Olivier Palombi

Guillaume Bousquet

David Jospin

Lionel Revéret

François Faure

Anonymous

My Corporis Fabrica : a Unified Ontological, Geometrical and Mechanical View of Human Anatomy

A new anatomical database, My Corporis Fabrica (MyCF), is presented. It extends the reference anatomical ontology FMA (the Foundational Model of Anatomy) with the possibility to coherently integrate 3D geometrical data, as well as biomechanical parameters. The purpose of this extension is to allow the user to intuitively create a patientspecific 3D representation from a formal description of anatomical entities and also to automatically export this description to test a physical simulation. The main contribution of MyCF consists in the formalization of a comprehensive database structure implemented in MySQL, linking canonical description of anatomical entities with reality-grounded instances of such description in terms of geometrical data and physical attributes. The principle of ontology modeling inherited from FMA is maintained in order to guarantee a close consistency between the two databases. A detailed illustration is given for the knee. In this example, the canonical description of anatomical entities is developed, including the location of attachment of ligaments as well as their stiffness, using an intuitive 3D GUI. Finally, an instance of the knee based on this description is automatically augmented with the canonical part information and then exported to a physical simulator. This example shows the benefit of coupling an exhaustive description of the anatomical structures with a physical simulation to study the dynamic effect of ligaments on the stability of the knee articulation.

Introduction

Anatomical models of human body are required in an increasing number of domains, from the teaching of anatomy to biomechanical simulation. Typically, accurate 3D representation of human anatomy is needed in application such as surgery simulator, realistic prototyping of cloth or representation of avatars in 3D virtual communities. The 3D representation of internal anatomy entities is primarily accessible from medical imaging such as CT or MRI tomography. These imaging techniques provide a raw volume and need to be segmented to identify individual anatomical entities. Even if some powerful methods have been proposed, this segmentation step is still time consuming as it cannot be fully automatic. An alternative approach to volumetric segmentation is to use 3D models designed by artists, but this involves a risk of loss in realism. Therefore, the need for a validated anatomical database of the human body linked with 3D representation is rapidly growing. This database should provide high-level information such as ontological relationships, as well as geometrical data for graphics applications and physical parameters for simulations. It should be extensible and anatomically relevant to integrate new conceptual entities, physical instances and characterization parameters. This is the goal of our approach with the design of the My Corporis Fabrica (MyCF) database. Specifically, MyCF extends FMA (the Foundational Model of Anatomy) [START_REF] Rosse | Motivation and organizational principles for anatomical knowledge representation: the digital anatomist symbolic knowledge base[END_REF] to provide additional data to create geometrical and mechanical models of anatomy. It is based on a novel database structure which considerably facilitates its extensibility. The remainder of this paper is organized as follows. In section 2, we review previous work in the domain. The design of MyCF database is presented in section 3. Implementation details are discussed in section 4. An overview of potential applications is presented in section 5. Concluding remarks are presented in section 6.

Previous work

Integrating patient-specific content into computational resources requires the use of formal and logical representation of anatomical knowledge. The field of computing science focused on structuring knowledge is called ontology. Ontologies of anatomical knowledge has been widely performed with a high level of details and complexity. The FMA can be presented as the most sophisticated anatomical ontology available today for the scientific community [START_REF] Rosse | A reference ontology for biomedical informatics: the foundational model of anatomy[END_REF]. Nonetheless, the technical complexity involved in formalizing the anatomy can not handle general variability in any easy way. Specific approaches have to be imagined to address this issue. This lacking link is the key point in the emerging research field of semantic 3D media [3] and more specifically for medical applications [START_REF] Magnenat-Thalmann | 3d anatomical modelling and simulation concepts[END_REF][START_REF]Webpage 3D Anatomical Human project[END_REF].

The Digital Anatomist Information System is based on FMA to link graphical and symbolic representations of anatomy [START_REF] Brinkley | The digital anatomist distributed framework and its applications to knowledge-based medical imaging[END_REF]. A static 3D content (pictures of 3D models) is available to illustrate the FMA's concepts. But 3D data are not completely integrated in the ontology. Starting from FMA to embed ontology into specific computational resources seems to be relevant regarding the level of compliance with modeling principles used in FMA [START_REF] Zhang | Law and order: assessing and enforcing compliance with ontological modeling principles in the foundational model of anatomy[END_REF]. FMA is implemented with an open source ontology editor and knowledge-base framework called Protégé which is the only way to manage the ontology. Data are stored in a MySQL table but direct access to query is not intuitive. A querying agent exists for FMA but the flexibility of use is restricted [START_REF] Mork | Oqafma querying agent for the foundational model of anatomy: a prototype for providing flexible and efficient access to large semantic networks[END_REF].

Only structural anatomy has been considered in FMA. Topology, geometry and functions are required to build an efficient ontology for physical simulation. Strategy to extend FMA for a specific purpose is presented in [START_REF] Rosse | A strategy for improving and integrating biomedical ontologies[END_REF]. An original approach, strictly limited to the muscoloskeletal system, has been proposed by Charbonnier et al. where FMA is extended to anatomical functional knowledge for physical simulation [START_REF] Charbonnier | A semantic-driven clinical examination platform[END_REF].

Generic concepts

An ontology is a specification of a conceptualisation [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. Common components of ontologies include classes, attributes and relations. Classes are sets, abstract objects, entities, concepts. Classes have attributes or properties. Relations make links between classes. Each relation has a conceptual meaning. For instance in the following sentence: "The patella is a part of the knee and is a bone organ", relations (is a and part of ) used between anatomical entities (patella, knee and bone organ) give a formal and logical description of canonical anatomy. Is a and part of are also called hierarchical relationships. The key point to build an ontology for specific application is the use of associative relationships (domainspecific) which must be clearly defined [START_REF] Smith | Relations in biomedical ontologies[END_REF].

The general purpose of an ontology is to provide a mean of classifying individuals. Patient-specific data are instances of canonical anatomical entities stored in the ontology. Three kinds of binary relations can be used to manage efficiently anatomical knowledge and individuals: < class, class > seen below, < instance, class > and < instance, instance >.

MyCF is focused on the managing of individuals through an intuitive graphic user interface for non-expert users in ontology. The goal is to allow anatomists to contribute to the canonical part of MyCF. By this way, anatomical content of MyCF will be validated by experts. On the other hand, an user of MyCF can add individual content and get correct anatomical information as terminologies, functions, localizations, interactions and so on.

MyCF is based on FMA for the ontological description of anatomy. An original MySQL schema is proposed to ease sharing and extension, as explained in the subsection 3.2 and to manage individuals. A SQL query is enough to retrieve data from one or more tables, allowing efficient implementation in a specific application.

Description of the mySQL database

The figure 1 shows the logical structure of MyCF's database. The entity relationship diagram (ERD) can be divided in two main parts: canonical anatomical knowledge on one hand and individual data on the other hand.

MyCF's canonical part: All canonical classes of MyCF are stored in the main table called Anatomical Entities. An ontology id, used as a primary key, is given for each one. Definition and favorite name are available as well.

In an ontology, anatomical entities are organized into a hierarchy named taxonomy. A taxonomy is a directed acyclic graph satisfying some conditions [START_REF] Zhang | Law and order: assessing and enforcing compliance with ontological modeling principles in the foundational model of anatomy[END_REF]. Our database schema allows to represent generic graphs, including directed acyclic graphs. Each entry of Relation Type defines a graph, which can be a taxonomy or a more general graph, where its nodes are contained in Anatomical Entities and its oriented edges in Relation. In practical terms, the table called Relation Type gives definition of relations. For instance, the relation is a is defined here with text definition and examples. The binary relations between anatomical entities are stored in the table called Relations where parents and son are defined using anatomical entity ids and where the type of relation is given.

The attributes are managed also with two tables. The table Attributes defines the properties of Anatomical Entities. The table Attributes Type ensures the genericity of the attributes.

With those five tables, it is possible to expand endlessly MyCF without adding tables. Furthermore, the simple organization allows users to write his own SQL queries to retrieve or update data.

MyCF's individual part: Instances are linked to anatomical entities. The 1:n relationship (bold line in figure 1) allows to have many instances of one anatomical entity. Instances have specific attributes, the position in a 3D scene for instance. The types of attributes are stored in the canonical part of the database. But the attribute's value of a given instance is saved in the table called Instance Attributes. Furthermore, general information can be stored in the canonical part of the database as, for instance, the yellow color conventionally used to depict bones in anatomical drawings. If the color is not given by the user for an instance, color stored in the canonical part can be automatically retrieved. By this way, a priori anatomical knowledge can be used to complete patient-specific content. The origin of patient-specific 3D data is a set of medical images (after segmentation). 3D models coming from a given acquisition will be used together to generate a 3D scene. The table called Acquisitions is used to manage it.

Instances

Geometry is one of the most challenging issues in anatomy modeling. Each anatomical entity can be represented using various types of geometric models (volumetric images, surface or volume meshes, analytical surfaces, implicit surfaces, etc.) at various resolutions or levels of detail. Consequently, we can not store one single geometrical description for each anatomical entity. So, we store, for each entity, a set of instances referenced by file names. We set no limitation on the file format and the type of geometric models. We have developed plugins for Blender [13], a free and open-source geometric modeler, to edit geometrical models of MyCF, see Fig 2 . It handles surface meshes and can be used to modify shapes, to displace and resize objects and to label areas such as ligament attachment places. Geometry is often modeled based on medical images which represent several anatomical entities, which can be consistently viewed or simulated together. We therefore gather the anatomical entities modeled from a given set of medical images in acquisitions, which allows us to retrieve all of them simultaneously from the database.

Implementation

To ensure flexibility and extension, the canonical part of the database is described by only five tables. The ontology contains many taxonomies, as defined in section 3.2. As a consequence, many graphs are spread into this five tables and information retrieval can be tedious. To easily exploit the MyCF database, we propose a library called libmyCF, written in python. It contains classes representing the anatomical entities, their relations and attributes and some useful algorithms are present.

In the canonical part, we automatically deal with the case of the attributes inheritance. So, when the attributes of an anatomical entity are required, all the attributes of this entity are returned, plus all the attributes of its parents, defined by the relation is a, if the parsed attributes are not yet defined.

If the value of an attribute is modified and this attribute is inherited, a new attribute is created for this anatomical entity. Instance attributes are shown in the figure 3. In the individual part, when the required instance does not exist, a new instance is created using the corresponding anatomical entity information. On the same way, when the required instance attribute is missing, it is automatically created from the attribute of the same type of the anatomical entity corresponding to the instance.

For instance, let us consider a knee modeled by two bones (femur and tibia) and a ligament, see Fig 2 . During the importation of the instances into Blender, we need to know where to insert extremities of the ligament on the bones. To do that, we just have to request the attribute named position. Usually, the value of this instance attribute is retrieved and the extremity is placed. Then, we can modify the position of the extremity on the bone in Blender and export this new value into the database.

The first time the extremity is imported into Blender, the instance attribute is not set in the database. When the import script treat the instance 'extremity of the ligament', the position value is requested. As there is no instance attribute, a new one is automatically created from the attribute of the anatomical entity corresponding to the instance of the extremity. Then, modifications can be done in Blender and exported.

Thus, the link with the knowledge contained in the canonical anatomical model is automically managed by the library, and plugins using MyCF information are easy to write.

Applications

Gathering ontological, geometrical and mechanical data consistently in a database allows new applications. We sketch some of them in this section.

Teaching

An obvious application of the database is the teaching of anatomy. MyCF can be used by students to handle anatomical knowledge within a consistent presentation. The graphical representations of anatomical entities allow a more intuitive way for knowledge-based navigation and understanding, as illustrated in Fig 3 .  Each anatomical entity can be displayed in 3D, while its logical relations with the other entities appear in the interface. Variability can be seen as well through 3D instances saved in the database. The Digital Anatomist Interactive Atlases integrates the FMA with 3D graphical illustrations but without real 3D interaction [START_REF] Wong | Semi-automatic scene generation using the digital anatomist foundational model[END_REF].

Mechanical models

Beside geometry, we store mechanical parameters in the database, which allows to export an acquisition to physical simulation engines. The user can select the anatomical entities to export and the mechanical models to physically simulate them. For instance, to simulate a knee, the user can decide to export the bones and the main ligaments. Various mechanical models can be used to represent a given object, depending on the purpose of the simulation. For instance, studying the kinematics of a given joint based on bones and ligaments requires rigid bone models. Alternatively, studying bone fractures requires deformable models such as finite elements. When the rigid model is applied to a given entity, its mass and inertia matrix are automatically computed based on geometry and density. Alternatively, the user can chose to model an entity as a Finite Element Mesh with Hooke material. In this case, the same process is used to compute the mass and stiffness matrices, depending on the geometry, the shape functions as well as material parameters.

Deformable models such as finite elements require volumetric meshes. Meshes are an important issue in simulation, and we can not propose all meshing algorithms using various complexity and quality parameters. When volumetric tetrahedral meshes are available in the geometric data, they can be easily exported to finite element simulation modules. The precision and the computation time of finite element simulations heavily depend on the geometrical quality and the resolution of the volumetric meshes. Nesme et al.'s grid-based finite elements use deformable regular grids embedding objects defined by arbitrary geometrical models [START_REF] Nesme | Preserving topology and elasticity for embedded deformable models[END_REF]. The grids have arbitrary resolution, which allows to trade off accuracy for speed according to the application, and the geometrical quality is not an issue since all cells are cubes. This approach allows to create finite element models for any virtual object at desired resolution.

Completion of a model

A priori knowledge is useful to complete models where all anatomical entities are not defined. For instance, bone models may be built based on a CT-scan acquisition, whereas the connective tissues are not easily visible on the same data. In such a context, performing the mechanical simulation of a knee requires adding ligaments to the model, based on the knowledge of the ligaments properties and insertions locations on the bones.

MyCorporisFabrica allows the automatic creation of missing anatomical entities, using geometrical and material knowledge. The current implementation allows the creation of the main ligaments of the knee. The skeletal ligaments are modeled as one-dimensional mass-spring networks attached to the bones. To manage attach points, we have subdivided each ligament (using the part of relation) in a proximal and a distal extremities. The areas on bones where skeletal ligaments are attached are clearly defined in the ontology. We use the relation is inserted on to inform where ligament's extremities are inserted. That exact position required for a given instance should be added manually. We're currenlty working on an automatic process to localise ligament insertions on instances using anatomical knowledge. Default values for spring stiffness and damping constants are stored in the database, as well as a default number of particles for the ligament. These values are modifiable by the user at creation time. Moreover, the 3D editor can be used to modify the attach points. Figure 4 illustrates this process.

Once the model is complete, it can be exported to a simulation engine, as illustrated in Figure 5. We use the SOFA simulation library [START_REF] Allard | SOFA -an open source framework for medical simulation[END_REF] because it handles a variety of mechanical models and their interactions, it is free, opensource and highly customizable. However, export plugins to other simulation engines could be easily added to My Corporis Fabrica.

Discussion

To the best of our knowledge, MyCF is the first multidisciplinary project focused on a close collaboration around anatomical physical simulations. The anatomits can find a formal frame for anatomical knowledge with interactvie 3D visualization and nonanatomist users can manage patient-specific content through confirmed anatomical concepts.

MyCF is a brand new project. The knee is presented as a case study and has been used to built the system. We plan to increase amount of anatomical enities in MyCF. To do so collaboration between users and curators must be strong and active. A collaborative web site around MyCF has been recently open for this purpose. Readers are invited to visit www.mycorporisfabrica.org. In the futur, MyCf sould be included in the OBO Foundry (www.obofoundry.org).

Fig. 1 .

 1 Fig. 1. Entity Relationship Diagram of MyCF's database. The canonical anatomy part is on the top and the individuals part is on the bottom. Black lines depict relationships (1:n) between tables. Two bold lines show the relationships between the canonical and the individual parts. The dotted line depicts a relationship used to force user to select an existing anatomical entity (restricted to anatomical segment) during insertion process of a new acquisition in the database.

Fig. 2 .

 2 Fig. 2. Blender is used to edit individual geometrical models. Blender objects are created from information stored in MyCF (import script). Positions, insertions of ligaments are editable and modifications can be saved in MyCF (exportation script). The tibial collateral ligament (MYCF:00031) is under edition. The ligament has two extremities (the proximal and the distal extremity) which are defined in MyCF.

Fig. 3 .

 3 Fig. 3. MyCF graphical user interface. a) Main window of MyCF. 1.the anatomical entities tree for the relation part of, 2.information about the anatomical entity (AE), 3.attributes for this AE, 4.VTK visualization of an instance example. b) Edition of an anatomical entity. All the attributes and the relations of this anatomical entity are listed and editable here.

Fig. 4 .

 4 Fig. 4. Completion of a model. a) Knee bones from an acquisition. b) Ligament models not visible in the acquision are added using anatomical knowledge.

Fig. 5 .

 5 Fig. 5. Snapshots of a SOFA simulation of the model shown in Figure 4, with four ligaments.


Christine Petr

Keywords: Participatory science, visitor, tourist, digital solutions, study design, research method

published or not. The documents may come     Participatory sciences to measure tourist flows : involving islanders and visitors in the creation and use of digital counting solutions

Introduction

The [START_REF] Unwto | Overtourism'? -Understanding and Managing Urban Tourism Growth beyond Perceptions[END_REF] defines overtourism as "the impact of tourism on a destination, or parts thereof, that excessively influences perceived quality of life of citizens and/or quality of visitor experiences in a negative way". In order to properly distribute tourist visits over the territory and avoid feelings of overtourism (especially in the context of the relationship between tourists and residents, cf. [START_REF] Favier | Vers une meilleure compréhension de l'échange social et des interactions résidents-touristes : une application à la destination du Lac d'Annecy en proie au tourisme de masse[END_REF][START_REF] Holmes | At Home or Abroad, Does Our Behavior Change? Examining How Everyday Behavior Influences Sustainable Travel Behavior and Tourist Clusters[END_REF], it is crucial to propose solutions for managing tourist flows. However, in order to envisage innovations that are conducive to good management of visitor flows, an essential prerequisite is to have objective, quantified and updated data, ideally continuous throughout the year and over a time spectrum of hours (or at least half-days) on these visits. However, to date, this data, available in barometric format, is still lacking, particularly in small island territories. However, local and regional managers keep insisting on the fact that they need a barometer-type measurement tool that meets the threefold objective of 1) measuring and auditing visitor numbers, 2) putting the various experiments and management choices of each island into perspective, and ideally, 3) providing a basis for the daily management of these numbers so as to be able to provide a real welcome to visitors. The aim is to allow the tourism actors of the islands to stop being subjected to a frequentation perceived as deeply volatile, but to be able to anticipate the situations at least on a halfday scale.

In order to assess the feasibility of this type of barometric tool and to envisage -or invent -the modalities for constructing such a tool for measuring visitor numbers, the creation of a counting method based on the participatory science approach is envisaged. In essence, the idea is that each individual is capable of contributing individually to the collection of data if they are rigorously trained and equipped by the researcher to act within an established scientific framework. In this way, on a voluntary basis, research can be informed by a multitude of individual observers. This approach significantly increases the efficiency of the count and the volumes of data collected at minimal cost. Thus, it is all the volunteer citizens (whether they are permanent inhabitants, residents, day-trippers, tourists or tourism professionals) who can form the first link in the collection system. By providing a complete kit (explanation of the scientific project and the issues at stake, presentation of the counting tools, explanation of the counting methodology and the use of the tools), the researcher is setting up the conditions for a new source of data that is massive, reliable and inexpensive. By illustrating the case of a project to measure tourist numbers on various French island territories, the aim here is to describe a scientific design that makes it possible to obtain data on tourist numbers and that is based on the principles of participatory science. After presenting the theoretical framework and the problem, we describe the main stages of the research design envisaged, before concluding with the obstacles to be overcome and the expected contributions.

Theoretical framework

Participatory science (citizen science) is defined as scientific research conducted with the active participation of the population (Cham 2021). The population is then solicited for all or part of the research: helping to collect data, helping to classify them, etc., which increases the power of the scientific community tenfold (Kullenberg & Kasperowski, 2016). Originally, participatory science was developed mainly in the context of biology and biodiversity (e.g. counting the birds of each species in one's garden over a given period, following a specific protocol and standards). Today it is used in many fields, including the most modern ones. For example, image recognition algorithms are trained on sets of images whose characteristics or nominalisation have been made by groups of volunteers.

The huge potential of participatory science has since been established [START_REF] Margaret | Assessing Data Quality in Citizen Science[END_REF] proven methods for obtaining quality data (training of volunteers, statistical adjustment tools, accessibility, standardisation and simplicity of protocols and analysis methods to facilitate their appropriation, etc.). Beyond its practical usefulness, participatory science also responds to a social and public policy issue: the relationship between citizens and science. According to [START_REF] Irwin | Citizen Science: A Study of People, Expertise and Sustainable Development[END_REF], science must respond to the needs and concerns of citizens, and citizens themselves are capable of producing scientific and reliable knowledge. It is therefore a public good to be opened to the public. Furthermore, in the same logic, [START_REF] Doyle | Using the Web for Science in the Classroom: Online Citizen Science Participation in Teaching and Learning[END_REF] argue that involving citizens in scientific projects improves their knowledge of the scientific process and fosters their awareness of the issues being studied.

The originality of this approach is that it makes it possible to involve the actors who are closest to the field, to federate them and get them to talk to each other around a shared activity and a common frame of reference (participation in the count, its purpose, the method). The aim of these approaches is to increase the chances of success of the project. On the one hand, it is a question of avoiding that certain actors feel cheated or abandoned, which can lead them to concretely oppose the project, or to slow it down and make it unviable because of voluntary resistance and misuse of principles and rules. It is also a question of enabling better adherence to projects. Rather than having to convince a posteriori of the interest of an approach and new infrastructures, it is a question of encouraging its appropriation and the collaboration of all. This participatory science approach also refers to the principle of stakeholders, which is well known and used in the context of territorial development projects, and is now also at the heart of sustainable tourism management [START_REF] Muzy | la gouvernance des organismes de gestion de sites, une approche par la théorie des parties prenantes, 9ième Conférence de l'AFMAT[END_REF]. Let us recall the line of conduct advocated by the World Tourism Organisation according to which sustainable tourism must take "full account of its current and future economic, social and environmental impacts, meeting the needs of visitors, professionals, the environment and host communities" (UNWTO, 2016).

Sustainable tourism for small islands: the importance of economic issues

Sustainable tourism is a subject which mobilises a lot of attention from researchers, as shown by the large number of publications and productions on this topic [START_REF] Bourelly | Identification, mobilisation, valorisation et régulation des compétences territoriales pour une tourisme durable au sein des PNR: le rôle des valeurs Parc sur le PNR de Brière, 9ième Conférence de l'AFMAT[END_REF][START_REF] François-Lecompte | L'engagement environnemental du secteur hôtelier : proposition d'un modèle conceptuel[END_REF][START_REF] François-Lecompte | Le tourisme durable : les sept clés du succès[END_REF][START_REF] Falduto | L'OGD Iintercommunal : un acteur Stratégique des destinations touristiques durable ? Étude comparative : Le cas de Dracénie Provence Verdon Agglomération et Méditerranée Porte des Maures, 9ième Conférence de l'AFMAT[END_REF]. This interest in sustainable tourism is stimulated by the observation of numerous issues related to preservation: the environment, heritage, cultural diversity [START_REF] Camus | Tourisme durable : une approche systémique[END_REF], and recent literature abounds on the consideration of the social and environmental components of sustainable tourism, underlining the key importance of these elements for tourism [START_REF] Couao-Zotti | Sur fréquentation des sites touristiques du littoral varois : cartographie des acteurs touristiques autour du parc national Port-Cros, 9ième Conférence de l'AFMAT[END_REF][START_REF] Pupion | Tourisme durable et valorisation de l'environnement du littoral aquitain : diversité et stratégies des acteurs[END_REF]. The tendency of researchers to focus attention on the two pillars of sustainable tourism, the environmental and the social/societal, should not lead to the economic pillar being forgotten. Thus, the economic aspect of sustainable tourism is essential, as the UNWTO definition reminds us. And destination managers are all vigilant about this economic dimension, which is vital for the development of territories and the maintenance of a local population. In this respect, little has changed since [START_REF] Swarbrooke | Sustainable Tourism Management[END_REF] defined the economic stakes of sustainable tourism. Today, the economic aspect of sustainable tourism still involves and concerns the optimisation of benefits, the reduction of costs, the equitable redistribution and the development of local businesses. In the case of islands, the tourism economy is often vital. Indeed, tourism allows the maintenance of an activity throughout the year, and to maintain services and infrastructures which are rare for their low density of inhabitants (doctor, pharmacy, school, etc.). Tourism places are first and foremost living places, and must be and remain so to continue to attract tourists [START_REF] Pendlebury | Urban World Heritage Sites and the problem of authenticity[END_REF]. Thus, islands need to receive a flow of visitors, tourists and day-trippers, acceptable to the different stakeholders (neither too much nor too little). This confirms the need to objectify the real and felt frequentation of the islands.

Issue

In order to carry out this project to obtain data and indicators on tourist numbers, we propose to adopt the participatory science approach in the construction of collection tools and then in the stages of collection, measurement and qualification of tourist numbers on the island territories.

The main stages in the design of the participatory research on island tourism

The approach is envisaged as follows -Definition of the numerical objectives, scope and methods of the studies with the stakeholders: data to be collected, reference period, quantitative objectives, survey sites, number of observers expected for each site, practical organisation of the training and support of observers (e.g. location of trainingflash sites, trainers of trainers, deployment of spots on the island, etc.), definition of the communication strategy -Definition of observer training protocols, collection protocols (scheduling, points of vigilance, checkpoints, classification, reporting) -Training of observers (a section presenting the project, a section presenting the tools, an accompanied observation phase with supervision by scientists to ensure the reliability of the methodology, and then the launch of the observers on their own) -Opening of the observation phase, with monitoring of observers, ad-hoc controls, by sampling and in the field to ensure the quality of data collection and the relevance of established protocols -Ongoing verification and curation of the data collected to ensure the quality of the knowledge accumulated -Close the observation phase, consolidate and analyse the results, actively communicate with participants to thank them for their participation -Provide feedback, update the protocols and create a turnkey kit that can be adapted for use by local authorities

Results and contributions

The managerial contributions are to propose a complete protocol for measuring tourist numbers with a participatory approach: the kit (turnkey and modular) provided to local authorities will enable them to deploy information collection systems that meet legal concerns and operational objectives, while being acceptable to all stakeholders. The contribution of the participatory science approach to data collection allows for the concrete initiation of a negotiation process based on the objectification and qualification of the use of the site. After the first phase of data collection and the updating of observation protocols (based on feedback from the field), the documents produced will serve as standard guides and methodological tools that can be reused with marginal adaptations. This process is envisaged from its conception and documented as being replicable for other island destinations that face the same challenges of sustainable tourism trade-offs between the need for economic spin-offs, the moral obligation of openness to tourists, and the strategy of preserving the environment and quality of life for residents. The conceptual contributions concern the identification of the blocking points underlying participation in observation, and more broadly in the creation of barometers for measuring visitor numbers and the qualification/distinction of tourist and day-tripper numbers. Amongst other things, it is a question of showing that the participatory science approach to the subject of data collection for tourism-related subjects implies understanding in greater detail what are the levers and motivations for participating in the collection of this data (e.g.: individual-citizenship, ecoresponsibility, service to the community, etc.). It is also a question of identifying the difficulties and obstacles to this participation, to this collection, to compliance with the methodology, and to the sharing of results. The perception of the sensitivity/intimacy of these data thus appears to be at the heart of possible blockages or refusals to participate on the part of the actors. If these individual stakeholders are reluctant to collect and oppose the dissemination of results or their scientific interpretation, this leads the visitor measurement project to invent and develop other accounting and identification strategies


Modesar Shakoor
email: modesar.shakoor@imt-nord-europe.fr

Chung Hae Park

A pure Stokes approach for coupling fluid flow with porous media flow

Keywords: porous media, Stokes, Darcy, Brinkman, Beavers-Joseph-Saffman

Most numerical approaches for coupling fluid flow with porous media flow rely either on Stokes equations in the fluid part of the domain and Darcy's law in the porous part, or on Brinkman's equation. In both cases, difficulties arise at the boundary between the two parts because the equations used in the porous part are not of Stokes type. In this paper, an alternative to Darcy's law is proposed for modeling flows in porous media. This alternative relies on equations of Stokes type where the permeability tensor is replaced by force and stress derivative tensors. Numerical procedures are presented to compute these tensors from simulations at pore scale. Simulations in domains containing both fluid and porous parts are finally conducted simply assuming continuity of velocity and pressure and hence without imposing any condition at the boundary between the two parts. Results

Introduction

Flows in porous media are of great interest for a wide range of applications such as oil engineering, civil engineering and composites engineering. Although Stokes equations can model these flows at the pore scale, they are not convenient for large simulation domains where the pore characteristic length is very small compared to the domain size.

Under such conditions, indeed, simulations discretizing and solving simultaneously both the coarse scale of the domain and the fine scale of the pores become intractable from a computational cost point of view. Darcy's law, fortunately, has been proposed to circumvent this difficulty. This law has first been established from experimental measurements [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF], and later on proven from a mathematical point of view in the steady case [START_REF] Whitaker | The equations of motion in porous media[END_REF][START_REF] Neuman | Theoretical derivation of Darcy's law[END_REF][START_REF] Whitaker | Flow in porous media I: A theoretical derivation of Darcy's law[END_REF]. The main advantage of this law is that an effective property of the porous medium called permeability tensor can be computed beforehand in a pre-processing step, and then be used for coarse scale simulations where pores are not discretized.

For single-phase flows in domains containing both a porous part and a purely fluid part (i.e., without pores), it seems straightforward to rely on Stokes equations in the fluid part and Darcy's law where pores are present. In practice, this is quite challenging due to intrinsic mathematical and physical differences between both models:

• Velocities have different meanings in both parts (Marco Discacciati and Alfio Quar-teroni, 2009), as the Darcy velocity v D is related to the fluid velocity v through the relation v D = φv, with φ the porosity (i.e., pore volume fraction).

• Darcy's law involves no differential operator for the velocity while Stokes equations involve a differential operator with second order derivatives of the velocity.

• It is common to eliminate completely the velocity from Darcy's law and solve with only the pressure as unknown, while this is not the case for Stokes equations. More generally, numerical methods for efficiently solving Darcy's law and Stokes equations might be different [START_REF] Chidyagwai | Numerical modelling of coupled surface and subsurface flow systems[END_REF].

These difficulties are usually dealt with using specific boundary conditions at the interface between the fluid and porous parts [START_REF] Jäger | Asymptotic Analysis of the Laminar Viscous Flow Over a Porous Bed[END_REF][START_REF] Layton | Coupling fluid flow with porous media flow[END_REF][START_REF] Urquiza | flow in a fluid-porous system including inertia and slip effects[END_REF][START_REF] Discacciati | Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation[END_REF][START_REF] Münzenmaier | First-Order System Least Squares for Coupled Stokes-Darcy Flow[END_REF][START_REF] Pacquaut | Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes-Darcy flow[END_REF][START_REF] Eggenweiler | Unsuitability of the Beavers-Joseph interface condition for filtration problems[END_REF]:

• Continuity of the normal fluid velocity to ensure mass conservation across the interface.

• Continuity of the fluid normal stress to ensure balance of normal forces across the interface.

• Some condition on the tangential fluid velocity such as the Beavers-Joseph or Beavers-Joseph-Saffman conditions, or even a simple no-slip condition.

Beavers-Joseph and Beavers-Joseph-Saffman conditions have been the subject of many research works in the last decades. As shown in Ref. [START_REF] Auriault | About the Beavers and Joseph Boundary Condition[END_REF], they can be used only in restrictive conditions where the pore characteristic length is not too small as compared to the domain size, and they involve an empirical coefficient that is "dedicated to the particular setup at stake". A numerical proof can be found in Ref. [START_REF] Eggenweiler | Unsuitability of the Beavers-Joseph interface condition for filtration problems[END_REF]. For the cases with a very small pore characteristic length, a boundary condition involving a permeability tensor specific to the interface has been proposed in Ref. [START_REF] Auriault | About the Beavers and Joseph Boundary Condition[END_REF]. Similarly, the boundary condition proposed in Ref. [START_REF] Marušić-Paloka | The effective boundary condition on a porous wall[END_REF]) features a so-called "permeability tensor of the porous wall".

An "interface permeability tensor" also appears in the higher-order boundary conditions introduced in Ref. [START_REF] Sudhakar | Higher-Order Homogenized Boundary Conditions for Flows Over Rough and Porous Surfaces[END_REF], but it is accompanied by other terms that can all be pre-computed by solving fine scale problems.

These boundary conditions, with their empirical or pre-computed terms, should be introduced independently from the numerical methods used to solve and couple the fluid and porous media flows. Unified or monolithic approaches based on least squares [START_REF] Münzenmaier | First-Order System Least Squares for Coupled Stokes-Darcy Flow[END_REF], the level-set method [START_REF] Pacquaut | Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes-Darcy flow[END_REF] or Brinkman's equation [START_REF] Payne | Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions[END_REF][START_REF] Auriault | On the domain of validity of Brinkman's equation[END_REF]) may be used, as well as other approaches where the two parts are decoupled and discretized separately.

It it important to point out that the interface between the fluid and porous parts is an artificial mathematical boundary and it can be given any arbitrary thickness as long as this thickness is small compared to the pore characteristic length [START_REF] Marciniak-Czochra | Effective Pressure Interface Law for Transport Phenomena between an Unconfined Fluid and a Porous Medium Using Homogenization[END_REF]. As an alternative, the so-called one domain approach consists in modeling this boundary as a new intermediary porous medium whose thickness is not arbitrary but determined iteratively by the numerical method (Valdés-Parada and Lasseux, 2021a,b).

In a recent work [START_REF] Blanco | Homogenization of the Navier-Stokes equations by means of the Multi-scale Virtual Power Principle[END_REF], an alternative approach for homogenizing flows in porous media has been proposed. First, the velocity in the porous medium is averaged only over the pores, and therefore carries the same physical meaning of fluid velocity in both fluid and porous parts. Second, the coarse scale model is of Stokes type, and hence matches the model used in the fluid medium. This model has been extended to the unsteady case and implemented with two-way coupling at each time increment between the two scales in Ref. [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF]. To ease the analogy with Darcy's law, the present work is restricted to the steady case where the pore scale model is given by Stokes equations, while the coarse scale model is shown to remain of Stokes type.

In Sec. 2, this pure Stokes approach for modeling flows in porous media is described.

Stokes equations for the coarse scale are shown to depend on some effective properties of the porous medium that can be computed beforehand in a pre-processing step. Two methodologies for computing those properties in a Finite Element (FE) implementation are presented in Sec. 3. It is then demonstrated through comparisons with single-scale simulations (i.e., simulations where pores are directly discretized at the coarse scale) in Sec. 4 that the same discretization can be used for both fluid and porous parts, and that no boundary conditions are necessary at the interface.

Multiscale model

The coarse scale domain is denoted Ω M ⊂ R d , with Dirichlet boundary conditions v M D on Γ M D and Neumann boundary conditions t M N on Γ M N . In a strong form while neglecting body forces, the equations to satisfy at the coarse scale are

               f M (x) -∇ x • σ M,dev (x) + ∇ x p M (x) = 0, x ∈ Ω M , ∇ x • v M (x) = 0, x ∈ Ω M , v M (x) = v M D (x), x ∈ Γ M D , σ M,dev (x) -p M (x)I • n(x) = t M N (x), x ∈ Γ M N , ( 1
)
where I is the identity matrix and n is the outgoing normal vector to ∂Ω M . In an equivalent variational form, the coarse scale problem is to find

v M ∈ V M , p M ∈ L 2 (Ω M ),
such that

Ω M f M (x) • δv M (x) + σ M,dev (x) : ∇ x δv M (x) dx + Ω M -p M (x)∇ x • δv M (x) -δp M (x)∇ x • v M (x) dx = Γ M N t M N (x) • δv M (x)dx, (2) ∀δv M ∈ V M , ∀δp M ∈ L 2 (Ω M ), (3)
with functional spaces

V M = w ∈ H 1 (Ω M ) d , w(x) = v M D (x), ∀(x) ∈ Γ M D , H 1 (Ω M ) = w ∈ L 2 (Ω M ), ∇ x w ∈ L 2 (Ω M ) d , L 2 (Ω M ) = w : Ω M → R, Ω M w(x) 2 dx < +∞ . ( 4
)
This is standard and easier to understand if incompressible Newtonian flow is modeled directly at the coarse scale. Stokes equations then lead to:

   f M (x) = 0, σ M,dev (x) = 2µ M (x)∇ S,dev x v M (x), ∀x ∈ Ω M , ( 5
)
where the deviatoric part is

∇ S,dev v M = D : ∇v M , ( 6
)
the fourth-order tensor D being defined by . The problem with this direct approach is that the computational cost is going to blow up if there are pores of a small size compared to Ω M , as the element or cell size should be smaller than the pore size. The alternative proposed herein is a multiscale approach, in which f M (x) and σ M,dev (x) are unknown functions of v M (x) and ∇ x v M (x), while p M is still a Lagrange multiplier.

D = δ i,k δ j,l + δ i,l δ j,k 2 - δ i,j δ k,l d ijkl , with δ i,j =    1, i = j, 0, i = j, (7)
Consequently, a fine scale domain The proposed multiscale approach is kinematic, which means averaging constraints are going to be imposed on the velocity field, while the expressions of f M (x) and σ M,dev (x) are going to be derived from a so-called principle of multiscale virtual power [START_REF] Blanco | Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models[END_REF][START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF].

Ω m = Ω m (x) ⊂ R d is introduced
For points x of the coarse scale domain Ω M that are inside the fluid part, there is no solid part at the fine scale. For those points which will correspond to integration points of the coarse scale mesh in the numerical implementation, it is possible to use either the single-scale Stokes model from Eq. ( 5), or a homogeneous fine scale domain where [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF].

Γ m O = ∅ (
The details on the principle of multiscale virtual power are given in the Appendix, in particular regarding the derivation of the system of variational equations to solve for each fine scale domain using the FE method. It can be written in a strong form as

∂Ω m Γ O m Γ O m Γ O m
                               -∇ y • 2µ m (y)∇ S,dev y v m (y) + ∇ y p m (y) -α = 0, ∀y ∈ Ω m , v m (y) = 0, ∀y ∈ Γ m O , 2µ m (y)∇ S,dev y v m (y) -p m (y)I • n m (y) = β • n m (y), ∀y ∈ ∂Ω m \Γ m O , ∇ y • v m (y) = 0, ∀y ∈ Ω m , 1 |Ω m | Ω m v m (y)dy = v M (x), 1 |Ω m | Ω m ∇ y v m (y)dy = ∇ x v M (x), (8)
where n m (y) is the outgoing normal vector at the fine scale domain boundary. It should be noted that this problem is linear as there is no auto-advection term and Newtonian flow is considered.

This problem involves Lagrange multipliers α and β to impose averaging constraints on the fine scale velocity and its gradient. The force per unit volume and deviatoric stress can be computed using f M = α and σ M,dev = D : β as proven in the Appendix.

To summarize:

• The coarse scale problem in Eq. ( 2) is similar to the well-known Stokes equations for unsteady incompressible Newtonian flows, except that it is written in terms of an unknown force per unit volume and an unknown deviatoric stress.

• Small pores are not represented directly within the coarse scale. Instead, they are embedded in fine scale domains, which are placed at each point of the coarse scale domain.

• For each fine scale domain, the boundary value problem in Eq. ( 8) should be solved.

The boundary conditions for this problem come from averaging constraints relating the fine scale velocity field and its gradient to their coarse scale counterparts.

• The force per unit volume and deviatoric stress for the coarse scale are computed from the solution of the fine scale problem.

Because the fine scale problem in Eq. ( 8) is linear, it is possible to compute the effective properties beforehand and use them for running coarse scale simulations. An FExFE (FE 2 ) strategy with two-way coupling between the scales is hence not necessary.

Numerical method

As in Ref. [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF], the Taylor-Hood P2/P1 pair is chosen at both scales. Quadratic interpolation is hence used for the velocity, and linear interpolation for the pressure.

Contrary to the unsteady case for which an FE 2 scheme was proposed in Ref. [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF], the steady case only requires the computation of the effective properties in a pre-processing step. These properties are the force derivatives ∂f M ∂v M and

∂f M ∂∇ x v M ,
and the stress derivatives ∂σ M,dev ∂v M and ∂σ M,dev ∂∇ x v M . They are independent of v M and ∇ x v M due to the linearity of the fine scale problem. They are necessary to compute the force and stress through the following relations:

f M = ∂f M ∂v M • v M + ∂f M ∂∇ x v M : ∇ x v M , σ M,dev = ∂σ M,dev ∂v M • v M + ∂σ M,dev ∂∇ x v M : ∇ x v M . ( 9
)
If the fine scale domain is the same for all points of the porous part, for instance if the pore geometry and distribution do not vary, then these properties can be computed once and for all. In the sequel, the FE solution of the fine scale problem is first detailed, and then two approaches based on the FE method are proposed to compute the force and stress derivatives.

Fine scale problem

The fine scale problem in Eq. ( 8) can be written as

               a v m v m (δv m , v m ) + a v m p m (δv m , p m ) + a v m α (δv m , α) + a v m β (δv m , β) = 0 a p m v m (δp m , v m ) = 0 a αv m (δα, v m ) = b α (δα) a βv m (δβ, v m ) = b β (δβ) (10)
with the bilinear forms

a v m v m (δv m , v m ) = 1 |Ω m | Ω m 2µ m (y)∇ S,dev y v m (y) : ∇ y δv m (y)dy, a v m p m (δv m , p m ) = - 1 |Ω m | Ω m p m (y)∇ y • δv m (y)dy, a v m α (δv m , α) = -α • 1 |Ω m | Ω m δv m (y)dy , a v m β (δv m , β) = -β : 1 |Ω m | Ω m ∇ y δv m (y)dy , a p m v m (δp m , v m ) = - 1 |Ω m | Ω m δp m (y)∇ y • v m (y)dy, a αv m (δα, v m ) = -δα • 1 |Ω m | Ω m v m (y)dy , a βv m (δβ, v m ) = -δβ : 1 |Ω m | Ω m ∇ y v m (y)dy (11)
and the linear forms

b α (δα) = -δα • v M (x) , b β (δβ) = -δβ : ∇ x v M (x) . ( 12
)
After FE discretization, the linear system has the following form:

        A v m v m A v m p m A v m α A v m β A T v m p m 0 0 0 A T v m α 0 0 0 A T v m β 0 0 0                 U v m U p m U α U β         =         0 0 B α B β         (13)
where the dimension of the vector U v m is the number of P2 nodes of the FE mesh times d, the dimension of the vector U p m is the number of P1 nodes of the FE mesh, the dimension of the vector U α is d, and the dimension of the vector

U β is d × d.
Rows and columns of the linear system in Eq. ( 13) that are associated to nodes on Γ m O are modified to set the velocity to zero at those nodes.

Computation of force and stress derivatives

The first approach to compute the derivatives for the coarse scale solve is based on the linear system in Eq. ( 13). It is, indeed, possible to eliminate the velocity from the first line:

A v m v m • U v m + A v m p m • U p m + A v m α • U α + A v m β • U β = 0, ⇔ U v m = A -1 v m v m • (-A v m p m • U p m -A v m α • U α -A v m β • U β ) , ( 14
)
and then inject it in the third line:

A T v m α • U v m = B α , ⇔ A T v m α • A -1 v m v m • (-A v m p m • U p m -A v m α • U α -A v m β • U β ) = B α , ⇔    -A T v m α • A -1 v m v m • A v m α • U α = B α -A T v m α • A -1 v m v m • (-A v m p m • U p m -A v m β • U β ) , -A T v m α • A -1 v m v m • A v m β • U β = B α -A T v m α • A -1 v m v m • (-A v m p m • U p m -A v m α • U α ) . ( 15
)
Since f M = α = U α and v M = -B α , it can be shown that:

∂f M ∂v M = ∂α ∂v M = - ∂U α ∂B α = A T v m α • A -1 v m v m • A v m α -1 (16)
and since σ M,dev = D : β = D : (P β • U β ), with P β = δ (i-1)×d+j,k ijk , it can be shown that:

∂σ M,dev ∂v M = D : ∂β ∂v M = -D : P β • ∂U β ∂B α = D : P β • A T v m α • A -1 v m v m • A v m β + , ( 17
)
where, for any matrix C, C + denotes its Moore-Penrose inverse. Similarly, it can be shown that

∂f M ∂∇ x v M = A T v m β • A -1 v m v m • A v m α + , ∂σ M,dev ∂∇ x v M = D : A T v m β • A -1 v m v m • A v m β -1 . ( 18
)
This approach might not be suitable in practice, as it involves inverting and multiplying several matrices, which might be prone to both human and numerical errors, especially regarding Moore-Penrose inverses. It has not been investigated in this work, in fact, whether these inverses always exist or not. As an alternative, solving the linear system d + d × d times could yield all derivatives. For instance, in two dimensions (2D),

with

v M =   1 0   , ∇ x v M =   0 0 0 0   , ( 19
)
components ∂α i ∂v M k i=1...d,k=1
of ∂α ∂v M and components

∂β ij ∂v M k i,j=1...d,k=1
of ∂β ∂v M would be obtained, and with

v M =   0 0   , ∇ x v M =   1 0 0 0   , ( 20
) components ∂α i ∂∇ x v M kl i=1...d,k=l=1
of ∂α ∂∇ x v M and components

∂β ij ∂∇ x v M kl i,j=1...d,k=l=1
of ∂β ∂∇ x v M would be obtained. Consequently, all components of the force and stress derivatives would be obtained from a total of d + d × d solves.

Coarse scale problem

The weak form of the coarse scale problem in Eq. ( 2) can be written as

                       Ω M ∂f M ∂v M (x) • v M (x) + ∂f M ∂∇ x v M (x) : ∇ x v M (x) • δv M (x)dx + Ω M ∂σ M,dev ∂v M (x) • v M (x) + ∂σ M,dev ∂∇ x v M (x) : ∇ x v M (x) : ∇ x δv M (x)dx - Ω M p M (x)∇ x • δv M (x)dx = Γ M N t M N (x) • δv M (x)dx, - Ω M δp M (x)∇ x • v M (x)dx = 0. ( 21
)
As pointed out in Ref. [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF], the linear system to solve after FE discretization is not different from the standard Taylor-Hood saddle point problem:

  A v M v M A v M p M A T v M p M 0     U v M U p M   =   B v M 0   . ( 22
)
In fact, provided that force and stress derivatives have been computed in pre-processing, there is no specific difficulty in assembling and solving this linear system as compared to standard Taylor-Hood FEs. This is also true for setting up the Dirichlet boundary conditions. As a consequence, although the computational complexity of the coarse scale problem is slightly higher than one-domain or two-domain methods in the sense that solving Stokes is more expensive than solving Darcy, it is still of the same order.

The complexity and the number of fine scale problems to solve in order to compute the force and stress derivatives are also of the same order. Since there are no interface conditions and no additional parameters, the proposed method does not involve any empirical coefficients. Contrary to the one-domain approach, moreover, it does not require the introduction of an inter-region between the fluid and porous parts.

Simulation cases and results

To keep the computational cost reasonable, only 2D simulations are considered in this section. All simulations are conducted using the FEMS software [START_REF] Shakoor | FEMS -Finite Element Modeling Software[END_REF][START_REF] Shakoor | FEMS -A Mechanics-oriented Finite Element Modeling Software[END_REF].

Flow through a coarse scale domain containing both a purely fluid part and a porous part is considered. This domain is the square of area 1 m 2 shown in Fig. 2(a). The fluid part is the light part, while the porous part is the dark one. Boundary conditions force the flow to occur along the horizontal direction, which gives the role of obstacle to the dark part. The fluid dynamic viscosity is fixed to 1 Pa • s and gravity is neglected.

In the fluid part, the homogeneous fine scale domain shown in Fig. 2(b) is used, which is equivalent to directly using Stokes equations as per Eq. ( 5) in this part. With this technique, the size of the fine scale domain has no influence on results [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF], and it is fixed to l = 1 m. In the porous part, two heterogeneous fine scale domains may be considered: the one with pore area fraction φ = 52% presented in Fig. 2(c), and the one with φ = 36% presented in Fig. 2(d). In both cases, the fine scale domain is a square of size l with a circular solid at its center. This solid's interior is not discretized.

The radius is R = 0.39l for φ = 52% and R = 0.45l for φ = 52%. Both components of the velocity are set to zero at this solid's boundary. Unless otherwise mentioned, the fine scale domain with φ = 52% is chosen and its size is set to l = 0.1 m.

Single-scale simulations

Single-scale simulations are first conducted in order to establish reference solutions. Two meshes corresponding to two different mesh sizes are used, as shown in Fig. 3. These meshes rely on a fine discretization close to solids, and a coarser discretization in the fluid part. The average edge length h is used to identify mesh size.

The results of pressure and velocity fields are presented in Fig. 4. On the one hand, the pressure changes mainly in the horizontal direction, with a perturbation due to the porous part of the domain. On the other hand, the horizontal velocity component changes mainly at the boundary between fluid and porous parts, as the flow is significantly slowed down in the porous part.

The evolution of the pressure along a horizontal line at y = 0.25 m is shown in Fig.

5 (a). The two curves are superimposed, which proves that the so-called coarse mesh is actually quite fine enough for this problem. It should be noted that the pressure is arbitrarily represented as zero inside solids. The velocity curve in Fig. 5(b) confirms this conclusion. A mesh size h = 4 µm can hence be used close to solids. For the fine scale domains with solids, this means a mesh size h = 0.04l.

Influence of coarse scale mesh size

The proposed multiscale approach relying on pre-computed force and stress derivatives is now considered. The mesh with only two elements shown in Fig. 6 Regarding the coarse scale mesh, two levels of discretization are considered, a coarse mesh with uniform mesh size h = 0.04 m, and a fine mesh with h = 0.02 m. These meshes are much coarser and easier to construct than those used in single-scale simulations.

They rely on a simultaneous discretization of fluid and porous parts, with an explicit A comparison between multiscale simulation results using the two levels of discretization at the coarse scale and the reference single-scale solution is presented in Fig. 8. Both pressure and velocity curves prove that the coarse scale mesh using h = 0.04 m is already fine enough.

1 1 v x =v y =0 v y =0 v y =0 v x =1 v x =1 (
The curves for the pressure field in Fig. 8(a) demonstrate that the pressure prediction from the proposed multiscale approach is accurate in the fluid part. The pressure in the porous part between solids is also accurate but only in an averaged sense. Regarding velocity curves in Fig. 8(b), there is a slight difference inside both fluid and porous parts.

As shown in Figs. 8(c,d), this seems to be due to a gap between the boundary of the porous part in the single-scale model and in the multiscale model. In the single-scale model, indeed, the porous part actually ends at the boundary of the last solid, while in the multiscale model it is defined by the geometry in Fig. 2(a). This will be investigated further in the sequel. x along a vertical line at x = 0.5 m, (c) zoom on (a), (d) zoom on (b). The multiscale pressure curve has been offset to ease the comparison.

Influence of fine scale domain size

Simulations with smaller solids are conducted to investigate the effect of fine scale domain size. The latter is set to l = 0.025 m, which results in solids four times smaller and a number of solids multiplied by 16. The size of the mesh close to solids, consequently, is divided by four in the single-scale mesh. In fine scale meshes, however, reducing l automatically reduces h, with no effect on the total number of elements. The number of elements in the coarse scale mesh also remains constant, as the mesh with h = 0.04 m is used.

Results are presented in Fig. 9. They are qualitatively similar to those presented in Figs. 4 and7. In the single-scale result, however, the fine pressure fluctuations are less visible due to the smaller solid size, and it seems that single-scale and multiscale simulation results are closer.

A quantitative comparison is presented in Fig. 10. It should be noted that these curves show velocity and pressure evolution along lines that do not cross solids, as compared to Fig. 8. As a consequence, the multiscale model appears to be much more accurate inside the porous part. This is also a consequence of reducing l.

It is more interesting to look at the gap between the two curves, especially in Fig. 10(d). It is reduced with a smaller fine scale domain size, which leads to a more accurate multiscale prediction also inside the fluid part.

More importantly, the proposed multiscale approach, without any boundary condition between fluid and porous parts, predicts very accurately both pressure and velocity values at the boundary of the porous part. This is particularly remarkable for this simulation where the pore characteristic length is very small compared to the simulation domain size.

It is quite advantageous as the computational cost of the proposed approach is actually comparable to the Darcy approach, since it involves the same number of velocity-pressure degrees of freedom. Moreover, it becomes more interesting than a single-scale approach from a computational cost point of view when the pore characteristic length decreases. This was already demonstrated for the unsteady case in Ref. [START_REF] Shakoor | Computational homogenization of unsteady flows with obstacles[END_REF], but it is more obvious in the present work where force and stress derivatives for coarse scale simulations are pre-computed from a few fine scale simulations. For the results shown in Fig. 9, indeed, the single-scale simulation involves 455974 velocity degrees of freedom while the multiscale simulation involves only 6566 coarse scale velocity degrees of freedom. Force and stress derivatives, moreover, are pre-computed using the fine scale mesh shown in Fig. 6

Influence of porosity

Investigating the influence of porosity is also interesting as decreasing it leads to larger solids and hence decreases the gap between the boundary of the porous part in singlescale and multiscale models. The porous fine scale domain shown in Fig. 2 and8(b). More precisely, the gap between the two curves is reduced in Figs. 11(c,d) as compared to Figs. 8(c,d). The proposed multiscale approach is therefore more accurate with a smaller porosity. This could be expected as the gap between the boundary of the porous part in single-scale and multiscale models is decreased with a smaller porosity.

Another possible explanation could be a capability of the multiscale model to accurately approximate a completely solid part with no porosity.

Application to a woven textile composite

Finally, simulations are conducted for a more realistic case inspired from the woven textile composite material characterized in Ref. [START_REF] Syerko | Benchmark exercise on image-based permeability determination of engineering textiles: Microscale predictions[END_REF]. As shown in Fig. 12(a), the rectangular domain of area 1 × 0.1m 2 contains a porous part which is an elliptical yarn of major radius 0.25 m and minor radius 0.025 m. This leads to an aspect ratio of 10, which is similar to what is observed in the experimental images [START_REF] Syerko | Benchmark exercise on image-based permeability determination of engineering textiles: Microscale predictions[END_REF].

Boundary conditions and fluid properties are identical to previous simulations. The fine scale domain is the one with pore area fraction φ = 52% presented in Fig. 2(c), with three sizes l = 10 mm, 5 mm, 2.5 mm. In this application, the holes in Fig. 2(c) actually represent fibers, and a yarn typically contains thousands of fibers whose diameter depends on the material (carbon, glass, etc.). Composite manufacturing processes involve liquid resin flow around yarns as well as within yarns around the fibers.

Although it has been refined close to the porous part boundary to improve the accuracy, the coarse scale mesh shown in Fig. 12(a) contains only 14190 elements. The domains used for single-scale simulations are presented in Fig. 12(b-d). They are discretized with meshes that contain up to 4118396 elements for l = 2.5 mm.

Results are presented in Fig. 13, where the velocity component v M

x is shown along a diagonal line. For this example with more realistic porous medium boundary topology, the accuracy of multiscale predictions is again increased when the pore characteristic length becomes very small compared to the simulation domain size. This is particulary relevant for this application where the most representative setup is the one with l = 2.5 mm and it features the greatest number of fibers.

Conclusions

An alternative approach relying on equations of Stokes type instead of Darcy's law has been proposed in this paper to model flow in porous media. These equations depend on force and stress derivatives tensors that can be computed from pore scale simulations.

The latter are based on a fictitious fine scale domain where pores are discretized. The equations to solve for the fine scale domain in order to compute the force and stress derivatives tensors have been presented, as well as two Finite Element (FE) procedures for the numerical implementation.

Similarly to the permeability tensor in Darcy's law, the derivatives tensors can be used for simulations at the coarse scale with discretizations that are no longer required to deal with very small pores. The main advantage of the proposed approach is that it eases greatly the task of modeling flows in domains containing both fluid and porous parts.

While with Darcy's law, specific boundary conditions including the well-known Beavers-Joseph-Saffman condition should be imposed at the boundary between fluid and porous parts, the proposed approach circumvents this difficulty. This advantage both in terms of modeling and numerical implementation is obtained thanks to the use of equations of Stokes type instead of Darcy's law, which means the equations used in both fluid and porous parts are of the same type in the proposed approach.

Numerical simulations considering a domain including a fluid and a porous part have been conducted. Particular attention has been given to pressure and velocity evolution across the boundary between the two parts. It has been shown by comparisons with single-scale simulations that the proposed approach is accurate, and that its accuracy is improved with a lower pore characteristic length or volume fraction.

The proposed pure Stokes approach for coupling fluid flow with porous media flow opens quite a range of interesting possibilities. Among them, the implementation of numerical methods relying on an implicit representation of the interface between fluid and porous parts (e.g., the level-set method) should be facilitated.

Appendix

This appendix details the derivation of the proposed multiscale model and in particular the governing equations for the fine scale problem.

Kinematic averaging

The average of the fine scale velocity field is constrained to be equal to the coarse scale velocity at the corresponding point x of the coarse scale domain:

v M (x) = 1 |Ω m | Ω m v m (y)dy. (A.1)
Similarly, the average of the fine scale velocity gradient field is constrained to be equal to the coarse scale velocity gradient at the corresponding point x of the coarse scale domain:

∇ x v M (x) = 1 |Ω m | Ω m

  at each point x of the coarse scale domain. This domain is exclusively composed of closed and/or open pores, where the fluid can flow. The solid rigid part where the fluid cannot flow is not represented. As shown in Fig. 1, ∂Ω m includes the boundaries between the fluid and solid parts: Γ m O = Γ m O (x) ⊂ ∂Ω m (x). Dirichlet boundary conditions v m = 0 are imposed on Γ m O . The implementation of other types of boundary conditions such as the Navier condition may be considered in the future.

Figure 1 :

 1 Figure 1: Example of square fine scale domain Ω m in light gray including three holes of different shapes corresponding to solid parts where the fluid cannot flow. The domain boundary ∂Ω m in red includes the boundary Γ m O where Dirichlet boundary conditions v m = 0 are imposed.

  (a) is used for the fine scale domain associated to the fluid part, as it is sufficient to model Stokes flow (Shakoor and Park, 2023). The mesh shown in Fig. 6(b) for the fine scale domain associated to the porous part relies on the same mesh size used close to solids in the single-scale mesh shown in Fig. 3(a).

Figure 2 :

 2 Figure 2: (a) Coarse scale domain and boundary conditions with a dark part and a light part, (b) homogeneous fine scale domain used in the light part, (c) porous fine scale domain with φ = 52% used in the dark part, (d) porous fine scale domain with φ = 36% used in the dark part. Lengths are in meters and velocities in meters per second.

Figure 3 :

 3 Figure 3: Single-scale meshes: (a) coarse mesh with h = 4 µm close to solids, (b) fine mesh with h = 2 µm close to solids.

Figure 4 :

 4 Figure 4: Results using the coarse single-scale mesh: (a) pressure field (in Pa), (b) velocity component v x (in m • s -1 ).

Figure 5 :

 5 Figure 5: Single-scale results: (a) pressure along a horizontal line at y = 0.25 m, (b) velocity component v x along a vertical line at x = 0.5 m.

Figure 6 :Figure 7 :

 67 Figure 6: Fine scale meshes: (a) homogeneous fine scale domain, (b) fine scale domain for φ = 52% and with h = 0.04l.

Figure 8 :

 8 Figure 8: Multiscale simulation results compared to the reference single-scale solution: (a) pressure along a horizontal line at y = 0.25 m, (b) velocity component v M x along a vertical line at x = 0.5 m, (c) zoom on (a), (d) zoom on (b). The multiscale pressure

  (b) through 2 + 2 × 2 = 6 solutions involving 1008 fine scale velocity

Figure 9 :Figure 10 :

 910 Figure 9: Results for a fine scale domain size l = 0.025 m: (a) pressure field (in Pa) from the single-scale simulation, (b) velocity component v x (in m • s -1 ) from the single-scale simulation, (c) coarse scale pressure field (in Pa) from the multiscale simulation, (d) coarse scale velocity component v M x (in m • s -1 ) from the multiscale simulation.

  (d) is used to compute force and stress derivatives. Mesh size is kept at h = 0.04l, with l = 0.1 m. The single-scale mesh is similar to that shown in Fig 3(a), except that it has larger solids. Pressure and velocity curves are shown in Fig. 11. Reducing the porosity improves the performance of the multiscale model. This is clearly demonstrated by comparing Figs. 11(b)

Figure 11 :

 11 Figure 11: Multiscale and single-scale results for a porosity φ = 36%: (a) pressure along a horizontal line at y = 0.25 m, (b) velocity component v M x along a vertical line at x = 0.5 m, (c) zoom on (a), (d) zoom on (b). The multiscale pressure curve has been offset to ease the comparison.

Figure 12

 12 Figure 12: (a) Coarse scale mesh with an elliptical porous part for the composite application, (b) single-scale domain for l = 10 mm, (c) single-scale domain for l = 5 mm, (d) single-scale domain for l = 2.5 mm, (e) zoom on (d).

Figure 13 :

 13 Figure 13: Multiscale and single-scale results for the composite application: (a) velocity component v M x along the diagonal line joining the lower left corner to the upper right corner of the coarse scale domain, (b) zoom on (a). The vertical line on each graph marks the porous medium boundary.

  ∇ y v m (y)dy. (A.2) These constraints are additional to the boundary condition v m = 0 at Γ m O .

Funding

This work was supported by the French Agence Nationale de la Recherche (ANR) through the MISSA projet (ANR-22-CE46-0002).

Kinematic admissibility

Kinematic admissibility requires the averaging constraints for the velocity field to be satisfied. These constraints will all be added using Lagrange multipliers except for the Dirichlet boundary conditions on Γ m O , thus the functional spaces for the fine scale unknowns are

(A.3)

Given that additional Lagrange multipliers are set, these functional spaces can also be used for virtual variations.

Virtual powers

The coarse scale virtual power P M is based on the coarse scale problem stated in Eq. ( 2):

The averaged fine scale virtual power P m is based on Stokes equations for incompressible Newtonian flow written at the fine scale:

The divergence constraint on the fine scale velocity gradient has been added through Lagrange multiplier p m ∈ L 2 (Ω m ) and the kinematic averaging constraints on the fine scale velocity field and its gradient have been added through Lagrange multipliers (α, β) ∈ R d × R d×d . Similarly, Lagrange multipliers have been added for the variations of the fine scale unknowns.

Principle of multiscale virtual power

The principle of multiscale virtual power establishes the balance between the two scales.

Therefore, the problem is to find

which is equivalent to

By taking arbitrary variations from Eq. (A.7), it is possible to derive the problem to solve at the fine scale with all necessary boundary conditions, and also to derive the expressions of the missing terms for the coarse scale problem in Eq. (2).

Fine scale problem

As the spaces for the variations are very large, it is possible to zero out δV M , δG M . This leads to the problem of finding

(A.8)

Coarse scale force per unit volume and stress

By zeroing out all variations except δV M in Eq. (A.7), one obtains

Similarly, it is possible to obtain

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Author Contributions


Christine Petr
email: christine.petr@univ-ubs.fr

Paul Caudan

touristique des îles et distinguer les publics touristes vs excursionnistes : Apports d

Keywords: Théorie des parties prenantes, fréquentation, touriste, excursionniste, data Stakeholder theory, frequentation, tourist, excursionist, data Petr C

Measuring tourist attendance on islands and distinguishing tourist vs.

Introduction

Si la notion de « tourismophobie » n'est pas nouvelle pour les territoires traditionnellement sous haute pression touristique, surtout depuis la pandémie de Covid19 et les contraintes des voyages internationaux, on a vu se généraliser un peu sur tous les territoires, des discours et des actes d'agacement vis-à-vis des touristes. Ainsi, il devient presque courant de voir, à chaque période de vacances scolaires, des reportages sur le sujet de la tourismophobie. Et dans des zones fortement touristiques, il y a des plaintes régulières de la part de certains résidents, souvent les propriétaires de résidences secondaires. Mais ce sont aussi parfois des manifestations affichant des messages antitouristiques (« Tourist : Go home »). Dans ce contexte, tout le monde s'accorde à dire qu'il est crucial de proposer des solutions de gestion des flux touristiques. Toutefois, pour envisager la mise en place et les expérimentations terrains sur des solutions de gestion des flux de fréquentations, un préalable essentiel est de disposer de données objectivées, chiffrées et actualisées, et idéalement continues sur toute l'année et sur un spectre temporel horaire (ou a minima à la demi-journée) sur ces fréquentations. Pourtant, à ce jour, ces données disponibles sous un format barométrique font encore défaut, tout particulièrement sur les territoires insulaires de petites tailles. Pourtant, les responsables locaux et régionaux ne cessent d'insister sur le fait qu'ils ont besoin d'un outil de mesure de type baromètre qui réponde aux triples objectifs de 1) la mesure et de l'audit des fréquentations, de 2) la mise en perspective des différentes expérimentations et choix de gestion de chaque île, et dans l'idéal, de 3) base pour le pilotage journalier de la gestion de ces fréquentations de manière à pouvoir réaliser un réel accueil des visiteurs touristes et excursionnistes. Il s'agit de permettre aux acteurs du tourisme des îles de cesser de subir une fréquentation perçue comme profondément volatile mais de pouvoir anticiper les situations à l'échelle au moins de la demi-journée. Afin d'évaluer la faisabilité de ce type d'outil barométrique et afin d'envisager -ou inventer-les modalités permettant de construire un tel outil de mesure de la fréquentation et de distinction entre les touristes et les excursionnistes, une démarche de concertation suivant la logique d'action de la théorie des parties prenantes a été proposée. Ainsi, c'est l'ensemble des acteurs, régionaux, départementaux et locaux, de plusieurs îles bretonnes proches du continent et appartenant au réseau associatif des îles du ponant (association d'îles françaises du littoral de la Manche et de l'océan Atlantique) à qui il a été proposé d'exprimer leurs besoins, leurs motivations, mais aussi leurs difficultés et leurs freins. Par l'illustration du cas d'un projet de mesure et de qualification des fréquentations touristiques et excursionnistes sur des territoires insulaires, la recherche présentée ici permet de montrer les apports d'une approche par la théorie des parties prenantes pour les projets basés sur l'obtention de data sur le tourisme. Après la présentation du cadre théorique et de la problématique, nous décrivons la méthode de recherche et les résultats attendus, avant de conclure par les contributions.

Cadre théorique

La théorie des parties prenantes comme cadre structurant des projets de gestion touristique La théorie des parties prenantes est née des débats sur la raison d'être de l'entreprise. Pour [START_REF] Friedman | The Social Responsibility of Business is to Increase its Profits[END_REF], l'objectif classique de l'entreprise est le profit pour satisfaire les actionnaires (shareholders). Ce sont [START_REF] Freeman | Stockholders and Stakeholders: A New Perspective on Corporate Governance[END_REF] qui élargissent cette idée à la satisfaction des intérêts de l'ensemble des "parties prenantes" (stakeholders). Ces parties prenantes sont définies comme « tout groupe ou toute personne qui peut affecter la réalisation des objectifs organisationnels ou qui est affectée par la réalisation des objectifs organisationnels ».

Petr C. et P. Caudan (2023), Mesurer la fréquentation touristique des îles et distinguer les publics touristes vs excursionnistes : Apports d'une approche par la théorie des parties prenantes, 10e Conférence de l'Association Française de MAnagement du Tourisme (AFMAT), AFMAT, 31 May-2 June 2023, Girona, Spain.

La gestion d'un projet suivant la théorie des parties prenantes suppose en premier lieu d'identifier les interlocuteurs que constituent les parties prenantes et d'accepter de négocier avec elles (names and faces approach). En second lieu, il faut veiller à questionner ses propres points de vue pour tenir compte des besoins de toutes les parties prenantes. En troisième lieu, il faut s'attacher à trouver un accord entre les parties, dans la mesure où cet accord est, selon Freeman, toujours possible et souvent sources d'innovations puisqu'il faut faire preuve de créativité pour inventer le compromis ou la solution qui pourra convenir à tous. Ce qui fait l'originalité de cette approche est qu'elle permet de fédérer et de faire se parler des acteurs qui n'ont parfois pas l'occasion d'échanger. Le but de ces approches collaboratives est de vouloir augmenter les chances de succès du projet. Il s'agit d'une part d'éviter que certains acteurs se sentent floués ou abandonnés ce qui peut les conduire à concrètement s'opposer aux projets, ou à le ralentir et le rendre peu viable en raison de comportements volontaires de résistance et de détournement des principes et règles. Il s'agit aussi de permettre une meilleure adhésion aux projets. Plutôt que de devoir convaincre a posteriori de l'intérêt d'une démarche et de nouvelles infrastructures, il s'agit de favoriser son appropriation et la collaboration de tous.

Cette approche holistique des parties prenantes, bien connue et utilisée dans le cadre de projets de développements de territoires, est aussi aujourd'hui au coeur du management du tourisme durable [START_REF] Muzy | la gouvernance des organismes de gestion de sites, une approche par la théorie des parties prenantes, 9 ième Conférence de l'AFMAT[END_REF]. Rappelons la ligne de conduite que l'Organisation Mondiale du Tourisme prône selon laquelle le tourisme durable doit tenir "pleinement compte de ses impacts économiques, sociaux et environnementaux actuels et futurs, en répondant aux besoins des visiteurs, des professionnels, de l'environnement et des communautés d'accueil" (UNWTO, 2016).

Le tourisme durable pour les petites îles : l'importance des enjeux économiques

Le tourisme durable est un sujet qui mobilise beaucoup l'attention des chercheurs comme le montre le nombre important de publications et productions sur ce thème [START_REF] Bourelly | Identification, mobilisation, valorisation et régulation des compétences territoriales pour une tourisme durable au sein des PNR: le rôle des valeurs Parc sur le PNR de Brière, 9 ième Conférence de l'AFMAT[END_REF][START_REF] François-Lecompte | L'engagement environnemental du secteur hôtelier : proposition d'un modèle conceptuel[END_REF][START_REF] François-Lecompte | Le tourisme durable : les sept clés du succès[END_REF][START_REF] Falduto | L'OGD Iintercommunal : un acteur Stratégique des destinations touristiques durable ? Étude comparative : Le cas de Dracénie Provence Verdon Agglomération et Méditerranée Porte des Maures, 9 ième Conférence de l'AFMAT[END_REF]. Cet intérêt pour le tourisme durable est stimulé par le constat de nombreuses problématiques en lien avec la préservation : l'environnement, le patrimoine, la diversité culturelle [START_REF] Camus | Tourisme durable : une approche systémique[END_REF], et la littérature récente abonde largement sur la prise en compte des volets sociaux et environnementaux du tourisme durable, soulignant l'importance clef de ces éléments pour le tourisme [START_REF] Couao-Zotti | Sur fréquentation des sites touristiques du littoral varois : cartographie des acteurs touristiques autour du parc national Port-Cros, 9 ième Conférence de l'AFMAT[END_REF][START_REF] Pupion | Tourisme durable et valorisation de l'environnement du littoral aquitain : diversité et stratégies des acteurs[END_REF]. La tendance des chercheurs à focaliser l'attention du tourisme durable sur les deux piliers que sont l'environnement et le social/sociétal ne doit pas faire oublier le pilier de l'économie. Ainsi, le volet économique du tourisme durable est essentiel comme le rappelle la définition de l'OMT. Les managers de destination sont tous vigilants à cette dimension économique qui est vitale pour le développement des territoires et le maintien d'une population locale. A ce titre, peu de choses ont changé depuis la période où Swarbrooke (1999) définissait l'enjeu économique du tourisme durable. Aujourd'hui encore, le volet économique du tourisme durable implique et concerne l'optimisation des bénéfices, la diminution des coûts, la redistribution équitable et le développement des entreprises locales.

Dans le cas des îles, l'économie touristique est souvent vitale. En effet, le tourisme y permet le maintien d'une activité tout au long de l'année, et d'entretenir des services et des infrastructures rares pour leur faible densité d'habitants (médecin, pharmacie, MAM, école, etc.). Les lieux de tourisme sont avant tout des lieux de vie, et doivent l'être et les rester pour continuer d'attirer les touristes [START_REF] Pendlebury | Urban World Heritage Sites and the problem of authenticity[END_REF]. Ainsi, les îles ont besoin de recevoir un flux de visiteurs, touristes et excursionnistes, acceptable pour les différentes parties prenantes (ni trop, ni trop peu). Cela pose la question de la typologie des visiteurs (cf. paragraphe suivant) et confirme le besoin d'objectiver la fréquentation réelle et ressentie des îles.

Les îles de proximité du continent : des destinations d'impulsion

Dans le cas des destinations insulaires de proximité, c'est-à-dire des iles souvent de petites tailles facilement atteignables par une courte traversée en bateau (moins d'une heure pour beaucoup), la Petr C. et P. Caudan (2023), Mesurer la fréquentation touristique des îles et distinguer les publics touristes vs excursionnistes : Apports d'une approche par la théorie des parties prenantes, 10e Conférence de l'Association Française de MAnagement du Tourisme (AFMAT), AFMAT, 31 May-2 June 2023, Girona, Spain. fréquentation est à la fois très volatile et très importante du fait de l'importance des visiteurs excursionnistes. En effet, les petites îles des territoires côtiers font face à des défis de gestion de flux et de préservation de l'environnement et de la qualité de vie des habitants qui sont d'autant plus importants qu'elles sont aisément accessibles à la journée. En raison de la facilité de leur accès, elles sont régulièrement soumises à des situations ponctuelles d'hyper fréquentation qui sont difficiles à prévoir. En effet, leur visite est souvent fortement liée à des conditions météorologiques favorables qui favorisent une décision de promenade chez les résidents des territoires proches. La configuration de leur proximité au continent et de la simplicité de leur accès (bateau bus sans réservation souvent) conduit à en faire une sorte de destination d'impulsion. Conséquence de cette proximité, les îles proches du continent se retrouvent à subir les effets négatifs de fortes pressions touristiques sans pour autant bénéficier effectivement des retombées économiques liées aux activités touristiques. En effet, les visiteurs à la journée ou à la demi-journée consomment nettement moins que les touristes classiques qui dépensent pour se loger et se nourrir sur le territoire d'accueil.

La fréquentation excursionniste des îles : un enjeu essentiel mais peu de mesures

Il existe bien en France la taxe Barnier qui est une taxe sur les passagers maritimes embarqués à destination d'espaces naturels protégés, cas le plus souvent des destinations insulaires visitées. Cette taxe a deux vertus. Elle permet aux collectivités insulaires de recevoir d'une part des retombées économiques de tous leurs visiteurs, même s'ils ne résident pas sur l'île et ne payent pas alors la taxe de séjour classique du visiteur touristique, cette taxe permet aussi aux gestionnaires touristiques et des îles d'avoir une évaluation du nombre de leurs visiteurs utilisant des transports maritimes. A cette mesure, ils ajoutent ensuite des estimations relatives aux venues par des plaisanciers et propriétaires privés de leurs bateaux. Malgré ces éléments de mesures, à ce jour, le niveau de précision et le délai d'obtention des données de fréquentations touristiques et excursionnistes sont insatisfaisants pour envisager un pilotage de l'ensemble de la fréquentation à l'échelle des îles. En effet, dans les outils et les données existantes, seules des données de passage et de présence sont disponibles. Finalement, le tout reste relativement approximatif en termes de périodes et heures d'affluence, et les données permettent mal de distinguer la part des visiteurs et la part des excursionnistes.

Problématique

Dans la perspective de pouvoir piloter la gestion des flux touristiques, de faire des tests et des expérimentations sur certaines îles, et pour mettre en perspective les différentes situations insulaires, il est important de disposer d'informations objectives et qualifiantes sur la fréquentation globale de ces îles. En préalable à ce projet de gestion des flux touristiques, les responsables de la gestion de ces territoires ont besoin de pouvoir disposer de mesures et d'indicateurs qui permettent d'une part d'objectiver la fréquentation effective et d'autre part de distinguer la part des excursionnistes au sein des visiteurs de l'île. Afin de mener à bien ce projet d'obtenir ces données et indicateurs sur la fréquentation touristique et excursionniste, nous proposons d'adopter les principes de la théorie des parties prenantes à ce projet de mesure et de qualification des fréquentations touristiques et excursionnistes sur des territoires insulaires.

Méthodologie

La méthode L'étude consiste en des entretiens individuels et de groupes auprès de représentants des différents acteurs concernés par ce projet de connaissance du tourisme via la data. Sauf cas particulier, il y a entre 2 à 4 représentants par catégorie d'acteurs.

Les matériels de collecte: les guides d'entretien

Après avoir identifié les différentes parties prenantes en opérant des typologies et catégories d'acteurs, la démarche méthodologique consiste à concevoir un guide d'entretien dont les thématiques sont communes mais qui sont exprimées et proposées à la discussion de manière adaptée à chaque catégorie d'acteur.

Les méthodes d'analyse

L'analyse du corpus de données textuelles est réalisée selon les étapes classiques de l'analyse verticale par entretien, et de l'analyse horizontale. Une restitution des conclusions est fournie aux participants afin qu'ils puissent réagir, corriger ou préciser ces conclusions. In fine, les conclusions validées sont mises à disposition à l'ensemble des parties prenantes à l'occasion d'une session de synthèse et de brainstorming pour envisager des solutions et compromis de mesure et de qualification acceptables par tous.

Discussion -Contributions

Les contributions managériales sont de proposer un déroulé de travail collaboratif qui permet aux gestionnaires de la destination touristique à l'échelle régionale et aux responsables et élus des différentes collectivités territoriales insulaires, de déployer des solutions de collecte d'information qui répondent aux préoccupations légales et aux objectifs opérationnels, tout en étant acceptables par toutes les parties prenantes. L'apport de l'approche par les parties prenantes permet d'initier concrètement un process de négociation basé sur l'objectivation et la qualification de la fréquentation. Après la première phase d'identification des différentes catégories d'acteurs qui se retrouvent dans toutes les destinations ayant des problématiques de gestion d'importants flux excursionnistes, les documents de travail construits pour accompagner les entretiens individuels et de groupe de chacune des catégories d'acteur représentent des guides types et outils méthodologiques réutilisables avec des adaptations ponctuelles. Ce process est envisagé dès sa conception et documenté comme devant être réplicable pour d'autres destinations insulaires qui connaissent les mêmes défis d'arbitrages de tourisme durable entre nécessité de retombées économiques, obligation morale d'ouverture aux excursionnistes et day-trippers, et stratégie de préservation de l'environnement et de la qualité de vie pour les résidents.

Les contributions conceptuelles portent sur l'identification des points de blocages sous-jacents à la création des baromètres de mesure de la fréquentation et de la qualification/distinction des fréquentations touristiques et excursionnistes. Entre autres, il s'agit de montrer que l'approche par les parties prenantes sur le sujet de la data à collecter pour des sujets de fréquentation touristique implique de comprendre plus en détails quels sont les leviers et motivations à collecter et à diffuser cette data. Il s'agit aussi de relever les difficultés et les freins à cette collecte et à ces diffusions sachant que la mesure et de la qualification de la fréquentation entre touristiques ou excursionnistes, implique pour les quatre acteurs individuels (les résidents permanents, les résidents temporaires, les excursionnistes et les touristes) d'accepter de partager des données sur leur géolocalisation et sur leurs pratiques de vie et de loisir en tant que résidents du territoire élargi à l'espace du continent. La perception de la sensibilité / l'intimité de ces deux données apparaît donc au coeur de possibles blocages ou refus de partager ces données de la part des acteurs individuels qui les produisent. Si ces acteurs individuels sont réticents à diffuser et volontairement partager ces données, cela conduit le projet de mesure de la fréquentation à inventer et développer d'autres stratégies de comptabilisation et d'identification qui peuvent être moins précises (e.g. estimations basées sur les consommations électriques) et parfois plus coûteuses (e.g. mise en place de compteurs). L'exposé des coûts en termes de perte d'efficacité du projet pourrait être une piste intéressante pour faire reconsidérer leurs positions.

Références

Petr C. et P. Caudan (2023), Mesurer la fréquentation touristique des îles et distinguer les publics touristes vs excursionnistes : Apports d'une approche par la théorie des parties prenantes, 10e Conférence de l'Association Française de MAnagement du Tourisme (AFMAT), AFMAT, 31 May-2 June 2023, Girona, Spain.