Olivier Pironneau
email: olivier.pironneau@upmc.fr

Computational Issues for Optimal Shape Design in Hemodynamics

A Fluid-Structure Interaction model is studied for aortic flow, based on Koiter's shell model for the structure, Navier-Stokes equation for the fluid and transpiration for the coupling. It accounts for wall deformation while yet working on a fixed geometry. The model is established first. Then a numerical approximation is proposed and some tests are given. The model is also used for optimal design of a stent and possible recovery of the arterial wall elastic coefficients by inverse methods.

Introduction

Hemodynamics, a special branch of computational fluid dynamics, poses many problems of modeling, data acquisition, computation and visualization. However even as of now it is a valuable tools to understand aneurisms, to design stents and heart valves, etc (see for example [START_REF] Usabiaga | Staggered schemes for fluctuating hydrodynamics[END_REF][START_REF] Formaggia | Cardiovasuclar Mathematics[END_REF][START_REF] Thiriet | Biomathematical and Biomechanical Modeling of the Circulatory and Ventilatory Systems[END_REF]).

In this paper we shall focus on aortic flow, its modelisation, numerical simulation and inverse techniques.

Blood in large vessels like the aorta is newtonian and flows in a laminar regime with Reynolds number of a few thousands. The Navier-Stokes equation for incompressible fluid is a good model for it.

A blood vessel on the other hand is a complex structure for which linear elasticity is only a first crude approximation and for which the Lamé coefficients do not have a universal value and can vary with individuals.

Nevertheless, like many authors ( [START_REF] Tambaca | Mechanical Behavior of Fully Expanded Commercially Available Endovascular Coronary Stents[END_REF][START_REF] Nobile | an effective fluid-structure interaction formulation for vascular dynamics by generalized robin conditions[END_REF] for instance) we shall use Koiter's linear shell theory.

Koiter's Shell Model for Arteries

The following hierarchy of approximations for the displacement d of the aortic wall will be made:

• Small displacement linear elasticity instead of large displacement (needed for the heart).

• No contact inequalities with the surrounding organs.

• Shell model for the mean surface,

• With reference to the mean surface, normal displacement of the walls only.

Let Σ be the shell surface representing the mean position of the blood vessel.

Let n(x) be the normal at x ∈ Σ. Let d(x, t) be the displacement of the wall at x at time t. Normal displacement implies d = η n.

In [START_REF] Nobile | an effective fluid-structure interaction formulation for vascular dynamics by generalized robin conditions[END_REF] it is shown that under such conditions, Koiter's model reduces to the following equation of η on Σ

ρ s h∂ tt η -∇ • (T∇η) -∇ • (C∇∂ t η) + a∂ t η + bη = f s , (1)
where ρ s is the density and h the thickness of the vessel, T is the pre-stress tensor, C is a damping term, a, b are viscoelastic terms and f s the external normal force, i.e. the normal component of the normal stress tensor -σ s nn .

As with all second order wave type equations two conditions must be given at t = 0, for instance

η |t=0 = η 0 , ∂ t η |t=0 = η ′ 0 Remark 1 When [h, T, C, a] << b, (1)
leads to the so-called surface pressure model

-σ s nn = bη, with b = Ehπ A(1 -ξ 2 ) ( 2
)
where A is the artery's cross section, E the Young modulus, ξ the Poisson coefficient.

Some typical values are (in the metric system MKSA) for a heart beat of one pulsation per second:

E = 3M P a, ξ = 0.3, A = πR 2 , R = 0.013, h = 0.001, ρ f = 9.81 10 6
leading to b = 3.310 7 ms -2 and giving displacements in the range of 0.1 10 -3 m and flow rates around 2 10 -5 m 3 s -1

Fluid Equations

The Navier-Stokes equations in a moving domain Ω(t) define the velocity u and the pressure p:

ρ f ( ∂ u ∂t + u • ∇u) + ∇p -µ∇ • (∇ u + ∇ u T ) = 0, ∇ • u = 0, (3)
where ρ f is the density of the fluid and µ its viscosity.

Continuity on Σ of fluid and solid velocities implies

u = ∂d ∂t := n ∂η ∂t , on Σ
Continuity of normal stresses implies

σ f nn := n • (µ(∇u + ∇u T ) -p) n = -σ s nn := bη
Notice that as a consequence of the hypothesis of normal displacements only of the structure, there is no provision to write the continuity of the tangential stresses.

For aortic flow there also an inflow and an outflow boundary Γ i and Γ o on which we will prescribe pressure and no tangential velocity.

If S = Γ i ∪ Γ o , then the boundary Γ is Γ := ∂Ω(t) = Σ ∪ S = Σ ∪ Γ i ∪ Γ o
In [START_REF] Nobile | Vergana an effective fluid-structure interaction formulation for vascular dynamics by generalized robin conditions[END_REF] and many other authors, the matching conditions on Σ are written on the boundary of a fixed reference domain ∂Ω 0 because Koiter's shell model works with a fixed mean surface Σ.

With the notations of [START_REF] Decoene | Moving Meshes with freefem++[END_REF], assume that the domain of the fluid is

Ω t = A t (Ω 0 ) with A t : x 0 → x t := A t (x 0 ). Let u τ (x, t) = u(A t (A -1 τ (x)), t), ∀x ∈ Ω τ (4)
Then in Ω t at t = τ , the Navier-Stokes equations are in ALE format

ρ f ∂ u τ ∂t + ( u τ -c τ ) • ∇ u τ + ∇p --µ∇ • (∇ u τ + ∇ u T τ ) = 0, ∇ • u τ = 0, with c τ (x) = - ∂A t (A -1 τ (x)) ∂t | t=τ (5)
3 Transpiration Conditions for the Fluid

Conservation of Energy

We begin with an important remark on the conservation of energy.

The variational formulation of (3)-divided by ρ s -is, ∀û, p

Ω(t) [û • (∂ t u +u • ∇u) + ∇p • û -p∇ • u + ν 2 (∇u + ∇u T ) : (∇û + ∇û T )] = Ω(t) f s • û (6)
An energy balance is obtained by taking û = u and p = -p,

∂ t Ω(t) u 2 2 + ν 2 Ω |∇u + ∇u T | 2 = Ω f s • û - ∂Ω pu • n (7) because ∂ t Ω(t) u • w = Ω(t) ∂ t (u • w) + ∂Ω v u • w Ω ((u∇u) • u) = ∂Ω u • n u 2 2 = ∂Ω v 2 u • u (8) when v = u • n, the normal speed of ∂Ω.
With transpiration conditions we intend to work on a fixed domain with zero tangential velocity but non zero normal velocity u • n = w. In that case, in order to preserve energy, [START_REF] Hecht | New development in freefem++[END_REF] needs to be modify into

Ω [û • (∂ t u + u • ∇u) + ∇p • û -p∇ • u + ν 2 (∇u + ∇u T ) : (∇û + ∇û T )] - ∂Ω w 2 u • û = Ω f s • û (9)
or equivalently into

Ω [û • (∂ t u -u × ∇ × u) +∇p • û -p∇ • u + ν 2 (∇u + ∇u T ) : (∇û + ∇û T )] = Ω f s • û (10)
Finally we recall an identity (see [START_REF] Costabel | Singularities of electromagnetic fields in polyhedral domains[END_REF] for instance) which shows that we can use several forms for the viscous terms:

Ω [∇ × u • ∇ × v + ∇ • u∇ • v] = Ω ∇u : ∇v = Ω [ 1 2 (∇u + ∇u T ) : (∇v + ∇v T ) -∇ • u∇ • v] (11)
Hence a variational formulation adapted to the problem is to find u with u×n = 0 and, for all p and all û with û × n = 0

Ω [û • (∂ t u -u × ∇ × u) -p∇ • û -p∇ • u + ν∇ × u • ∇ × v] + ∂Ω pu • n = Ω f s • û (12)

Transpiration

As the wall vessel is {x + η n : x ∈ Σ} and as, by Taylor,

u(x + η n) = u(x) + η∇ u • n(x) + o(η)
matching the velocities of fluid and structure may be written as

u + η ∂u ∂n = n ∂η ∂t + o(η) on Σ, u × n = 0 (13)
On a torus of small radius r and large radius R, at a point of coordinates (R+r cos θ) cos ϕ, (R+r cos θ) cos ϕ, r sin θ), a straightforward calculation shows that

u × n = 0, ∇ • u = 0 ⇒ n • ∂u ∂n = (1 + r R cos 2 θ) u • n r So (13) becomes u • n = ∂ t η 1 + η r (1 + r R cos 2 θ) -1 , u × n = 0 (14)
Similarly the normal component of the normal fluid stress tensor is

σ f nn = p + 2(1 + r R cos 2 θ) µ r u • n
Therefore for a quasi toroidal geometry, for large R, (1) is

ρ s h∂ tt η-∇ • (T∇η) -∇ • (C∇∂ t η) + a∂ t η + bη = p + 2(1 + r R cos 2 θ) µ r ∂ t η 1 + η r (1 + r R cos 2 θ) -1 (15)
So, in fine, the domain Ω no longer varies with time but on part of its boundary

u • n = ∂ t η 1 + η r (1 + r R cos 2 θ) -1 , u × n = 0, ρ s h∂ tt η -∇ • (T∇η) -∇ • (C∇∂ t η) + a∂ t η + bη = p ( 16
)
where a is a non linear function of η.

Remark 2 Notice that η << r, i.e. large vessels, allows us to eliminate η and write everything in terms of ∂ t p and u n := u • n. It suffices to differentiates the last equation with respect to t and use the first one and integrate in time:

p = p 0 + L(u • n) := t 0 ρ s h∂ tt u n -∇ • (T ∇u n ) -∇ • (C∇∂ t u n ) + a∂ t u n + bu n (17)

Variational Formulation and Approximation

Coming back to (4) and using (17): Continuous Problem Find u with u × n = 0 and, for all p and all û with û × n = 0

Ω [û • (∂ t u -u × ∇ × u) -p∇ • û -p∇ • u + ν∇ × u • ∇ × v] + Σ p 0 + L(u • n) u • n = - S p Γ û • n

Spatial Discretization with Finite Elements

The easiest is to use penalization to enforce u×n = 0 by adding to the boundary integral

1 ǫ Σ u m+1 × n • û × n.
Then we may use conforming triangular or tetrahedral elements P 2 or P 1 +bubble for the velocities and P 1 for the pressure.

A freefem++ implementation (see [START_REF] Hecht | New development in freefem++[END_REF]) is shown on Figure 1 5 Optimization and Inverse Problems

Optimal Stents with the Surface Pressure Model

A stent is a device to reinforce part of a cardiac vessel and/or to change the topology of the flow by its rigidity. This results in a change of the coefficient b. So with a first order scheme in time we can consider

min b(x) J = Σ×(0,T ) F (p)dxdt : Subject to Ω [û • ( u m+1 -u m δt -u m+1 × ∇ × u m ) -p m+1 ∇ • û -p∇ • u m+1 ] + Ω ν∇ × u m+1 • ∇ × û + Σ (u m+1 bδt + p m n) • û = - S p Γ û • n ∀û ∈ V h , p ∈ Q h with û × n| Γ = 0 (19)
For instance F = |p| 4 will minimize the time averages pressure peak on Σ.

First order discretization and adjoint

Consider the adjoint state

Ω [v • v m -v m+1 δt -v × ∇ × u m-1 • v m -u m+1 × ∇ × v • v m+1 +ν∇ × v m • ∇ × v + ∇q • v m -q m ∇ • v] + Σ δtbv m • v = Σ F ′ (p m )q (20)
for all v, q such that v × n = 0 on ∂Ω.

Letting v = δu m , q = δp m and summing in m, from 1 to M gives

M 1 Σ F ′ (p m )δp m δt = M 1 δt Ω δu m • v m-1 -v m δt + M 1 δt Ω -δu m × ∇ × u m-1 • v m-1 -u m+1 × ∇ × δu m • v m +ν∇ × v m • ∇ × δu m + M 1 δt Ω (∇δp m • v m -q m ∇ • δu m ) + Σ δtbv m • δu m (21)

Optimality Conditions

As δu 0 = 0 and by choosing v M = 0 it is also

δJ = M -1 0 δt Ω v m δu m+1 -δu m δt + ν∇ × v m+1 • ∇ × δu m+1 - M -1 0 δt Ω δu m+1 × ∇ × u m • v m + u m+1 × ∇ × δu m • v m - M -1 0 δt Ω [δp m+1 ∇ • v m+1 + q m+1 ∇ • δu m+1 ] + Σ (δtbδu m+1 + δp m+1 n) • v m+1 (22)
The same is found by linearizing (19) and taking û = v m , q = q m , except that there is an additional term due to δb. In fine

δJ = δt 2 Σ δb M -1 0 u m+1 • v m (23)

Preliminary Computer Experiments

Experiment 1 This is only a feasibility test with F = p 4 ; The geometry is a quarter of a torus with R=4 and r=1. It is discretized with 1395 vertices and 6120 elements. The number of unknown of the coupled system [ u, p] is 23940 with the P 1 -bubble/P 1 element and Crank-Nicolson implicit scheme. The viscosity is ν = 0.01; we chose ǫ = ν. The final time is T = 1, the time step is dt = 0.1 and the pressure difference imposed at Γ i (top) and Γ o bottom is 6 cos 2 (πt).

The flow is stored on disk at every iteration ready to be reused backward in time for the adjoint equations.

Starting with b=200, after 3 iterations of steepest descent with fixed step size, the cost function is decreased from 1200 to 900. But as there is no constraint b is much reduced at the top near Γ i . Consquently the vessel wall becomes fragile as shown by a simulated wall motion by x → x + u m • ndt at every time step, as shown on Figure 2. The results are shown on figure 4. Because of the computing cost, we made only an initial study; the target is not reached, but 5 iterations go into the right direction. To do better one would have to used a varying step size gradient method and a better computer (this being done on a macbook pro, takes about 15 min).

Figure 1 :

 1 Figure 1: An implementation using freefem++ for problem (18)

Figure 2 :Experiment 2

 22 Figure 2: Top left: Optimization criteria versus iteration number. Top right: the coefficient b(x) after 3 iterations. Bottom Left: effect of the change of b on the dilatation of the vessel and some iso surfaces of constant pressure. Bottom right: a snap shot of the adjoint pressure and some iso surfaces.

Figure 3 :

 3 Figure 3: Left: Optimization criteria Σ×(0,T ) p 4 versus iteration number. Right: the coefficient b(x) after 4 iterations. Right: effect of the change of b on the dilatation of the vessel.

5. 2

 2 Identification of bFinally we run an identification test of b from the observation of the wall displacement, ideally, u • n. However the formulation does not allow it becausze the extra integral in the adjoint variational formulation is in competition with a similar term from the surface pressure model, so we used p/b. For this first test the criteria isJ = Σ×(0,T ) |pp d | 2 dxdtwhere p d is obtained from a reference computation (introduction of b in the criteria makes the problem harder) with b = 200 + 100 cos x cos y cos z.

Figure 4 :

 4 Figure 4: Left: Optimization criteria Σ×(0,T ) (pp d ) 2 versus iteration number. Right: the coefficient b(x) after 5 iterations. middle: The target b.

Acknowledgement: Special thanks to Frédéric Hecht for his help with freefem++ and Marc Thiriet and Suncica Canic for helpful discussions.

Approximation in Time

From now on, for clarity, we consider only the case of the surface pressure model, i.e. h = T = C = a = 0, L(u • n) = bu • n. However everything below extends to the full model.

So define

u(s)ds and use the integration rule U m+1 = U m + u m+1 dt.

Then Time discrete Problem p(t) = p 0 + bU (t) and we seek

where

Convergence

A convergence analysis was done in [START_REF] Chacon Rebollo | Analysis of a Simplified Coupled Fluid-Structure Model for Computational Hemodynamics[END_REF]; we recall the results. We denote u δ the linear in time interpolate of {u m } M 1 on (0, T ) = ∪ M 1 [ (m -1)δt, mδt]. For clarity let's assume that S = ∅.

Theorem 1 The solution of the time discretized variational problem satisfies

Theorem 2 If Ω is simply connected, there is a subsequence (u δ ′ , p δ ′ ) which converges to the continuous problem in L 2 (W) × H -1 (L 2 ) where


Keywords: spatial modelling, spatial simulation, geography, Internet, knowledge dissemination

Sharing and disseminating knowledge of advanced spatial modelling Presentation of an action carried out by the European research group S4

As in many sciences, the conception and the use of spatial modelling in geography has increased during the last ten years. Such a modelling approach is called "theoretical and quantitative geography". It aims to characterise and to study the spatial distribution of different kinds of phenomena using spatial data analysis and modelling. Studied phenomena can be for example population, activities, diseases, road networks, types of landscapes... Belonging to this field of research, the European research group S4 (Spatial simulation for social sciences) gathers researchers in geography as well as in geographical information sciences coming from about 30 European research centres1 . It has been created in January 2006, but its informal existence goes back 1978 (date of the first European colloquium on theoretical and quantitative geography).

Even if some parts of research belonging to the field of theoretical and quantitative geography could be very useful for urban and regional development and planning, at the moment, they remain mostly in the field of research. A fact is that few results are applied for answer practical concerns, in particular in France. The conclusions of a study realised in the framework of the PREDIT (French program dedicated to the promotion of research and innovation for transportation systems) illustrates this statement. The study underlines the real urgency both of a better legibility of spatial modelling research and of an increased coherence of the results of this research in order to better integrate them in the field of the territorial management. It emphasizes in particular the interest of the use of renewed and advanced mobility models for urban transportation management 2 .

Regarding this statement, the European research group S4 decided to develop an action entitled "Sharing and disseminating knowledge of advanced spatial modelling". The first aim of this action is to improve the diffusion of the results of research in spatial modelling, particularly in direction of regional and urban management and planning. The second aim is the development of tools and actions to improve coherence of knowledge and experiences that is especially required in those fields characterised by a rapidly developing research as it is the case for spatial systems analysis and modelling.

Ensuing from these aims, 4 objectives govern the presented action.

First aim: diffusion of the results of the research in spatial modelling

Objective 1 Diffuse the research results to the whole scientific community interested in spatial simulation: geography, urban and regional planning, but although economy, informatics, physics, biology... Objective 2 Disseminate methods, outputs and results to stakeholders. Objective 3 Promote main findings among the public through a "Science and Society" approach.

Second aim: development of tools and actions to improve coherence of knowledge and experiences. Objective 4 Ensure an optimal sharing of knowledge between the members of the S4 research group.

To answer these 4 objectives, several actions have been programmed. Some of them are in the process of being realised.

ACTION 1: DEVELOPMENT OF AN INTERACTIVE PLATFORM FOR GEOGRAPHY AND SPATIAL MODELLING

The S4 Interactive Platform is a web site with a double key access: for the members themselves and for an interested public (i.e. the whole scientific community, stakeholders, end-users and students).

http://www.spatial-modelling.info/ From a technical point of view, the S4 Interactive Platform is an ergonomic website characterized by a high level of user-friendliness and the ease of navigation. The Web-designers (grafactory.net, Besançon) have developed a customised Content Management System, which allows a dynamic site by preserving the autonomy of the authors for the content (articles, models, links…): modifications and news are set on line in real time by the members of the S4 research group themselves without needing to go through a web master.

The S4 Interactive Platform is organised on the basis of seven main entries.

• Who are we -This first heading offers basic information about the ERG S4.

• If you want contribute to the S4 platform -This second heading gives practical information about how to do this.

• Directory of links -This third heading is dedicated to the search of other web sites that deal with spatial analysis and modelling.

• Activities -This fourth heading offers a set of information about the network events. The announcement of seminars, conferences and general meetings appears here.

• Electronic Publication Platform -This fifth heading gathers two publications. Firstly, the European Journal of Geography Cybergeo 3 is a purely electronic journal founded in 1996. It was the first electronic journal in geography to appear in the world and one of the first in social sciences. The journal addresses the whole domain of geography, with a special emphasis on interfacing with other sciences and developing theoretical and applied modelling. Cybergeo plays a central role in the S4 Research Group, enhancing both its experience and reputation through a wider and more efficient dissemination of knowledge. Secondly, the Electronic Encyclopaedia Hypergeo is dedicated to the definition of basic terms used in geography. The aim of Hypergeo is to improve the diffusion of the concepts and the theories of the actual geography, particularly in direction of students and teachers.

• S4 Private Room -This sixth heading is dedicated to the exchanges between the members of the ERG S4. It allows them to share information about work in progress and results obtained in each working group. Facilities for shared writing is provided. 3h t tp://www.cybergeo.presse.fr/

• Spatial Modelling Park -In the framework of the CAENTI, this last heading represents the most interesting part of the S4 Interactive Platform. It requires a more detailed description.

Spatial Modelling Park

Several initiatives already exist, which aim to increase exchanges and communication about research on spatial systems modelling via Internet. As examples, we chose to quote the inventory of web sites realised by the CSISS (Centre for Spatially Integrated Social Sciences)4 of the University of Santa Barbara; and also the on line disposal software for spatial analysis and simulation as proposed by the AI-GEOSTATS, a Web service for geostatistics and spatial statistics (Joint Research Centre -European Commission) or by the Global Warming Resource Centre of the US Environmental Protection Agency. Although interesting, such initiatives only deal with one precise aspect (which could be either methodological, technical or centred on a given topic) of the whole field of spatial modelling of social systems. Moreover, most part of the research developed by the S4 members is not presented on these web sites. These statements led to the decision to create the Spatial Modelling Park.

Several years before the creation of the S4 Interactive Platform, two actors of the ERG S4 have already been interested in developing comparable initiatives.

•

The French Research Group Libergeo has developed an Interactive database on spatial modelling5 which contains 21 models (1999)(2000)(2001)(2002)(2003).

•

The web site of the Centre for Advanced Spatial Analysis (CASA) of London contains a set of pages dealing with the recent advances in the research on spatial simulation models (in particular, cellular automata and multi-agent systems) developed for studying and planning urban systems6 .

At the moment, the S4 Spatial Modelling Park comprises four on-line services:

• The section Spatial simulation models is a database on spatial modelling which comprises standard description of simulation models including tools for analysing the contained data.

• The Spatial simulation software repository contains downloadable software dedicated to the construction of different types of simulation model.

• The section Spatial analysis tools contains the description of a set of tools most of them being downloadable.

• The section Educational modules is dedicated to the presentation of basic scientific models. The aim of this section is to promote and popularise research in advanced spatial modelling by giving simple modelling demonstrations and applications.

It has also been planned to develop two other services:

• Consultancy, assessment, valuation of spatial simulation tools, which will allow the users to set the performance of the models, in particular by defining their field of application.

• Technology watch, which will help users by giving information about new advances as well as about the evolution of the available contents of other resources centres. The performance of the technology watch will be ensured by using an automatic news search engine.

Even if those two services are not yet developed, they are perhaps the most important ones. Actually, the concept of the S4 Interactive Platform is very close to existing Internet realisations, which aim to help the development of innovative firms by providing innovation tools and user reviews about each of them. In particular, the Virtual Environment for Innovation Management Technologies (developed in the frame of the European VERITE network) and the ONLI (On-Line Innovation) project (also supported by the European Commission) were an inspiration for the Spatial Modelling Park. A fact is that the assessment and the valuation of spatial simulation models is a broad and rather unexplored field of investigation, but several members of the ERG S4 already began to work in that direction.

The actual content of the S4 Modelling Park is as following (see tables 1 and 2).

• 11 simulation models are described on line, 8 of them being downloadable. 13 other simulation models will soon be published on line. At the moment, their description is not yet completed and only available via the S4 private room.

• 6 spatial analysis tools are described on line, 5 of them being downloadable. 2 other tools will soon be published on line.

• 1 software dedicated to the development of a given type of simulation model (i.e. cellular automata) is downloadable on line.

• 6 educational modules are described on line and 1 more module will soon be published on line.

Table 1

The S4 Interactive Platform: on line spatial simulation models (July 2006)

Table 2

The S4 Interactive Platform: on line spatial analysis tools (July 2006) Considering the general orientation of the Spatial Modelling Park, the development of a network policy for the preservation of intellectual property is required in order to avoid the plagiarizing of the know-how and to assure the platform's durability. A thought to this topic is work in progress. Some of the contributors to the S4 platform are in favour of the open source and free ware distribution of their software. Other contributors are not. A first possibility to preserve the intellectual property of their applications has been explored, which is the registration of a software using an InterDeposit Digital Number. The InterDeposit Digital Number is an international system for the identification of works offering the possibility of protecting any type of digital creation (music, sound, photographs, moving and still pictures, logos, texts, software, data bases, web sites, etc.) regardless of format. At the moment, one software of the S4 Platform (named Fractalyse) has been protected via an IDDN. The registration of the software Fractalyse is successful. Indeed, Fractalyse was potentially interesting for a scientific community not very numerous, widespread all around the world and not specific to a given scientific field. But, without any protection, the rights holders of the software would not have allowed many people to download it. Since Fractalyse is available on line, it is downloaded by people from all around the world belonging to different scientific fields (geographers, computer scientists, physicists...).

The S4 platform is in course of development. It means that it is the beginning of its life. Much is still to be done in order to add more contain, more description of applications, more spatial analysis tools and more software. But, the development of the platform is seen as a long time project, which should go on for several years.

An increased sharing of knowledge and know-how as a result of the development of the S4 Interactive Platform should generate important benefits.

Especially in terms of research in the field of spatial dynamics of social systems, the consequences should be:

• enhancing the initiatives dealing with the coupling of several models, • increasing knowledge of spatial processes through the comparison of numerous simulation results, • the diffusion of the models throughout the different research teams of the S4 network as some other tools circulate via Internet.

The expected results are the improvement of the capabilities and the performance of the future models that will be built by research teams of different countries.

ACTION 2: DISSEMINATING METHODS, INSTRUMENTS, OUTPUTS AND RESULTS TO STAKEHOLDERS AND END-USERS

To attain this objective some preliminary work is required, which will involve the definition of categories of stakeholders: national governments, local and regional authorities, regional development agencies, consultancy firms specialised in regional development and spatial and urban planning, corporate location professionals, professional and business organisations, etc. Then, the transition from stakeholders to end-users comes down the definition of a potential market of end-users.

Once the preliminary work of definition has been completed, it will be possible to define actions aiming to disseminate the S4 knowledge. This requires to look at the simulation methods and models themselves. Actually, as regards the needs of stakeholders and end-users, it is of crucial importance to test the operational use of models and to improve them. Ensuing from this is a very close link with the development of the Spatial Modelling Park to the S4 Interactive Platform. The goals are indeed converging in the same direction: sharing the knowledge between communities of actors of different level (searchers, stakeholders, end-users), promoting the understanding of the models and their reuse, striving towards a better understanding of the modelling process for spatial simulation in social sciences. Hence, testing the operational use of models and improving it represents a way of both exploiting the contents of the Spatial Modelling Park as well as improving it.

Another kind of reflection is also required which aim is to evaluate and better define the real possibilities of the operational use of the developed modellings. Such a field of investigation is at the heart of a doctoral research currently undertaken in the research centre ThéMA (Besançon, France) 7 .

S. Grandjean "Simulations spatiales intra-urbaines -

The research proceeds from the statement that several factors may occur in the fact that the spatial simulation models developed by researchers can mostly not be directly used for operational purposes. Firstly, existing modellings are characterised by a lack of conceptual framework, which consequence is the very few possibilities of combining the models the ones with the others. Secondly, the user interfaces are not enough oriented toward a general use. Thirdly, the applications are operational considering a given case (either fictitious or real), but they rarely allow the introduction of data different from those originally used.

The central question of the discussed research is about the identification of the possible transfers of the scientific knowledge as well as the possible diffusion of simulation tools to the social and economic world. As a result, the way of performing such transfers is of well interest. Three steps have been defined to tackle this question.

The first step aims to determine the relevance of different types of spatial simulation modellings considering operational goals (i.e. urban and regional planning).

The second step deals with the modelling process itself: its formal description will give a conceptual and reusable framework, which could be used to generalise the existing modelling tools and applications. Such a reusable framework would be helpful for:

-the share of knowledge between several types of users and actors (researchers, planners, consultancy firms...), -the promotion of the modelling tools and their reuse, -a better knowledge of the modelling process itself considering the goal of the simulation of spatial systems.

Within the field of Information Systems Development and Knowledge Representation, some methods have been developed which could helpfully be applied to move in such direction. One of these methods is the definition of an ontology which aims to identify and to represent the knowledge in the field of spatial modelling. The second method is the formalisation of the knowledge through reusable patterns that will serve as a basis for building Product Models. Here, a pattern represents a solution advocated to solve a recurrent problem defined in a given context. Now, regarding the previously mentioned problem of the plagiarizing of the S4 know-how, combining these two approaches (ontology and reusable patterns for building product models) will lead to the development of a shadowing method for the models.

The third step is dedicated to the application of the tools and methods developed during the two previous steps in order to create and apply operational spatial simulation modellings.

ACTION 3: EVALUATION OF EFFICIENCY OF THE ACTIONS 1 AND 2

Two benchmarking actions should be usefully undertaken, but at the moment, they remain under consideration. The first action would aim to evaluate sharing of knowledge as a crucial part of the efficiency and effectiveness of the ERG S4. Its realisation could be based on the definition of quantitative and qualitative indicators and face-toface interviews and meetings during the conferences and workshops. The second action could consist in ex-ante and ex-post evaluation of the dissemination actions also through the definition of quantitative and qualitative indicators.

Conclusion

The action "Sharing and disseminating knowledge of advanced spatial modelling", carried out by the European research group S4, participates in the development of research in geography seen as very active regarding its opening toward other scientific fields as well as the evident potentiality of using spatial modellings for urban and regional planning. But, although very promising, much is still to be done to ensure the general use of spatial modelling for operational purposes.

Table 1

 1 The S4 Interactive Platform: on line spatial simulation models(July 2006)

			Creation date
	Name of the model	Author(s)	of the	Downloadable
			presented
			version
	CalCom: Calculating the Attractiveness of Retail Clusters ThéMA, Besançon, France	2000	yes
	SimNoise: Simulating the Impact of Noise Annoyance on	ThéMA, Besançon, France	2003	yes
	Intra-urban Residential Migration
	Environmental and Urban Systems CamDeus: Cellular Automata Models to Design	France Image et Ville, Strabourg,	2002	no
	Growth Peuplement: Simulation of Size Differenciated Urban	Provence, France ESPACE, Aix-en-	2001	yes
	Project 3D-IMA: Simulation of the Visual Impact of a Planning	l'Université de Tours, polytechnique de CITERES, Ecole	2001	yes
		France
	Spatial Diffusion of an Innovation: a Multi-agent	ESPACE, Avignon, France	2003	no
	Formalisation of the Hägerstrand's modelling
	Urban Growth Modelling using Cellular Automata: the	IDEES, Rouen, France	2003	no
	example of the urban area of Rouen
	coastal zone management RamCO: An integrated assessment tool for sustainable	Maastricht, Netherlands for Knowledge Systems), RIKS (Research Institute	1999	yes
	in St. Lucia SimLucia: climate change and dynamic land use planning	Maastricht, Netherlands for Knowledge Systems), RIKS (Research Institute	1996	yes
	Waddenzee WadBOS: A policy support system for the Dutch	Maastricht, Netherlands for Knowledge Systems), RIKS (Research Institute	2001	yes
	Newfoundland Lobster-IBM: individual based model of lobster fishing in	Maastricht, Netherlands for Knowledge Systems), RIKS (Research Institute	2003	yes

European Research Groups are research networks, with no legal identity, made of public or private European laboratories, working on the same scientific theme. The European Research Group S4 proposes to develop spatial dynamic modelling as an integrative tool for understanding, discussing and helping to manage the evolution of our complex societies. It is directed by Denise Pumain.

2B a y e E . et al., Bilan de compétences des laboratoires en recherche français en matière de modélisation des déplacements de voyageurs et de marchandises, Octobre 2002, Rapport PREDIT.

http://www.csiss.org/

http://www.mgm.fr/libergeo/modele

http://www.casa.ucl.ac.uk/research/index.htm

Rendre opérationnels les modèles issus de la recherche fondamentale", directed by P. Frankhauser and C. Tannier.

Table 2

The S4 Interactive Platform: on line spatial analysis tools (July 2006)


Ronan Gaugne

Valérie Gouranton

Georges Dumont

Alain Chauffaut

Bruno Arnaldi

Immersia, an open immersive infrastructure: doing archaeology in virtual reality

come

INTRODUCTION TO VIRTUAL REALITY

Virtual reality is often considered solely as a field of computer science in relation to interactive digital 3D worlds. It actually holds a special position in the usual scientific scheme by coupling humanities sciences with engineering. According to Fuchs et al. [START_REF] Fuchs | Virtual Reality: Concepts and Technologies[END_REF] « The purpose of virtual reality is to make possible a sensorimotor and cognitive activity for a person (or persons) in a digitally created artificial world, which can be imaginary, symbolic or a simulation of certain aspects of the real world. » This proposition positions the man and its activity in the centre of virtual reality. A technical and literal definition of virtual reality [START_REF] Fuchs | Virtual Reality: Concepts and Technologies[END_REF] attempts to characterise the domain in one compact and sufficiently consensual sentence so that the practitioners of the domain can relate to it. "Virtual reality is a scientific and technical domain that uses computer science and behavioural interfaces to simulate in a virtual world the behaviour of 3D entities, which interact in real time with each other and with one or more users in pseudonatural immersion via sensorimotor channels."

Virtual reality historic

Virtual reality is a relatively young domain. Ivan Sutherland is considered as a great precursor, through results published in his MIT PhD in 1963 [START_REF] Sutherland | A Man-Machine Graphical Communication System Garland Publishing[END_REF]. He contributed shape editing and manipulation tasks with the SketchPad tool, which foreshadowed our current to industrial CAD tools. Shortly after, in 1965, Ivan Sutherland published "The Ultimate Display" [START_REF] Sutherland | The Ultimate Display[END_REF] where, in particular, he discussed the key elements related to immersion, interaction, capture of user activity and realism of 3D environments. This is to be considered and assessed in a context where the practical implementation of these concepts was impossible given the available technology. Known as the "inventor" of the term Virtual Reality, Jaron Lanier founded in the 80's the company VPL Research, the first SME to industrialize software and hardware solutions for the production of virtual reality applications. In particular, among its solutions, this company integrated the first Datagloves.

In any virtual r e a l i t y a p p l i c a t i o n , t h e u s e r i s i m m e r s e d a n d i n t e r a c t s w i t h a v i r t u a l e n v i r o n m e n t . Immersion is defined in [START_REF] Pimentel | Virtual Reality: through the new looking glass[END_REF] as a (perceptive, mental, emotional) state of a subject when one or several senses are isolated from outside and only stimulated with information from computers.

In the field of immersive visualization, the early 90s was marked by the development of immersive visualization systems with very wide screens, as the CAVE ™ in EVL (Electronic Visualization Laboratory: University of Illinois) [START_REF] Cruz-Neira | The CAVE: audio visual experience automatic virtual environment[END_REF], by Fanti and Cruz-Neira. This concept was then extended to less cumbersome concepts of workbench.

In 1990, Poupirev et al [START_REF] Poupyrev | Egocentric Object Manipulation in Virtual Environments: Empirical Evaluation of Interaction Technique[END_REF] proposed to categorize the interaction into two major families:

• Exocentric interactions: the user is in an external position in relation to the virtual environment so that the environment can be seen as a whole. Among these techniques, we find in a way Worldin-Miniature [START_REF] Stoakley | Virtual Reality on a WIM: Interactive Worlds in Miniature[END_REF], Metaphor resizing [START_REF] Mine | Moving Objects in Space: Exploiting Pro-prioception in Virtual Environment Interaction[END_REF], and the technique of Voodoo-Dolls [START_REF] Pierce | Voodoo dolls : seamless interaction at multiple scales in virtual environments[END_REF], all providing a means to manipulate objects by miniature representations; • Egocentric Interactions: the user is inside the virtual world. Among these techniques, we find the technique of virtual hand and its many variants [START_REF] Mine | Virtual Environment Interaction Techniques[END_REF]: linear extension of the arm when the virtual user's hand is close to his body, then exponential expansion from a given distance [START_REF] Bowman | An Evaluation of Techniques for Grabbing and Manipulating Remote Objects in Immersive Virtual Environments[END_REF].

Haptic perception when interacting (tactile feedback [START_REF] Massimino | Sensory Substitution for Force Feedback in Teleoperation[END_REF] or force feedback [START_REF] Sreng | Spatialized Haptic Rendering: Providing Impact Position Information in 6DOF Haptic Simulations Using Vibrations[END_REF]) is a very strong trend in the scientific community, and contributes to multimodal feedback [START_REF] Zhang | The Use of Visual and Auditory Feedback for Assembly Task Performance in a Virtual Environment[END_REF] (visual, auditory and haptic).

Concerning local or distant collaborative interaction [START_REF] Fleury | A Generic Model for Embedding Users' Physical Workspaces into Multi-Scale Collaborative Virtual Environments[END_REF], the problem is essentially to permit several users to interact simultaneously on the same environment. Collaborative work in Virtual Reality addresses the problem of the perception of other users and their activities [START_REF] Fraser | Supporting Awareness and Interaction through Collaborative Virtual Interfaces[END_REF]. Several models of synchronization and communication architectures [START_REF] Duval | PAC-C3D: A New Software Architectural Model for Designing 3D Collaborative Virtual Environments[END_REF], (client/server, P2P) can ensure the coherence of perceptual interactions and the validity of the collaborative interaction.

Virtual reality and archaeology

Introduced by Reilly in 1990 [START_REF] Reilly | p u t e r A p p l i c a t i o n s i n A r c h a e o l o g y 1 9 9 0 , B r i t i s h Archaeological reports[END_REF], virtual archaeology was initially presented for excavation recording and virtual re-excavation using multimedia technologies. In a similar way, Krasniewicz [START_REF] Krasniewicz | Immersive imaging technologies for archeological research[END_REF] proposed a 360° visualization infrastructure to help archaeologists in their research work. In this case, virtual archaeology was not used to restore knowledge, but to acquire new knowledge. More recently, Pujol Tost et al. [START_REF] Tost | Vers une Réalité Virtuelle véritablement interactive[END_REF], contributed to the interaction, perception and simulation in VR for archaeology, and not only for the visualization of 3D models.

Le Cloirec [START_REF] Cloirec | Les atouts de l'imagerie 3D pour l'archéologie de terrain[END_REF], focused on 3D models, and further extended their use in immersive structures as a perception tool to evaluate the functional or symbolical role of some architectural elements and spaces. Vergnieux [START_REF] Vergnieux | Archaeological research and 3D models (Restitution, validation and simulation)[END_REF], considered the simulation for the movement and gesture validation, the physical coherency and the technical feasibility of constructions.

According to new trends in the domain, Forte [START_REF] Forte | Cyber-Archaeology: Notes on the simulation of the past[END_REF], one of the mainstays of virtual archaeology, suggests to replace the terminology related to a "reconstitution of the past", by the expression "Cyber-Archaeology" relying on a "simulation of the past".

Christou et al. [START_REF] Christou | A versatile large-scale multimodal vr system for cultural heritage visualization[END_REF] used an immersive CAVE-like structure combined with haptic devices and 3D sound for pedagogic purposes in a museum exhibition, but also as a tool for research. The Archave project [START_REF] Vote | Discovering petra: Archaeological analysis in vr[END_REF] integrated also a CAVE-like structure and proposed tools for archaeologists to study historical sites. Another interesting work [START_REF] Forte | Cyberarchaeology: Experimenting with teleimmersive archaeology[END_REF] presents a tool where archaeologists collaborate remotely on shared virtual objects through realistic avatars reconstructed from 3D cameras. Their virtual reality framework proposed a rich toolkit of interaction features, including navigation, measurement, lighting and dragging. However, due to real-time 3D capture and rendering of users, the visualisation was restricted to small image resolution (320x240 pixels) to ensure fluid rendering (25 FPS).

All of these contributions were based on models of sites manually build in 3D modelling tools. Another trend, when dealing with existing buildings or sites, is to consider photomodeling tools. This technique allows to automatically build textured 3D models from sets of photos of a site or object. An interesting example is presented in [START_REF] Drap | Towards a Photogrammetry and Virtual Reality Based Heritage Information System: A Case Study of Shawbak Castle Int[END_REF] with the provision of an immersive virtual reality interface for Shawback Castle. The immersive environment consists of a large stereoscopic projected wall, with infrared tracking. The work focuses on semantic annotations of blocks to ensure consistency with a knowledge database associated to the model.

In parallel, during the last decade, many contributions in the domain focussed on 3D restitution of past structures or environments v i s u a l i z a t i o n . In particular, there has been a great deal of effort to communicate archaeological and historical heritage to a large public through various media such as web [START_REF] Pescarin | Back to 2nd AD AVR on-line experience with Virtual Rome Project[END_REF], second life environments [START_REF] Morgan | re)building catalhoyuk : Changing virtual reality in archaeology[END_REF], or museum exhibitions [START_REF] Bale | Linking Evidence with Heritage Visualization using a large Scale Collaborative Interface[END_REF].

GOING FURTHER

We believe that fields of virtual reality and archaeology may enjoy mutual benefits triggered by questions such as:

1. How can virtual reality, considered in its whole scope from science and technology to human sciences and natural sciences, benefit to archaeology? 2. What are the specific challenges brought by archaeology to virtual reality?

Virtual reality for archaeology Immersion, and particularly immersion in scale one environments, should allow archaeologists to access to the evaluation of symbolical or cultural roles of architectural buildings. Moreover, she or he archaeologist can be placed in situations to validate specific activities, ranging from displacements within environments to more complex interactions by using haptic devices to evaluate the physical feasibility or the coherency of a task. This aspect is strongly pertained to ergonomics and musculo-skeletal activity [START_REF] Pontonnier | Designing and Evaluating a Workstation in Real and Virtual environment: from Digital Mock-up to Realization[END_REF]. Figure 1 illustrates an experiment performed in our virtual reality platform. In order to evaluate ergonomics, the user was immersed in a virtual environment mimicking the real environment. Motion capture, electromyographic (EMG) electrodes, force sensors and subjective questionnaires were used to evaluate sensory-motor aspects.

Collaborative virtual reality can be of a great help to build or modify shared the 3D environments through the collaboration of multiple users. In such a context, we proposed the Collaviz open-source software platform [START_REF] Dupont | Collaborative Scientific Visualization: The COLLAVIZ Framework[END_REF] dedicated to real-time remote collaboration for the design and visualization of scientific simulation-based applications. We connected two immersive infrastructures, Immersia and UCL VR platform, enabling two distant users to collaborate on the same digital world, a 3D model of the underground propagation of a seismic wave, using shared tools [START_REF] Fleury | Evaluation of Remote Collaborative Manipulation for Scientific Data Analysis[END_REF](Figure 2). It is worth noting that a lot of work is produced on integrating digital 3D models of large landscapes such as countries (http://www.territoire3D.com) in virtual reality. In a similar way, reconstitutions of archaeological sites in their real geographical context are certainly a great challenge.

From archaeology to virtual reality

Until now, virtual reality has been essentially focussed on industrial applications. In this context, the user interacts with recent, smooth and clean, manufactured objects modelled by well-known CAD tools. In the context of archaeology, manipulated objects may be closer from nature, less steady, thus represented by more complex geometric models. The management of this new kind of data requires the study of new methodologies. Since reconstitution of historical sites is an empirical activity, subject to assumptions, approximations and modifications, virtual reality will have to propose methods to handle this uncertainty and to enable dynamic modification of the digital universe. Realism and objective credibility of the rendering are diametrically opposed issues. Objective credibility is an intrinsic quality of the model depending of the archaeologist perception. As an X-ray picture has a functional credibility for a medical practitioner, virtual archaeological models and universes must be designed to ensure this credibility. This will lead the development of generic methods that should push the boundaries of virtual reality. Scale-one immersion is a key characteristic proposed by virtual reality to archaeology, it is also a real issue in virtual reality while dealing with the concept of immersion within an environment at human size.

IMMERSIVE VIRTUAL REALITY IN IMMERSIA

We present the immersive platform Immersia, its role in the European project Visionair, and two projects related to archaeology that illustrate the possibilities such an environment.

Immersia platform

The immersive platform of the Irisa/Inria computer science laboratory is a large virtual-reality room dedicated to real-time, multimodal (vision, sound, haptic, BCI) and immersive interaction. It hosts experiments using interactive and collaborative virtual-reality applications that have multiple local or remote users. Images are rendered on four glass screens: a front one, two sides and a ground. Dimensions are 9.6 m wide, 2.9 m deep and 3.1 m high. Over 15 millions pixels are displayed. The visual reproduction system combines thirteen HD projectors. A tracking system, composed of 16 infrared cameras, enables real objects to be tracked within the scene. Images are recomputed as the user moves to fit his point of view, together with high-resolution rendering, active stereoscopy a n d h o m o g e n e o u s c o l o u r i n g , thereby delivering a visually realistic experience. Spatial sound is rendered by speakers with 10.2 format or by headsets with virtual 5.1, controlled by the user's position.

Visionair project

Immersia is a key node of the FP7 European Project Visionair [START_REF] Kopecki | VISIONAIR: VISION Advanced Infrastructure for Research[END_REF] which goal is to create a European infrastructure that should be a unique, visible and attractive entry towards high-level visualisation facilities for Virtual Reality, Scientific Visualisation, Ultra High Definition, Augmented Reality and Virtual Services. These facilities, distributed across about twenty countries in Europe, are open and easily accessible to a wide set of research communities. Both physical access and virtual services are provided by the infrastructure. Full access to visualization-dedicated software is offered through call for projects, while physical access to high level platforms will be partially accessible to other scientists, free of charge, based on the excellence of the project submitted.

Archaeology research projects in Immersia

Two projects started in Immersia, in collaboration with two different archaeology research laboratories. The first one, with the Creaah, a laboratory of archaeology, archeoscience and history, relies on immersive reconstitutions of several paleolythical existing buildings. Associated with interactive tools, the project enables archaeologists to extract new knowledge ex situ. The second one, with the French preventive archaeology research institute INRAP, focuses on the immersive reconstitution of one Gallo-Roman ruined site, the Bais domain.

The Carn monument

This project has with three objectives: (i) propose a common workflow to reconstruct archaeological sites as 3D models in fully immersive systems, (ii) provide archaeologists with tools and interaction metaphors to exploit immersive reconstitutions, and (iii) develop the use and access of immersive systems to archaeologists. In a first contribution [START_REF] Gaugne | Large-scale immersive reconstitution of a Neolithic corbel dome[END_REF], we proposed a procedure to model the central chamber of the Carn monument, and compare several softwares to deploy it in an immersive structure (Figure ). We then proposed two immersive implementations of the central chamber, with simple interaction tools, and evaluated them with the help of archaeologists. The results obtained in this work validated the possibility and relevance of working in an immersive virtual reality structure.

The gallo-roman Bais site

Following a different approach, we started from a 3D model proposed by an archeologist of INRAP based on the excavation of a gallo-roman villa in Bais, France [START_REF] Cloirec | Le bourg St Père, propoition de restitution des volumes, dans Dominique Pouille -"La villa gallo-romaine du bourg St Père à Bais (35)[END_REF]. We then built an immersive application from this model, focusing on the navigation inside the site (Figure ). The goals of this work, still under progress, are (i) to validate at scale-one, different assumptions made by the archaeologists during the modelling step, (ii) to study the possible human displacements in the site, (iii) to better understand the life of the people with the immersive perception of the place.

Virtual reality provides in this case tools for argumentation, reasoning and understanding to the archaeologists, by placing them at the centre of the simulation.

CONCLUSION

This paper focussed on cross-domain mutual enrichment between archaeology and virtual reality. On the one hand, Virtual reality could be a real improvement for archaeological field by ensuring the ability to propose collaborative modelling and analysis of archaeological objects in their real geographical context. On the second hand, the application field of archaeology could provide complex models that will lead to propose new generic methods for complexity management and interaction in an immersive context for large environments.

Figure 1 A

 1 Figure 1 A Manipulation task in real (left), virtual (centre), virtual with haptic (right).

Figure 2

 2 Figure 2 Collaborative work between Immersia and UCL VR platform.

Figure 3

 3 Figure 3 Immersive work in the virtual chamber of Carn

Figure 4

 4 Figure 4 Exploring the Bais Gallo-Roman villa

Université de Rennes 1; Ronan.Gaugne@irisa.fr

INSA de Rennes;Valerie.Gouranton@irisa.fr

Ecole normale supérieure de Cachan, antenne de Bretagne ; Georges.Dumont@irisa.fr

INRIA; Alain.Chauffaut@inria.fr

INSA de Rennes; Bruno.Arnaldi@irisa.fr