efficient planning and management of new urban developments and retrofitting of existing ones. For this purpose, it promotes the implementation of blue and green infrastructures to maximize ecosystem services and increase resilience to climate change. Authors would like to thank Jean-Baptiste Abbes who initially implemented the Multi-Hydro on the Loup catchment.


Bernard Anselmetti

Robin Chavanne

Jian-Xin Yang

Nabil Anwer

Robin Chavannea

Jian-Xin Yanga

Nabil Anwera

Quick GPS: A new CAT system for single-part tolerancing

Keywords: ISO tolerancing standards, Functional Dimensioning and Tolerancing, CAT systems, specification synthesis, tolerancing rules, positioning table

published or not. The documents may come

INTRODUCTION

The tolerancing data reported by the designer on technical drawings are becoming nowadays very critical when formalizing the contractual relationships between companies. In industry, functional tolerancing of mechanisms is today more and more based on ISO GPS (Geometrical Product Specification) and ASME Y14.5-2009 standards. The designer's activity consists also on optimizing parts so as to increase mechanism quality while reducing costs. This tolerancing activity requires time, a good knowledge of standards and the use of CAD systems (Computer Aided Design), while the designer must have expertise and skills for selecting appropriate tolerancing specifications. Within a classical approach towards tolerancing, the designer determines the geometrical functional requirements of the mechanism according to function analysis. For each requirement, the designer must identify the influent parts and the 3D tolerance chain, choose tolerances and verify all the results of tolerance chains computation. Current industrial CAD software's do not have specific tolerancing workbenches that cover all the above tasks and only few research works allow quick functional dimensioning and tolerancing based on ISO GPS concepts [START_REF]Geometrical product specification (GPS) -Masterplan[END_REF][START_REF]2: Geometrical Product Specifications (GPS) --General Concepts--Part 2: Basic tenets, specifications, operators and Uncertainties[END_REF][START_REF] Srinivasan | An Integrated View of Geometrical Product Specification and verification[END_REF] and tolerance editing controlled by syntactic analysis [START_REF] Lu | A Host System for Form Tolerance Specification in the Next-generation GPS based on AutoCAD[END_REF]. FT&A (Functional Tolerancing and Annotation) workbench of CATIA V5 (Dassault Systèmes) [START_REF]CAA V5 Encyclopedia[END_REF] uses a specific GD&T (Geometric Dimensioning and Tolerancing) editor and enables syntactic analysis. FT&A is based on TTRS (Technologically and Topologically Related Surfaces) approach [START_REF] Gaunet | 3D Functional Tolerancing & Annotation: CATIA tools for Geometrical Product Specification, Geometric Product Specification[END_REF]. The designer must select datum reference frame and toleranced surface. A dialog box proposes only acceptable specifications and modifiers. Several CAT systems are dedicated to tolerance analysis such as 3DCS ® (Dimensional Control Systems), CETOL ® (Sigmetrix), VSA ® an eM-TolMAte (Siemens PLM). Assembly of parts is simulated (Monte Carlo simulation approach) considering the geometrical defects limited by the chosen tolerances. In Tolerance Manager ® (PCO technologie), the designer must define the set of tolerance chains and the tolerance values. A dashboard presents the results of tolerance chains with worse case or statistical approach. In Mecamaster [START_REF] Clozel | 3D tolerance analysis, from preliminary study[END_REF], the designer creates an isostatic model of the assembly and the system determines the influence of each part deviation on one requirement. SolidWorks DimXpert [START_REF] Therrien | DimXpert for parts: Introduction for Resellers, SolidWorks[END_REF] is used to tolerance a part using either automatic or manual mode based on GD&T Symbols and according to ISO 1101 and ISO 16792 and also ASME Y14.41-2003 standards. Under-constrained, iso-constrained and over-constrained conditions of a part are considered and highlighted by a set of colors. DimXpert GD&T layout forms the basis for Tolerance Analysis using SolidWorks TolAnalyst module. The modular system FROOM (Features and Relations used in Object Oriented Modeling) developed by Salomons and al is based on ACIS ® kernel modeller environment. The first module generates geometrical specifications with TTRS approach [9]. Designer chooses tolerances. The second module performs tolerance analysis with torsor concept [10]. The resulting specifications are very basic and didn't cover for example common zones and modifiers.

In order to provide the designers with complete and reliable solutions, the CLIC system (French acronym for "Cotation en Localisation avec Influence des Contacts") developed by Anselmetti [START_REF] Anselmetti | Generation of functional tolerancing based on positioning features[END_REF][START_REF] Anselmetti | Bases de la cotation fonctionnelle[END_REF][START_REF] Anselmetti | [END_REF] proposes the automatic generating of functional requirements, functional specifications in accordance with ISO tolerancing standards and 3D tolerance chains considering worst case and statistical approaches [START_REF] Anselmetti | Part optimization and tolerances synthesis[END_REF] (Fig. 1). Today, CLIC system has been applied successfully and can process simple mechanisms while producing detailed tolerancing data in a short time (10 minutes for a 5 parts mechanism and 100 generated specifications).

Figure1: Tolerancing in CLIC

When considering complex mechanisms with many parts, the overall design process involves many designers and each of them is responsible for single part tolerancing. The Quick GPS method proposed here, that conforms to ISO GPS Tolerancing standards, is a new approach for tolerancing single parts that ensure consistent tolerancing for the entire mechanism (Fig. 2). This method requires information sharing among the designers and due to its simplicity it is intended to achieve only limited objectives. Quick GPS can also be considered as the first stage of CLIC system, without tolerance chains. In the following sections, we will focus on specification synthesis of a single part, in accordance with the ISO GPS standards. In Section 2, a case study mechanism will be described, while Section 3 will develop the tolerancing rules for specifying all contact surfaces based on positioning requirements. Section 4 will then describe an alternative method for the classical 3D chain transfer. Section 5 will illustrate an application of the Quick GPS system in a CATIA V5 CAD environment using both VBA and CAA procedures.

MECHANISM DEFINITION

The studied mechanism

The Quick GPS method will be illustrated in this paper through a case study: "air control of a gearbox" (Fig. 3), the approach can be used for all mechanisms consisting of rigid parts without loss of generality. The shaft has three positions for moving an arm connected to the gearbox that modify the air input. To ensure consistent tolerancing, the designers must define the positioning plan.

Left

The positioning graph

Tolerancing a mechanism depends on its part positioning. For each configuration of the mechanism, the set-up of the parts is represented by one positioning graph (Fig. 4). The most left part of the graph is called a "base". One after another, each part on the right is positioned on the part located to its left on the graph. It is also possible to assemble a subset, called a "block", along with its proper base. A block is composed of parts without mobility. Figure 4 presents the positioning graph in the left configuration. The body is the base of the body block. The flange is setting-up in the body. The spool is fixed in the flange. The shaft is centered in the body block and is in contact with the body. The piston is centered on the shaft and is in contact with the body. The probe finger is setup by the shaft and the spool. Whereas in the centered and right control configurations, the piston is blocked on the shaft and this shaft block move between these two configurations.

The positioning tables

The positioning of each part (and each block) on positioning graph (except for the bases) must be described using a positioning table. For this purpose, designers must consider degrees of freedom blocked by each positioning feature. The feature that removes the maximum of the degrees of freedom in orientation is the primary feature. The secondary and tertiary features must then impose at least one degree of freedom [START_REF] Anselmetti | Bases de la cotation fonctionnelle[END_REF][START_REF] Anselmetti | Computer Aided Tolerancing using Positioning Feature[END_REF]. The positioning table (Fig. 5) describes the flange setting up. The Plane A of the flange is in contact with the plane D of the body in order to define the primary feature. The Cylinder B of the flange is centered in the cylinder E of the body, with clearance to constitute the secondary feature. The Set C containing three holes of the flange is in front of the set F of three tappings of the body. Three screws fitted in the tappings locate the flange with clearance.

Basic positioning surfaces are either a cylinder or a plane, yet very often a set of surfaces constitutes a composite positioning feature as well as a group of parallel cylinders. The three classical interfaces are the contact, the clearance, and the interference.

During this step, designers take into account the parts defects and forces applied to the part before making key decisions regarding product quality and potential failure in the case of surface deviations. In Figure 5, plane D locates and orientates the flange. This orientation cannot be performed by cylinder E because the common length of both the body and flange cylinders is very short and moreover a clearance lies between these two parts. D therefore is a primary surface. In addition, this approach serves to verify compliance of the positioning face dimensions with respect to the preponderance order. The third row of the table defines the main datum reference frame A|B|C of the flange, according to the datum reference frame definition found in ISO tolerancing standards. The last row depicts the auxiliary datum reference frame D|E|F to be created on the body in relation to the flange.

The positioning tables are defined for each part and are made available to each designer. With this common database, team of designers can work independently but generate a consistent and robust tolerancing of the mechanism.

The assembly or specification tree currently used in CAD systems cannot provide the same data. For example, designer can center any cylinder of the flange on any cylinder of the body. There is no precedence order between plane and cylinder and no information on the clearance or interference between parts.

Other approaches group all data in one complex graph (nodes represent the parts, and edges represent the links between surfaces of the parts). M. Linares [START_REF] Linarès | Tolerancing by functional group[END_REF] defined functional positioning features in the node. In their work, A. Ballu, L. Mathieu [START_REF] Ballu | Choice of functional specifications using graphs within the framework of education, Integrated Design and Manufacturing in Mechanical Engineering[END_REF] and D. Tessandier [START_REF] Teissendier | Three-dimensional functional tolerancing with proportional assembly clearance volume (UPEL): Application to setup planning[END_REF] added the type of kinematic joint as an attribute to the edge (e.g. planar joint, revolute joint). M. Samper [START_REF] Samper | Models for tolerancing process by considering mechanism flexibility[END_REF] noted the type of contact on the edge (e.g. fixed, sliding, floating, forbidden), whereas B. Marguet oriented edges from the base to their end parts [START_REF] Marguet | Method for Geometric Variation Management from Key Characteristics to Specification[END_REF].

Positioning features recognition

A CATIA V5 based interactive procedure has been developed to define the positioning surfaces of the body. In a three-step sequence (Fig. 6), the designer selects the set of primary, secondary and tertiary surfaces. A dialogue box enables the designer to detail the interface between the various positioning surfaces (contact, clearance, interference, etc.). With the current VBA procedure, it is not possible to detect the thread on a hole. This dialogue box serves also to define tapping as an interface for the set of threads. This interactive selection is straightforward; nonetheless, Armillotta [START_REF] Armillotta | Tolerance specification through automated generation of assembly requirement[END_REF] proposed a method for automatically recognizing these contact surfaces on the CAD assembly model. The precedence order is based on surface area.

For each selection, CATIA VBA procedures recognize the elementary geometry features: plane, cylinder, sphere, cone, torus (Fig. 7). Geometric characteristics can then be extracted from this recognition phase (point, vector, radius, etc.). If the positioning feature is composed of several elementary surfaces, then the combinations between geometric characteristics would enable determining the type of feature that allows differentiating coaxial cylinders from, for example, a set of parallel cylinders. Moreover, each positioning feature is associated with a TTRS class, so as to apply the set of rules listed above. Figure 8 indicates the set of available positioning features in the current version of Quick GPS.

Surface features

Simple

Set-up verification

During each step, a set of rules serve to verify junction consistency. The first rule is based on TTRS theory and ensures that each datum reference frame created is indeed consistent. The primary feature must define one of the six classes (plane, cylinder, revolute, prismatic, spherical and complete). Any reclassification of the primary and secondary features must result in a class different from the primary class. Figure 9 shows an example with primary cylinder A and secondary plane B. This case is well accepted by TTRS theory and the syntax control on the CATIA FTA workbench. The Quick GPS procedure takes into account the size of surfaces and detects that the length of the cylinder is very small with respect to its diameter. If ever Length < 0.5 × diameter, a warning is displayed.

Other functional surfaces

To complete the mechanism definition, the designer must select other isolated functional surfaces from CATIA part model (e.g. surface 13b corresponding to the piston external cylinder, and plane 14b in contact with the face of the piston face and set of fixture holes 22,23,24 and 25b) (Figure 19).

DATUM SYSTEM SPECIFICATIONS

Positioning requirements

Part tolerancing can now be established. These specifications must ensure both part assembly and contact quality (Fig. 10).  Primary surfaces in contact need specifying the form of surfaces in order to control the maximum distance between surfaces lying in contact.

 Secondary or tertiary surfaces in contact impose an orientation specification with respect to the primary reference.

 Clearance or interference of a primary cylindrical joint imposes part diameters and the envelop requirement.

 Clearance of a secondary or tertiary cylindrical joint necessitates to control the maximum material boundary.

Tolerancing of positioning features

The functional specifications of the corresponding positioning requirements are depicted in patterns for each type of positioning feature and for the primary, the secondary and the tertiary cases as well (Fig. 11). In these patterns, the perpendicularity symbol indicates orientation specification while the location symbol identifies the positioning specification, but this symbol must be in accordance with rule 2 (section 3.4).

t A t CZ A t A B t CZ A L±t/2 E A B t A t A B C t A B t CZ A C t CZ t CZ A B A B L±t/2 E B t A t A M M M M C A B A B t CZ A t CZ B A t CZ C A B 15 15 15 (2) (2) (2)
(1)

D±t/2 E D±t/2 E t A B A t A A B C A B A t CZ M d±td/2 E d±td/2 E t M 2x 50 A B C (1)
(1) (2) This tolerancing method is very efficient and can be easily used by the designer. Clement [START_REF] Clement | Cotation tridimensionnelle des systèmes mécaniques: Théorie et pratique[END_REF] and Salomons [9] have a similar approach using TTRS concept, but their table cannot distinguish features in such as simple planar, coplanar planes, parallel planes, and symmetrical planes. The specifications are then very basic and do not take into account clearance or interference in joint in order to add modifier in specifications.

(1) (1)

Cylinder

Coaxial cylinders

Set of cylinders

20°t t A B M14 x 2, 6g -6g A on thread M14 x 2, 6g -6g B t A M14 x 2, 6g -6g C t A B t A t A B A st A M A B (2) (2) (2) (2) (2)
t (2) A t A (2) B t A B (2) C Continuous surface Discontinuous surface L±t/2 E L±t/2 E t M t M D±t/2 E D±t/2 E M M M M t t M M M M D±t/2 E D±t/2 E D±t/2 E D±t/2 E D±t/2 E D±t/2 E t CZ A t CZ A M M M M d±td/2 E d±td/2 E A B A B t CZ t CZ M M M M A E t M 50 B d±td/2 2x A B E t M 50 C d±td/2 2x 20°A  t A B C (2) 20°  (2) (2) D±t/2 E D±t/2 E D±t/2 E D±t/2 E D±t/2 E D±t/2 E st M

Set of Threads

Rule 1: Pattern selection

The selection of positioning surfaces of the flange on the body (Fig. 6), the feature recognition and the definition of the interfaces are presented in Figure 5. The corresponding pattern must now be added to the CAD model. Figure 12 shows tolerancing for the auxiliary reference frame of the flange, and automatically generated by Quick GPS in 30 seconds. Datum D is a primary plane, and secondary datum E is a cylinder perpendicular to D. The set of three tappings F must be located with respect to datum reference frame D|E.

Rule 2: Specification determination

In Figure 11, some patterns for secondary and tertiary planes or cylinders suggest one of two solutions: orientation or position for any feature. Most of these cases require the orientation specification, yet the position specification must be applied if a dimension must be defined between the toleranced surface and a datum, e.g. if the feature is parallel to a datum.

To formalize the choice decision, TTRS theory is used to identify all possible cases, but with more details in order to determine the symbol and if the tolerance zone is cylindrical () or not. Figure 13 sets forth the specifications to be applied to these features, depending on whether the toleranced surface and datum system. This datum system belongs to one class among: planar, cylindrical, prismatic, revolution, spherical and complex. The primary surface is plane A. Figure 11 suggests a simple type of flatness ①.

The secondary feature is composed of two coaxial cylinders of direction u ②. The primary datum is a plane normal to v ③. For both a planar reference surface and a cylindrical toleranced surface, the test ④ u.v=±1 is true.

The specification is perpendicularity with the  symbol ⑤.

The tertiary feature is a simple cylinder of direction u ⑥. The datum system A|B belongs to a revolution class with the v axis ⑦. The test ⑧ u.v=±1 is true. The specification is a location with the  symbol.

The table in figure 13 is intended to control the positioning table. The reclassification result for both toleranced surface and datum should not be in the same class as the datum. A star indicates that this case is impossible for positioning specifications (but usable for the other requirements described in Section 4).

Rule 3: Maximum material modifier

The clearance is obviously favorable for the mechanism assembly. The maximum material modifier should be introduced on both a toleranced surface and a datum surface, provided the availability of clearance at the corresponding interface (Fig. 14).

Theoretically speaking, this rule can be applied to all surfaces. At the present time, ISO standards do not accept this modifier for a free form surface with profile specification. Figure 14 depicts a shaft composed of three free surfaces. Quick GPS displays a comment "MMC only" that merely states the given boundary under the maximum material condition. ASME Y14.5-2009 accepts this modifier with this type of profiles. In the simple case shown in Figure 5, the flange is set up on just one part: the body. The auxiliary datum reference frame can be completely defined on the body. If one positioning feature is split on different parts, the problem can be solved with CLIC approach. The primary positioning feature of the shaft is made of surfaces 1s, 2s and 3s (Fig. 15). 1s is fitted to the body. 2s and 3s are fitted to the flange. According to Figure 11, the first step of the CLIC method proposes a straightness specification on both subsets defined by the positioning graph, the shaft and the body block. The tolerancing of the body block is currently defined between different parts and requires a second step. The flange positioning table (Fig. 4) reveals that the flange is set up on the body. With CLIC method, a specification in common zone must be broken down into both location and orientation of each surface with respect to a common datum reference frame. The straightness therefore is broken down into location and orientation specifications of body surfaces (1b, 2b) and flange surface 1f, with respect to the datum reference systems defined on the junction between flange and body (Fig. 16). The clearance between flange and body imposes an LMC modifier on the datum reference frame. In this case, by using Quick GPS, during the interactive definition of the primary auxiliary positioning features of the body, the designer is only able to select the two body cylinders (1b,2b) (Fig 15) and must notify in a dialog box that this feature is incomplete. In addition, the designer must indicate the name of the reference frame link to the other surfaces of the shaft positioning feature. Also in this case, the datum reference frame to be used is the auxiliary frame D|E named A1f, corresponding to the flange. Rule 4: if all surfaces of one positioning feature do not belong to the active part, surfaces of active part must be located and oriented with respect to the designed reference frame corresponding to the other surfaces.

t D t D 15 E E E b1±t/2 D E t M L L D E t M L L D t E A t 15 20 E E E A B t M L A B t M L t A B f2±t/2 f1±t/2 b2±t/2

Results

Figure 17 presents body specification results for the positioning requirements. Tolerance values are, by default, set at basic values, e.g. defined at 0.02 for flatness, and must be adjusted by the designer.

Figure 17: Results for the positioning requirement specifications A|B|C represents the main reference frame (R1b) of the body defined in the body positioning table (Fig. 12). D|E|F constitutes the auxiliary reference frame for the flange, denoted A1f according to the flange positioning table (Fig. 4). G represents a portion of the primary auxiliary datum of the shaft, denoted A1s (in Fig. 14), and requires both orientation and location specifications with respect to the auxiliary reference frame D|E of the flange. The pairs (J,K), (L,M) and (N,P) correspond to the auxiliary reference frames of nipples (A1n, A2n, A3n).

RELATIVE POSITION OF REFERENCE FRAMES 4.1 Links between reference frames

In the CLIC method, the synthesis of specifications is based on a set of functional requirements between functional surfaces. For each functional requirement, the contact loop links functional surfaces, thus allowing for the generating of functional specifications on influent surfaces as well as for the relationship between tolerances for these specifications [START_REF] Anselmetti | Generation of functional tolerancing based on positioning features[END_REF]. In this manner, the 3D tolerancing chain creates links between the various datum reference frames of each part.

According to the Quick GPS approach, the designer must indicates the set of useful links between the various datum reference frames (R1b, A1f, A1s, A1n, A2n and A3n) created in Section 3.7 (Fig. 18). In a basic approach, the designer would identify all mechanism requirements and determine the contact loop that links functional surfaces. This loop serves to link the influential reference frames of each part. As an example, in figure 18, requirement C1 imposes positioning the auxiliary reference frame A1f with respect to the main reference frame R1b. C2 requires a minimum distance between the piston and surface 4b of the body, which in turn imposes a link between A1s, A1f and functional surfaces 4b. This method is still not very efficient due to the difficulty involved in identifying all mechanism requirements. Furthermore, many contact loops do not relate to the active part, and a considerable number of these loops actually link the same reference frames.

Specification corresponding to links

This second method is more efficient. The designer must analyze the influence of the displacement of each auxiliary datum reference frame, which directly yields the other target frames and allows detecting the functional surfaces of the active part. For example, if the hole of datum reference frame A1s is placed too high, interference will arise between the piston and surface 4b of the body. The displacement of A1f exerts the same effect, hence A1s and A1f must display a good relative position.

Figure 19 depicts the various links considered useful for this body.

-Datum reference frames A1f, A1s must be located with respect to main frame R1.

-Surfaces 13b and 14b (corresponding to interference with the piston) must be located with respect to A1f.

-Set of holes (fixture of the body) must be located with respect to R1. This example offers a complementary approach: three nipples are fixed in three tappings. To avoid interference between nipples, a minimal distance is required between tappings. In this case, the Quick GPS method proposes grouping these positioning features A1n, A2n and A3n into a single group G1, which is then located with respect to main frame R1. The third phase displays the matrix (see Table 1), which contains all frames, all isolated functional surfaces, all groups and a single specific column for general tolerancing. The cells shown in gray on this table are impossible to obtain and therefore locked. The designer must now place the letter "P" in order to localize a frame or functional surface with respect to a given datum reference frame. The letter "O" indicates an orientation specification and F for a form specification.

P P A1#ba A1n G1 G1 A2n G1 G1 A3n G1 G1 G1
Table 1: Relative position between datum reference frames

Tolerancing rules

The next phase generates the tolerancing corresponding to these links.

Rule 5:

For each link displayed in Table 1, all surfaces of the specified frame must be located or oriented with respect to the reference frame. 2 describes in detail the condition employed in figure 20 to determine whether the reference frame is indeed sufficient for a given position specification. A similar table exists for the orientation specifications. This frame is associated with a TTRS to be calculated by means of reclassification [START_REF] Dantan | A System for Modeling Geometric Tolerances for Mechanical Design[END_REF]. In choosing the positioning surface specification, Maeda [START_REF] Maeda | A Toleranced feature modelling by constant of degree of freedom for assignment of tolerance[END_REF] proposed a model to express the behavior of a toleranced feature by using the degree of freedom for this feature. TTRS theory is used to model surface associations and then, based on these associations, proposes standardized tolerancing [START_REF] Clement | Cotation tridimensionnelle des systèmes mécaniques: Théorie et pratique[END_REF]. O.W. Salomons applied the same method with minor changes [10].

Results

* *

Figure 21: Complete tolerancing based on Quick GPS Figure 21 indicates the complete tolerancing, including locations and orientations obtained by the links defined in Table 1.

The grouping of the three tappings allows for just one location specification of the planes (datum Q) in the common zone and one location of the group of tappings (datum R); otherwise, without any such grouping, six independent specifications would be created. This tolerancing may be excessive. For example, the location of the group of three tappings (denoted with a star in Fig. 21) proves useless. However, such a location step would have been necessary if a hole in the flange had been positioned in front of a hole of the body in order to pass air through the flange. The designer must therefore analyze the relevance of the proposed specifications and possibly delete some of them. Experience demonstrates that this case is quite rare.

IMPLEMENTATION OF QUICK GPS

General structure

Tolerancing annotations are created using the functional tolerancing and annotations (FT&A) workbench of CATIA V5. The Quick GPS tolerancing method has been implemented within a CATIA VBA environment (Dassault Systèmes), with data stored in a simple Excel spreadsheet. The application comprises 3 steps (Fig. 22). The first step enables an interactive dialogue between the designer and the computer-aided tool. For each link with the neighboring parts, the designer selects the primary, secondary and tertiary positioning features through the use of the CATIA interface. The designer must also select isolated functional surfaces. Surface characteristics are stored in the Excel spreadsheets as well.

During the second step, VBA procedures generate tolerancing data in accordance with the previous set of tolerancing rules (Fig. 22). Each specification listed in an Excel spreadsheet can be modified or deleted, and tolerances can be adjusted. Finally in step 3, all ISO functional specifications are transferred into the CAD model with a CAA procedure, by means of the FT&A semantic language. Views and captures are created automatically.

CATIA MODEL OF PART

Generation of tolerancing

Generation of annotations

Figure 22: Interface with the CAD system

Comments

Each specification is described in an Excel spreadsheet, including comments to explain the specification and/or function performed by the specification or possible failure should the specification not be respected. In CAA V5, the CATTPSInterfaces framework provides interfaces to describe the Technological Product Specification (TPS) domain; moreover, the CATTTRSInterfaces framework provides interfaces to link TPS with surfaces. The object model of tolerancing components is represented in Table 3, with each TPS object adhering to various interfaces, which define the set of specific behavior that the object is required to support. As an example, for a given annotation set object, the CATITPSSet interface allows retrieving standards and the TPS components included in this set, while the CATITPSFactoryElementary interface provides the basic factory for annotation creation. The client application focuses on the object solely through the interface.

A new CAA procedure has been developed and integrated into the FT&A (Functional Tolerancing & Annotation) workbench of CATIA (Figure 23). Tolerancing data are retrieved from an Excel spreadsheet using straight C++ automation code. User surfaces and group of user surfaces (TTRS) are created on topological cells that correspond to the stored generic naming labels, with all ISO functional specifications then being generated in the following strict order: -Creation of datums -Creation of datum reference frames -Creation of semantic dimensions and geometric tolerances.

With a number of interfaces implemented by a single annotation object, each annotation can be characterized by various parameters, such as: dimension limits (CATITPSDimensionLimits), tolerance zone form (CATITPSToleranceZone), envelop condition modifier (CATITPSEnvelopCondition), and material condition modifier on the tolerance or datum reference frame (CATITPSMaterialCondition). A view plane is created for several annotations. The following rules should be respected for each annotation:

-A specification concerning a cylinder or coaxial cylinders requires a cross-section view plane passing through the axis (diameter, coaxiality, location, etc.); -A specification concerning a plane requires a view plane perpendicular to the toleranced plane (flatness, location, etc.); -A specification concerning a set of holes requires a front view plane perpendicular to the axis of the holes. This procedure will create a minimum number of view planes and set each annotation into a single view plane, which is also helpful for creating a generative drawing sheet.

Grouping of annotations

The previous procedure merely creates isolated annotations. For a master 3D annotation, the following slave 3D annotations, regarding the same surface or set of surfaces, can be grouped under this annotation through the CATITPSAssociativeGroup interface. Another CAA procedure has been developed to automatically group all annotations in order to offer a clear presentation. Final tolerancing results are shown in Figure 21.

Limitations

Due to the principle behind Quick GPS, this software only recognizes one part. It is impossible therefore to manage the tolerances and lengths of projected zones. Restricted zones corresponding to the surface actually in contact between two parts cannot be delimited, leading to confusion if two different parts are in contact with the same face, e.g. the left face of the piston in contact with both the circlips and the body in the central configuration.

The tolerancing generated by current rule 3 is not guaranteed if the set-up of one part is provided by more than two parts, since it is impossible to analyze the degrees of freedom blocked by an unknown part. Free surfaces are considered as complex surfaces with six degrees of freedom blocked. This point is currently under development.

The tolerancing is still incomplete, as regards for example growing, tapping and pinion. The proposed tolerancing is limited to assembly requirements and links between reference frames. The current CAA V5 development tools available in CATIA V5 are very complex for programming applications. Due to the limitation of APIs, the CAA procedures developed remain limited, for example, for framed basic dimension and projected zone.

CONCLUSION

This paper has demonstrated that a consistent tolerancing of a single part may be achieved with knowledge of both the positioning tables and the links between the various contact surfaces. The Quick GPS method provides an efficient solution for many industries, in cases where each designer is responsible for just a single part. The positioning table offers a way to specify all joints so as to respect positioning requirements. The table summarizes all links into the part and then serves as an alternative to the classical 3D chain transfer approach for the contextual study of a single part.

Figure 2 :

 2 Figure 2: Tolerancing in Quick GPS

Figure 3 :

 3 Figure 3: Three configurations of the mechanism

Figure 4 :

 4 Figure 4: Two positioning graphs for the three configurations

Figure 5 :

 5 Figure 5: Positioning table of the flange

Figure 6 :

 6 Figure 6: Selection of positioning features of the flange

Figure 7 :

 7 Figure 7: Elementary surfaces identified from the CAD model

Figure 8 :

 8 Figure 8: List of positioning features

Figure 9 :

 9 Figure 9: Part set-up verification

Figure 10 :

 10 Figure 10: Positioning requirements and corresponding tolerancing

Figure 11 :

 11 Figure 11: Patterns for positioning feature tolerancingThis tolerancing method is very efficient and can be easily used by the designer. Clement[START_REF] Clement | Cotation tridimensionnelle des systèmes mécaniques: Théorie et pratique[END_REF] andSalomons [9] have a similar approach using TTRS concept, but their table cannot distinguish features in such as simple planar, coplanar planes, parallel planes, and symmetrical planes. The specifications are then very basic and do not take into account clearance or interference in joint in order to add modifier in specifications.

Figure 12 :

 12 Figure 12: Creation of an auxiliary reference frame for the flange

Figure 13 :

 13 Figure 13: Specifications according to the types of toleranced surfaces and datum system

Figure 14 :

 14 Figure 14: Clearance on both a cylinder and free surface

Figure 15 :

 15 Figure 15: Straightness of both body block and shaft

Figure 16 :

 16 Figure 16: Breakdown of straightness into locations and orientations

Figure 18 :

 18 Figure 18: Search for links

Figure 19 :

 19 Figure 19: Links between datum reference frames

Rule 6 :

 6 For each specification, the algorithm summarized in Figure20enables determining whether one, two or three references are needed.

Figure 20 :

 20 Figure 20: Test of datum frame combinations

2 :Rule 7 : 8 :

 278 Datum system test for positioning The choice of specification symbol respects Figure13. Rule If the toleranced surface or datum is a fitting feature with clearance, it becomes necessary to add a minimum material modifier.

Figure 23 :

 23 Figure23: The object model of tolerancing components[START_REF]CAA V5 Encyclopedia[END_REF]

Table 3 : Comments table 5 .3 Creation of annotations in the 3D CAD model with a CAA procedure

 3Comments5 The main development introduced in this part is based on the CATIA V5 Component Application Architecture (CAA). A CAA application is composed of references to functions available in the API libraries and contained in different CATIA V5 components.

This method has now been implemented within a VBA environment, in interaction with the CATIA environment. The module has proven to be very effective: the selection of set-up surfaces takes just 5 minutes, and body tolerancing generates 42 specifications in only 20 seconds. As opposed to Armillotta's findings [21], the contact surfaces are not detected automatically, yet the subsequent goal of this application is to enhance the assembly method on a CAD model by integrating the positioning table concept. This approach is not sufficient for tolerance optimization, a process that requires tolerance chains and a system of complex equations. This is still open to investigate using other commercial software, such as 3DCS and MECAmaster, in order to determine the effect of these tolerances on functional characteristics. Finally, one major advantage of the work presented here is the introduction of junction analysis between parts within the designer methodology.


Laurent Pierre
email: laurent.pierre@lurpa.ens-cachan.fr

Bernard Anselmetti
email: bernard.anselmetti@lurpa.ens-cachan.fr

Comparison of analysis line and polytopes methods to determine the result of a tolerance chain

Keywords: Functional tolerancing, tolerance analysis, Analysis line, Polytope, ISO standard of tolerancing

Functional tolerancing must ensure the assembly and the functioning of a mechanism. This paper compares two methods of tolerance analysis of a mechanical system: the method of "analysis lines" and the method of "polytopes".

The first method needs a discretization of the ending functional surface according to various analysis lines placed on the outer-bound of the face and oriented along the normal of the surface. The second method uses polytopes. The polytopes are defined from the acceptable limits of the geometric deviations of parts and possible displacements between two parts. Minkowski sums and intersections polytopes are then carried out to take into account all geometric variations of a mechanism.

INTRODUCTION

Functional tolerancing context

Now the evolution of digital tools allows designers to define their entire mechanism in a CAD environment. The parts are described as perfect shape with nominal dimensions. To ensure interchangeability and allow the production of parts at lower cost, it is very important to allocate the widest possible tolerances, while ensuring that the functional requirements of the mechanism are respected. With this approach, the designer chooses the geometric specifications and tolerances. He must ensure that the combination of the variations permitted by the tolerances will be compatible with each requirement. This step is called tolerance analysis.

State of art

Tolerancing process consists in specification synthesis, tolerancing analysis, tolerancing synthesis and tolerancing verification [START_REF] Salomons | A computer aided tolerancing tool i: Tolerancing specification[END_REF][START_REF] Salomons | A computer aided tolerancing tool ii: Tolerancing analysis[END_REF][START_REF] Chase | Tolerance allocation methods for designers[END_REF].

The specification synthesis determines the set of functional specifications to impose on parts with respect to a functional requirement [START_REF] Islam | Functional dimensioning and tolerancing software for concurrent engineering applications[END_REF][START_REF] Anselmetti | Generation of functional tolerancing based on positioning features[END_REF].

Traditionally, tolerance analysis is provided by a tolerance chart model in only one direction which is simply to sum of tolerances of influential parts. Soon as there is an angular effect in the chain of dimensions, designers are still quite poor. Some analytics methods calculates the result for each requirement using analytical tolerance chains [START_REF] Johannesson | Structure and matrix models for tolerance analysis from configuration to detail design[END_REF][START_REF] Ballot | Geometrical behavior laws for computer aided tolerancing[END_REF][START_REF] Mehdi-Souzani | Integration of automatic functional tolerancing into the design process[END_REF] or a statistic methods [START_REF] Nigam | Review of statistical approaches to tolerance analysis[END_REF][START_REF] Shan | Genetic algorithms in statistical tolerancing[END_REF][START_REF] Kuo | An analytical computation method for statistical tolerance analysis of assemblies with truncated normal mean shift[END_REF] and Monte Carlo methods [START_REF] Skowronski | Using Monte Carlo variance reduction in statistical tolerance synthesis[END_REF][START_REF] Lin | Study of an assembly tolerance allocation model based on Monte Carlo simulation[END_REF].

The variational approach of geometric tolerancing differs from parametric approaches [START_REF] Shah | Navigating the tolerance analysis maze[END_REF]. The variational approach oftolerancing consists of characterising the relative position of two surfaces from two different parts of a system by intersections and Minkowski sums of sets of geometric constraints defined by half-spaces. The domains [START_REF] Giordano | Mathematical representation of tolerance zones[END_REF] and T-Maps [START_REF] Davidson | A new mathematical model for geometric tolerances as applied to round faces[END_REF] manipulate half-spaces of which the boundaries are generally not linear. In contrast, other methods use half-spaces of which the boundaries are linear like polyhedral objects [START_REF] Roy | Representation and interpretation of geometric tolerances for polyhedral objects. II.: Size, orientation and position tolerances[END_REF] and polytopes [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF][START_REF] Pierre | Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine[END_REF].

Indeed, there are very few commercial software tools for the analysis of tolerances in 3D. The principle is based on the simulation of assemblies:

-Mécamaster [START_REF] Clozel | MECAmaster: a tool for assembly simulation from early design, industrial approach[END_REF] simulates the assembly of nominal parts and introduces deviations between parts corresponding to tolerances of the support surfaces or clearance in joint.

-Cetol [21], 3DCS [START_REF] Dcs | Variation Analysis and Tolerance Analysis[END_REF] generate, by Monte Carlo methods, parts with position, orientation or form defects corresponding to the ISO specifications. Such software measures the characteristic of the requirement on the set of assemblies obtained with simulated parts.

-Anatole [START_REF] Falgarone | An innovative Design Method and Tool for Structural and Functional Analysis[END_REF] characterizes defects of surfaces with small displacement torsor.

The equations expressing the displacement of the parts relative to their nominal positions are based on contact between parts to give a formal relationship of part tolerances.

Few scientific studies have been devoted to the allocation of tolerance. Some purely mathematical approaches have been derived, and these essentially develop cost functions to varying levels of sophistication [START_REF] Loof | An efficient solution to the discrete Least Cost tolerance allocation problem with general loss functions[END_REF][START_REF] Kurlarni | Allocation of tolerances to the components of an assembly for minimum cost[END_REF][START_REF] Anselmetti | Part optimization and tolerances synthesis[END_REF].

Functional tolerancing must be including in product life management [START_REF] Khanafer | Tolerancing assistance methodology in a product life cycle perspective[END_REF].

Position of proposed work

Since 1997, B. Anselmetti developed the CLIC method (a French acronym for "Localization Tolerancing with Contact Influence") [START_REF] Anselmetti | Generation of functional tolerancing based on positioning features[END_REF]. This system proposes the automatic generation of functional requirements and functional specifications in accordance with ISO tolerancing standards. The result of each tolerance chart is described by an EXCEL formula according to tolerances of influential parts. This structure allows optimizing the nominal dimensions of CAD model to maximize tolerances [START_REF] Anselmetti | Part optimization and tolerances synthesis[END_REF].

In this software, tolerance analysis is ensured by the analysis line method. The ending functional surface is discretized by various analysis lines placed on the outerbound of the face and oriented along the normal to the surface. The principle consists in the sum of the influence of defects in each junction on analysis line using relationship pre-established for all classical types of junctions. The calculation is very fast and expresses the result with a formula function of tolerances with worst case or with a statistical approach [START_REF] Benichou | Thermal dilatation in functional tolerancing[END_REF]. The method of operations on polytopes in tolerance analysis is a variational approach based on the work initiated by Flemming in 1988. It is based on the operations of sets of geometric constraints [START_REF] Fleming | Geometric relationships between toleranced features[END_REF]. The set of possible positions of a surface within a tolerance zone is characterized by a set of geometric constraints. This allows characterizing the geometric variations of a part according to orientation or position ISO specifications [START_REF] Giordano | Mathematical representation of tolerance zones[END_REF][START_REF] Davidson | A new mathematical model for geometric tolerances as applied to round faces[END_REF][START_REF] Roy | Representation and interpretation of geometric tolerances for polyhedral objects. II.: Size, orientation and position tolerances[END_REF][START_REF]Geometrical Product Specifications (GPS), Geometrical tolerancing, Tolerances of form, orientation, location and run-out[END_REF]. Similarly, a set of geometric constraints characterises all relative positions between two distinct surfaces potentially in contact [START_REF] Fleming | Analysis of Uncertainties and Geometric Tolerances in Assemblies of Parts[END_REF]. Fleming established the correlation between the accumulated defects limits on parts and the Minkowski sum of finite sets of geometric constraints [START_REF] Fleming | Geometric relationships between toleranced features[END_REF]. A summary of these issues is detailed in [START_REF] Srinivasan | Role of Sweeps in Tolerancing Semantics[END_REF]. Giordano showed that the relative position between two parts in contact resulting from a number of potential contacts can be formalized by an intersection operation of a set of geometric constraints [START_REF] Giordano | Clearance Space and Deviation Space[END_REF]. More generally, the variational tolerancing approach characterises the relative position between two surfaces of two parts of any system by intersections and Minkowski sums of sets of geometric constraints. This sets of constraints result from ISO specifications expressed in parts and contact characteristics between two parts [START_REF] Teissandier | Operations on polytopes: application to tolerance analysis[END_REF]. Minkowski sum algorithms applied to the problem of tolerance analysis have been developed [35 36]. A variational method of tolerance analysis using the method based on operations on polytopes is proposed by [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF][START_REF] Pierre | Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine[END_REF]. This method was developed to take into account thermomechanical strains.

The aim of this paper is to compare these two methods on an example to show the similarities and limitations.

Definition of requirement

Mechanism description

The mechanism is composed of two rigid parts with two bores (Fig. 1). The shaft is represented by a single cylinder which is mounted with clearance in a bore of the cover and in a bore of the housing. The cover is assembled and screwed onto the housing, before introducing the shaft. It must therefore be ensured that the shaft can be fitted in the mechanism in any case.

Functional tolerancing

Functional tolerancing was performed with the Quick GPS software developed by B. Anselmetti [START_REF] Anselmetti | Quick GPS: A new CAT system for single-part tolerancing[END_REF] in the CATIA environment with Functional Tolerancing Annotation Workshop.

The junction is described by a positioning table of the cover on the housing (Fig. 2).

The housing is the base of this mechanism. The cover is setting-up on the housing by a primary plane and two bores which receive two pins blocked in the housing. There is clearance between the pins and the cover.

A simple matrix shows all the links in the part studied (Fig. 3). The hole of the cover has to be positioned relative to the junction with the housing. These data are sufficient for the QUICK GPS software that automatically generates annotations in the 3D CAD model (Fig. 4). The designer can adjust the tolerances.

Each part has a main datum reference frame AB and a location of the bore.

Functional requirement

This paper examines more specifically the assembling of the shaft in the sub system composed by housing and cover. The assembling is difficult when the shaft and the bores are at maximum material condition, when the cover and housing are shifted due to the clearance between the pins and the cover (Fig. 5).

The shift is maximal when the distances between the pins are the same on both parts and the pins and the holes in the cover are at minimum material condition with a maximum clearance of 0.04.

The problem is really three-dimensional, because the holes are not in the plane containing the two pins. But to simplify the Fig. 5, the pins are shown in the plane of bores.

Methodology

At maximum material condition, the shaft is considered a perfect cylinder of diameter 29.96. The envelop diameters of the bores are 29.98 mm. If the straightness in common zone of the two holes is r, then the free space for the shaft is 29.98 -r. The straightness has to be less than 0.02 in all cases.

Tolerancing and significant dimensions

Fig. 1 depicts the tolerancing corresponding to straightness requirement.

Analysis line method

Principle

The analysis lines method was developed in 2004, to calculate the resulting 3D with a statistical approach. The challenge is to globalize the influence dues to translation and angular deviations, whatever the tolerance values. Calculation equations show that the accumulation occurs naturally in certain points and in particular directions called analysis points and analysis directions.

The analysis line method is based on transfer relations that have been established for ten classical junctions (Primary plane / secondary plane / tertiary plane, Primary plane / secondary cylinder / tertiary, Primary plane / 2 secondary holes, Primary plane / n secondary holes; Primary cylinder / secondary plane / tertiary, straightness or flatness in common zone ...).

The linear relationship gives the influence of a given junction in the worst case, directly on the functional requirement. This requires a discretization and successive studies in a limited number of directions. The advantage of this method is that for each study, the analysis direction is known. It is very easy to define the position of the part in the worst case permitted by clearance in the junction.

Other junctions can often be modelled by 6 points of contact with a linear relationship. Some complex connections cannot be treated directly. The solution proposed by Robin Chavanne [START_REF] Chavanne | Contribution au tolérancement fonctionnel 3D des mécanismes complexes : Synthèse des spécifications et Analyse de tolerances[END_REF] is to use a solver to find the worst case situation and to find the 6 contact points to establish the linear relationship.

Transfer of straightness

The requirement studied thus reduces to straightness between two coaxial bores. The deviation from straightness r is the diameter of a cylinder containing both real axis of the two bores.

The problem is three-dimensional. The analysis line method proposes to study in 8 radial directions 1 f to 8 f . Fig. 8 can be applied to all cases of straightness in common zone of two coaxial cylinders (Fig. 8a).

A location and an orientation specifications are placed on each bore with respect to a common datum reference frame fixed on the housing (Fig. 8b). The relationship of transfer is therefore:

( ) ( ) ( ) ( ) , , ; max , , C H C C BC i H H C H BC E d C d C E L r E d B d B E L     + -     +   =       + -     +   fi fi fi fi (1)
These displacements ( )

, H d B fi , ( ) , H d C fi , ( ) , C d B -fi ; ( ) , C d C -fi
( ) ( ) 3 4 OB OC C t 2 t +L E , L C d C + ⋅ = fi (3)
The calculation is similar to the points H B and H C of the housing axis with the tolerance 1 t in location and 2 t in orientation.

( ) (

)

1 1 2 BC H t 2 , , t 2 t L E H H d C d B + ⋅ = = fi fi (4)
By symmetry, these relationships are independent of analysis directions fi.

Displacements of points B and C of the nominal cover axis must now be determined according to the housing.

Influence of clearance in junction housing/cover

The problem is to calculate the displacements of the nominal axis of cover in C B

and in C C

The cover is setting-up on the housing with a primary plane and two pins. The datum reference frame of the housing is AB. Datum plane A of the housing and datum plane A of the cover are supposedly confused.

The nominal axis of the holes being perpendicular to the primary plane, there is no angular deviations between both nominal axis of the housing and cover.

The shift of the nominal axis is due only to the mobility permitted by the clearance J between the housing and the cover. Fig. 10 shows both axis E and F of the housing holes, and both axis E' and F' of the cover holes in the cover.

The connection between these bores is provided by the pins blocked in the housing. Displacement is maximal when the distances between the holes are identical in housing and the cover and when the pins and the cover holes are at least material condition with a maximum clearance J.

If the analysis line ( ) ( )

,
arctan C OE OB ϕ = .
In this case, the displacement of B C is cos J ϕ ⋅ y . The maximal displacement in fi direction is:

( ) cos cos , C B J d θ ϕ ⋅ = fi (5)
The maximal clearance between pins and hole of cover is: ( )

max_hole min_pin 5 6 - J D D t t = = + (6)
, 2 C d B J = fi (7)
For f3 and f7 , cos 1

θ = ± : ( ) ( ) cos , 2 .56 2 1 C d B J J ϕ = = ⋅ ⋅ fi (8)
For f2 , f4 , f6 and f8 , 4 θ π = :

( ) ( )

co , 2 s 1.10 2 2 4 C d B J J ϕ = ⋅ ⋅ = ⋅ fi (9)

Full condition

The full condition giving the straightness in the worst case is the maximum value of 8 relationships in 8 directions fi .

(

,

C H C C BC i H H C H BC E d C d C E L r E d B d B E L     + -     +   =       + -     +   fi fi fi fi (10) With ( ) (
)

1 1 2 BC H t 2 , , t 2 t L E H H d C d B -= - + ⋅ = fi fi (11)
Maximal displacements of C B and C C depend on fi direction. The displacement is maximal for f3 and f7 directions in plane ( )

, y z : ( ) ( ) ( ) ( ) ( ) ( ) 3
C C d C d B ϕ ϕ = ⋅ + ⋅ + ⋅ + + + fi fi ( 12
)
The straightness is obtained in plane ( )

,
C C BC i H H BC E E L r E E L ϕ ϕ     + +     + ⋅     =       +     + + + ⋅ + + +  ⋅ ⋅    (13) with:
( )

arctan C OE OB ϕ =
These relationships are linear and simple to integrate into a system of equations for the synthesis of tolerances.

Method by operations on polytopes

Transfer of geometric deviations on parts

A real surface resulting from the manufacturing is modelled by a surface of substitution that has the same type as the nominal surface. The deviations between the surface of substitution and nominal surface are called geometric deviations. The surface of substitution i of the part j is noted , i j . Geometric deviations between a surface of substitution 1,1 and a surface of substitution 1,2 are formalized by a small displacement torsor 1,1/1,2 d     , where the vector 1,1/1,2 ρ is the rotation vector of 1,1 with respect to 1,2 while the vector B-1,1/1,2 ε is the translational vector of 1,1 with respect to 1,2 at point B:

B 1,1/1,2 1,1/1,2 B-1,1/1,2 d     =       ρ ε (14)
According to the specification of the location of axis 1,1 of the bore of the housing 1 (Fig. 6), the axis of the cylindrical surface of substitution 1,1 is within a tolerance zone ZT . ZT is a cylinder of diameter 1 t ∅ orthogonal to the plane A and centred on nominal axis relating to AB datum reference frame. To ensure that the axis 1,1 of the surface is located in the zone of tolerance, it is necessary to write, to both points C and D, the following relationship where i θ n is a unitary vector orthogonal to the axis z and i θ is the discretization angle around the axis z :

( )

1 1 -1,1/ 1 1 -1,1/ 2 2 2 2 cos sin with
, 0 and ,

C AB i D AB i i i i i t t t t i i n i n n θ θ θ θ θ π θ   -≤ ⋅ ≤         -≤ ⋅ ≤     = ⋅ + ⋅    = ≤ < ∈   ε n ε n n x y ℕ (15)
The relation ( 15) is expressed according to the rotation vector and the translation vector of the deviation of the point P middle of the bounded by line segment B and C (Fig. 5):

( ) ( ) ( )

1 1 P-1,1/ 1,1/ 1 1 P-1,1/ 1,
AB AB i AB AB i i i i i t t t t i i n i n n θ θ θ θ θ π θ   -≤ + ∧ ⋅ ≤         -≤ + ∧ ⋅ ≤     = ⋅ + ⋅    = ≤ < ∈   ε CP ρ n ε CD ρ n n x y ℕ (16)
The relations ( 16) define a bounded intersection of finite number of closed halfspaces whose boundaries are hyperplanes of 4 ℝ [34] intersection. In general, it can formalize the h-representation of a polytope called geometric polytope, noted g 1,1/AB D .

, y P x ρ ε , .
The orthogonality of the housing axis 1,1 with respect to the datum A can be characterized by a geometric polytope g 1,1/ A D . The axis of the cylindrical surface of substitution 1,1 is within a tolerance zone TZ . TZ is a cylinder of diameter 2 t ∅ orthogonal to the plane A . To ensure that the axis of the surface 1,1 , is within the orientation tolerance zone TZ , the orthogonal distance to the axis y between the points C and D must be less than the dimension of the tolerance zone:

( ) ( ) 2 -1,1/ -1,1/ 2 cos sin with
, 0 and ,

C A D A i i i i i t t i i n i n n θ θ θ θ π θ -≤ - ⋅ ≤ = ⋅ + ⋅    = ≤ < ∈   ε ε n n x y ℕ (17)
The relation ( 17) is expressed according to the rotation vector and the translation vector of the deviation of the point P middle of the bounded by line segment B and C:

( ) ( ) 2 -1,1/ -1,1/ 2 cos sin with
, 0 and ,

C A D A i i i i i t t i i n i n n θ θ θ θ π θ -≤ - ⋅ ≤ = ⋅ + ⋅    = ≤ < ∈   ε ε n n x y ℕ (18)
Relations ( 18) formalize the geometric polytope, noted g 1,1/ A D , corresponding to the location of 1,1 with respect to the reference A. This is a 2-polytope whose graphical representation is given in Fig. 12 in ( )

x y ρ ρ , .
Both specifications expressed on the axis 1,1 of the housing 1 must be respected simultaneously. The polytope characterizing the position of the axis 1,1 with respect to the datum reference frame AB is the polytope resulting from the intersection between the polytopes

Transfer of contact deviations between the parts

Previously geometric deviations of parts were formalised by polytopes. In the following, deviations due to mobility and clearance in the joints will be formalised by polytopes. The joint cover/housing is composed of three elementary joints: a plane pair contact and two ball and cylinder pair contacts. To model the full joint, each of three joints is modelled by a contact polytope.

The planar pair contact between the plane 1,2 cover 2 and the plane 2,2 the housing 2 is without clearance (Fig. 6). The relation [START_REF] Pierre | Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine[END_REF] characterised the contact without clearance on the whole contour of the contact surface:

P-1,2/2,2 1,2/2,2 1,2/2,2 0 0 0  ⋅ =    ⋅ =     ⋅ =   ε z ρ x ρ y (19)
Relations [START_REF] Clozel | MECAmaster: a tool for assembly simulation from early design, industrial approach[END_REF] characterize the limits of displacements between surfaces 1,3 and 2,3 and between surfaces 1,4 and 2,4 of the both ball and cylinder pair contacts (Fig. 6):

( )

E-1,3/2,3 F-1,4/2,4 2 2 2 2
cos sin with , 0 and ,

i i i i i i J J J J i i n i n n θ θ θ θ θ π θ   -≤ ⋅ ≤         -≤ ⋅ ≤     = ⋅ + ⋅    = ≤ < ∈   ε n ε n n x y ℕ (20)
The maximum clearance corresponds to cases where the diameters of the shafts are smaller and diameters of holes are larger:

max_hole min_pin 5 6 - J D D t t = = + (21)
Relation ( 20) is expressed according to the rotation vector and the translation vector of the deviation of the point P:

cos sin with , 0 and ,

i i i i i i J J J J i i n i n n θ θ θ θ θ π θ   -≤ ∧ ⋅ ≤         -≤ ∧ ⋅ ≤     = ⋅ + ⋅    = ≤ < ∈   EP ρ n FP ρ n n x y ℕ (22)
The polytope of contact characterising the relative position between the housing 1 and the cover 2 results of the intersection between the polytopes defined in [START_REF] Pierre | Variational tolerancing analysis taking thermomechanical strains into account: Application to a high pressure turbine[END_REF] and [START_REF] Dcs | Variation Analysis and Tolerance Analysis[END_REF]. This polytope called C / AB CD D is a 3-polytope which a projection in ( )

P, P, ,
x y ε ε is given in Fig. 15.

Full condition

Geometrical deviations in parts and in contacts were respectively determined in sections 4.2 and 4.3. The relative position between the bore of the housing and the cover will be characterized by the calculated polytope defined by the following Minkowski sum:

g c c 1,1/2,1 1,1/ / /2,1 AB AB CD CD = D D + D + D (23)
The functional polytope f 1,1/2,1 D characterizing the straightness between the bore axis of the cover and the housing is defined by the following relation:

f f f 1,1/2,1 1,1/ZT /2,1 ZT = D D + D (24)
The Fig. 16 illustrates the result of the sum of Minkowski [START_REF] Loof | An efficient solution to the discrete Least Cost tolerance allocation problem with general loss functions[END_REF] D is shown. To characterize the straightness, the polytope must be included in calculating the functional polytope.

If the calculated polytope is included in the functional polytope, the straightness of the two bores is inferior to the value defined in the polytope functional. To determinate the value of straightness of the two bores is necessary to minimize the size of the functional polytope. When the size of the functional polytope is minimal, each vertex of the calculated polytope must be included into each half-spaces of the functional polytope. The straightness is defined by these inequalities:

( ) ( ) ( ) ( )
( ) C C BC C H C H BC H H BC C H H C BC E k a E L E E r k b E E L E r k c E L E E r k d r E E L   ≥ + + ⋅   +     + ≥ + + ⋅   + +     ≥ + + ⋅   +  + ⋅ + + ⋅ + + ⋅ + + ⋅ +    + ≥ + + ⋅   + +   (25)
In these four inequalities ( 25 ( ) ( )

E k E L E E L r r k   ≥ + + ⋅   +     ≥ + ⋅ + + + ⋅ + + +   ⋅   (26)
With this method, the straightness value defined into the relation ( 26) is the same as the previous method defined into relation (13).

Conclusion

Both methods integrate the influence of the geometrical defects of location and orientation, and the mobility allowed by the clearance in the joints. But they consider directly the situation in the worst case defined by the minimal material condition in the joints and the maximal material condition in the bearings.

For primary planar joint, the analysis line method considers that the datum reference of one part remains in the tolerance zone of the support part. The polytope method neglects the flatness of planes in contact for superimposing the datum reference and the support surface. These two ways of expressing the assumptions lead to the same behaviour.

This study shows that geometric models are identical, and that the analysis lines means studying the polytope in particular directions. The polytope is more complex, but gives a global view of all degrees of freedom. The method of analysis line is more direct, but requires considering each requirement in several directions by discretization of the ending surface.

Analysis line method directly gives the result as a linear formula of tolerances of influential parts, which subsequently allows to globally optimize the whole system of inequalities to maximize tolerances.

Both methods require a lot of rigor and expertise. A specific tool in CAD system is necessary. The setting-up of parts and functional requirements are described by the designer. The tolerance analysis application collects geometric data and the influential annotations. The calculations can be fully automated to display the result in the form of the numerical value of the result to be compared with the limit value of the requirement.

NOMENCLATURE

Fig. 8c shows

  Fig. 8c shows that in each plane in fi direction, straightness is proportional to the sum of the maximum displacement ( ) , H d C fi of point H C of the housing axis in fi direction and the maximal displacement ( ) , C d C -fi of point C C of the cover axis

CdB

  fi cut segment EF , the maximum displacement in the direction fiis equal to 2 J . If the analysis line ( ) , C d B fi does not cut the segment EF , the displacement of C B is maximal when E move in E' and when F move in F' in ϕ direction. The displacement of the point C B depends of the angle ϕ. The derivation shows that the displacement is maximum, irrespective of the angle θ value, when the points C B , E and E′ are aligned, either for

DFig. 11 .

 11 Fig. 11. The Fig. 11a and the Fig. 11b represent two projections of 4-polytope respectively in ( ) , x P y ρ ε ,

  a 4-polytope in which two graphical representations are shown in Fig. 13. Fig. 13a and Fig. 13b represent the projections of the 4-polytope in ( ) manner to the location and orthogonality specifications of the axis 1,1 of the housing 1, the geometric polytope characterizing the position of the axis 2,1 with respect to the datum reference frame CD result from the intersection of the 4-polytope in which two graphical representations are shown in Fig. 14. Fig. 14a and Fig. 14b represent the projections of 4-polytope in

  ), r is greater into the relation (a) than into the relation (b) and r is greater into the relation (c) than into the relation (d).

Figure 1 DFig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .

 112345678910 Figure Captions List

Fig. 15 . 1 VFig. 16 . 1 D in the functional polytope f 1 , 1 / 2 , 1 D

 1511611121 Fig. 15. Polytope of contact C / AB CD D


Stéphane Le Corre

Dark Matter And Planes Of Corotating Satellite Galaxies

Keywords: galaxies, dwarf -galaxies, evolution -galaxies, formation -Local Group -galaxies, star formation -cosmology, dark matter

We propose a new model to explain the reported excess of corotating satellite pairs located near diametric opposition with respect to their host (i.e. at opening angles of α < 8 • ) detected in the Sloan Digital Sky Survey (SDSS) data. This model also allow explaining the precipitous drop in the corotating fraction over 8 • < α < 28 • and the no overabundance of corotating satellite pairs for greater angles. Furthermore, this model predicts two other unexpected features that could be tested now in the SDSS data, the correlation between the distance of hosts' galaxies and the angle of the planes of corotating satellites and the spatial extent of this correlation. This new model has been proposed to explain dark matter without modifying the gravitation's laws, without introducing new matter and in the frame of general relativity. The verification of these two new predictions would be a very strong validation of this model of "dark matter" and would put very strong constraints on the other models.

Introduction

A recent published paper (Phillips et al. 2015) tested the hypothesis that a large fraction of satellites corotate in coherent planes. This hypothesis comes from the planar distribution exhibited by M31 and from recent observational data that has been interpreted in some papers as a possible ubiquitous corotating planes of satellites. In this recent paper, the authors analyze several models and compare them with the observational data. It leads to the following results.

Data analysis

In term of observational data, they confirm the previously reported excess of corotating satellite pairs located near diametric opposition with respect to their host (i.e. at opening angles of α < 8 • ). They also noticed a precipitous drop in the corotating fraction over 8 • < α < 28 • and no overabundance of corotating satellite pairs for greater angles.

In term of models, they demonstrate that no viable alternative models can explain this distribution. The best fit is obtained with dumbbell model . But this model is not physically acceptable and it can't explain the M31 distribution. The more physical model seems to be an isotropic velocity distribution. But this model can't explain the precipitous drop in the corotating fraction over 8 • < α < 28 • . Without other better models, they conclude that the corotation signal for α < 8 • is likely the product of random chance.

A new model which predicts these unexpected observational data

Caution must be taken with this conclusion. This conclusion is one possible "explanation" (by interpreting the tendency of corotating satellite pairs for small angle α as a random noise), but one cannot exclude the possibility of a new model that could explain this distribution. One can also note that if the distribution is the product of random chance, it can't explain the distribution for M31. Furthermore, and it is certainly the main weak point of this explanation, it is very astonishing (and very unlikely) that M31 and these new observational data provide by chance the same tendency of corotating satellite pairs for the same small angle α. One can also cite the authors who write "repeatedly resampling 400 satellite pairs placed randomly in phase space (as in our 'isotropic" model) will frequently produce satellite samples with excess corotating fractions at random opening angles". The important words are "at random opening angles", to compare with the observation data that produce by two times (M31 and SDSS data) the same small angle around 0 • . So another conclusion (also in agreement with their study) cannot be rejected, to find a new model that can explain M31 and the current observational data.

In fact, a recent paper published in ArXiv (Le Corre 2015) had predicted this distribution. This paper emits an assumption to explain the dark matter without modifying the gravitation laws and without new matter. This model implies very strong constraints. These constraints allow making several sine qua none predictions. In particular the two following ones (with -→ k 0 a gravitomagnetic vector approximatively uniform on large areas):

The movement of satellite dwarf galaxies should be in a plane perpendicular to a vector

- → k 0
Statistically, inside a cluster, the satellite dwarf galaxies' planes should be close to the supergalactic plane These two predictions translated with the words of the paper (Phillips et al. 2015) means that the movement of satellite galaxies should be in planes with small angle α (because close to their supergalactic plane). These planes should lead to a precipitous drop in the corotating fraction for slightly greater angle (because the planes are perpendicular to one vector -→ k 0 ). And finally one should have no overabundance of corotating satellite pairs for greater angles (because the vector -→ k 0 is uniform on large areas). These predictions match very well with the three previous observational data characteristics.

New unexpected predictions of the new model on observational data

This explanation of the dark matter also implies two others predictions that could be tested with these observational data:

Statistically, smaller is the distance between two galaxies; smaller is the difference of orientation of their satellite dwarf galaxies' planes.

In the terms of previous paper, it means that in the sample of corotating satellite pairs (i.e. for small angles), more the distance between two hosts galaxies is small, more the difference of the angle between the corotating satellite galaxies' plane of these two hosts should be small. In other words, the planes of the corotating satellites pairs between two hosts tend to be more and more parallel with the proximity of the two hosts but it doesn't mean that the angle α becomes smaller (Fig. 1). The other prediction can be translated in terms of a spatial extension of this correlation:

Only the clusters [. . . ] seem to be able to generate the expected vector -→ k 0 on large areas.

It means that this unexpected correlation on the angle of the planes of corotating satellites of close hosts' galaxies should be detectable on hosts' distances of the order of magnitude of the cluster's size (i.e. at least until around 5Mpc).

Conclusion

In this work, we propose a new model (Le Corre 2015) which explains the main features of the observational data about the planes of corotating satellite galaxies (Phillips et al. 2015).

1. An excess of corotating satellite pairs located near diametric opposition with respect to their host (i.e. at opening angles of α < 8 • ).

2. A precipitous drop in the corotating fraction over 8 • < α < 28 • .

3. No overabundance of corotating satellite pairs for greater angles.

This new model predicts two others features:

1. A correlation between the distance of hosts' galaxies and the angle of their planes of corotating satellites.

2. This correlation should be maintain on very large areas (at least until 5Mpc).

One can recall that the main goal of this new model is to explain the dark matter without modifying the gravitation's laws and without new matter. Furthermore, with another unexpected assumption (soon tested at CERN), this new model could also explain the dark energy. The verification of these two new predictions would be a very strong validation of this model of "dark matter" and would put very strong constraints on the other models. This could be a major flaw in the ΛCDM model.

Figure 1 .

 1 Figure 1. Smaller is the distance between the H1 and H2 hosts, smaller should be the difference of angles between the corotating pairs | β2 -β1 | of each host.