Imen Bizid
email: imen.bizid@univ-lr.fr

Nibal Nayef
email: nibal.nayef@univ-lr.fr

Patrice Boursier
email: patrice@iumw.edu.my

Sami Faiz
email: sami.faiz@insat.rnu.tn

Antoine Doucet
email: antoine.doucet@univ-lr.fr

Identification of Microblogs Prominent Users during Events by Learning Temporal Sequences of Features

Keywords: H.3.3 [Information Storage and Retrieval]: Information Search and Retrieval-selection process, retrieval models; H.3.1 [Information Systems]: User/Machine Systemshuman factors, human information processing Prominent microblogs users identification, Learning temporal user behavior, MoG-HMM classification

During specific real-world events, some users of microblogging platforms could provide exclusive information about those events. The identification of such prominent users depends on several factors such as the freshness and the relevance of their shared information. This work proposes a probabilistic model for the identification of prominent users in microblogs during specific events. The model is based on learning and classifying user behavior over time using Mixture of Gaussians Hidden Markov Models. A user is characterized by a temporal sequence of feature vectors describing his activities. The features computed at each time-stamp are designed to reflect both the on-and off-topic activities of users. To validate the efficacy of our proposed model, we have conducted experiments on data collected from Twitter during the Herault floods that have occurred in France. The achieved results show that learning the time-series of users' actions is better than learning just those actions without temporal information.

INTRODUCTION

Microblogging platforms represent a rich source of information indispensable to manage specific events. These platforms are seen as the perfect ground to mine relevant and exclusive information during such events. Although these microblogs such as Twitter provide many characterizing features about their content -such as the number of retweets or favorites etc. -, such features generally refer to popular users content rather than reflecting the real importance of that content. Therefore, in the context of a specific eventsuch as a disaster -, it is more logical to associate the relevance and quality of messages with the user's prominence in that specific event [START_REF] Wagner | It's not in their tweets: Modeling topical expertise of twitter users[END_REF][START_REF] Liao | Understanding experts' and novices' expertise judgment of twitter users[END_REF][START_REF] Zhou | Topic-sensitive probabilistic model for expert finding in question answer communities[END_REF].

We define prominent user identification in microblogging platforms as identifying key users who provide fresh and relevant information during specific events regardless of their popularity in the platform. To the best of our knowledge, this task has never been studied in this context. However, the related problems of finding influencers have been widely explored in the state-of-the-art [START_REF] Romero | Influence and passivity in social media[END_REF][START_REF] Chen | Efficient influence maximization in social networks[END_REF]. Other works have dealt with the identification of topical authorities and domain experts [START_REF] Pal | Identifying topical authorities in microblogs[END_REF][START_REF] Xianlei | Finding domain experts in microblogs[END_REF][START_REF] Ghosh | Cognos: Crowdsourcing search for topic experts in microblogs[END_REF]. In these related contexts, key users were identified by analyzing either the network structure using time consuming centrality algorithms such as PageRank [START_REF] Kwak | What is twitter, a social network or a news media?[END_REF][START_REF] Weng | Twitterrank: Finding topic-sensitive influential twitterers[END_REF][START_REF] Zhang | Identifying influential users of micro-blogging services: A dynamic action-based network approach[END_REF] and HITS [START_REF] Agichtein | Finding high-quality content in social media[END_REF], or using the historic topical users activities independently of their temporal characteristics and off-topic activity [START_REF] Romero | Influence and passivity in social media[END_REF][START_REF] Xianlei | Finding domain experts in microblogs[END_REF].

These general purpose existing approaches would give a misleading image of users behavior in real scenarios. Users are mainly evaluated according to their on-topic activity while neglecting their off-topic one. Such practice would promote official media channels toggling between several topics and which are not necessarily sharing fresh information. Moreover, users are typically represented in terms of the quantity of their produced and forwarded information independently of the temporal distribution of this information. This would give the same description for users interacting at an early stage of the event by sharing fresh information and other users posting the same information at its end.

This work is thus designed to alleviate these shortcomings. More specifically, we present the following contributions: (1) a novel representation of microblogs user behavior as a temporal sequence of features that characterize both the onand off-topic user activities, (2) a probabilistic model for the identification of prominent users in microblogs during specific events. This model is based on learning and classifying the mentioned representation using a Mixture of Gaussians Hidden Markov Model (MoG-HMM).

The rest of this paper is organized as follows. Section 2 presents problem formulation. Section 3 discusses our proposed approach of modeling user behavior. Section 4 describes the used HMM learning model for classifying prominent users. Experimental evaluation is presented in Section 5. Finally, we conclude with directions for future work in Section 6.

PROBLEM FORMULATION

We formulate the problem of prominent users identification in the context of specific events as two main subproblems. The first is representing users so that to reflect their temporal behavior during an event. Each user u has to be represented by a temporal sequence Vu = (V 1 u , V 2 u , ..., V m u ) where m is the length of the sequence describing the user behavior over m time stamps. V (i) u represents user description at each time interval i, and can be any set of features characterizing the user. The second sub-problem is learning to classify users' temporal sequences of features as either belonging to prominent c1 or non-prominent c2 user classes. Thus we need to learn two probabilistic models Hc 1 and Hc 2 by training the continuous temporal sequences describing each class of users. Given these models, we need to estimate the likelihood L(V |Hc1) and L(V |Hc2) of each user sequence.

MODELING USERS AS TEMPORAL SE-QUENCES OF ACTIVITY FEATURES

In order to model users consistently with their realistic behavior in microblogs during events, we propose a temporal sequence representation approach. The behavior of users is represented according to their observed on-and off-topic activities at different temporal stages during an event.

Temporal Sequence Representation

The time-line of an event is divided into equispaced intervals at m time stamps. During each interval, users activities are characterized by a set of features rather than a single feature as there are several types of activities in microblogs. The user activity is represented by the feature vector V t i u calculated based on t1, t2, t3, ..., tm time stamps. Those features -discussed in the next subsection -describe the user behavior regarding an event (on-topic activity) and also regarding other topics (off-topic activity) during each time interval. Figure 1 illustrates -in its upper part -such a user representation.

Using this user behavior modeling approach, we provide a more enriched representation of microblogs users by considering both on-and off-topic user activeness levels over time. Such representation is capable of characterizing similarities and regularities of users behaviors.

User Activity Features

We define a set of new features inspired by the features proposed by Pal et al. [START_REF] Pal | Identifying topical authorities in microblogs[END_REF]. For each user u and each time stamp tm, we compute the following features by taking into account both the on-and off-topic user activities: Topical Attachment: indicates the involvement rate of the user regarding the analyzed event by referring to the number of his original on-topic tweets (T 1on) and the number of his on-topic shared links (T 2on) adjusted by the offtopic metrics (T 1 of f ) and (T 2 of f ).

F 1 (t) u = (T 1 (t) on + T 2 (t) on ) -(T 1 (t) of f + T 2 (t) of f ) (1)
Topical Influence: estimates the value (or worthiness) of the user's original tweets regarding the event according to the number of favorites (T 3on) and the number of retweets (R2on) of his produced tweets.

F 2 (t) u = T 1 (t) on × log(T 3 (t) on + R2 (t) on + 1) (2)
Retweeting Rate: measures the impact of the shared eventrelated-tweets on the user retweeting activity. This feature adjusts the number of the user's retweets (R1on) by the number of unique users that he has retweeted (R4on).

F 3 (t) u = R1 (t) on * log(R4 (t) on + 1) (3)
Retweeted Rate: calculates the impact of the original tweets produced by the user on the other users. This feature adjusts the number of users who have retweeted the user's on-topic tweets (R3on) and the total number of retweets (R2on) by the corresponding off-metrics (R2 of f ) and (R3 of f ).

F 4 (t) u = R2 (t) on * log(R3 (t) on + 1) -R2 (t) of f * log(R3 (t)
of f + 1) (4) Incoming Mention Rate: measures the diversity of mentions related to the user. This feature adjusts the number of received on-topic mentions (M 1on) by the number of unique users mentioning the user (M 2on).

F 5 (t) u = M 1 (t) on * log(M 2 (t) on + 1) (5)
Outcoming Mention Rate: represents the mentioning activity of the current user by computing the number of his on-topic produced mentions (M 3on) adjusted by the number of unique mentioned users (M 4on).

F 6 (t) u = M 3 (t) on * log(M 4 (t) on + 1) (6)
Centrality Degree: promotes users having more on-topic followers and followees than off-topic ones. This feature is measured by adjusting the number of on-topic followers (G1on) and followees (G2on) of each user with the number of his off-topic relations (G1 of f ) and (G2 of f ).

F 7 (t) u = log( G1 (t) on + 1 G1 (t) of f + 2 ) -log( G2 (t) on + 1 G2 (t) of f + 2 ) (7)
Once these features are computed, each user u can be represented by the following feature vector at each time stamp tm.

V t i u = (F 1 (t) u , F 2 (t) u , F 3 (t) u , F 4 (t) u , F 5 (t) u , F 6 (t) u , F 7 (t) u ) (8)
The set of concatenated feature vectors computed at all the time stamps represent the temporal sequence of the user behavior.

LEARNING TO CLASSIFY USER TEM-PORAL SEQUENCES

In order to classify the time-series of feature vectors V describing each microblogs user, we train two models for prominent and non-prominent users classification using MoG HMMs. There are various types of continuous HMM: leftright, parallel left-right and ergodic. To learn our MoG HMMs, we use the ergodic model as the user activity level state at a period of time ti can change to every other state at the period of time ti+1 through a single transition. Figure 1 shows a 3-state ergodic model describing how the sequence of feature vectors representing a given user can be transformed into a sequence of discrete states.

To learn the parameters for our MoG-HMM ergodic models, we use the Baum-Welch algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF]. This algorithm is based on the EM algorithm to search for the maximum probability of the HMM models parameters that better fit The transformation of these feature vectors into discrete states is processed by the construction of a continuous observation probability density function (PDF) matrix B. This matrix is represented as a mixture of Gaussians in order to associate the sequence of a user's feature vectors into the different finite states using equation 10.

bi(V

t ) = M k=1 c ik N [V t , µ ik , W ik ] ( 10
)
where c ik is the mixture weight, N is the normal density, µ ik is the mean vector and W ik is the covariance matrix for the k th mixture component in state Si.

Once the models parameters Hc1 and Hc2 are set through training, we can compute the probability of any microblogs user to belong to each class P (Vu|Hc1) and P (Vu|Hc2) given the two learned models. These probabilities are obtained using the forward-backward algorithm [START_REF] Baum | An inequality with applications to statistical estimation for probabilistic functions of markov processes and to a model for ecology[END_REF]. If the model Hc1 gives a higher probability to a represented user compared to P (Vu|Hc2), then this user is classified as prominent.

PERFORMANCE EVALUATION 5.1 Dataset

To conduct experimental performance evaluation on real data, we have tracked the Twitter users who have shared at least one on-topic information about the floods that have occurred from 29 th to 30 th September 2014 in the Herault area, situated in the south of France using ou multi-agent system MASIR [START_REF] Bizid | Masir : A multi-agent system for real-time information retrieval from microblogs during unexpected events[END_REF]. 3332 users have been tracked during the two days of the disaster.

To create the ground-truth, we conducted a subjective user study in order to label each user in the dataset with c1 and c2 indicating respectively whether the user is prominent or not. The tracked users were evaluated according to the relevance and exclusivity of their tweets during the disaster. According to this study, 90 users have been classified as c1, and 3242 users have been classified as c2.

Evaluation Set-up and Metrics

For experimental set-up, we randomly sampled 60% of both prominent and non prominent labeled users datasets as training data for building the Hc1 and Hc2 classifiers, and the remaining 40% as test data. Features characterizing user behavior were extracted sequentially at each time interval of 90 minutes from the beginning of the event to its end. Thus each user is represented by a sequence of 32 feature vectors. We have also extracted features using different interval lengths.

Following the standard evaluation criteria used in the context of prominent users identification, we use precision, recall and F1-score measures to evaluate the performance of our approach. In order to choose the coherent parameters for the representation of microblogs users behavior through Hc1 and Hc2 models, we have tested different values of "number of states" Ns (from 1 to 6) and "number of multivariate Gaussian" NG (from 1 to 5) with the training dataset. The experimental results are shown in Table 1. The best results are obtained when Ns = 3 states and NG = 1, yielding F1-score value to 64%.

Experimental Results

Importance of Time-series Representation

To demonstrate the effectiveness of our temporal sequence representation approach for the identification of prominent users, we test the performance of our model by decreasing the length of the feature vectors sequence m( from 32 to 2) (e.g. m=2 users activities features are recorded at each 720 minutes). In other words, users behavior are represented during longer periods of time. Figure 2 shows how the sequence length variation affects the performance of our model. According to these results, we find that larger sequence length characterizing detailed users activities over time works significantly better than smaller ones. Thus, user behavior is better characterized with more time stamps.

Comparison with Different Models

We compared our HMM temporal sequence classification model with two baseline models: Baseline 1: refers to an SVM model learned using our Figure 3 shows the results of comparing the three different models, where our proposed model significantly outperforms the models which are trained on users activities in the whole event duration (i.e independently of the activity distribution over time). We also note that our features are more effective in the context of prominent users identification in specific events than Pal et al. [START_REF] Pal | Identifying topical authorities in microblogs[END_REF] features which are based on analyzing only the on-topic user activity.

CONCLUSION AND FUTURE WORK

This paper has presented a novel microblogs users representation according to their behavior over time in specific events. This representation has been used for building and training a MoG-HMM model to identify prominent users. In our model, users are characterized by a temporal sequence composed of feature vectors recorded in different periods of time during the event. These features characterize the ontopic and off-topic users activities at each time interval. Our experiments show that a longer length of the sequence characterizing the user behavior over time, produces a better identification model. We have additionally compared the performance of our model to traditional machine learning SVM models using our features and state-of-the-art features.

The achieved results show the importance of learning time sequences of users activities as compared to learning those activities independently of their temporal characteristics.

For future work, we aim to predict prominent users at different event phases. We would like also to analyze the correlation between temporal activities of prominent users and the time of occurrence of the trending sub-events.

Figure 1 :

 1 Figure 1: A time-series representation of user activities during an event using a 3-state ergodic HMM

Figure 2 :

 2 Figure 2: Time-series sequence length effect on the model performance

Figure 3 :

 3 Figure 3: The different ergodic MoG-HMM models appropriated to each phase

Table 1 :

 1 Prominent users identification performance for different Ns and NG in terms of F1-score.

	Ns/NG	1	2	3	4	5
	1	0.5201 0.5615 0.6035 0.5643 0.6039
	2	0.5500 0.5755	0.01	0.01	0.01
	3	0.6455	0.01	0.01	0.1337 0.2026
	4	0.5855	0.01	0.1100	0.01	0.1020
	5	0.6156	0.01	0.01	0.01	0.01
	6	0.01	0.4700	0.01	0.01	0.01


Imen Bizid

Sami Faiz

Patrice Boursier

Jawahir Che

Mustapha Yusuf

Integration of Heterogeneous Spatial Databases for Disaster Management

Keywords: Heterogeneous Spatial Databases, Data Integration, Disaster Management, Ontologies

The response phase in a disaster case is often considered to be the most critical in terms of saving lives and dealing with irreversible damage. The timely provision of geospatial information is crucial in the decision-making process. Thus, there is a need for the integration of heterogeneous spatial databases which are inherently distributed and created under different projects by various organizations. The integration of all relevant data for timely decision making is a challenging task due to syntactic, schematic and semantic heterogeneity. This paper aims to propose a framework for the integration of heterogeneous spatial databases using novel approaches, such as web services and ontologies. We focus on providing solutions for the three levels of heterogeneity, in order to be able to interrogate the content of the different databases conveniently. Based on the proposed framework, we implemented a use case using heterogeneous data belonging to La Rochelle city in France.

Introduction

Disaster management is a scope of great relevance as it aims to reduce the negative impact and consequences of a disaster and to provide a timely and clear rescue plan. Proposing powerful tools to generate efficient strategies for the decision makers gives rise to the delivery of disaster relief in the right moment. Two types of disasters are of interest natural and man-made. Natural disasters often strike without warning, such as earthquakes, tsunamis and tornados. Man-made disasters include every action due to technological hazards, sociological hazards or transportation hazards among others. Despite the difference between these two categories in specifics and complexity, both instances can cause irreversible damage if the rescue plan is not executed at time. In order to effectively manage such disasters, it is important to coordinate between the police, Red Cross, and other first responders to work together and extract the needed information to maintain a rescue plan. However, the data stored in these distinct heterogeneous sources located in different departments, are maintained under heterogeneous formats, structures and are not semantically rich enough to express the emergency responders ' needs. Experience suggests that the bottlenecks of emergency responses are, in most cases, the difficulties in making intelligent search and integration of heterogeneous geospatial information stocked in different departments [START_REF] Zhang | Automatic search of geospatial features for disaster and emergency management[END_REF]. This paper has been working on the heterogeneity levels that can block emergency responders to automate the search of geospatial data stored in different spatial databases. Although there were many works that all address important pieces of the major challenge to automatically search geospatial information from heterogeneous spatial databases, many issues still remain to be solved to fully realize this goal. This paper is structured as follows: In section 2, we describe the three heterogeneity levels. In section 3, we present briefly the recent progress made on the integration of heterogeneous spatial databases, Section 4 proposes a new solution represented in a framework dealing with the three heterogeneity levels. In section 5, we describe the different interoperability modules designed in our framework. Afterward, we analyze the evaluation results related to the framework in Section 6. Finally, we present our conclusions in section 7.

Spatial Databases Heterogeneity Levels

Each emergency response community has developed its own database using its personalized infrastructure and its own requirements. Hence, in order to maintain a rescue plan, emergency responders have to interrogate at the same time the overall data stored in heterogeneous databases through complex queries. This gives a rise to the problem of access to the required information in a critical time. As mentioned in [START_REF] Bishr | Overcoming the semantic and other barriers to gis interoperability[END_REF][START_REF] Stuckenschmidt | Sharing on the Semantic Web[END_REF] that illustrate databases heterogeneity through three levels, i.e., semantic schematic and syntactic.

Syntactic heterogeneity refers to data format differences and problems related to the implementation of databases in different storage paradigms: relational or object orientated, and the geometric representation of objects: raster or vector. Many organizations contributing in a disaster case rely on a range of storage solutions. However, some organizations are motivated to make their data accessible through Open Geospatial Consortium (OGC) formats that provide syntactically unified formats and services such as Geography Markup Language (GML) [START_REF] Cox | Opengis geography markup language (gml) implementation specification[END_REF] and Web Feature Service (WFS) [START_REF] Zhang | The roles of web feature and web map services in real time geospatial data sharing for time-critical applications[END_REF].

Schematic heterogeneity regards the differences in data models between emergency response communities. Each spatial database schema reflects each community 's abstracted view of data. Therefore, different hierarchical and classification structures are used by each community to refer to identical or similar objects. Schematic heterogeneity can be classified into three main schemes:

• Entity versus Entity, i.e., the same entity can exist in two different databases with different name or structure;

• Attribute versus Attribute, i.e., an attribute related to a class in one database can exist in another class related to another database;

• Entity versus Attribute, i.e., a class in one database can be designed as an attribute in another database.

Semantic heterogeneity can be divided into two types: the naming heterogeneity and the cognitive heterogeneity. The naming heterogeneity arises when the same data objects are named in a different manner or when different semantic data objects are named identically. The cognitive heterogeneity refers to the different domains assigned to each organization. Each community has different cognitive views of a word, which means that they describe similar real word objects from different perspectives.

It is against these complex backdrops that our paper applies. Hence, we offer a comprehensive approach providing an access to the whole information consistently and quickly.

Related Works

The integration of heterogeneous databases constitutes an old challenge for computer information systems communities; it has been extensively studied in the literature. Therefore, various approaches have been proposed that we divide into two broad categories; traditional approaches and novel approaches. For the first category, two main approaches can be cited: the virtualized approaches that include the Federated Databases approach [START_REF] Heimbigner | A federated architecture for information management[END_REF] and the Mediator / Wrapper approach [START_REF] Leclercq | Amun: An object oriented model for cooperative spatial information systems[END_REF], and the Data Warehouse (or materialized ) approach [START_REF] Inmon | Building the Data Warehouse[END_REF][START_REF] Kimball | The Data Warehouse Toolkit: Practical Techniques for Building Dimensional Data Warehouses[END_REF]. Novel approaches deal with interoperability issues, and include Web Services [START_REF] Zhang | Study on gml spatial interoperability based on web service[END_REF], Ontologies [START_REF] Xu | Ontologies for disaster management response geomatics solutions for disaster management[END_REF][START_REF] Mustapha | Analyzing the ontology approaches and the formation of open ontology model: A step for organisational ontology employment[END_REF] and Semantic Web approaches [START_REF] Zhang | Geospatial data sharing based on geospatial semantic web technologies[END_REF]. Due to the large volumes of spatial data, the complexity of handling and analysing data and its importance in various domains, different researchers working on geographical information systems all over the world have focused on the adaption of the existing integration solutions to spatial databases. Thus, by using different techniques that integrate the spatial component, solutions have been proposed, such as Spatial Data Warehouse, Federated Spatial Databases, Geo-web services, Spatial Data Infrastructure (SDI) [14] among others.

Table 1. The interoperability position of the different integration approaches.

Approaches Syntactic Schematic Semantic Spatial datawarehouses X Federated Spatial Databases X X Mediation systems X X Spatial Data Infrastructure X Geo-Web Services X Ontologies X X Semantic Web X X
Traditional and innovative approaches presented in Table 1 provide partial solutions for the integration of heterogeneous spatial databases; they only afford one or two interoperability levels and most of them do not support the specication of the data semantics. However, the area related to heterogeneous databases and disaster management is still asking for new solutions and approaches. Currently, there exist some software packages, such as ArcGIS4 and Sahana5 , that are used in real disaster scenarios in order to help emergency actors. However, they do not take into account the semantic interoperability and the entities and naming conventions in different databases are duplicated, ambiguous and imprecise. Hence, urgent solutions are required in order to solve these ambiguities that result from the heterogeneous spatial databases stored in diverse organizations.

A Hybrid Approach for Providing Intelligent Search in a Disaster Framework

The previous sections have introduced the different heterogeneity levels that could block emergency responders when they query various heterogeneous spatial databases. In response to these difculties, we design a conceptual framework using Geo-web services and ontology based data integration approach to support the three heterogeneity levels. This framework is based on SOA (Service-Oriented Architecture), which is composed of a service provider, a service broker, and a client service as shown in Figure 1. Service Provider affords the syntactic interoperability, the data models are conceptualized in UML 6 (Unified Modeling Language) and converted afterward into GML for sharing data. OGC WFSs (Web Feature Services) were used for publishing feature-level data from heterogeneous databases. For the schematic and semantic interoperability, two types of mappings are required: information source to local ontology mapping and local ontologies to global ontology mapping. Information source to local ontology mapping is obtained by translating GML (Geographic Markup Language) data to RDF (Resource Description Framework) using an automatic conversion system. Local ontologies to global ontology mapping is used for merging the different local ontologies built into one global ontology.

Service Broker is the main component of the architecture that contains the global ontology, a mapping directory, and a binding technique to apply the ontology hybrid approach by connecting the local ontology to the global ontology. The global ontology is used to create mappings of equivalent classes and attributes in the local ontologies. Hence, it describes the global model that represents local ontologies.

Client Service employs the RDF SPARQL query language to query the global ontology that contains all the properties defined in the different local ones. The results of SPARQL queries will be presented in a map using Google Visualization API and SPARQL (SPARQL Protocol and RDF Query Language) proxy.

The framework modules

In this section, we explain the two modules developed in our framework in order to ensure the three interoperability levels. Firstly, we describe the syntactic interoperability module that includes the conversion of the UML models into GML and the integration of these GML models into web feature services. Then, we analyze the schematic and semantic module used to build the different local ontologies and to merge them into a global one.

Syntactic Interoperability Module

This module deals with the syntactic heterogeneity problems. it contains an automatic converter from UML application schema to a common GML format. The application schema is expressed in UML, and the chosen exchange format is GML. A set of conversion rules have been identified and implemented in a tool named Shapechange that reads UML class diagrams and writes corresponding GML code. Therefore, in order to ensure the GML conversion, we have firstly to conceptualize the different UML models for each spatial database. Then, the UML application schemas have to be converted into XMI representations. Once the XMI documents have been designed using UML, conversion to GML schema specified by the OGC can be done easily by defining some conversion rules. The UML to GML Application Schema mapping tool supports this conversion automatically. After standardizing the different databases into GML files, they are published and integrated in WFS.

Schematic and Semantic Interoperability Module

In our framework, we opt to use the hybrid ontology approach for its various advantages [START_REF] Mustapha | Analyzing the ontology approaches and the formation of open ontology model: A step for organisational ontology employment[END_REF]. In order to apply this approach, two types of mappings are required: information source to local ontology mapping and local ontologies to global ontology mapping.

Information Source to Local Ontology Mapping

The local ontologies corresponding to each database are built using two languages, namely the Resource Description Framework (RDF) and the Web Ontology Language (OWL) for representing triples and modeling the relations between them. Such a representation is obtained by using the GML2RDF automated conversion system [START_REF] Bulen | A program for the conversion of the national map data from proprietary format to resource description framework[END_REF]. Each entity in the database is represented by its own configuration which contains a list of all the attributes. The following steps present briefly the building of the different local ontologies:

• Each spatial database is separately converted into its local ontology.

• Each entity in the database is converted to the class name of the local ontology. Each component related to a class is described using a set of triples. This triple comprises three parts: subject, predicate, and object.

• The subject is the result of the conversion of each class instance in the dataset.

• The predicate is the conversion result of each non-prime attribute. It expresses a relationship between the subject and the object.

• The object is obtained by converting each attribute value corresponding to an attribute.

• If two classes C1 and C2 take part in formal corresponding relationships, then class C2 will be declared as the subclass of class C1. The foreign key relationship between the classes is shown as the predicate where C2 will be the domain.

Local Ontologies to Global Ontology Mapping After building the different local ontologies, we have to map them in order to determine the semantic and schematic correspondences between the concepts of two ontologies. If two concepts correspond, then they mean the same thing or closely related things. The mapping process proposed in our research is based on the two approaches presented below. [START_REF] Cavnar | N-gram based text categorization[END_REF] is a method used to compute lexical similarity between two words. Generally, the N-Grams algorithm takes as input two strings and computes the number of common substrings between them. An n-gram is a sequence of n characters; for each string, the set of all possible n-grams, are computed. Then, a pre-processing step is executed to normalize both strings.

N-Grams match

The third step is to extract the n-grams from each string. Finally, the algorithm counts the matching percentange of the two strings.

Synonym match [START_REF] Lin | A survey of exploiting wordnet in ontology matching[END_REF] is a technique for detecting semantic similarity between two words. In many cases, two strings may appear as different words but may represent the same meaning in a particular domain. To identify such semantically related words, we use the ontology and lexical database Wordnet.

Ontology Matching Steps To map the local ontologies and merge them into a global one, all the possible combinations of local ontologies should be detected and mentioned in a list. Then, we extract and match the different classes in each local ontology.

Class Name matching. The different classes are represented in prefixes. Hence, we extract all the names of classes from there without considering the predifined prefixes list. Then, we apply the class name match, each class in the first local ontology is paired with all other entities in the second one. The N-grams and the Synonym match scores are used to validate each step. So, the N-grams score should be greater than a threshold and the Synonym match returns a boolean result synonym [START_REF] Zhang | Automatic search of geospatial features for disaster and emergency management[END_REF] or not synonym (0). If the matching result is validated by the N-gram or the Synonym match, then we can proceed to next matching steps using the same approaches and techniques.

Attribute Type matching. This step is executed only if the previous one was validated by a matching approach. At this stage, the attribute type of the matching class is extracted from the list created for each class name containing the attributes name and their corresponding types. This list is obtained by the extraction of the attributes name from the predicate. The data type attribute is paired in such a way that one data type attribute related to a class in one local ontology is matched with all the data types of the second class in the other local ontology. If two types of attributes related to one pair are detected as similar then we proceed to the next step to compare their related names.

Attribute Name matching. This phase could be employed only if the pair of attribute names that will be matched have the same data types. In this case, the attributes of the matching entity names are executed.

Attribute Value matching. Until now, this approach is relatively susceptible to homonym-like pairs of strings: although syntactically very similar, they could mean entirely different things, N-Grams and N-Synonym will give them a high similarity score. To deal with this problem, we apply an attribute value match. This matching is executed only if the attribute name matching of one pair was validated, then three instances related to each attribute are matched to detect if the two classes are similar or not. Moreover, this solution is extensible and we could make an attribute-class matching, relations matching and many others using the same principle.

Local Ontologies Merging To merge the different local ontologies, we analyze the matching results using algorithm 1. If two classes are matched as similar, then one of these classes will be added to the global ontology. Else, the two classes are saved in the global ontology. The same method is applied for the couple attribute-class and attribute-attribute merging. After constructing our global ontology, we need to store it in an ontology server in order to be able to query it. Therefore, we stored the resulting global ontology in the joseki7 ontology server.

Experiments and Results

Based on the proposed framework, we have implemented a prototype for allowing emergency responders to automatically search from heterogeneous datasets. Therefore, we have used three heterogeneous datasets belonging to La Rochelle city in France. The first one represents the office of mayor data stored in KML files, the second one contains the emergency service data stored in MIF files and the third dataset belongs to the police service data stored in Shapefiles. Then, we designed the different UML models conceptualized for each dataset using Rational Rose Software. The Unisys XMI tool is used to store the different UML models in a single XMI document. At that time, conversion to GML schema specified by the OGC can be done easily using the UML to GML mapping tool Shapechange 8 . To inte-grate the different GML files, we use OGC web services specially WFS. Thus, we use Geoserver 2.2.1 and the GML plugin to publish the generated GML datasets. To build the local ontologies for each dataset, a conversion program named GML2RDF is used which makes use of two predeveloped open source APIs for parsing GML files and handling the creation of the RDF models. The GeoTools API is used for reading the data from GML files, and the Jena API is used to contain the converted RDF data in a memory model and eventually write the data into a N3 file. Similarly, the Jena semantic web framework for java is integrated in our application to access ontology definitions, analyze the Word-Net thesaurus and to realize the different necessary mappings. To evaluate the mapping approaches applied in our framework, we visualize the different results provided by the mapping algorithm. Hence, these different results are analyzed using our three datasets and 90% of them are matched correctly. A spatial ontology server was developed based on Joseki to support the global ontology N3 file and to interrogate it using SPARQL forms. The query results are displayed into a satellite image using Google Maps API. Our experiments yield to promising results and show that, with the implemented prototype, it is possible to directly search geospatial information from heterogeneous sources using our framework. Therefore, to deliver the needed information in a short time, we only have to interogate our global ontology using a simple query that returns the response as a map.

Conclusion and Future Work

In this paper, we propose a framework for the integration of heterogeneous spatial databases that enables emergency responders to easily query spatial heterogeneous databases and facilitates the automatic search of geospatial information in order to make quick and correct decisions and prompt actions. Our framework uses novel techniques maintained under SOA architecture to deal with three major tasks: Syntactic interoperability is analyzed by adopting GML as a common language; Schematic (Structural) interoperability is construed by the ontology mapping process and Semantic interoperability is assured by using the hybrid ontology approach. Furthermore, by the extensible and autonomy characters hidden behind the ontology approach and OGC web services many interesting functionalities can be added. We plan to treat information from texts and comments posted in social networks. This will be helpful in the case of a lack of information.

Fig. 1 .

 1 Fig. 1. The framework architecture

http://www.esri.com

http://sahanafoundation.org

http://www.uml.org

http://www.joseki.org

http://www.shapechange.net/


The 4-parameter Compressible Packing Model (CPM) for sustainable concrete.

G. Roquier 1 1 Laboratoire Navier, 77420 Champs-sur-Marne, France, gerard.roquier@laposte.net ABSTRACT: To reduce the environmental impact due to the CO 2 -emission, it is necessary to optimize the concrete mix-design by reducing the amount of cement. In this case, analytical packing models are necessary to predict the packing density of a pile of grains. A new version of the Compressible Packing Model (CPM) (de Larrard et al.), the 4-parameter CPM, is introduced to predict the solid fraction of maximally dense disordered packings of bidisperse particles. It is apt to account for the loosening effect on big particles by interstitial small ones, and for the wall effect within assemblies of small particles near a big one, with moderate size ratios. The new theory is based on a specific treatment of configurations of one secondary class particle surrounded by dominant class neighbours. It has four parameters : wall effect and loosening effect coefficients, critical cavity size ratio and compaction index. The model proves its efficiency when compared to 780 results on various tested materials. The correlation coefficients between predicted and measured packing densities are very high: 99% for frictionless glass beads, 98,7% for spherical particles numerically simulated, 97,8% for natural aggregates and 96,4% for crushed aggregates. To predict the viscosity of the same grains in a high concentrated suspension compound of spherical, inert and rigid particles suspended in a Newtonian fluid, we resort to the iterative approach advocated by Farris and to the Krieger-Dougherty power-law relation for the relative viscosity. The theory was developed to highlight a new relation between relative viscosity and the solid volume fraction, compatible with Einstein's relation.

INTRODUCTION.

A sustainable concrete mix-design implies both the substitution of cement by various cementitious materials, the optimization of the aggregate skeleton to minimize the interparticular volume of voids to be filled with the cement paste and the reduction of the concrete quantity needed for a civil engineering structure by increasing its mechanical performance. The latter can be easily linked to the packing density which itself depends on the concrete formulation. This one is divided in two stages. In the first one, concrete can be considered as a high concentrated suspension that we want to maximize the packing density for a given workability. This one can be linked, on a simplified basis, to the viscosity. The second stage concerns the search of the best sustainability while ensuring that the concrete continues to meet requirements: reducing the environmental cost of the material while playing with the proportioning of superplasticizer to obtain the desired workability. This ambitious goal leads to use models. This article deals with the particle packing and the concrete rheology by calling upon the last generation of the Compressible Packing Model (CPM): the 4-parameter CPM. To achieve its sustainable revolution, the concrete calls upon science to become a high-tech material. THE REFERENCE FRAME: THE COMPRESSIBLE PACKING MODEL (CPM).

Introduced by the works of Stovall, de Larrard, Buil (1986) and Sedran, de Larrard, Angot (1994), the CPM, developed by de Larrard (2000), is a tool used to predict the packing density of multicomponent mixtures, while taking into account placing conditions.

2.1

Virtual packing density of a binary mixture.

The real packing density corresponds to that obtained for a randomly dense packing. It depends on process used to fill and to compact the material inside the mould. If we consider a perfect placing process where each particle is placed one by one in its ideal location, the compaction index tends to infinity and the packing density reaches the virtual packing density. For a binary mixture "without geometrical interaction", the latter is calculated by distinguishing two domains: "large grains dominant" and "small grains dominant". In the first case, the small component is fine and mobile enough to be introduced in small quantities in available cavities between large particles according to the insertion mechanism. In the second case, the method consists in substituting some fine particles forming the matrix by a small quantity of spread large particles: this is the substitution mechanism. For a binary mixture "with geometrical interactions", the virtual packing density γ obtained by the CPM can be written as follows:

γ = Min γ 1 = β 1 1 -1 - β 1 β 2 a 12 $ % & ' ( ) y 2 ; γ 2 = β 2 1 -1 -β 2 + b 21 β 2 1 - 1 β 1 $ % & ' ( ) $ % & & ' ( ) ) y 1 $ ' ( ) ) ) ) ) (1)
where γ 1 and γ 2 are respectively virtual packing densities of the binary mixture in the case "large grains dominant" and in the case "small grains dominant", β 1 and β 2 the virtual packing densities of the large size class 1 and of the small size class 2, y 1 and y 2 their volume fractions by reference of the total solid volume, a 12 the loosening effect coefficient, b 21 the wall effect coefficient. When the size ratio x=d 2 /d 1 (fine/large) is equal to 1, a total interaction occurs:

b 21 1 ( ) = 1 and a 12 1 ( ) = 1

(2)

2.2

Real packing density of a binary mixture: compaction index.

The real packing density φ* is calculated by introducing the compaction index K:

K = y 1 β 1 1 φ * - 1 γ 1 + y 2 β 2 1 φ * - 1 γ 2 (3)
Because K is a representative value of the placing process, the expression (3) is an implicit equation of φ*, with one and only one positive root. It can easily be solved numerically.

WALL EFFECT, LOOSENING EFFECT, INTERFERENCE EFFECT.

In figure 1 representative of "void ratio" (e=1/φ*-1) versus "volume fraction of small particles" for a binary mixture, the insertion mechanism is represented by the straight line AM and the substitution mechanism by the straight line FM. The wall effect is highlighted by FH section which is located above FO. Now, we are going to introduce the concept of interference effect wich is locally a loosening effect.

If we consider the "large grains dominant" area, the loosening effect is only localized around a particle insufficiently fine to insert into a cavity created by some touching larger particles.

Interference effect is more global. If we consider the "small grains dominant" area, the effect of large particles is reduced to a wall effect if they are in small quantities troughout the matrix of fine elements. If their fraction increases, their relative position will have an influence: their walls will be too close to each other and they are going to interact on the arrangement of small particles. An interference will occur between the two components. Interference effect is materialized by the IKA section. It is locally called loosening effect in the AK section.

4

WALL EFFECT THEORY FOR SPHERES.

The wall effect has been the object of many experimental and theoretical studies: Ben Aim (1970), Stovall, de Larrard, Buil (1986), de Larrard (1988), de Larrard (2000). Ben Aim was a precursor in this domain. He considered that a planar wall effect is almost totally localized at the interface between the large and small spheres in a layer d/2 thickness (d: particle diameter). He generalises this way of reasoning to a curved wall. When he studies a large sphere of diameter d 1 surrounded by small ones of diameter d 2 , he considers the wall effect disturbance in a portion delimited by two concentric spheres. The first one has a diameter d 1 . The second hypothetical one is chosen in such a way that contact points between each small sphere pressed against the wall are located on its surface area. However, the boundary condition b 21 (1)=1 is not respected. That's why we propose an adjustment of the Ben Aim reference cell:

d 1 d 2 d1 (1+2 kP2 x) 1/2
Compactness β'2 of small spheres in the disturbed area by wall effect.

Delimitation of the disturbed area by wall effect. The diameter of the sphere delimiting the disturbed volume by the wall effect around a large sphere is:

d hyp = d 1 1+ 2 k P2 x (4)
Let α 2 be the real packing density of the fine class 2. We can deduce its virtual packing density β 2 by involving the compaction index K:

β 2 = α 2 1+ K ( ) K (5)
The number of small spheres against a large one is calculated from the spherical square model, not shown here, as part of the dense virtual packings:

N 12,SSM dense x ( ) = π 1+ x ( ) x arcsin x 1+ x " # $ % & ' (6)
The packing density β' 2 of the small particles in the disturbed volume by the wall effect is deduced from this expression:

β' 2 x ( ) = π 1+ x ( ) 4 x 1+ 2 k P2 x ( ) 3 2 -1 $ % & & ' ( ) ) arcsin x 1+ x $ % & ' ( ) . . 2 1+ 2 k P2 x ( ) 3 2 -3 1+ 2 k P2 x ( ) 1+ k P2 x 1+ x $ % & ' ( ) + 1+ k P2 x 1+ x $ % & ' ( ) 3 -x 3 + 3 k P2 x 3 1+ x + x 3 1 - k P2 1+ x $ % & ' ( ) 3 $ % & & ' ( ) ) (7)
In the case where β 1 =β 2 , the wall effect coefficient is expressed by:

b 21 x ( ) = β 2 -β' 2 x ( ) ( ) 1 -β 2 ( ) 1+ 2 k P2 x ( ) 3 2 -1 # $ % & ' ( (8)
It remains to be determined k P2 by respecting the boundary condition b 21 (1)=1, leading to the numerical solution, by a spreadsheet program, or the analytical solution of the following equation:

β 2 -6 ( ) 1+ 2 k P2 ( ) 3 2 + 9 2 k P2 2 + 18 k P2 + 5 = 0 (9)
The equation to be solved being of the 3 rd degree, the value of k P2 which is coherent with those presented in the following table should be kept:

LOOSENING EFFECT AND INTERFERENCE EFFECT THEORY FOR SPHERES.

Stovall, de Larrard, Buil (1986) introduced the concept of critical cavity size ratio x 0 : below this value, the intrusion of small quantities of smaller spheres does not disturb the bed of larger spheres; beyond this value, small spheres cannot be placed without disturbing this one. Furthermore, when x→1, all particles become identical: calculated packing densities in the frame of the "small grains dominant" and in the frame of the "large grains dominant" must be equal when volume fractions of fine and large particles are 0,5. In addition to these two hypothesis, the concept of local isotropic expansion (de Larrard ( 1988)) of the large particle skeleton around a small one is adopted when x>x 0 . But contrary to the original publication, a spherical reference cell is used for that, allowing to be consistent with the wall effect theory. The diameter of the sphere delimiting the disturbed volume by the loosening effect is:

d hyp = d 2 1+ 2 k P1 x (10)
Let α 1 be the real packing density of the large class 1. We can deduce its virtual packing density β 1 :

β 1 = α 1 1+ K ( ) K (11) THE 4-PARAMETER COMPRESSIBLE PACKING MODEL (CPM).
The compaction index K and the critical cavity size ratio x 0 need now to be calibrated by an analysis of 780 values on binary mixtures. x 0 is considered as a function of the type of aggregate, of its shape and of its finish surface. *The expression "optimized filling" corresponds to the use of a mixing chamber to fill the container by glass beads.

Table 3. Critical cavity size ratios x 0 (for natural aggregates, x: size ratio).

Type of aggregate

Crushed aggregate Round natural aggregate

Frictionless glass beads Angular very rough

Angular rough Angular A model intended to predict the viscosity of a bidisperse concentrated suspension with spherical, inert and rigid particles suspended in a Newtonian fluid is now presented. It takes into account geometrical and hydrodynamic interactions between particles. The fractal concept initially proposed by Farris ( 1968) is modified to include geometrical interactions used in the 4parameter CPM. We have chosen a power-law relation (Krieger-Dougherty type) for the relative viscosity (Pabst ( 2004)). In the case β 1 =β 2 =β (equal residual packing densities), the viscosity is calculated by:

η = η 0 1 - ψ 1 ψ 1 MAX $ % & & ' ( ) ) -C E β 1 - ψ 2 ψ 2 MAX $ % & & ' ( ) ) -C E β with d 1 > d 2 (17) ψ 1 = φ y 1 ψ 2 = y 2 1 φ -y 1 (18) ψ 1 MAX = β 1 φ - 1 γ 1 & ' ( ) * + + y 1 1 -β ( ) 1 φ - β γ 1 ψ 2 MAX = β 1 φ - 1 γ 2 & ' ( ) * + + y 2 1 -β ( ) 1 φ - β γ 2 -1 -β ( ) y 1 (19)
η is the suspension viscosity of the bidisperse suspension and η 0 the viscosity of the Newtonian suspending fluid.

φ is the volume fraction of the suspended spheres in a total volume unity, y 1 and y 2 the volume fraction of classes 1 and 2 by reference of the total solid volume, ψ 2 the volume fraction of the class 2 in a total volume unity, ψ 1 the volume fraction of the class 1 considering the presence of the finer class 2, ψ 2 MAX the maximal volume fraction of the class 2 in a total volume unity, ψ 1 MAX the maximal volume fraction of the class 1 considering the presence of the finer class 2, γ 1 and γ 2 respectively virtual packing densities of the binary mixture in the case "large grains dominant" and in the case "small grains dominant". C E is equal to 2,5 in agreement with Einstein's relation for dilute suspensions.

This theory allows to highlight a new relation between relative viscosity and the volume fraction of the suspended spheres. When the latter reaches its critical value, the suspension is jammed and the mixture reaches the packing density of the solid skeleton.

CONCLUSIONS.

In twenty years, progress in basic and applied research has revolutionized concrete. The 4parameter CPM is now able to analyze the concrete constituents packing at the level of particles which can be the micro level for the silica fume. It represents a useful tool in order to improve performance concrete such as mechanical strength by organizing the components so as to produce a denser material and to increase durability. One of these strongest qualities is that it is the result of a progression leading to a new theory on the wall effect and on the loosening effect. It includes the introduction of a critical cavity size ratio depending on the type of materials. This reveals that there would not be a single curve for geometrical interactions. Limitations of the proposed model are of two types. Firstly, the loosening effect appears, strictly speaking, from a threshold volume fraction. Secondly, the interaction coefficients are determined by considering the same virtual packing density for each granular class. However, the results obtained suggest that efforts will not have been in vain. By introducing science in concrete, concrete has become a high-tech material. Pabst W. ( 2004) "Fundamental considerations on suspension rheology", Ceramics -Silikáty 48 (1), p. 6-13. Sedran T., De Larrard F., Angot D. (1994) "Prévision de la compacité des mélanges granulaires par le modèle de suspension solide", Bulletin des laboratoires des ponts et chaussées, n°194, p.59-86. Stovall T., De Larrard F., Buil M. (1986) "Linear Packing Density Model of Grain Mixtures", Powder Technology, vol. 48, n° 1, p. 1-12.

Figure 1 .

 1 Figure 1. Illustration of wall effect, loosening effect, interference effect.

Figure 2 .

 2 Figure 2. Definition of the spherical reference cell for studying the wall effect.

Figure 3 .

 3 Figure 3. Representative example of the wall effect coefficient b 21 for a real packing density of the fine class α 2 =0,641, a compaction index K=9 and a residual packing density β 2 =0,712.

Figure 4 .

 4 Figure 4. Definition of the spherical reference cell for studying the loosening effect.

  value is very close to 0,2247 obtained for a tetrahedral cavern(Ben Aim (1970)). Now, we are going to evaluate the 4-parameter CPM when compared to 780 results on binary mixtures. Data concerning glass beads come from the thesis of Ben Aim(1970) in which we can find results of Ben Aim; Mc Geary; Westman and Hugill; Tickell, Mechem and Mc Curdy; Naar, Wygal and Henderson; Yerazunis, Bartlett and Nissan and from experiments of Kwan,Chan, Wong (2013). Round natural aggregates are coming from the Loire (France)(de Larrard (2000)) and from the Seine (France): experimental results of Joisel given in the thesis ofde Larrard (1988). Crushed aggregates are from the Pont-de-Colonne quarry at Arnay-le-Duc (France) (de Larrard (2000)), a soft limestone from Lorraine (France) (Lecomte and Zennir (1997)), a limestone aggregate from the northeast of Tlemcen (Algeria) (Hanini (2012)), a granite rock aggregate (Asia)(Kwan, Wong, Fung (2015)). Results of numerical simulation will be published in the thesis of Roquier at a later date.The 4-parameter CPM is evaluated by comparison of the original CPM and of the original 3parameter model(Kwan, Chan, Wong (2013)) or extended(Kwan, Wong, Fung (2015)).

Figure 6 .

 6 Figure 6. Experimental data compared with the 4-parameter CPM for frictionless glass beads.

  the Association Universitaire de Génie Civil (AUGC, France) and from the chaire École des Ponts Paristech / Lafarge for the conference participation, are gratefully acknowledged.10REFERENCES. Ben Aim R. (1970) Étude de la texture des empilements de grains. Application à la détermination de la perméabilité des mélanges binaires en régime moléculaire, intermédiaire, laminaire, Thesis attented at the Faculty of Sciences of the University of Nancy, France. De Larrard F. (1988) "Formulation et propriétés des bétons à très hautes performances", Rapport de recherche LPC, n°149. De Larrard F. (2000) "Structures granulaires et formulation des bétons", Études et recherches des laboratoires des ponts et chaussées, Ouvrages d'art OA34. Farris R.J. (1968) "Prediction of the viscosity of Multimodal Suspensions from Unimodal Viscosity data", Trans. Soc. Rheol., 12, (2), p.281-301. Hanini W. (2012) Étude de l'incidence de la granularité sur la compacité des mélanges granulaires, Magister's dissertation in Civil Engineering. University Aboubekr Belkaid, Algeria. Kwan A.K.H., Chan K.W., Wong V. (2013) "A 3-parameter particle packing model incorporating the wedging effect", Powder Technology 237, p.172-179. Kwan A.K.H., Wong V., Fung W.W.S. (2015) "A 3-parameter packing density model for angular rock aggregate particles", Powder Technology 274, p.154-162. Lecomte A., Zennir A. (1997) "Modèle de suspension solide et formulation de bétons calcaires en Lorraine", Bulletin des laboratoires des ponts et chaussées n°211, p.41-52.

Table 1 .

 1 Values of k P as a function of the residual packing density β.

	β	0,65	0,66	0,67	0,68	0,69	0,70	0,71	0,72	0,73	0,734
	k P	0,4466 0,5854 0,6729 0,7543 0,8369 0,9253 1,0252 1,1476 1,3284 1,4729
	Here is a representative example of the wall effect coefficient b 21 as a function of the size ratio
	x:

Table 2 .

 2 Compaction index K.

	Packing processes	Pouring	Vibration	Vibration + compression	Optimized filling* + vibration + compression	virtual
	K	4,7	5,6	9	15	∞

Table 4 .

 4 Mean deviation ξ and correlation coefficient r on packing densities for various materials.

	Models / Materials	Original CPM	Original or extended 3-parameter model	4-parameter CPM
		ξ	r	ξ	r	ξ	r
	Frictionless glass beads (300 values)	0,012	0,9754	0,009	0,9863	0,007	0,9904
	Spherical particles
	numerically simulated (20	0,012	0,8783	0,008	0,9598	0,006	0,9877
	values)
	Round natural aggregates (125 values)	0,009	0,9619	0,012	0,9534	0,007	0,9788
	Crushed aggregates (335 values)	0,013	0,9408	0,013	0,9455	0,010	0,9642

& & & & &

The number of large spheres against a small one is calculated from the spherical square model:

The packing density β'' 1 of the large spheres in the reference cell is:

The packing density of the large spheres in a more global interference zone is:

When x=x 0 , β' 1 (x 0 )/β 1 =1 : loosening effect and interference effect do not occur. When x=1, β' 1 (1)/β 1 =0,5 : the continuity between "large grains dominant" and "small grains dominant" is provided when their volume fractions are equal and when all particles become identical. The volume fraction of small particles at the "eutectic" point, corresponding to the crossing point into the "small grains dominant" area, is:

The loosening effect coefficient is deduced from this expression: . Representative example of the loosening effect coefficient a 12 for α 1 =0,641, K=9, β 1 =0,712 and critical cavity size ratios x 0 =0, x 0 =0,2 and x 0 =2x/5.