Martín G Marchetta

Frédérique Mayer

Raymundo Q Forradellas

A reference framework following a proactive-product approach for Product Lifecycle Management

Keywords: PLM, Proactive Product, Intelligent Agent, Virtual Enterprise, Concurrent Engineering

Product Lifecycle Management (PLM) has been identified as a key concept within manufacturing industries for improving product quality, time-to-market and costs. Previous works on this field are focused on processes, functions and information models, and those aimed at putting more intelligence on products are related to specific parts of the product lifecycle (e.g. supply chain management, shop floor control). Therefore, there is a lack of a holistic approach to PLM, putting more intelligence on products through the complete lifecycle. In this paper, a PLM framework supported by a proactive-product approach based on intelligent agents is proposed. The developed model aims at being a first step toward a reference framework for PLM, and complements past works on both product information and business process models (BPM), by putting proactivity on product's behavior. An example of an instantiation of the reference framework is presented as a case study.

Introduction

The need to compete at a global level has pushed enterprises to improve their operations and business practices and even to collaborate [START_REF] Nof | Collaborative control theory for e-work, e-production, and e-service[END_REF], breaking down physical barriers and creating what are nowadays called virtual enterprises [START_REF] Ming | Collaborative process planning and manufacturing in product lifecycle management[END_REF][START_REF] Croxton | The supply chain management processes[END_REF]. In this context, Product Lifecycle Management (PLM) has been considered a key concept in order to achieve efficiency and maintain consistency along the complete product lifecycle, from early stages of the development process of new products to their disposal [START_REF] Saaksvuori | Product Lifecycle Management[END_REF][START_REF] Schuh | Process oriented framework to support plm implementation[END_REF].

PLM involves the management of product information and the integration of business processes [START_REF] Sharma | Collaborative product innovation: integrating elements of cpi via plm framework[END_REF]. Initially, integration and collaboration were implemented mostly along the supply chain (with suppliers and customers [START_REF] Croxton | The supply chain management processes[END_REF]), but currently peer enterprises are also partners [START_REF] Bhandarkar | Step-based feature extraction from step geometry for agile manufacturing[END_REF], as shown in figure 1.

Figure 1: Current peer-to-peer and supply chain enterprise integration schema Partner enterprises share much information concurrently. Additionally, product information now needs to be shared across heterogeneous systems. Since business processes go beyond a single enterprise, they involve people, information and other resources with different ownership and cultures.

Previous research works have focused on information models, business processes integration and interoperability issues. Only recently, the need for putting more intelligence on products making them proactive [START_REF] Morel | Manufacturing plant control challenges and issues[END_REF][START_REF] Baã¯na | New paradigms for a product oriented modelling: Case study for traceability[END_REF] during their own development and management, to better support business processes, has been recognized. Additionally, there is lack of a holistic and generic (i.e. independent of the industry/activity) treatment of PLM aspects, including business processes definition, product information models and information exploitation approaches [START_REF] Schuh | Process oriented framework to support plm implementation[END_REF].

In this paper, a framework based on a proactive product approach to PLM is proposed. The model aims at being a first step toward a reference PLM framework, rather than a model for a specific application or industry. The framework includes a Business Process Model (BPM), a Product Information Model (PIM) and an architecture of applications based on intelligent agents for information exploitation.

Related research works

At least three main topics have been researched in PLM: Business Process Models (BPM), Product Information Models (PIM) and product information exploitation for supporting business processes.

PLM business process modeling

Integration of product information, project management, processes and procedures has been addressed through a product digital mock-up in [START_REF] Danesi | P4lm: A methodology for product lifecycle management[END_REF]. Collaborative frameworks and technological requirements for their implementation have also been discussed [START_REF] Osã³rio | Distributed process execution in collaborative networks[END_REF]. In this context, usefulness of enterprise modeling and reference models to achieve enterprise interoperability and collaboration has been highlighted [START_REF] Panetto | Enterprise integration and interoperability in manufacturing systems: Trends and issues[END_REF].

Both technical (i.e. low-level communication) and information/knowledge interoperability issues have also been studied [START_REF] Chituc | A framework proposal for seamless interoperability in a collaborative networked environment[END_REF]. Centralized approaches concentrating PLM information, accessible through communication infrastructures for supporting collaboration in the supply chain, have also been proposed [START_REF] Trappey | Applying collaborative design and modularized assembly for automotive odm supply chain integration[END_REF].

The efficiency vs. innovation dilemma in PLM has been addressed [START_REF] Chen | Developing new products with knowledge management methods and process development management in a network[END_REF], by proposing intensive use of Knowledge Management (KM) to involve buyers and suppliers in innovation processes. There is a need for collaborative innovation management within PLM, which can be achieved by integrating collaboration, product development and innovation by means of a technology framework [START_REF] Sharma | Collaborative product innovation: integrating elements of cpi via plm framework[END_REF].

Product information modeling

PLM involves management of product information along its complete lifecycle. Several aspects of product models have been researched, including handling of evolution of documents in a Product Data Management system [START_REF] Eynard | Uml based specifications of pdm product structure and workflow[END_REF], a product data framework for logistics planning activities [START_REF] Gimenez | Product ontology: Defining product-related concepts for logistics planning activities[END_REF], and an ontology based on the Core Product Model (CPM) of the National Institute of Standards and Technology (NIST) [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF][START_REF] Fenves | Cpm2: A core model for product data[END_REF].

In [START_REF] Rachuri | Information sharing and exchange in the context of product lifecycle management: Role of standards[END_REF], the information flows and the relevant existing standards within PLM were evaluated, highlighting current issues regarding interoperability and information exchange. Model-driven approaches for achieving interoperability have also been proposed [START_REF] Baã¯na | New paradigms for a product oriented modelling: Case study for traceability[END_REF]. This kind of interoperability, called "product-oriented interoperability", is based on a meta-model of the product representation, and a mechanism for make a mapping between it and specific models.

Product information exploitation for business process integration

In previous works, exploitation of information has been applied to different activities throughout the product lifecycle. Support to individual activities (e.g. CAPP) within a single enterprise has been proposed [START_REF] Tong | Product lifecycle oriented digitization agile process preparation[END_REF]. Extension of this schema for supporting collaboration among customers, suppliers and partner enterprises was also suggested [START_REF] Ming | Collaborative process planning and manufacturing in product lifecycle management[END_REF]. Change propagation within a single company [START_REF] Ma | Change propagation algorithm in a unified feature modeling scheme[END_REF], as well as for maintaining consistency of design documents between partners [START_REF] Shiau | A distributed control workflow for collaborative design network[END_REF], were also proposed.

The idea of intelligent products is to connect the physical products with their counterparts within information systems. In this context, classification dimensions for product intelligence, application domains and obtainable benefits have been proposed [START_REF] Meyer | Intelligent products: A survey[END_REF]. The application of the "holon" concept was suggested in [START_REF] Valckenaers | Intelligent products: Agere versus essere[END_REF]. Intelligent agents, RFID and other data acquisition technologies were applied to intelligent products in post-sales phases in [START_REF] Yang | Intelligent products: From lifecycle data acquisition to enabling product-related services[END_REF].

The use of different kinds of agent technologies (holons, multiagent systems, etc.), to support different aspects of PLM (control, production management, etc.) has been explored [START_REF] Marik | Industrial applications of agent technologies[END_REF]. In [START_REF] Contreras | Industrial application integration using the unification approach to agent-enabled semantic soa[END_REF] composition of services through semantic interoperability was suggested.

Multi-agent systems have been applied to distributed planning along the supply chain [START_REF] Forget | Multi-behavior agent model for planning in supply chains: An application to the lumber industry[END_REF], project coordination and monitoring [START_REF] Trappey | Development of an intelligent agent system for collaborative mold production with rfid technology[END_REF] and coordination of engineering activities in new product development projects [START_REF] Huk | An e-engineering framework based on service-oriented architecture and agent technologies[END_REF]. Creation of instant virtual enterprises and their optimization have also been supported by the use of agents [START_REF] Grefen | Dynamic business network process management in instant virtual enterprises[END_REF][START_REF] Kaihara | A multiagent-based complex systems approach for dynamic negotiation mechanism in virtual enterprise[END_REF].

Business process model for PLM and SCM

Traditionally, within PLM the product has concentrated all the data required by different activities, and the approaches considering products as active entities have been restricted in different ways (e.g. focusing on specific parts of the product's lifecycle). Thus, these systems are able to answer users' questions regarding the product reactively, and they can implement limited proactive behaviors. Moreover, despite the variety of works on information exploitation, there is a lack of an application-independent framework. Additionally, PLM business processes have also been researched, but there has been a separation between BPM and information exploitation.

In this section, a business process model for PLM and SCM is presented. Special emphasis is put on the new product development business process, through a detailed model of its activities. Additionally, a taxonomy of interactions is described, which highlights the different kinds of information exchange and assistance opportunities that exist in the described scenario.

Business process model

Several processes must be carried out during a product's lifecycle. Figure 2 depicts an integrated business process model, including development of new products, production and logistics processes 1 . The model is based on two complementary views of business processes in logistics [START_REF] Croxton | The supply chain management processes[END_REF][START_REF] Frazelle | Supply Chain Strategy[END_REF]. Processes of a product development/manufacturing company are shown inside the rectangle. The supplier, the customer and the partner enterprises are external actors. Partners are enterprises which design other components, or that manufacture products designed by the company. Two types of customer orders may be received: for development new products and for existing products. Additionally, at any moment the customer may ask for information about its order's 1

The models in this article were created with the MEGA suite state. All these information flows are handled by the Customer Service Management process (called Customer Response in [START_REF] Frazelle | Supply Chain Strategy[END_REF]). The Order Delivery process takes care of coordinating all activities to put products in the location accorded with the customer. Warehousing activities includes picking, put away, storage, etc. The New Product Development (NPD) process is concerned with turning customer needs into requirement specifications, designing the product, creating manufacturing specifications, etc. This process handles consistency and coordination both inside the company and with the external actors.

The Production Planning process accommodates the demand to the available manufacturing resources. The result is a production plan which is provided to Manufacturing. Demand/Inventory Management forecasts demand in order to support procurement and production planning. Procurement handles the buying orders. Finally, the Manufacturing process is in charge of products creation using manufacturing resources.

New Product Development process

The New Product Development process involves a great number of interactions of different types (along the supply chain, with partner enterprises, and also inside the process itself). Figure 3 presents a functional diagram for this process. The process is composed of 5 main activities: Product Specification Definition, Conceptual Design, Product Detailed Design, Product Engineering and Process Planning. In a concurrent engineering environment, these activities are carried out in an overlapped way, rather than sequentially.

Activities within New Product Development

The objective of the Product Specification Definition activity is to turn customer needs into a product specification. Customer needs represent the "problems" the product must solve, e.g. functional features, durability, etc, while a product's requirement specification is a formalized specification of customers' requirements, considering also performance, cost constraints. The product requirement specification is created by the Conception/Design Team, in collaboration with the partner enterprise's design team and the customer.

The Conceptual Design activity defines the first product's features. It also involves participation of customers, in order to assure that the product matches the needs. Product Detailed Design defines the main features of the product regarding its form, geometry and proposed materials. Detailed design creates a mapping from design features to requirements, including the rationale behind design decisions. Exchanging information relevant to all the partners is a good practice in order to avoid later problems during downstream activities. The resulting design undergoes engineering evaluations (Product Engineering), and it is also used by Process Planning. After the final design is obtained, it contains information that allows Manufacturing to produce the product.

The Product Engineering activity assures the accomplishment of engineering constraints by the product design (i.e. determining if geometry, structure, materials are able to cope with stress, durability, efficiency, etc.). Process Planning is the definition of manufacturing and assembly operations needed to produce one instance of each part, their sequence, along with the machines, tools and fixturing required for these tasks [START_REF] Kalpakjian | Manufacturing Engineering and Technology[END_REF].

If design changes are required as a result of these activities, these modifications are sent to the internal design team, which in turn makes decisions and articulates change impacts with corresponding internal and external affected stakeholders.

In figure 3, the product is shown as an actor (Product Agent entity), because it interacts with other products (in partner companies), as well as with people, in a proactive way, detecting inconsistencies, propagating changes and giving automated assistance when relevant events arise.

Taxonomy of product-automated support along product lifecycle

PLM in a concurrent engineering/virtual enterprise paradigm involves interactions between several types of entities and at different abstraction levels. Since each interaction type entails different issues (e.g. information ownership, concurrency, etc.), a structured categorization of these interactions is needed. Both the "business processes" and the "virtual enterprise" dimensions are considered here for the classification. As a result, a taxonomy containing four interaction types, depicted in figure 4, is proposed here as follows: 3 Inter-activity interactions: Inside a single business process, there exists the need for coordinating activities which contribute to its value chain. This is especially true when activities whose results impact on the others are executed concurrently. 4 Intra-activity interactions: Within a single activity the number of people involved may be very small, and so the information exchanged; thus, the most important thing an active product may contribute with, is exploitation of information in order to optimize results of that activity. Within our framework, the active product is similar to an expert advisor who actively integrates all the information across the product's lifecycle, resulting in a PLM system of PLM inter and intra systems [START_REF] Maier | Architecting principles for systems-of-system[END_REF]. For inter-process situations (types 1 and 2), the most important benefits include change impact analysis, propagation/notification of changes to (and only to) relevant people, global optimization, risk assessment, feedback on project evolution, etc.

Considering inter-activity interactions (type 3), similar benefits can be obtained, but restricted to activities within a single business process. Finally, inside a single activity (type 4), local multi-objective optimization can be supported by the product, as well as other aids such as know-how acquisition through machine learning [START_REF] Marchetta | Supporting interleaved plans in learning hierarchical plan libraries for plan recognition[END_REF], plan recognition for identifying user's intentions, task completion, automated generation of alternative solutions [START_REF] Marchetta | Artificial intelligence planning for generative computer aided process planning[END_REF], etc.

Applications architecture

Software applications that support product development have emerged independently from the needs of individual activities (CAD, CAM, scheduling, etc). This has generated issues related to interoperability, data formats, redundant information exchanges, much rework for interconnecting systems and lack of coordination and collaboration support. Figure 5 shows a typical applications architecture within virtual enterprises.

Figure 5: IT applications architecture typically used within virtual enterprises for product development Interoperability between existing applications has turned to be a key issue in PLM research [START_REF] Shen | Special issue on multi-agent and holonic systems in manufacturing[END_REF], because companies have made investments in solutions to individual problems, and partners are likely to use different software solutions. A way in which PLM systems provide interoperability is by being an information backbone linking applications. The problem is that much information must be converted to other formats in order to pass it from one system to another, which produces some semantic information loss. Additionally, a "pull strategy" is used for product information exploitation (reactive system), and coordination is made manually. Figure 6 depicts the global IT architecture of our framework. For simplicity, only entities relevant to the NPD process are shown. The architecture aims at providing a structured integration infrastructure to: In a distributed environment, there is no single system having all the relevant product information. Thus, global optimization can be achieved by means of collaborative optimization techniques. In our framework, the product is at the core of its lifecycle management, having both reactive and proactive behaviors. Therefore, it is capable of identifying opportunities to be exploited and to suggest how capitalize them. The Product Agent of figure 6 is an Intelligent Agent, which plays the role of an active product. Its environment is composed of both reactive (e.g. other applications) and proactive entities (e.g. human users or other artificial agents) in PLM.

The Product Agent acts as an automated expert connected to all the applications supporting PLM activities, which is capable of identifying events that take place on the environment and to act as a consequence. It has also the ability to communicate with other agents when not enough information is available to make (or suggest) decisions. This also allows managing information property issues, since there is no single agent having all the information but a set of agents, each one having access to a part of it, which communicate with each other to exchange data.

Thus, the whole PLM system moves from a set of isolated automation islands toward a set of systems which interact with each other. In our framework, the notion of system of systems is considered as a potential artifact [START_REF] Mayer | Is system of systems a candidate rationale artifact for entreprise information-intensive system modeling?[END_REF] in order to design a PLM system having proactive behaviors, supported by the underlying applications, information models and processes. Such a system must be able to autonomously exploit information scattered across the virtual enterprise and to communicate with other autonomous and non-autonomous systems representing other's interests and intentions. Therefore, intelligent agents are a suitable technology for implementing the framework, since they entail 4 properties [START_REF] Wooldridge | Intelligent agents: Theory and practice[END_REF]: autonomy, social ability, reactivity and proactivity.

Product information model to support PLM integration

A suitable product information model is the structure on which automated and semi-automated support to business processes is based. It contains product metadata defining the semantics of each product aspect to be registered within PLM databases. A product ontology is useful for identifying relevant data to be recorded during the product's lifecycle, it allows automated processing of product information and enables the interoperability with other systems.

It is very difficult to include every single piece of information concerning a product in all cases, due to the huge range of existing product types and industries. In this work, an extension of the model proposed in [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF][START_REF] Fenves | Cpm2: A core model for product data[END_REF] is presented. Relationships added to the original model are shown in figures with thick black lines. The model reported in [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF][START_REF] Fenves | Cpm2: A core model for product data[END_REF] includes information about product's structure, but it does not consider other aspects of PLM, such as logistics, process planning or manufacturing, aspects covered by two additional models, namely the Process Planning and Logistics models. The most important change made to this portion of the model is the inclusion of a FeatureModel. The original model associated Feature classes directly to the product. However, usually different activities have different views of the same product information, and each one of these points of view is represented by one or more feature models, so features are grouped according to the particular view they belong to. The consideration of feature models as explicit entities allows providing the most suitable model to each user or team, supports the assessment of change impacts and it enables automated reasoning for several purposes (e.g. improvement suggestions, identification of crossactivity optimization opportunities, etc).

Extensions to the Core Product Model

Another addition is that features may also be related to each other (e.g. dependency, mapping between different feature models, etc), through the FeatureFeatureRelation class. This class enables many information exploitation opportunities, since it allows linking features belonging to different feature models. Thus, different teams in charge of different activities may be coordinated, notified (of changes) and advised (about optimization opportunities), as the product development advances. The CustomerNeed entity was also added, in order to model needs as expressed by the customer.

Assembly, Part and LogisticsModel classes are further presented in following sections. The CPM presented here contains only the portion related to the changes we introduced (for further information see [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF]).

Proposed extended Open Assembly Model

The assembly model extends the core product model by adding more information about how components and parts are assembled into subassemblies and final products. Figure 8 shows part of the NIST Open Assembly Model [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF], and some modifications we introduced. In the original model, AssemblyFeature classes were not directly associated with each other, but as a transitive relation with AssemblyFeatureAssociation.

In our model, AssemblyFeatureAssociation is a subtype of FeatureFeatureRelation. This small modification has the advantage of preserving the same general structure defined in the core product model, customizing it across the different specific models.

Manufacturing Process Planning Model

Figure 9 shows the Manufacturing Process Planning Model. For making a product, non-decomposable parts are manufactured separately and then (if needed) they are assembled. Hence only the Part class (a "monolithic" product) is directly "manufacturable". Each operation uses a set of resources (tools, machines, etc.), represented by ManufacturingResource. Manufacturing resources, in turn, have configuration parameters represented by Setup. A single process may not completely produce a feature, either its geometry or its finishing characteristics. Thus, each manufacturing operation is associated to a feature by means of a ManufacturingContribution, which represents the effect of applying the process to produce the feature.

Logistics model

Figure 10 shows a basic logistics model, part of our framework. This model allows the product agent to support logistics activities, such as coordinating supply chain activities, determining lead times and costs, computing raw materials needed for production, etc. An artifact has one or more LogisticsModel, which conveys all this information. The product may have more than one model since it may have several assembly alternatives, or it can be produced in different ways (e.g. buying some components instead of making them). The former aggregates data related to assemblies and subassemblies and the latter is composed of logistics data of a single artifact (either a part or a subassembly). AssemblySubassemblyLogisticData represents information for the relationship of an assembled product with its subassemblies, and PartSubassemblyLogisticData for the relationship between a monolithic part and a subassembly. These data include the Bill of Materials (BOM), costs of assembly operations, lead times, etc.

ProductLogisticData and ComponentLogisticData include logistics information about a single product, assembly or part, as opposed to the previously mentioned classes which apply to relationships between components. Information considered in this case includes component's total cost, total lead time, warehousing information (component's volume, weight and security stocks), quantities available in the warehouse, production state, etc. The model is intended to be generic, so any relevant data may be added according to the needs of a specific application through subtypes of ProductLogisticData and ComponentLogisticData.

Case study

In order to illustrate how the described framework may be applied to specific applications and industries, a particular instantiation for supporting change propagation in the automotive industry is presented as a case study. This case aims at exemplifying the use of the general framework described in this paper, but the framework may be used in other applications and industries.

Consider the product shown in figure 11.a, which is complex in itself, and also because it interacts with other components (e.g. a clutch, an engine, etc). In this example, we assume that some components (shifter and cover, figure 11.b) and the internal mechanism of the product (figure 11.c) are designed by teams belonging to different (partner) companies. In addition to designers, another actor involved in the development of the product is the process planner, who is in charge of creating the manufacturing plan for each part. Considering a concurrent engineering environment, both design teams and the process planner would work in parallel. In this schema, changes introduced by either actor are likely to have impacts on the artifacts handled by the others. Figure 12 shows a collaboration diagram of typical design/process planning interactions within a virtual enterprise and a concurrent engineering setting.

Figure 12: Change propagation collaboration within a virtual enterprise/concurrent engineering environment Each (human) actor or team is assisted by an instance of the product agent, which is also modeled as an actor. Product agents may reside in different companies, and may communicate with each other through network links between partners. In such a setting, these agents have proactive capabilities which reduce the complexity and number of tasks that must be performed by humans in different situations (e.g. when a design change is introduced). For example, if a designer changes the diameter of the primary shaft, changes on the cover will be needed (that must be handled by another designer), as well as on the gears attached to it (which will affect the process plans of these components).

In this situation, the group of messages number 1 starts when the designer changes the primary shaft. This is detected by the corresponding product agent (message 1.1), who can then assess the impact of changes, going from the modified entities to the affected ones (gears, cover, etc), and determining the actors involved (external designer, process planner, etc).

In this way, these actors may be notified of the changes through their corresponding agent instances (messages 1.2, 1.2.1, 1.3 and 1.3.1). These latter agents, in turn, may carry out further assistances exploiting additional information of the specific task. For example, the product agent of the external designer may provide not only the impact of changes, but also some advice on how to overcome the inconsistencies they may have caused. Similarly, the agent assisting the process planner may propose an alternative process plan considering the new state of the product's design, using patterns learned from past experiences. According to user preferences, these change proposals may be automatically executed (message 1.3.4), or just suggested (message 1.3.2). Finally, design adaptations introduced by the external design team may have the same effect on the other one, yielding more interactions in the opposite sense (group of messages number 2). As shown in figure 13, the PrimaryShaft (a Part) has an assembly feature model (OAMFeatureModel), which in turn contains an AssemblyFeature (ShaftSurface). Similarly, the FirstGear part has its own assembly feature model (FirstGearAssemblyModel) containing the GearHole feature. These assembly features (ShaftSurface and GearHole) are in turn related by a mating relation, called PrimaryShaftFirstGearMating. When the primary shaft is changed, the product agent can navigate the model through assembly relations in order to find other components (the FirstGear) and people (the ProcessPlanner actor) affected. In a complete PIM there would be many of these relations between entities (figure Erreur ! Source du renvoi introuvable. is a small partial example). Thus, information may be propagated to the relevant people and actions may be carried out by both humans and intelligent agents for handling these changes.

Conclusions and Future work

In this paper, a first step towards a reference framework for PLM, including a business process model, a product information model and an architecture of applications based on modern technologies, particularly intelligent agents, was presented. The framework aims at providing a suitable model to expand the participation of the product along its complete lifecycle.

One contribution of this work is the integrated treatment of the virtual enterprise setting seen from the product development stages, and the Supply Chain Management integration. Another contribution is the holistic treatment of the three PLM aspects mentioned above. Additionally, the framework includes a generic applications architecture for PLM based on an intelligent product agent, which may be instantiated for specific applications. In addition to that, a hierarchy of interactions that an active product agent must face was also identified. As far as we know, such a structured hierarchy has not been previously defined. An extension of the product information model presented in past works was also presented, including two additional models, namely process planning and logistics.

Future work will include a refinement on the business process model. Additionally, application of the same active-product concept to previous and later stages along manufacturing and logistics administration will be explored, including coordination of production in the shop floor, storage, transportation, procurement, etc. In this approach a product would have an immaterial existence (product knowledge), and a material one (physical product). Finally, a refinement on the applications architecture is needed for assuring the interoperability and product-independence properties of the framework.

Figure 2 :

 2 Figure 2: Overview of Business Process Model for PLM including SCM and Product Development

Figure 3 :

 3 Figure 3: Functional diagram of the Individualized Product Definition

Figure 4 :

 4 Figure 4: Hierarchy of active-product's interventions along its lifecycle 1 Inter-process/multiple company interactions: It involves information exchanges, coordination and information exploitation between processes in different companies within the virtual enterprise. Information ownership as well as high concurrency must be taken into account, since interactions cross a single enterprise barriers. 2 Inter-process/single company interactions: It is related to interactions among business processes within a single company.

Figure 6 :

 6 Figure 6: Proposed IT applications architecture for supporting activeproducts 1 Support development of applications with proactive behaviors. 2 Support enhanced interoperability between heterogeneous systems. 3 Reduce information exchanged in non-structured formats. 4 Improve collaboration and coordination. The model is intended to support automated reason on design and manufacturing information, make improvement suggestions, identify impact of distributed changes and support global optimization. Systems (CAD, CAPP, ERP, etc.) can be physically separated and have different owners. Within our model coordination, collaboration and information exchanges are supported through interfaces exposed as services.In a distributed environment, there is no single system having all the relevant product information. Thus, global optimization can be achieved by means of collaborative optimization techniques. In our framework, the product is at the core of its lifecycle management, having both reactive and proactive behaviors. Therefore, it is capable of identifying opportunities to be exploited and to suggest how capitalize them. The Product Agent of figure 6 is an Intelligent Agent, which plays the role of an active product. Its environment is composed

Figure 7

 7 Figure7shows a UML class diagram representing an ontology for the core part of the information model. The main class is Artifact, which represents a product or component. An artifact has Form and Function, which define some of its physical (Material and Geometry) and behavioral properties. The artifact is also composed of a number of Features, which represent engineering characteristics associated to portions of the product, and that may include function, form and other aspects (e.g. design rationale through the mapping between features and requirements). Mapping to requirements is made through the relationship between the CoreProperty and the Specification classes.

Figure 7 :

 7 Figure 7: Core Product Model and proposed modifications

Figure 8 :

 8 Figure 8: Assembly Model and proposed extensions As in the product model presented in the previous section, the most important modification is the addition of the OAMFeatureModel

Figure 9 :

 9 Figure 9: Manufacturing Process Planning information model A part has one or more ManufacturingFeatureModel, which are specializations of the generic FeatureModel. Each manufacturing feature model is composed of ManufacturingFeature, a subtype of Feature which conveys manufacturing aspects (e.g. tolerances, finishing qualities, etc). Each part has one or more associated ProcessPlan, including manufacturing processes, their sequence, tools and machines to be used, etc. ManufacturingOperation represents

Figure 10 :

 10 Figure 10: Product Logistics information model Each logistics model is composed of AssemblyLogisticData and ProductLogisticData. The former aggregates data related to assemblies and subassemblies and the latter is composed of logistics

Figure 11 :

 11 Figure 11: Solid model of a complex product (gearbox), and its components

Figure 13 :

 13 Figure 13: Partial instance of the Product Information Model for the gearbox Figure 13 shows a partial instantiation of the core product and assembly models of the PIM. This model allows product agent instances to assess the impact of changes, and determine the people and other agents affected by these changes. The gearbox is modeled as an assembly, composed of several parts and subassemblies. The cover is a Part since it is manufactured as a single piece, while the shifter and the internal mechanism are composed of several parts. For simplicity reasons only a few entities and relations are shown in the figure. Actors may be associated to parts or assemblies, which allows identifying people responsible for each component. Three actors are shown in the PIM instance model, namely Designer1, Designer2 and ProcessPlanner.As shown in figure13, the PrimaryShaft (a Part) has an assembly feature model (OAMFeatureModel), which in turn contains an AssemblyFeature (ShaftSurface). Similarly, the FirstGear part has its own assembly feature model (FirstGearAssemblyModel) containing the GearHole feature. These assembly features (ShaftSurface and GearHole) are in turn related by a mating relation, called PrimaryShaftFirstGearMating. When the primary shaft is changed, the

Acknowledgements

This research was jointly developed by the School of Engineering (UNCuyo, Argentina), National Council on Scientific and Technical Research (CONICET, Argentina) and École Nationale Supérieure en Génie de Systèmes Industriels (INPL, France), and partially financed by the PREMER F-599, ARFITEC ARF-08-04 programs, and CONICET PhD fellowship.


Guylaine Le-Jan
email: guylaine.lejan@inria.fr

Yannick Benezeth

Guillaume Gravier

Frédéric Bimbot

A study on auditory feature spaces for speech-driven lip animation

Keywords: features selection, description of acoustic units, lipsync

We present in this paper a study on auditory feature spaces for speech-driven face animation. The goal is to provide solid analytic ground to underscore the description capability of some well-known features with relation to lipsync. A set of various audio features describing the temporal and spectral shape of speech signal has been computed on annotated audio extracts. The dimension of the input feature space has been reduced with PCA and the contribution of each input feature is investigated to determine the more descriptive. The resulting feature space is quantitatively and qualitatively analyzed for the description of acoustic units (phonemes, visemes, etc.) and we demonstrate that the use of some low-level features in addition to MFCC increases the relevance of the feature space. Finally, we evaluate the stability of these features w.r.t. the gender of the speaker.

Introduction

The visual component of speech provides valuable information that undoubtedly increases the intelligibility of speech. Visible speech can compensate for a substantial loss in the speech signal and can also compensate a degraded audio component [START_REF] Benoît | Audio-Visual Intelligibility of French Speech in noise[END_REF]. The intelligibility of natural speech can also be increased with synthetic faces. It is also important to note that a synthetic face with a realistic lip animation increases the human-computer interaction. In order to obtain natural looking and realistic animation of virtual characters, the lip movements shall be consistent with the corresponding speech. It is possible to obtain such an animation manually by adjusting, frame after frame, the control parameters of the synthetic face lips. The results are then very realistic, but this method requires a substantial workload and does not allow real time avatar animations. Usually, methods to automatically generate lip movements are based on speech recognition approaches. In its basic form, the synchronization of lip movements with the speech signal implies the detection of the phoneme sequence (e.g., [START_REF] Park | Real-Time Continuous Phoneme Recognition System Using Class-Dependent Tied-Mixture HMM With HBT Structure for Speech-Driven Lip-Sync[END_REF]). Phonemes are then mapped to a set of visemes. A viseme is a lip configuration of the animated character. The recognition system can also be trained to directly detect visemes (e.g., [START_REF] Foo | Recognition of Visual Speech Elements Using Hidden Markov Models[END_REF]). In both cases, the mouth movements are monitored with a discrete set of positions: the visemes.

Because of the large number of applications that require speech-lip synchronization, it is now a well-studied problem. In human-computer interactions, virtual characters with realistic lip animation allow to interact with users in a friendly and natural way. Video games, eLearning, instant messaging or 3D animation productions (movies, commercials, etc.) need also to generate lip animations. All these lipsync applications have different constraints in terms of computational resources, real-time or animation quality. A virtual character in mobile device interfaces can obviously tolerate an approximate lip animation but needs real-time animations with high computational contraints. The opposite observation can be drawn concerning the 3D animation productions.

As explained previously, a lipsync system is often based on the recognition of a speech acoustic unit, i.e., determining a sequence of phonemes or visemes. In that case, the recognition system can be broadly divided into two main components: the feature extraction step and the classification step. HMM [START_REF] Park | Real-Time Continuous Phoneme Recognition System Using Class-Dependent Tied-Mixture HMM With HBT Structure for Speech-Driven Lip-Sync[END_REF], neural networks [START_REF] Zorić | Real-time language independent lip synchronization method using a genetic algorithm[END_REF], Bayesian estimation, mixture models of Gaussian or vector quantization can be used to map acoustic vectors to the corresponding sequence of acoustic units. This classification step can employ a variety of feature spaces. This representation shall characterize the information carried by the acoustic signal in a convenient form. The well-known mel frequency cepstral coefficients (e.g., [START_REF] Park | Real-Time Continuous Phoneme Recognition System Using Class-Dependent Tied-Mixture HMM With HBT Structure for Speech-Driven Lip-Sync[END_REF], [START_REF] Zorić | Real-time language independent lip synchronization method using a genetic algorithm[END_REF]), the LPC, the Rasta-PLP, or even low-level features such as the zero crossing rate or the energy [START_REF] Malcangi | Audio Based Real-Time Speech Animation of Embodied Conversational Agents[END_REF] can be used. In this paper, we are interested in measuring the contribution of low-level features to lip-position in order to design an optimal feature space for lipsync. The separability of the desired classes (set of visemes or phonemes) in a given feature space is not evaluated based on a recognition rate obtained with a classification but the feature space is directly analyzed.

Feature selection methods have been proposed in the literature (e.g., [START_REF] Bocchieri | Discriminative feature selection for speech recognition[END_REF], [START_REF] Saon | Minimum Bayes error feature selection for continuous speech recognition[END_REF]). However, these methods aim at selecting features for speech recognition applications. As far as we know, no such study is dedicated to lipsync systems. This experimental study will helpfully contribute to the acoustic feature selection during the design process of a speech-driven face animation system. We present in this paper a selection of the more descriptive features with PCA. The quality of the determined feature space is then evaluated with regard to the separability of the given classes. The stability of these features considering the gender of the speaker is also considered.

We first present in this paper the experimental protocol with the description of our speech database and the features analyzed. Then, we present a method to identify a new feature space and we evaluate qualitatively and quantitatively the selected feature space.

Experimental protocol 2.1. Speech database

The auditory features validity has been performed on 6 continuous speeches of French native-speakers (3 male and 3 female speakers) extracted from the CPROM corpus [START_REF] Avanzi | C-PROM. An annotated corpus for french prominence studies[END_REF]. Two phonetic experts have annotated this corpus following a strict protocol. The duration of the corpus is about 11 minutes for male and female speakers. The reason for choosing audio extracts from the CPROM corpus rather than from larger databases relies in the high accuracy of the phonemic annotation.

We use a set of 33 phonemes to describe the French language. These phonemes are listed in Tables 1 and2 using the X-SAMPA notation. We also use a set of visemes composed of 17 classes (16 plus the neutral viseme). This viseme classification has been performed by Benoît et al. [START_REF] Benoit | A set of French visemes for visual speech synthesis[END_REF] and is illustrated in Table 1.

Table 1: Phonemes aggregated into 16 visemes

Visemes Phonemes V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 a i y, u, 2, 9, o, o~, @, H e, E, U~ O a~ p,b,m t,d,n k,g f,v s,z S,Z L R w J
We also present in Table 2, nine classes of phonemes aggregated according to the mode of articulation (oral or nasal vowels, voiced or voiceless plosives, voiced or voiceless fricatives, liquids, nasal occlusion and mid-vowels). This classification will be used in the results analysis section.

Acoustic speech features

We present in this part the list of features analyzed. A large set of features has been used. They characterize the temporal or the spectral shape of the speech signal, and also the energetic and perceptual features:

-Temporal features:

-Zero crossing rate (ZCR), -Envelope shape statistics: centroid, deviation, skewness and kurtosis (ESS i ), -Spectral features:

-Fundamental frequency (f 0 ), -The 4 first formants (f i ) and bandwidths (b i ), -Ratio of formant frequencies:

f 2 /f 1 , f 3 /f 1 , f 4 /f 1 ,
-39 MFCC (13 static coefficients, and their first and second order derivatives), -Spectral shape statistics: centroid, deviation, skewness and kurtosis (SSS i ), -Spectral slope (SS), -Spectral variation (SV), -Spectral flux (SFX), -Spectral roll-off (SR), -Spectral decrease (SD), -Energy features:

-Energy (en), -Log of energy (logen), -Perceptual features:

-Perceptual sharpness (PSH).

More information about these auditory features can be found in [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF]. Because of the large number of auditory features, it was interesting to propose a new feature space with a lower space dimension and select the more relevant one for acoustic unit description.

Feature space definition

We present in this part a selection of an optimal feature space to describe acoustic units. For this, a PCA has been performed in order to analyze the description capability of each feature and reduce the multidimensional data set. At the opposite of other data analysis method, such as LDA, PCA disregards the information on the class of each example. This method is more appropriate for our study because the analysis is not dedicated to the specific problem of phoneme recognition. PCA find the projection that maximizes the variance of the projected data and provide principal components defined as linear combination of the original variables. Each extracted factor can be interpreted through the loadings of the initial variables. The first factors are the most representative of the total variance of the original data. In order to reduce the feature space dimension, we retain only the most contributive factors. In the next part of the study, we retain the first twenty factors such as they explain near 70% of the total variance.

Then, to determine the best set of auditory features, the contribution (CT) of each feature is calculated on the set of selected principal factors (such as 20 factors). The contribution CT i, of the feature i to the N selected principal factors is defined as follows:

     N N i i x CT 0 0 2      , (1)
where x i,α is the coordinate of feature i on the axis α and λ α the eigenvalue of the principal factors α.

Features with a high CT show a significant contribution to the realization of the new feature space and also to the acoustic unit description.

Feature space analysis

We have defined in the previous section a feature space composed of the first twenty factors of the PCA. In order to evaluate the relevance of this feature space, we first present in this part a quantitative comparison of this feature space with the original feature space composed of all auditory features listed in section 2.2 and the usual MFCC-based feature space. Then, we present a qualitative analysis about the observations disposition in the feature space. The objective is to detect outliers and potential problems. We also evaluate the stability of this feature space w.r.t. the gender of the speaker

Quantitative analysis

First, we compare the capacity of the contributive factors to aggregate observations into phonemes, articulation modes and visemes. To do that, we compute features on the annotated speech database. We obtain several observations for each acoustic unit and conserve only the median value to characterize it. Consequently, we have a distribution of elements that belong to various classes (phonemes, articulation modes or visemes). We use the compactness value α and the separation value β defined as:

              k i n j i j i i i x x n k 0 0 2 , 1 1  , (2)            k i k i j j x x j i e k k 1 , 1 2 1 1  , ( 3
)
where k is the number of classes, n i the number of observations in class i, x i,j the observation j in class i and x i the centroid of the class i. The compactness α is a generalized definition of the variance of a vector dataset and the separation β is the pair-wise distance among the class centroids. As the objective is to minimize the distances among the data points in individual classes and to maximize the distances between classes, the smaller α is, and the bigger β is, the easier it is to separate acoustic observations into the desired classes. We present in Table 3 the compactness values of the phoneme, viseme and articulation mode classes in different feature spaces. We present in Table 4 the separations values. We compare the original feature space of dimension 68 (called FS1) composed of all features listed in section 2.2, the usual feature space composed of the 39 MFCC (called FS2), and the feature space composed of the twenty most representative factors (called FS3).

The smaller the compactness is, and the higher the separation is, the easier it is to separate acoustic observations into appropriate classes. Consequently, few observations can be drawn from the Tables 3 and4. First, it is relevant to note that the visemes are better distributed than the phonemes within each space. This observation has already been demonstrated (e.g., [START_REF] Bozkurt | Comparison of phoneme and viseme based acoustic units for speech driven realistic lip animation[END_REF]) from the recognition error rate of phonemes and visemes. Here, we validate this observation from the feature space analysis. Secondly, the articulation mode representation obtains good separability and compactness values. This result is intuitive because observations in the same articulation mode class are likely to share the same acoustic characteristics. Nevertheless, the major differences between Table 1 and 2, show that the articulation mode information is not sufficient for lip animation. This information can still be used to constraint the mouth position with regard to the current articulation mode. Then, another interesting observation is that the use of low-level features in complement to the usual MFCC does not directly improve the feature space representation quality. Actually, the separatbility of the analyzed acoustic units in Table 3 and 4 presents better result in the MFCC-based feature space. However, the dimension reduction with PCA maximizes the variance of the projected data and consequently conserves information while the dimension reduction optimizes the feature space representation.

Qualitative analysis

In order to analyze in depth the disposition of the observations in the feature space, we perform a hierarchical clustering onto the reduced feature space. We aggregate phonemes classes with a bottom-up agglomerative hierarchical clustering [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF] with the average linkage strategy. The distance between each phoneme is the Euclidian distance. The goal is twofold: first we want to evaluate if the phoneme aggregation is related to the articulation modes and then graphically detect if some phonemes have an unexpected behavior. We present in Figure 1 the dendrogram illustrating the phoneme clustering in the reduced feature space. Clearly, phoneme aggregation is closely related to the articulation modes. Observations in the same articulation mode class are likely to share the same acoustic characteristics. Phonemes aggregate to form clusters composed of vowels, voiced and voiceless plosives, voiced and voiceless fricatives and nasal occlusions. Nevertheless, the distinction between oral and nasal vowels is not clear. Then, it is possible to observe that liquids are spread into several groups of phonemes and specific phoneme v is especially ill represented in the distribution. Then we present in Table 5 and in Figure 2, an analysis on the influence on the speaker gender. The Table 5 presents the compactness and the separation values of the phoneme classes on the reduced feature space while the Figure 2 presents the clustering of phonemes also on the reduced feature space. First, the compactness and separation values show a slight influence of the speaker gender. While the compactness value is nearly equal, the separation value is higher with female speakers. With a male speaker, mid-vowels are spread into several phoneme classes and phoneme f is unexpectedly associated with the voiceless plosives.

This qualitative analysis permits to draw some generalities about the design of speech-driven lip animation systems. First, we again confirm the relevance of the reduced feature space. A feature space composed of the usual MFCC with some complementary low-level features, after dimension reduction, characterizes in a coherent form the phoneme groups. Then, we identify some specific phonemes that are inconsistently represented in the feature space. The resulting lip animation would be consequently unrealistic for these occurrences. Nevertheless, the a priori knowledge of these difficulties should influence the design process of lip animation systems. For example, the weight of the phonetic context of these phonemes could be a priori increased to reduce the negative impact on the lip trajectories.

Conclusions

We have presented in this paper a study on feature spaces for speech-driven face animation. A set of various audio features describing the temporal and spectral shape of speech signal has been computed on annotated audio extracts. The dimension of the input feature space has been reduced with PCA and the relative contribution of the input features is investigated. The resulting feature space is quantitatively and qualitatively analyzed in a speech-driven animation context. We quantitatively and qualitatively demonstrate that the use of some low-level features in addition to MFCC increases the relevance of the feature space when the dimension of this feature space is reduced. We also evaluate the stability of these features w.r.t. the gender of the speaker.

Figure 1 :

 1 Figure 1: Dendrogram illustrating the clustering of phonemes in the reduced feature space.

Figure 2 :

 2 Figure 2: Dendrogram illustrating the clustering of phonemes in the reduced feature space with male speakers (on the left) and female speakers (on the right).

Table 2 :

 2 Phonemes aggregated into 9 articulation modes.

	Classes	Phonemes
	oral vowel (OV)	a, O, o, u, 9, 2, @, y, E, e, i
	nasal vowel (NV)	a~, o~, U~
	voiced plosive (VP)	b, d, g
	voiceless plosive (VLP)	p, t, k
	voiced fricative (VF)	v, z, Z
	voiceless fricative (VLF)	f, s, S
	liquid (L)	R, l
	nasal occlusion (NO)	m, n
	mid-vowel (MV)	j, H, w

Table 3 :

 3 Compactness values

		Phonemes	Visemes	Articulation
				modes
	FS1	0.010	0.004	0.003
	FS2	0.007	0.003	0.002
	FS3	0.005	0.002	0.001

Table 4 :

 4 Separation values

		Phonemes	Visemes	Articulation
				modes
	FS1	0.167	0.210	0.275
	FS2	0.312	0.354	0.448
	FS3	0.376	0.422	0.498

Table 5 :

 5 Compactness and separations values illustrating the influence of the speaker gender.

		Males	Females
	Compactness 0.004	0.005
	Separation	0.383	0.404


Piotr Biler
email: biler@math.uni.wroc.pl

Cyril Imbert
email: imbert@ceremade.dauphine.fr

Grzegorz Karch
email: karch@math.uni.wroc.pl

Partial Differential Equations Barenblatt profiles for a nonlocal porous medium equation

We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit selfsimilar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem.

Résumé Solutions auto-similaires pour une équation des milieux poreux non locale

Cette note est consacrée à l'étude d'une généralisation non locale de l'équation des milieux poreux. Plus précisément, on obtient des formules explicites de solutions auto-similaires à support compact qui ressemblent fortement aux solutions de type Barenblatt. On donne aussi un argument formel qui permet d'obtenir des estimations L p des solutions faibles du problème de Cauchy.

Version française abrégée

Nous considérons le problème de Cauchy pour l'équation non locale suivante

avec m > 1, x ∈ R d , α ∈ (0, 2), t > 0, à laquelle on ajoute une condition initiale u(0, x) = u 0 (x). Ici ∇ β est un opérateur intégral singulier généralisant le gradient usuel (β = 1) et lié au laplacien fractionnaire. Le résultat principal de cette note sont des formules explicites de solutions auto-similaires qui se propagent à une vitesse finie.

Théorème 1 (Solutions auto-similaires) La fonction

(ainsi que ses translations en x) est une solution auto-similaire de l'équation (1) pour R > 0 quelconque et une constante C > 0 convenable.

Ensuite, nous présentons un calcul formel qui permet d'obtenir des estimations en norme L p (R d ) des solutions faibles du problème de Cauchy, en particulier celles construites par Caffarelli et Vázquez [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF] for all t > 0.

Introduction

We study a nonlocal generalization of the porous medium equation

∂ t u -∇ • u∇ α-1 |u| m-1 = 0, (1)
where m > 1, α ∈ (0, 2), x ∈ R d , t > 0, supplemented with an initial condition u(0, x) = u 0 (x).

(2)

The pseudodifferential (vector-valued) operator ∇ β in ( 1) is defined via the Fourier transform as ∇ β u = F -1 (iξ|ξ| β-1 F u). This definition is consistent with the usual gradient: ∇ 1 = ∇; the components of ∇ 0 are the Riesz transforms; moreover we have

∇ • ∇ α-1 = ∇ α 2 • ∇ α 2 = -(-∆) α 2 , where (-∆) α 2 denotes the fractional Laplace operator: (-∆) α 2 u = F -1 (|ξ| α F u).
It can also be defined by real analysis tools as follows ∇ α-1 u = ∇I 2-α u, where I β for β ∈ (0, d) is an integral smoothing operator, called the Riesz potential (see [12, Ch. V])

I β (u)(x) = -C β u(x + z) |z| d-β dz with some C β > 0. Note that then ∇ α-1 u(x) = ∇I 2-α (u)(x) = (d + α -2)C 2-α (u(x + z) -u(x)) z |z| d+α dz, α ∈ (0, 2).
Eq. ( 1) can be interpreted as a transport equation of the type ∂ t u = ∇ • (uv) for some velocity vectorfield v which is a potential; more precisely, v = ∇p where p = I 2-α (|u| m-1 ). It can be interpreted as a nonlocal pressure in the case of nonnegative initial data. Then, of course (-∆) 2-α 2 p = |u| m-1 , see [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF], [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF] for that notation.

Notice that for α = 2 and nonnegative initial data we recover the Boussinesq equation (m = 2), and the usual porous media equation (m > 1):

∂ t u = ∇ • u∇ u m-1 , t > 0, x ∈ R d .
Recently, L. Caffarelli and J. L. Vázquez ([3], [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF]) studied equation [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF] in the case m = 2. They proved the existence of weak solutions for nonnegative bounded integrable initial data with exponential decay at infinity. They also treat the case of bounded and compactly supported initial data, which propagate with finite speed. It is shown in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF] that self-similar solutions can be constructed by considering an obstacle problem for the fractional Laplace operator. In this note, we contribute to those results constructing explicit compactly supported self-similar solutions. Moreover, we show a kind of hypercontractivity estimates, i.e. the optimal decay in L p of general solutions with u 0 ∈ L 1 (R d ).

Results on equation [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF] in this note are multidimensional generalizations of those obtained in [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF] for a model of the dynamics of dislocations in crystals (for the integral of u when d = 1 and m = 2). The structure of [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF] suggests that it should enjoy the conservation of mass and some comparison properties as was shown in [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF]. For an analysis of a related nonlocal equation, see [START_REF] Imbert | A higher order non-local equation appearing in crack dynamics[END_REF].

Complete proofs of all results announced in this note will be published in [START_REF] Biler | Nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF].

Self-similar solutions

The equation ( 1) has the following scaling property:

if u(t, x) is a solution, so is ℓ dλ u(ℓt, ℓ λ x)
for each ℓ > 0 and λ = 1 d(m-1)+α . We look for nonnegative solutions that are invariant under that scaling, i.e. of the following form

u(t, x) = 1 t dλ Φ α,m x t λ , where λ = 1 d(m -1) + α , (3)
for a function Φ α,m : R d → R + satisfying the following nonlinear and nonlocal equation in

R d -λ∇ • (yΦ α,m ) = ∇ • Φ α,m ∇ α-1 Φ m-1 α,m . (4)
Before stating our main result, we recall the definition of weak solutions for (1) introduced in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF] in the case m = 2.

Definition 2.1 (Weak solutions) A function u : (0, T ) × R d → R is a weak solution of (1) in Q T = (0, T )×R d submitted to the initial condition u(0, x) = u 0 (x) if u ∈ L 1 (Q T ), I 2-α (|u| m-1 ) ∈ L 1 (0, T ; W 1,1 loc (R d )), u∇I 2-α (|u| m-1 ) ∈ L 1 (Q T ) and u(ϕ t -∇I 2-α (|u| m-1 ) • ∇ϕ) dx dt + u 0 (x)ϕ(0, x) dx = 0 (5)
for each test function ϕ ∈ C 1 (Q T ) such that ϕ has compact support in the space variable x, and vanishes near t = T .

Theorem 2.2 (Self-similar solutions) For each α ∈ (0, 2], m > 1 and R > 0, the function

Φ α,m (y) = k(R 2 -|y| 2 ) α 2 + 1 m-1 with k = d d(m -1) + α Γ d 2 2 α Γ 1 + α 2 Γ d+α 2 (6)
is a solution of (4). Consequently, the function

u(t, x) = t - d d(m-1)+α k R 2 -|x| 2 t - 2 d(m-1)+α α 2 + 1 m-1 (7)
is a weak solution of (1), satisfies the equation in the pointwise sense for |x| = Rt When α = 2, we recover the classical Kompaneets-Zel'dovich-Barenblatt-Pattle formulas, see, e.g., [START_REF] Vázquez | The Porous Medium Equation. Mathematical Theory[END_REF]. Note also that given M > 0 there exists a unique R > 0 such that Φ α,m (y) dy = M . Remark 2.3 As mentioned above, self-similar solutions of (1) have been proved to exist in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF] by studying the following obstacle problem for the fractional Laplacian

P ≥ Φ, V = (-∆) α 2 P ≥ 0, either P = Φ or V = 0,
with α ∈ (0, 2) and Φ(y) = C -a|y| 2 . The novelty of our approach is that we exhibit the explicit solution of this obstacle problem: P (y) = I α (Φ α,2 ) (y), where I α is the Riesz potential, and Φ α,2 is defined in ( 6) with m = 2 and a suitable R > 0.

Those explicit self-similar solutions express one of the most important features of the porous medium equation: the property of finite propagation speed. In the case of the classical porous medium equation (α = 2), this property is established using comparison with suitably large self-similar solutions, cf. [START_REF] Vázquez | The Porous Medium Equation. Mathematical Theory[END_REF]. For the generalized porous medium equation ( 1) with m = 2, special supersolutions have been used for comparison, cf. [START_REF] Caffarelli | Asymptotic behaviour of a porous medium equation with fractional diffusion[END_REF].

The proof of Theorem 2.2 is based on an application of the following fundamental technical fact.

Lemma 2.4 For all β ∈ (0, 2) and γ > 0, we have

I β (1 -|y| 2 ) γ 2 + =        C γ,β,d × 2 F 1 d -β 2 , - γ + β 2 ; d 2 ; |y| 2 for |y| ≤ 1, Cγ,β,d |y| β-d × 2 F 1 d -β 2 , 2 -β 2 ; d + γ 2 ; 1 |y| 2 for |y| ≥ 1, (8)
with

C γ,β,d = 2 -β Γ( γ 2 +1)Γ( d-β 2 ) Γ( d 2 )Γ( β+γ 2 +1) and Cγ,β,d = 2 -β Γ( γ 2 +1)Γ( d-β 2 ) Γ( d 4 )Γ( d+γ 2 +1
) , where 2 F 1 is the hypergeometric function. The verification of (8) consists in passing to Fourier transforms and calculating certain integrals (the so called (Sonine-)Weber-Schafheitlin integrals) involving Bessel and hypergeometric functions 2 F 1 , cf. [START_REF] Magnus | Formulas and theorems for the special functions of mathematical physics[END_REF].

Proof of Theorem 2.2. Let φ α (y) = 1 -|y| 2 α 2 + . Observe that φ α ∈ L 1 (R d ). We next show that I 2-α (φ α ) ∈ W 1,1 loc (R d )
. By Lemma 2.4 with γ = α ∈ (0, 2), β = 2 -α, and 2 F 1 (a, -1; c; z) = 1 -a c z, we get

I 2-α (φ α )(y) =        C α,2-α,d 1 - d + α -2 d |y| 2 if |y| ≤ 1, Cα,2-α,d |y| 2-(d+α) 2 F 1 d + α 2 -1, α 2 ; d + α 2 ; 1 |y| 2 if |y| ≥ 1,
which is a locally integrable function. Recalling that ∇ α-1 = ∇I 2-α , we then deduce by the chain rule that for

y ∈ B 1 , ∇ α-1 (φ α )(y) = -(d K α,d ) -1 y where K α,d is defined in Corollary 2.5 below. For |y| ≥ 1, one uses ∂ ∂z 2 F 1 (a, b; c; z) = ab c 2 F 1 (a + 1, b + 1; c + 1; z), hence ∇ α-1 (φ α ) is locally integrable. To conclude, we remark that φ 1 m-1 α (y)∇ α-1 φ α (y) = -φ 1 m-1 α (y) (K α,d d) -1 y, for all y ∈ R d , because φ α (y) = 0 for |y| ≥ 1.
Hence, scaling the variables, we immediately obtain that the function Φ α,m defined in ( 6) satisfies

∇ α-1 (Φ m-1 α,m ) = -λy for |y| < R. Now, for all y ∈ R d the identity -λyΦ α,m = Φ α,m ∇ α-1 Φ m-1 α,m
follows because Φ α,m (y) = 0 for all |y| ≥ R, so (4) holds with λ = 1 d(m-1)+α and k = dλ K α,d defined in [START_REF] Getoor | First passage times for symmetric stable processes in space[END_REF]. It is straightforward to verify using (4) that u given by formula ( 7) is a weak solution of (1) in each strip (t 0 , T ) × R d , 0 < t 0 < T < ∞. Moreover, the Hölder continuity is easy to check.

The following known result (with an important probabilistic interpretation) proved by Getoor [6, Th. 5.2], see also [START_REF] Landkof | Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften. Band[END_REF]App.] for a related calculation, is an immediate consequence of Lemma 2.4. For its proof, it suffices to use the relation (-∆)

α 2 = ∆I 2-α . Corollary 2.5 For each α ∈ (0, 2], the identity K α,d (-∆) α 2 1 -|y| 2 α 2 + = -1 in B 1 holds with the explicit constant K α,d = Γ( d 2 ) 2 α Γ(1+ α 2 )Γ( d+α 2 )
.

The Cauchy problem and asymptotics

We now briefly discuss the questions of the existence of weak solutions and their uniqueness in the case m = 2. In [START_REF] Biler | A nonlinear diffusion of dislocation density and self-similar solutions[END_REF], viscosity solutions have been considered which permitted the authors to prove regularity and uniqueness of solutions to (1) in one space dimension. In higher dimensions, a construction of mild solutions is achieved in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF] through a parabolic regularization of (1) with a suitable cutoff of the singular kernel of I 2-α , and then a passage to the limit. The uniqueness of weak solutions and, a fortiori, the validity of the full comparison principle seem to be difficult questions, cf. a discussion in [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF]. Another construction of weak solutions to (1) with m > 1 as limits of mild solutions of parabolically perturbed equation (1) will be published in [START_REF] Biler | Nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF].

Formal computation 3.1 (Decay of solutions for the Cauchy problem) Suppose that for 0

≤ u 0 ∈ L 1 (R d )∩L ∞ (R d ). Then L p (R d ) norms (1 ≤ p < ∞)
of any sufficiently regular and global in time nonnegative weak solution u of (1)-( 2) such that u(t, x) dx = u 0 (x) dx decay algebraically

u(t) p ≤ C(d, α, m, p) u 0 α+d(m-1)/p α+d(m-1) 1 t - d d(m-1)+α (1-1/p) for all t > 0. (9)
Our computation below can be applied, for example, to weak solutions constructed by Caffarelli and Vázquez [START_REF] Caffarelli | Nonlinear porous medium flow with fractional potential pressure[END_REF], who studied problem (1)-( 2) with m = 2 and with initial conditions satisfying 0 ≤ u 0 (x) ≤ Ce -c|x| for some constants c, C and all x ∈ R d .

Formal proof of [START_REF] Landkof | Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften. Band[END_REF]. In order to prove the announced L p estimates of solutions (similar to those for degenerated partial differential equations like the porous medium equation in e.g. [START_REF] Vázquez | Smoothing and Decay Estimates for Nonlinear Diffusion Equations[END_REF], [5, Ch. 2]), we recall the Stroock-Varopoulos inequality for q ≥ 1 |w| q-2 w(-∆)

α 2 w dx ≥ 4(q -1) q 2 ∇ α 2 |w| q 2 2 dx ( 10
)
valid for each w ∈ L q (R d ) such that (-∆) α 2 w ∈ L q (R d ). Note that the constant in [START_REF] Liskevich | Some problems on Markov semigroups, Schrödinger operators[END_REF] is the same as for the usual Laplacian operator -∆ (i.e. α = 2). The proof is given, e.g., in [START_REF] Liskevich | Some problems on Markov semigroups, Schrödinger operators[END_REF]Prop. 1.6] and [10, Th. 2.1 combined with (1.7)].

We will also need the following Nash inequality

v 2(1+ α d ) 2 ≤ C N ∇ α 2 v 2 2 v 2α d 1 (11)
valid for all functions

v with v ∈ L 1 (R d ), ∇ α 2 v ∈ L 2 (R d ) with a constant C N = C(d, α
). The proof of ( 11) for d = 1 can be found in, e.g., [START_REF] Karch | On the convergence of solutions of fractal Burgers equation toward rarefaction waves[END_REF]Lemma 2.2], and this extends easily to the general case d ≥ 1.

Moreover, we will need the Gagliardo-Nirenberg type inequality: for p > 1 and p ≥ m -1, The computation, which lead to [START_REF] Landkof | Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften. Band[END_REF], consists in getting a differential inequality of the form d dt |u| p dx ≤ -K u a p u -b 1 for some positive constant K and where a and b appear in [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]. Multiply (1) by u p-1 with p > 1, integrate by parts, and use the relation

u a p ≤ C N ∇ α 2 |u| r 2 2
∇ • ∇ α-1 = -(-∆) α 2 to get 1 p d dt u p dx = -(p -1) u p-1 ∇u • ∇ α-1 (u m-1 ) dx = - p -1 p u p (-∆) α 2 u m-1 dx ≤ - 4(p -1)(m -1) (p + m -1) 2 ∇ α 2 u p+m-1 2 2 2
after applying the Stroock-Varopoulos inequality [START_REF] Liskevich | Some problems on Markov semigroups, Schrödinger operators[END_REF] with w = u m-1 and q = p m-1 + 1. To estimate the right hand side of the above inequality, we use [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]. We finally have: .

Of course, this is sufficient to get the conclusion of Theorem 3.1 for each 1 ≤ p < ∞ since 1

α+d

  dans le cas m = 2 et telle que |u 0 (x)| ≤ Ce -c|x| pour deux constantes C et c. Calcul formel 1 (Asymptotique du problème de Cauchy) Étant donnée une fonction 0≤ u 0 ∈ L 1 (R d ) ∩ L ∞ (R d ), les normes L p (R d ), 1 ≤ p < ∞,des solutions faibles u tendent vers 0 quand t → ∞ avec le taux algébrique suivant u(t) p ≤ C(d, α, m, p) u 0

1 d

 1 (m-1)+α , and is min α 2(m-1) , 1 -Hölder continuous at the interface |x| = Rt 1 d(m-1)+α .

1 ) 2 , 2 , 1 and u r 2 ≤ u δ p u 1 -δ 1 ,

 122111 and r = p + m -1. This inequality is a consequence of the Nash inequality[START_REF] Magnus | Formulas and theorems for the special functions of mathematical physics[END_REF] written for v = |u| r and two Hölder inequalities for the L q norms: u p ≤ u γ r u 1-γ which hold with γ = pthe above three inequalities, we get[START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF].

d dt |u| p dx ≤ -K u a p u -b 1 5 with K = 1 CN 2 . 1 ap - 1 .

 51211 The above inequality leads to the differential inequalityd dt f (t) ≤ -KM -b f (t) a pfor the function f (t) = u(t) p p , M = u 0 1 , and a/p > 1, which immediately gives the algebraic decay of the L p norms for p ≥ m -1:f (t) ≤ K a p -1 M -b t-Finally, we obtain the desired estimate with C(d, α, m, p)

a p - 1 =

 1 d(p-1) d(m-1)+α , and interpolating between L 1 (R d ) and L p (R d ) with p sufficiently large.Remark 3.2 In our recent work[START_REF] Biler | Nonlocal porous medium equation: Barenblatt profiles and other weak solutions[END_REF], we present a more subtle iterative argument, which allows us to show that also u(t) ∞ ≤ C u 0 for all t > 0.

Acknowledgments. The preparation of this paper was supported by the Polish Ministry of Science (MNSzW) grant N201 418839, and a PHC POLONIUM project 0185, 2009-2010, and the Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007-2013 funded by European Regional Development Fund (Ph.D. Programme: Mathematical Methods in Natural Sciences). The authors wish to thank Juan Luis Vázquez, Régis Monneau and Jean Dolbeault for fruitful discussions they had together.