KUNSOAN N B David S Usami
email: davideshingo.usami@uniroma1.it

Noella B Kunsoan
email: noellabajia.kunsoan@uniroma1.it

Luca Persia
email: luca.persia@uniroma1.it

Brayan Gonzalez Hernandez

Eleonora Meta
email: eleonora.meta@uniroma1.it

Maria Rosaria Saporito
email: &mariarosaria.saporito@uniroma1.it

Govert Schermers
email: govert.schermers@swov.nl

Laurent Carnis
email: laurent.carnis@ifsttar.fr

Joël Yerpez
email: joel.yerpez@ifsttar.fr

Nesrine Bouhamed
email: nesrine.bouhamed@ifsttar.fr

J Cardoso
email: joao.cardoso@lnec.pt

L Kluppels
email: ludo.kluppels@vias.be

la diffusion de documents scientifiques de niveau

Keywords: Road Safety, Safety System, Vision Zero, Africa, Decade of Action for Road Safety

recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Road safety is a major health concern in society. In fact, around 1.35 million people die and 50 million are injured in road crashes every year [START_REF]Global status report on road safety[END_REF]. Road traffic crashes are estimated to be the ninth leading cause of death and projections reveal that it will be the third leading cause of death by 2020 [START_REF] Peden | World report on road traffic injury prevention[END_REF]. Relative to their level of motorization, Low-and Middle-income countries (LMICs) bear a disproportionately high burden of road deaths, when compared to more motorized high-income countries. Approximately 90% of the related deaths resulting from road traffic crashes (RTCs) occur in LMICs, while these countries account for 82% of the world's population and their level of motorization stands at only 54% of the world's registered vehicles [START_REF]Global status report on road safety[END_REF]. In the same way, the accident risk (number of accidents per traffic exposure, such as vehicle-km) is generally higher in developing countries, with far more severe consequences, as well ( [START_REF] Nordfjaern | Perceptions of traffic risk in an industrialised and a developing country[END_REF] and [START_REF] Sobngwi-Tambekou | Road traffic crashes on [16] 26th World Road Congress the Yaoundé-Douala road section, Cameroon[END_REF]).

Africa is the worst performing continent in road safety. In 2013, the mortality rate in this Continent (26.6 fatalities/10 5 population) was almost three times that of Europe, where the number of road fatalities represented 31% of the relevant global picture [5]. However, the most disturbing concern is the fact that the disparity in road safety results seems to be increasing. Specifically, according to the World Health Organisation [START_REF]Global status report on road safety[END_REF], fatality rates increased in Africa from 26.1 per 100,000 population in 2013 to 26.6 per 100,000 population in 2016.

In any case, progress has been made by some countries in mitigating the number and severity of road accidents [START_REF] Adminaite | Ranking EU progress on road safety: 10th Road Safety Performance Index Report[END_REF], but the situation in most LMICs is alarming and worsening [START_REF] Bliss | Meeting the management challenges of the Decade of Action for Road Safety[END_REF]. Overall transport policies in LMICs are often poorly designed and implemented. Infrastructure investments are lagging and price instruments are rarely used [START_REF] Foster | Africa's Infrastructure[END_REF]. In Africa, several actions are already ongoing and important high level documents are already in place, paving the way for road safety improvements, such as the African Road Safety Action Plan 2011-2020 [START_REF]African Union and United Nations Economic Commission for Africa[END_REF], a result from the common effort of the African Union (AU) and the United Nations Economic Commission for Africa (UNECA).

Europe could play an important role in supporting African countries to improve their road safety and traffic management performance, due to the improvements achieved in this area. These considerations are addressed through the SaferAfrica project; a joint effort of 17 partners from Africa and Europe, aiming to create favorable conditions and opportunities for the effective implementation of road safety and traffic management actions in the African countries, by setting up a Dialogue Platform between both continents [START_REF] Mavromatis | Assessing road safety data collection systems and definitions in Africa[END_REF].

In order to improve road safety performance in African countries, many barriers must be overcome [START_REF] Mavromatis | Assessing road safety data collection systems and definitions in Africa[END_REF]. The adoption of the Safe System approach to road safety will be a valuable tool to implement effective countermeasures that may significantly improve road safety in Africa. In fact, the Safe System approach to road safety has been successful in a number of countries [START_REF] Mooren | Australasian College of Road Safety Conference 'A Safe System: Making it Happen![END_REF] and has been adopted in the United Nations Plan for the Decade of Action for Road Safety 2011-2020 [START_REF]World Health Organization, Global Plan for the Decade of Action for Road Safety 2011-2020[END_REF].

Towards this direction, the objective of the present paper is to outline the results of the process used in the SaferAfrica project to define Safe System projects in the four mentioned African countries (Cameroon, Burkina Faso, Tunisia, and Kenya), which are suitable to their existing contexts. This process included the development and application of a transferability assessment tool, within a "participative" process involving all possible national interested parties, from the institutions to NGOs.

SAFE SYSTEM APPROACH

The Safe System approach evolved from the visions that emerged in Sweden and The Netherlands in the mid-1990s [START_REF] Tingvall | The Swedish'Vision Zero'and how Parliamentary approval was obtained[END_REF]. In Spring 1995, work on the development of a safe system approach started within what was formerly the Swedish Road Administration. The results of this development work were documented in a memorandum entitled 'Vision Zero -An idea for a road transport system without unrecoverable health losses' [START_REF] Vägverket | Nollvisionen -En idé om ett vägtransportsystem utan hälsoförluster[END_REF]. Vision Zero entails a shift in the road safety planning paradigm. Instead of starting from an existing problem situation, Vision Zero departs from an absolute state of the futuresafe road traffic [START_REF] Belin | Vision Zeroa road safety policy innovation[END_REF]. In the Netherlands, a similar policy was developed in the 1990s by the Dutch Institute for Road Safety Research (SWOV) in an effort to promote "inherently safe road traffic". This vision was named "Sustainable Safety" [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF].

At the time, scientists and policy makers began to question the prevailing view that the safety of road users was, in the last instance, a matter of their own responsibility, and that the task of road safety policy was thus primarily to influence road users' behaviour so they would act safely at all times. As the decades-long decreases in the number of road fatalities and severe injuries were levelling out, it became clear that a predominant focus on education, information, regulation and enforcement was no longer delivering progress.

The Safe Systems approach represents a "paradigm shift" [START_REF] Tingvall | The Swedish'Vision Zero'and how Parliamentary approval was obtained[END_REF], [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF], [START_REF] Rechnitzer | Crashworthy Systems-a paradigm shift in road safety design[END_REF], [START_REF] Itf | Ambitious Road Safety Targets and the Safe System Approach[END_REF] in road safety. The shift is from treating road injury factors as notionally equal with the underlying assumption that there will always be injury risks inherent in road travel, to conceptualising and pursuing the development and management of a road traffic transport system that is inherently safe for human users. The Safe System approach calls for road, vehicle, cyclists, pedestrians and management design parameters consistent with human fallibility and vulnerability, and places both human biomechanical injury tolerance criteria and consideration of human fallibility as the central governing principles underpinning all road safety policy decisions [START_REF] Mooren | Australasian College of Road Safety Conference 'A Safe System: Making it Happen![END_REF].

The Safe System approach is an effective way to achieve the vision of zero road fatalities and serious injuries, and requires the road system to be designed assuming and accommodating human errors. A Safe road System has the following characteristics:

• It recognises that prevention efforts notwithstanding, road users will remain fallible and crashes will occur. • It stresses that those involved in the design and operation of the road transport system need to accept and share responsibility for the safety of the system, and those that use the system need to accept responsibility for complying with the rules and constraints of the system. • It aligns safety management decisions with broader transport and planning decisions that meet wider economic, human and environmental goals. • It shapes interventions to meet the long term goal, rather than relying on "traditional" interventions to set the limits of any long term targets. The basic strategy of a Safe System approach is to ensure that in the event of a crash, the impact energies on human beings remain below the threshold likely to produce either death or serious injury. This threshold will vary with the crash scenario, depending on the level of protection offered to the road users involved [START_REF] Itf | Ambitious Road Safety Targets and the Safe System Approach[END_REF].

The concept of a Safe System emerged in countries that have been most successful in reducing road trauma in past decades but saw progress becoming more and more difficult to achieve. Yet it is highly relevant too for LMICs and fast-growing cities, that see increased numbers of road deaths and injuries in the wake of rapid motorisation. Unlike many other public health issues, strong economic growth correlates not with less road crash injuries but, often enough, with more victims. Safe System thinking offers LMICs and cities that face a deteriorating road death and serious injury epidemic an [5] 26th World Road Congress opportunity to take a bold step forward, towards convergence with performance in pioneer countries [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF].

In fact, it is recommended that all countries, regardless of their level of road safety performance, adopt the Safe System approach to road safety. This approach builds on existing road safety interventions but reframes the way in which road safety is viewed and managed by the community. It addresses all elements of the road transport system in an integrated way, with the aim to ensure crash energy levels are not sufficient to produce unrecoverable injuries. It requires acceptance of shared system safety responsibilities and of accountability between system designers and road users. It stimulates the development of innovative interventions and of the new partnerships necessary to achieve ambitious long term targets [START_REF] Itf | Ambitious Road Safety Targets and the Safe System Approach[END_REF].

Safe System projects are stand-alone, multisectoral initiatives targeting high-risk corridors and areas, with outcomes large enough to be reliably measured. These projects should address three broad components, namely: 1) institutional capacity strengthening priorities, 2) targeted interventions in high-risk corridors and areas, and 3) policy reforms where weaknesses have been identified. Moreover, consideration of safety performance monitoring in these projects and the evaluation of their results are also recommended [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF].

METHODOLOGY

This section gives an overview of the methodology used for developing safe system projects in selected African countries.

The process started with a Road Safety Management Capacity Review (RSMCR) in each of the four countries, that were selected as representing different UN geographic areas of Africa.

Based on the findings of the capacity reviews, a list of desirable road safety intervention projects was developed for each country and these was prioritised on the basis of a number of criteria such as ease of implementation, as well as costs and time to implementation.

Next, for each selected project a Terms of Reference (ToR) list was compiled, and, finally, these were reviewed in cooperation with each country's stakeholders, with the aim of acknowledging and mitigating removing any potential local implementation barriers.

ROAD SAFETY MANAGEMENT CAPACITY REVIEWS (RSMCRS)

Road Safety Management Capacity Reviews (RSMCR) were conducted by reviewing and analysing (the development of) road safety and traffic management in the four selected countries and on-site interviews with high level major road safety stakeholders.

Dedicated review teams were drawn from the SaferAfrica project partners and assisted by two internationally recognised experts (Martin Small and Jeanne Breen). To ensure that the reviews would follow international best practice, this task was explicitly based on the World Bank guidelines [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF], and attended, as well, the ISO 39001:2012 [START_REF] Iso | Road traffic safety (RTS) management systems -requirements with guidance for use[END_REF] and the policy frameworks as set by Sustainable Safety [START_REF] Wegman | Sustainable safety in The Netherlands: the vision, the implementation and the safety effects[END_REF], [START_REF] Wegman | Advancing sustainable safety: National Road Safety Outlook for 2005-2020[END_REF] and Vision Zero [START_REF] Tingvall | The Swedish'Vision Zero'and how Parliamentary approval was obtained[END_REF], known generically as Safe System [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF]. Furthermore, recommendations on road safety management provided by important EC-funded projects (such as DaCoTa) were considered. Importantly, international experiences and, specifically, experiences related to the institutional framework of policy making and the relationship between road safety policy and science, were also considered in this process. The overall objectives of a road safety and traffic management capacity review, based on engagement with senior management of the key agencies, were the following:

• systematically assess the state of road safety and traffic management;

• summarise the strengths and weaknesses of institutional capacities to significantly improve road safety results; • reach consensus amongst the key agencies about next steps, and sustainable activities; • fundamentally improve road safety and traffic management by proposing a longterm headline Safe System strategy and a project concept for activity to launch it.

This last objective is intended at developing a qualitative and long term investment strategy covering the three traditional product development stages: establishment; growth; and consolidation phases [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF].

PRIORITISATION ACTIVITIES

Based on the results of the RSMC reviews, this step served to prioritise and develop specific future implementation projects for each of the selected countries, taking into account the project concept defined in those reviews. These projects are intended to help accelerate the transfer of road safety knowledge and strengthen the capacity of local road safety stakeholders.

To assess the feasibility of further developing these project components and their related enabling projects, a number of approaches, not strictly related to road safety, were explored to look for further criteria that could be included in the transferability tool. Various of these approaches have a different scope, the most frequent one is project complexity used to understand the difficulty of managing a specific project. One of these is the tool called CIFTER (Crawford-Ishikura Factor Table for Evaluating Roles) which identifies seven factors that affect the management complexity of a project (Table 1). Each factor is rated from (1) to ( 4) using a natural values scale which expresses a qualitative metric as a quantitative value. These values are then totaled to produce a management complexity rating for the project [START_REF] Gapps | A Framework for Performance Based Competency Standards for Global Level 1 and 2 Project Managers[END_REF].

TERMS OF REFERENCES PREPARATION

For each selected project a list of terms of reference was compiled, following SMART (Specific; Measurable; Achievable; Realistic and Time bound) project objectives and criteria. Following the World Bank guidelines [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF], [START_REF] Bliss | Road Safety Management Capacity Reviews and Safe System Projects Guidelines[END_REF], the terms of reference prepared, include the following sections:

• The objectives of the required technical assistance services • The outputs of the required technical assistance service

• The scheduling of the required technical assistance services • Professional skills and experience required

IDENTIFICATION OF THE POSSIBLE BARRIERS TO THE IMPLEMENTATION

The success of a road safety intervention is influenced by many factors. The specific context in which an intervention is applied plays a crucial role for its applicability. Experience shows that the successful application of a road safety intervention in a given country or region does not ensure that its implementation will be equally successful in a different context. Therefore, in SaferAfrica, a method for ex-ante evaluation of the applicability of proposed interventions had to be developed, namely to identify and understand potential barriers influencing the results and effects of the proposed. This type of assessment is based on a range of data, but mainly on interviews with local stakeholders. Therefore, for each project a detailed analysis of the barriers for the implementation needs to be carried out, providing the stakeholders of relevant information about the project. The method adopted is based on the concept of Road Safety Space proposed by [START_REF] King | Case studies of the transfer of road safety knowledge and expertise from western countries to Thailand and Vietnam, using an ecological road safety space model: elephants in traffic and rice cooker helmets[END_REF] [START_REF] King | Case studies of the transfer of road safety knowledge and expertise from western countries to Thailand and Vietnam, using an ecological road safety space model: elephants in traffic and rice cooker helmets[END_REF] to organize the factors influencing a road safety intervention (Figure 1). According to [START_REF] King | Case studies of the transfer of road safety knowledge and expertise from western countries to Thailand and Vietnam, using an ecological road safety space model: elephants in traffic and rice cooker helmets[END_REF], in a given country each road safety issue exists in a threedimension space defined by economic, institutional, and social and cultural conditions of the country they are applied to and which factors which influence it. The dimensions include both broad and specific influences. The position of a road safety issue in that space is unique and varies from country to country, although some dimensional factors may be shared across road safety issues or across countries [START_REF] King | Case studies of the transfer of road safety knowledge and expertise from western countries to Thailand and Vietnam, using an ecological road safety space model: elephants in traffic and rice cooker helmets[END_REF].

Figure 1: Model of the road safety space [START_REF] King | Case studies of the transfer of road safety knowledge and expertise from western countries to Thailand and Vietnam, using an ecological road safety space model: elephants in traffic and rice cooker helmets[END_REF] Within SaferAfrica WP7 "Sharing of good practices", a Problem Priority Matrix (PPM) is adopted to assess the transfer process (i.e. mainly the applicability) of international road safety good practices to an African country. To some extent this entails improving road safety outcomes, but the main objective is to improve the transfer process in the expectation that better outcomes will follow. The tool is based on the methodology adopted in SaferBrain project where the transfer process of interventions improving vulnerable road user safety were assessed for India and Brazil [START_REF] Appelt | Innovative guidelines and tools for vulnerable road users safety in India and Brazil[END_REF].

The basic task is to assess whether a given road safety intervention may contain a problem within one (or more) of the assessment dimensions provided by the Road Safety Space (i.e. Society/Culture, Economy and Institution) describing the receptor context.

Economy

-Costs affordability -Technical skill availability [START_REF] Foster | Africa's Infrastructure[END_REF] 26th World Road Congress To this aim, 6 factors are proposed in SaferBrain related to the 3 main road safety space dimensions. Namely these are: People; Environment; Availability of regulation; Political commitment; Design, implementation and maintenance costs affordability; and Technical skill availability (Table 2). A number of possible questions related to each factor were derived from existing literature to help assess the applicability of a road safety intervention [START_REF] Appelt | Innovative guidelines and tools for vulnerable road users safety in India and Brazil[END_REF].

The transferability evaluation of each measure was made using a matrix in which country stakeholders rated each factor according to its difficulty (if the implementing the intervention is challenging from that factor perspective) and weight (if the factor is important for the implementation of the intervention in the country). A final aggregated score was then produced for each road safety intervention, which allows serializing the transferability of the analysed interventions.

Technical skill availability

Does the provider of the intervention in the local setting have the skills to deliver this intervention?

RESULTS

SUMMARY OF THE KEY FINDINGS FROM THE RSMCR

The four RSMCS [START_REF] Cardoso | F Vandemeulebroek Road Safety Management Capacity Review -Burkina Faso[END_REF][31] [START_REF] Schermers | Road Safety Management Capacity Review -Kenya[END_REF][33] highlighted both existing issues and provided recommendations at the three levels of the road safety management system model (i.e. Institutional management functions, Interventions, Results). Based on these, it was possible to define strategic priorities to be undertaken during the three development phases (establishment, growth and consolidation phases) of a road safety management implementation plan. For the selected African countries the challenges in building capacity in road safety management have been initiated and the initial steps to establish the organisational structures and procedures have been taken. However, as was evident in [START_REF]African Union and United Nations Economic Commission for Africa[END_REF] 26th World Road Congress countries in Europe, this process will take time. Moreover, as recommended by the World Bank guidelines [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF]. it is crucial that the political will is channelled into long term investment in road safety improvements across all sectors.

In the following the main evidence from RSMCR are reported together with examples of the recommended priorities for the establishment phase.

Results focus at system level: leadership, goal and target-setting

There are relevant differences in how road safety is planned and managed at local and central levels in each country. Considering the framework adopted for undertaking road safety management capacity reviews within SaferAfrica project, a road safety lead agency should be mandated to promote road safety, set strategies and targets for road safety improvement, and perform the seven institutional management functions that produce road safety interventions: Results focused approach, Coordination, Legislation, Funding and resource allocation, Promotion, Monitoring and evaluation, Research and development.

Availability of reliable and complete data on fatalities and serious injuries is a major issue for the definition of quantitative targets and strategies. According to the RSMCRs, a national road safety strategy has been adopted in three of the four countries, namely: Burkina Faso, Cameroon and Kenya. However, it can be said that a lead road safety agency, if established, is frequently missing the legal power and/or dedicated financial and human resources to be effective in defining a comprehensive national road safety strategy and coordinating responsible stakeholders for its implementation.

Interventions

Safe roads and roadsides. Factors affecting infrastructure safety can be related to road planning and design, construction (e.g. work zone) and maintenance procedures. In all the addressed countries most of the roads are not paved and, in some countries, like Cameroon and Tunisia, the paved roads are in poor condition. The existent of these problems, together with the lack of maintenance, signage, lighting and design errors, imply that in these countries the roads are not safe and characterized by a high risk of road accidents. Though speed limits are set for all roads, the levels of non-compliance to speed limits are not being measured and documented. Safe Vehicles. There is in general a high percentage of powered-two-wheeled vehicles (ranging from 35-40% in Kenya to more than 80% in Burkina Faso) but a little proportion of them is insured (10% in Tunisia). Some countries (Kenya, Burkina Faso and South Africa) have regulations on the safety standards of the vehicles in use, but the standards are limited, and the regulations are weakly enforced. Mandatory vehicle inspections are present in all countries however, these seem not to be carried out periodically (as established by law) and in some cases, not all vehicles are required to be submitted to them. Road users. According to WHO, there are five main behavioural risk factors for road traffic injuries: speed, driving under the influence, failure to use motorcycle helmets, seat-belts and child restraints [5]. Regarding speed limit laws, these are present in all the 5 countries (with related limits) but drivers rarely respect them and rather speeding is one of the main causes of road accidents. Regarding the driving under the influence of alcohol or drugs each of the 4 countries has a law regulating it with their respective limits; only Burkina Faso does not to have a law that regulates driving under the influence of drugs. Finally, regarding the laws of helmet, seat belts and child restraints each of the 4 treated countries has a law that regulates these aspects, except for child restraint law, which is present only in Burkina Faso. Emergency services and post-crash care. The quality and coverage of the existing medical services are one of the biggest obstacles to an adequate post-crash care.

SAFE SYSTEM PROJECTS FOR CAMEROON

In the following it is reported a description of application of the proposed methodology to Cameroon.

Road Safety Management Capacity Reviews (RSMCRs)Prioritisation activities

Based on the results of the road safety management capacity review (RSMCR) undertaken in Cameroon [START_REF] Baja | Road Safety Management Capacity Review -Cameroon[END_REF], it is reported that Cameroon's road safety management system is in a lower phase of development. Drivers, vehicles and roads are generally of low standard and a lack of enforcement and supporting road safety infrastructure, all contribute to a declining road safety situation.

The RSMCR allows the development of a strategic action plan in which the strategic priorities are highlighted per development phase. The RSMCR of Cameroon identified the following projects for the establishment phase (Table 4): A project concept is defined to address weaknesses in the key institutional arrangements at the national levels. The overall objective of the project is building road safety management capacity through institutional reform and accelerating knowledge transfer through "learning by doing". The focus is to hasten the process of shifting from a weak to a strong institutional management capacity to govern the evolution of improved road safety results. The project will encourage agencies to work together constructively to deliver and evaluate a set of well-targeted, best practice multi-sectoral interventions. The successful implementation of the project will hinge on the transfer of road safety knowledge, strengthen the capacity of the participating partners and stakeholders, and rapidly produce results in the country that will provide benchmarks to apply to the next stage of investment.

The project is structured in 16 improvement projects organised in four main components (Table 5):

• Project leadership and management,

• Multi-sectoral interventions in demonstration corridors,

• Monitoring and Evaluation System,

• Policy reviews. • Police action on speed and drinking driving against best practice deterrence principles;

• Legislative, regulatory and data system needs;

• Institutional delivery by the ministry of transport/lead agency and its partners of the necessary key interventions to achieve road safety results.

[13]

26th World Road Congress

Assessment methodology and draft Terms of References preparation

A CIFTER scale was used to determine the most viable improvement projects in the country. Each project was assessed by the RSMCR team considering the 7 CIFTER criteria.

In order to select those projects to be considered for a further assessment by the stakeholders, two criteria have been considered. The first criterion is the total score gained by each project, given by summing all the points across the seven factors. According to CIFTER methodology higher scores identify complex projects, especially in terms of management, so priority should be given to those projects with the lowest rates.

The second criterion is related to the overall scope of the group of selected projects. To this aim, the group of projects should possibly cover the four project component categories: Demonstration projects in targeted high-risk corridors and areas, Policy reviews, Project leadership and management and Monitoring and evaluation systems.

The CIFTER ratings have indicated that the majority of projects are rated either as highly complex or complex. It will therefore be essential that skilled project teams led by experienced road safety project managers are appointed in the execution phase. Based on the results of the assessment the following projects were considered for Cameroon:

Identification of the possible barriers to the implementation and ToRs refinement

A stakeholders' consultation to further assess the feasibility of the selected projects was undertaken. A checklist with possible barriers to project implementation was prepared based on the transferability tool defined in WP7 of SaferAfrica project. Six stakeholders from the government, NGOs and research were involved. Each interviewed stakeholder received the used the checklist to assess any possible social, cultural, institutional, economical factor representing an obstacle to the selected projects.

Society/Culture related barriers Some of the proposed interventions (projects n. 1, 2, 6) could be accepted with some resistance by the general public due to low awareness of the possible impact of the projects on the population. The potential solution to this will be to create a period of sensitization and awareness within which the importance and impact of the project will be made known to the target population.

The target population (especially for project 3) is aware of the road safety problem but might not have enough educational level to comprehend the contents of the intervention, as such there should be awareness raising and sensitization tailored to their level of education to enable clarity and comprehension of the contents of the interventions.

Institution related barriers

In project nr. 1, Create and clearly define governmental road safety roles, there is conflict of interest among key stakeholders resulting from lack of clear definition of roles. For example, on the roads the Ministry of Transport and the Ministry of Defense are not working in synergy. The gendarmes are supposed to be there for enforcement while the operational staff should be from the Ministry of Transport but that is not the case. Each Ministry functions independent of the other.

The legislation relevant to the implementation of project nr. 2 is partially available, because legislation is available for Police action on speed and drink driving, however, the roles of the key players are not clearly defined; agreements or memoranda should be considered at the design stage of the intervention.

Economy related barriers

Weak capacity was highlighted for a number of projects (nr. [START_REF] Nordfjaern | Perceptions of traffic risk in an industrialised and a developing country[END_REF]5,[START_REF] Adminaite | Ranking EU progress on road safety: 10th Road Safety Performance Index Report[END_REF]. This has been explained by shortage of trained personnel (e.g. human resource to manage data collection), lack of motivation for the gendarmes and police on the roads, insufficient equipment available (e.g. radars to control speeds).

The skills are dispersed and need coordination. There is need for synergy and coordination to have the right people working at the right departments. Skill exist in dispersed ranks and need synergy and coordination for efficiency. More police officers need training. For definition of roles, the capacity of stakeholders in the local setting need to be reinforced to meet up with the roles. The skills are dispersed and need coordination. Lack of clarity with respect to who is responsible for financing the project

DISCUSSION AND CONCLUSIONS

This paper has presented the results of the process used in the SaferAfrica project to define Safe System projects in the four African countries: Cameroon, Burkina Faso, Tunisia, and Kenya, which are suitable to their existing contexts. Its applicability in a context like the Africa region, make the procedure very attractive for Low-and Middleincome countries (LMICs). This process included the Road Safety Management Capacity Reviews (RSMCR), prioritisation activities according to CIFTER tool, Terms of References preparation and identification of the possible barriers to the implementation.

The effective implementation of the Road Safety Management Capacity Review must be supported by recognized road safety specialists with successful strategic management experience at country and international levels [START_REF] Bliss | Country guidelines for the conduct of road safety management capacity reviews and the specification of lead agency reforms, investment strategies and safe system projects[END_REF]. In this study, the RSMCR were conducted by recognised international experts who are part of the partners of the SaferAfrica project, assisted by two internationally recognised experts (Martin Small and Jeanne Breen). Regarding to this issue, it is very important to build knowledge in the region, so that later local experts can carry out this kind of work with the support of international experts.

Regarding to the tool CIFTER used to understand the difficulty of managing a specific project, that is to say the project complexity. There is the limitation of depending on the concept and experience of the expert who performs the analysis. The CIFTER is a subjective tool, in this way, ratings on individual factors may vary for the same project.

There is no single pathway for the adoption, establishment and implementation of a Safe System. Moving to a Safe System is a learning-by-doing process best described as a journey which presents opportunities, hazards and challenges along the way. The experiences of the pioneering countries show that each follows its own journey, shaped by the cultural, temporal and local context [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF]. Regarding these aspects, it is important to take them into account for the African context.

The methodology was validated in the four African countries: Cameroon, Burkina Faso, Tunisia, and Kenya. But nevertheless, this first phase only includes the process to define Safe System projects in Africa. Therefore, the implementation phase is still pending and it will be necessary to take into account other factors not analysed in this study. In any case, this first step allows a great advance in the objective of reducing road traffic crashes and improving road safety in developing countries and especially in the Africa region.

Cameroon, Kenya and Tunisia are classified as Lower-Middle-income countries, while Burkina Faso belongs to Low-income countries. According to RSMCR findings, in all these countries the initial steps to establish the organisational structures and procedures have been taken already. However, a well-defined road safety investment strategy still needs to be developed, to build capacity and move these countries from the early establishment phase "Focus on driver interventions" to a long-term consolidation phase "Focus on system-wide interventions, long-term elimination of deaths and serious injuries and shared responsibility".

In Burkina Faso and Kenya, for instance, the RSMCR highlighted a road safety approach focussed "on driver interventions". This was the approach used in high income countries in the period 1950-1960, when safety management was characterized by a set of uncoordinated decisions and actions [START_REF] Koornstra | SUNflower: a comparative study of the development of road[END_REF] and the emphasis was laid on the errors of road users, leading to preventive measures focused on road user training and education [START_REF] Mcdermott | Control of road trauma epidemic in Australia[END_REF], [START_REF] Mcdermott | Reduction in road fatalities and injuries after legislation for compulsory wearing of seat belts: experience in Victoria and the rest of Australia[END_REF]. This approach deprived the authorities of complete responsibility to road safety and proved of limited value in preventing serious injuries and deaths, especially among nonmotorized road users. Finally, the Safe System approach represent a substantial shift in how road safety problems and solutions are conceived. Strong and sustained leadership to initiate and see through the shift to a Safe System is vital [START_REF] Itf | Road Deaths and Serious Injuries: Leading a Paradigm Shift to a Safe System[END_REF]. This requires actions from all fronts: political leaders, leaders of companies and public agencies, policy makers and academics in a position to influence change in a local, regional or national government, corporate or social setting. However, it is too early for a full critical analysis of the successes and failures of the Safe System projects in African countries. Thus, further studies should be carried out in this direction.

-

  Availability of regulation -Political commitment

••

  Vehicle registration, licensing and roadworthiness • Driver licensing and penalties • Vehicle safety regulations Enforcement • Equipment, standards, protocols, stategies (training and capacity building) Emergency centres and numbers

Table 1 CIFTER

 1 Assessment matrix[START_REF] Gapps | A Framework for Performance Based Competency Standards for Global Level 1 and 2 Project Managers[END_REF]

	1. Stability of the overall project context	Very	High	Moderate	Low
		High (1)	(2)	(3)	(4)
	2

. Number of distinct disciplines, methods, or approaches involved in performing the project

	Low	Moderate	High	Very
	(1)	(2)	(3)	High (4)
	3

. Magnitude of legal, social, or environmental implications from performing the project

	Low	Moderate	High	Very
	(1)	(2)	(3)	High (4)
	4

. Overall expected financial impact (positive or negative) on the project's stakeholders

	Low	Moderate	High	Very
	(1)	(2)	(3)	High (4)
	5.

Strategic importance of the project to the organization or organizations involved

	Very Low	Low	Moderate	High
	(1)	(2)	(3)	(4)
	6.

Stakeholder cohesion regarding the characteristics of the product of the project

	High	Moderate	Low	Very Low
	(1)	(2)	(3)	(4)
	7.

Number and variety of interfaces between the project and other organizational entities

	Low	Moderate	High	High
	(1)	(2)	(3)	(4)

Table 2

 2 Questions addressing intervention transferability[START_REF] Appelt | Innovative guidelines and tools for vulnerable road users safety in India and Brazil[END_REF]

	Component	Factors	Questions to assess a factor
	Society/Culture	People	Would the general public and the targeted population
			accept this intervention? Does any aspect of the
			intervention go against local social norms? Is it ethically
			acceptable? Can the contents of the intervention be
			tailored to suit the local culture? Does the target
			population in the local setting have a sufficient
			educational level to comprehend the contents of the
			intervention? Is the target population aware of the road
			safety problem?
		Environment	Is it possible to change the built environment in order to
			accommodate the proposed practice?
	Institution	Availability of	Legislation relevant to the transferability of the
		regulation	intervention available (standards of service and safety,
			…)
		Political	Does the political environment of the local society allow
		commitment	this intervention to be implemented? Is there any political
			barrier to implementing this intervention?
	Economy	Design,	Are the essential resources for implementing this
		implementation	intervention available in the local setting? (list of essential
		and maintenance	resources would help answer this question)
		costs affordability

Table 4

 4 Examples of Level 1 priorities defined for the Establishment phase in Cameroon and Kenya

	Results focus at system level	Interventions	Institutional	management
								functions
	Review	and	strengthen	Establish	road	infrastructure	The new accident data collection
	appropriate lead agency functions,	safety management procedures	and analysis system should be
	organizational	structures	and	on a selected high-risk corridor	rendered operational in the two
	processes					and in the two main cities of	main cities and in the selected
						Yaoundé and Douala	corridor(s)
	Manage, monitor and evaluate	Review	and	internationally	Set quantitative targets for the two
	road safety results in the two	benchmark safety policies and	main cities
	major cities Yaoundè and Douala	interventions	and	commence
	and on a selected inter-urban	implementation of reforms
	corridor
	Define	institutional		roles,
	responsibilities			and
	accountabilities for the national
	goal
	Define annual provision for road
	safety expenditure in budgets of
	local governments of Yaoundè
	and Douala
	Establish	an	appropriately
	resourced road safety strategy
	unit

Table 5

 5 Cameroon project components and related improvement projects

	Project Component	Improvement projects
	1. Project leadership	•	Create and clearly define governmental road safety roles
	and management,	•	Training programs for road safety institutions
			•	Institutional arrangements
			•	Project promotion
	2. Multi-sectoral		•	Infrastructure safety improvement
	interventions demonstration	in	•	Enforcement program
	corridors,		•	Publicity and awareness campaign
			•	Post-crash care improvement
	3. Monitoring	and	•	Project performance targets definition
	Evaluation System	•	Survey for project performance measuring
			•	Accident data collection operational in Yaoundè, Douala and selected
				corridor(s)
			•	Analysis and reporting
	4. Policy reviews		•	Road planning and design standards against Safe System principles;

26th World Road Congress

ACKNOWLEDGEMENTS

The study presented in this paper is part of the research project titled "SaferAfrica -Innovating dialogue and problems appraisal for safer Africa", funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 724029.

Recommendations to improve emergency care can include the development of a digital trauma registry, the introduction of trauma training for healthcare workers and the development of trauma teams.

Institutional management functions

Coordination. Weak horizontal and vertical coordination.

Legislation. There exists an abundance of road safety legislation (laws, decrees and circulars) some of which need revision (e.g. Vehicle registration, licensing and roadworthiness, Driver licensing and penalties, Vehicle safety regulations). In Burkina Faso Laws are not adapted to the local situation.

Funding and resource allocation. In some countries there is a sustainable source of funding (e.g. the Road Fund in Cameroon). However, there is a lack of resources for implementation and resource allocation procedures are missing.

Promotion. Promoting was evaluated as being ineffective compared to international best practice. Statistics on institutional outputs are not published and made available to stakeholders.

Monitoring and evaluation. Sustainable systems are not operational to collect and manage data on road crashes and mobility. Even if the majority of countries regularly investigate and record road accidents, road accident data are likely to be underestimated. A critical aspect for road safety management in Africa is the lack of a reliable data collection system and the problem of underreporting. This is confirmed by all the RSMCRs. Accident data are often incomplete, and it is unknown what proportion of road accident have been reported and recorded in the official databases used. Modernizing the road safety data collection process seems to be a high priority to enable effective and efficient monitoring and evaluation of road accidents.

Research and development and knowledge transfer.

There is very little existing research capacity in road safetyhas limited research capability in the area of road safety and this capacity will need to be built or sourced from other (international) organisations

Summary

Based on these evidences, a list of common issues/opportunities is reported in Table 3.


Matteo Astorino -Jean-Frédéric

Gerbeau -Olivier Pantz -Karim-Frédéric Traoré

Matteo Astorino

Jean-Frédéric Gerbeau

Olivier Pantz

Karim-Frédéric Traoré

Fluid-structure interaction and multi-body contact. Application to aortic valves

Keywords: fluid-structure interaction, contact, cardiac valves interaction fluide-structure, contact, valves cardiaques

We present a partitioned procedure for fluid-structure interaction problems in which contacts among different deformable bodies can occur. A typical situation is the movement of a thin valve (e.g. the aortic valve) immersed in an incompressible viscous fluid (e.g. the blood). In the proposed strategy the fluid and structure solvers are considered as independent "black-boxes" that exchange forces and displacements; the structure solvers are moreover not supposed to manage contact by themselves. The hypothesis of non-penetration among solid objects defines a non-convex optimization problem. To solve the latter, we use an internal approximation algorithm that is able to directly handle the cases of thin structures and self-contacts. A numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme.

Introduction

We are interested in the interaction of several elastic bodies immersed in an incompressible viscous fluid. Cardiac valves are the main motivation of the present study, but our algorithms can address more general configurations.

The numerical simulation of cardiac valves offers many challenges: the constitutive laws of the valves are very complex, the blood interacts with the valves and the wall, the valves are submitted to kinematic constraints like contact between leaflets or attachments to the chordae tendineae (for the mitral valves). In this paper we are interested in the management of fluid-structure interaction in presence of contacts. More precisely, we propose a general strategy to address this problem with existing fluid and structure solvers, assuming that the structure solvers do not include contact capabilities.

The mechanical properties of the valve is an important topic which is beyond the scope of the present study. Different approaches have been proposed in literature. We refer to [START_REF] Weinberg | On the constitutive models for heart valve leaflet mechanics[END_REF] for an overview of different constitutive models. For example, the fiber-reinforcement of the leaflets has been modeled as two layers of fibers in [START_REF] Hart | A computational fluidstructure interaction analysis of a fiber-reinforced stentless aortic valve[END_REF]. Recently, a multiscale approach, including cells, tissue and organ models, has been proposed in [START_REF] Weinberg | Transient, three-dimensional, multiscale simulations of the human aortic valve[END_REF].

A number of articles has been devoted to fluid-structure interaction around artificial or natural cardiac valves. They can be roughly divided in three groups: the approaches based on the Immersed Boundary (IB) methods (see in particular [START_REF] Griffith | An adaptive, formally second order accurate version of the immersed boundary method[END_REF] and the references therein), those based on the Arbitrary Lagrangian Eulerian (ALE) formulation (see e.g. [START_REF] Jianhai | ALE finite element analysis of the opening and closing process of the artificial mechanical valve[END_REF][START_REF] Morsi | Transient fluid-structure coupling for simulation of a trileaflet heart valve using weak coupling[END_REF]), and those based on Fictitious Domains (FD) formulations (see e.g. [START_REF] Baaijens | A fictitious domain/mortar element method for fluid-structure interaction[END_REF][START_REF] Hart | A three-dimensional computational analysis of fluid-structure interaction in the aortic valve[END_REF][START_REF] Hart | Evaluation of a fictitious domain method for predicting dynamic response of mechanical heart valves[END_REF][START_REF] Diniz Dos Santos | A partitioned fluidstructure algorithm for elastic thin valves with contact[END_REF]). The present paper belongs to the third group.

The most common approach in mechanical contacts is known as the master/slave formulation (see [START_REF] Laursen | Formulation and treatment of frictional contact problems using finite elements[END_REF], [START_REF] Laursen | Computational contact and impact mechanics. Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis[END_REF] and [START_REF] Laursen | A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems[END_REF]). Initially designed to prevent a deformable body (the slave) to penetrate a rigid foundation (the master), it has been extended to the case of contacts between different deformable bodies. In this article, we consider the more complicated case of contacts and self-contacts between deformable thin structures. The master/slave approach can be adapted to this case (see [START_REF] Heinstein | A generalpurpose contact detection algorithm for nonlinear structural analysis code[END_REF]), though it is no more completely consistent from a mathematical viewpoint. Ad hoc modifications have therefore to be added in order to correctly handle the inconsistent situations in which the standard master/slave approach may fail. In this article, we propose to follow a totally different path, which allows us to consider contact, self-contact between thin or not thin structures in a single setting. For more details about the state of the art in contact mechanics, the reader is referred to [START_REF] Hallquist | Sliding interfaces with contact-impact in large-scale Lagrangian computations[END_REF], which contains many references, and to the monographs [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF] or [START_REF] Wriggers | Computational Contact Mechanics[END_REF] (see also [START_REF] Wriggers | Finite element algorithms for contact problems[END_REF][START_REF] Klarbring | Large displacement frictional contact: a continuum framework for finite element discretization[END_REF]).

A few works have considered both fluid-structure interaction and the contact problem between the leaflets. For example, in [START_REF] Van Loon | A fluid-structure interaction method with solid-rigid contact for heart valve dynamics[END_REF], contact is taken into account with a rigid wall (convex constraint) and the algorithm is monolithic: fluid, structure and contact are governed by a unique ad hoc solver. In [START_REF] Diniz Dos Santos | A partitioned fluidstructure algorithm for elastic thin valves with contact[END_REF], the same kind of "simple" contact has been investigated but with a partitioned algorithm. In [START_REF] Carmody | An approach to the simulation of fluid-structure interaction in the aortic valve[END_REF], the contact is handled directly in the structure solver. In [START_REF] Tezduyar | Modeling of Fluid-Structure Interactions with the Space-Time Finite Elements: Solution Techniques[END_REF], the "SENCT" contact algorithm has been introduced to preserve the quality of the fluid mesh between the structural surfaces coming into contact. Compared to the existing studies, the main characteristics of the present work are the following: (i) the solvers are kept independent; (ii) the structure solvers are not supposed to manage contact by themselves; (iii) the contact occurs between several leaflets (non-convex constraint), which can be thin structures; (iv) selfcontact is automatically managed.

In Section 2, we briefly present the models and their discretizations. The fluid is governed by the incompressible Navier-Stokes equations. The solids are modeled by nonlinear shell elements. The fluid and solid meshes are independent: the continuity of the fluid and solids velocities is imposed through Lagrange multipliers (FD formulation).

In Section 3, we present the general algorithm. The fluid-structure coupling is handled with a standard fixed-point accelerated by an Aitken extrapolation. The constraint of non-penetration among the immersed structures defines a nonconvex optimization problem which is solved following an algorithm proposed in [START_REF] Pantz | A frictionless contact algorithm for deformable bodies[END_REF]. This approach is in particular able to manage the cases of thin structures and self-contacts. The proposed strategy allows to consider the fluid and structure solvers as "black-boxes" which only exchanges forces and displacements.

In Section 4, the algorithm is applied to the simulation of an idealized aortic valve. The proposed test case is far from the complexity of the real problem. Several simplifications should be relaxed to address the problem with more realism. In particular the constitutive laws should be improved and the elasticity of the aorta should be taken into account. The purpose is only to illustrate the algorithm in a configuration which is not trivial, in spite of all the simplifications.

Modeling and discretization

Fluid model

The fluid is governed by the incompressible Navier-Stokes equations. The approximation is performed with the finite element method. The structure and the fluid meshes are independent. The continuity of the displacement of the fluid and the structures is enforced through Lagrange multipliers, as it was proposed for example in [START_REF] Baaijens | A fictitious domain/mortar element method for fluid-structure interaction[END_REF]. We refer to [START_REF] Diniz Dos Santos | A partitioned fluidstructure algorithm for elastic thin valves with contact[END_REF] for the details of our approach.

Solid model

In view of the ratio thickness/size of the leaflets, it is necessary to consider robust structural models in order to avoid the well-known locking phenomena (see [START_REF] Chapelle | The Finite Element Analysis of Shells -Fundamentals[END_REF]). In this study we consider the MITC4 general shell element. This element is known to be reliable and effective in the two asymptotic states (membrane and bending) [START_REF] Bathe | Finite Element Procedures[END_REF][START_REF] Chapelle | The Finite Element Analysis of Shells -Fundamentals[END_REF] and can handle large displacements.

For the sake of simplicity, we use a generalized Hook law. It cannot be considered as a good model for biological valves. It is nevertheless sufficient to illustrate our algorithms. The internal stored energy W in the reference configuration of the solids Ωs is given by :

W = 1 2 ΩS C αβλµ e αβ e λµ + D αλ e αz e λz dV, (1)
where e = (e αβ ) denotes the nonlinear Green-Lagrange strain tensor. In equation (1) the Greek symbols varying from 1 to 2 are used for the tangential INRIA components to the surface, z is the third direction, and

C αβλµ = E 2(1 + ν) g αλ g βµ + g αµ g βλ + 2ν 1 -ν g αβ g λµ , (2)
D αλ = 8E t 2 s (1 + ν) g αλ , ( 3
)
where E is the Young modulus, ν the Poisson ratio, t s the thickness and g αλ the contravariant components of the metric tensor. The MITC4 finite element has 5 degrees of freedom per node (the three components of the displacement and the two parameters which define the variation of the unit vector). This element is almost free of locking. This desirable feature is obtained by using a particular interpolation strategy for the different components of the strain tensor. We refer to [START_REF] Bathe | Finite Element Procedures[END_REF] for more details.

Contact model

The contact is assumed to be frictionless and soft. In addition, we do not apply any specific treatment due to the presence of the fluid (no lubrication forces are added). In spite of these simplifications, the problem is quite complicated since the contact constraints are non-convex, as will be shown in the sequel.

General algorithm

We give in this section the details of the general algorithm which handles fluidstructure interaction and contact. We denote by M the family of immersed solids M = (M 1 , M 2 , . . . ) and by T h a P 1 finite element mesh of M :

X = {ϕ ∈ C 0 (M ; R d ), ϕ| T ∈ P 1 , ∀T ∈ T h },
where d = 2 or 3. The quantity ϕ(x i ) is the current position of the i th node of the structure. We denote by ϕ Σ the restriction of ϕ to the fluid-structure interface Σ.

The structure "discrete energy" is denoted by J. The energy J includes in particular the terms resulting from the discretization of the acceleration and the load exerted by the fluid. The deformation ϕ : ∪ i M i → R d is determined by solving at each time step the following minimization problem:

inf ϕ∈U J(ϕ), (4)
with

U = {ϕ ∈ X, dist(ϕ(T 1 ), ϕ(T 2 )) ≥ ε g , ∀T 1 , T 2 ∈ T h such that T 1 ∩ T 2 = ∅},
where ε g denotes a gap between the solids. Note that the set U defining the constraints is non-convex which makes the minimization problem (4) difficult. This difficulty will be circumvented by transforming the problem with nonconvex constraints into a sequence of problems with convex constraints. The proposed algorithm is made of three nested loops. The external loop (loop 1) solves the fluid-structure coupling. The first inner loop (loop 2) build a sequence of convex sets C(ϕ k ) which are used in place of U. The purpose of the most inner loop (loop 3) is to solve problem (4) on the convex sets C(ϕ k ). Figure 1 summarizes this algorithm. In the three next sections, we give the details of each loop.

Loop 1: Fluid-Structure Iterate on j until ϕ j+1 Σ -ϕ j Σ ≤ ε f si : 1.
Solve the fluid problem: given the structures deformation ϕ j Σ , compute the fluid velocity and pressure (u j , p j ).

2.

Compute the load exerted by the fluid σ j f .

Loop 2: Sequence of convex minimization problems

Iterate on k until ϕ j,k+1ϕ j,k ≤ ε T :

3.1. Definition of a convex neighborhood C(ϕ j,k ) of ϕ j,k .

Loop 3: Minimization with convex constraints

Uzawa algorithm (tolerance ε cvx ) to solve the structure problem: Several techniques have been proposed to solve the mechanical interaction between blood flow and arterial walls (among many references see e.g. [START_REF] Gerbeau | A quasi-Newton algorithm based on a reduced model for fluid-structure interactions problems in blood flows[END_REF][START_REF] Fernández | A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid[END_REF][START_REF] Fernández | A Newton method using exact Jacobians for solving fluid-structure coupling[END_REF][START_REF] Badia | Fluid-structure partitioned procedures based on Robin transmission conditions[END_REF][START_REF] Torii | Computer Modeling of Cardiovascular Fluid-Structure Interactions with the Deforming-Spatial-Domain/Stabilized Space-Time Formulation[END_REF][START_REF] Figueroa | A coupled momentum method for modeling blood flow in three-dimensional deformable arteries[END_REF]). The effort to devise good coupling algorithms can be explained by the fact that naive partitioned schemes can be either unstable or very inefficient. For example, in typical configurations, an accelerated fixed point algorithm needs up to 40 iterations to converge. Explanations of this fact have been provided in [START_REF] Causin | Added-mass effect in the design of partitioned algorithms for fluid-structure problems[END_REF][START_REF] Förster | Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible flows[END_REF].

J(ϕ j,k+1 ) = inf J(ψ) under the convex constraint ψ ∈ C(ϕ j,k ).
In the present study, we are not interested in the coupling with the wall but with an immersed valve. In this specific case, we observed that an accelerated fixed point algorithm typically converges in about 5 iterations. Thus, we adopted this simple algorithm for the fluid-valve interaction.

The acceleration of the fixed-point method is based on the Aitken formula which has been applied to FSI problems in [START_REF] Mok | Accelerated iterative substructuring schemes for instationary fluid-structure interaction[END_REF]. The algorithm reads:

Loop 1 : Fluid-Structure (i) Initial guess ϕ 0 Σ (prediction of the position of the interface).

(ii) For j ≥ 0, solve the fluid problem, given an interface deformation ϕ j Σ .

(iii) Compute the load σ j f exerted by the fluid on the structure.

(iv) Solve the structure to obtain a new deformation φj+1 .

INRIA

(v) Correction of the position of the interface:

ϕ j+1 Σ = ω j φj+1 Σ + (1 -ω j )ϕ j Σ ,
with

ω j = (ϕ j Σ -ϕ j-1 Σ ) • (ϕ j Σ -φj+1 Σ -ϕ j-1 Σ + φj Σ ) |ϕ j Σ -φj+1 Σ -ϕ j-1 Σ + φj Σ | 2 . ( 5
) (vi) While ϕ j+1 Σ -ϕ j Σ > ε f si go to (ii).
If the interaction with the aorta was also taken into account (which is not the case in this work), it would be necessary to use more sophisticated algorithms to avoid prohibitive computational costs. For example, the method proposed in [START_REF] Fernández | A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid[END_REF] could be extended to deal with both types of interaction (wall and valve). A step in this direction is presented in [START_REF] Diniz | Numerical methods for fluid-structure interaction problems with valves[END_REF].

Definition of a convex neighborhoods (loop 2)

Loop 2 is based on an original idea proposed in [START_REF] Pantz | A frictionless contact algorithm for deformable bodies[END_REF]. Its purpose is to replace the non-convex optimization problem (4) with a sequence of convex ones. For the sake of clarity, we drop the index j related to the FSI iteration in Figure 1. Suppose the current deformation of the structure is ϕ k . To compute the state ϕ k+1 , we solve the structure problem (4) replacing U with a convex set denoted by C(ϕ k ). Each convex set C(ϕ k ) contains the element ϕ k and is included in the initial admissible set U. Moreover, if ϕ k belongs to the interior of U, the set C(ϕ k ) is a convex (closed) neighborhood of the element ϕ k . In the following, C(ϕ k ) will be often referred to a "neighborhood" of ϕ by language abuse. The precise definition of the convex neighborhood in 2D and 3D is given in the two following sections. Here is a sketch of the algorithm:

Loop 2: Sequence of convex minimization problems

(i) Initial guess: ϕ 0 . (ii) For k ≥ 0, solve J(ϕ k+1 ) = inf ψ∈C(ϕ k ) J(ψ),
where

C(ϕ k ) is a convex neighborhood of ϕ k defined below. (iii) While ϕ k+1 -ϕ k > ε C go to (ii).
As C(ϕ k ) always contains ϕ k , the sequence J(ϕ k ) in the loop 2 of the algorithm is non-increasing, and therefore convergent if bounded from below. The resolution of the new minimization problem (where the solution is searched in C(ϕ k )) is the purpose of the third loop and will be explained later on.

Note that, at convergence, the optimality conditions of the original nonconvex problem are not exactly satisfied. Nevertheless, it can be proved that they are satisfied up to an error O(h), where h is the discretization step in the structure (see [START_REF] Pantz | A frictionless contact algorithm for deformable bodies[END_REF]).

Definition of C(ϕ k ) in 2D

In 2D, the convex neighborhood is defined as follows:

C(ψ) = ϕ ∈ X, min xe∈e n e,x (ψ) • (ϕ(x e ) -ϕ(x)) ≥ ε g ,
for all edges e and all nodes x ∈ e , where ε g > 0 and n e,x (ψ) is defined by:

min xe∈e n e,x (ψ) • (ψ(x e ) -ψ(x)) = dist(ψ(e), ψ(x)).
Loosely speaking, n e,x (ψ) is the normal to the edge e pointing to the node x. See Figure 2 for two typical configurations. all the couples edge/vertex can be separated by a straight line (dashed-line), with a gap ε g .

ψ(e) ψ(xe) ψ(x) ne,x(ψ) ψ(e) ψ (xe) ψ(x) ne,x(ψ)
We denote by e + and e -the vertices of an edge e. It is convenient to notice that the convex neighborhood can also be rewritten as:

C(ψ) = ϕ ∈ X, F - e,xi (ϕ) ≤ 0, F + e,xi (
ϕ) ≤ 0, for all edges e and all nodes x i ∈ e ,

where

F ± e,xi,ψ (ϕ) = ε g -n e,xi (ψ) • (ϕ(e ± ) -ϕ(x i ))
, Under this form, we see that the convex constraints consist in imposing that, after deformation, any edges and vertices can be separated by a straight line, with a gap ε g (see Figure 3).

INRIA

If loop 2 converges to ϕ ∈ X, then there exist λ + e,i ≥ 0 and λ - e,i ≥ 0 (for all e and i such that node i does not belong to edge e) such that ∀ξ ∈ X:

             J ′ (ϕ), ξ - e xi ∈e n e,xi • (λ - e,xi + λ + e,xi )ξ(x i ) -λ - e,xi ξ(e -) -λ + e,xi ξ(e + ) = 0, λ - e,xi F - e,xi,ψ (ϕ) = 0, λ + e,xi F + e,xi,ψ (ϕ) = 0. ( 7
)
The Lagrange multipliers λ ± e,xi represent the contact pressure acting on the nodes of the solids mesh and are added to the hydrodynamic force acting on the structure. The computation of λ ± e,xi will be explained in Section 3.3. Note that self-contact is automatically handled since the non-penetration condition is tested among two generic independent elements that can also belong to the same solid.

Definition of C(ϕ k ) in 3D

In 3D, two possible contacts can occur: Therefore the convex neighborhood C(ϕ k ) is defined as the set of admissible deformations subjected to the constraints associated to each couple edge/edge and triangle/vertex:

C(ψ) = ϕ ∈ X, min xa∈a,x b ∈b n a,b (ψ) • (ϕ(x a ) -ϕ(x b )) ≥ ε g ,
for all edges a and b of the mesh T such that a ∩ b = ∅ and min

xT ∈T n T,x (ψ) • (ϕ(x T ) -ϕ(x)) ≥ ε g , for all
triangle T and all vertex x of the mesh T such that x ∈ T ,

where n a,b (ψ) and n T,x (ψ) are defined by:

min xa∈a,x b ∈b n a,b (ψ) • (ψ(x a ) -ψ(x b )) = dist(ψ(a) -ψ(b)) RR n°6583
and min

xT ∈T n T,x (ψ) • (ψ(x T ) -ψ(x)) = dist(ψ(T ) -ψ(x)).
For the sake of completeness, we now give a few details on the computations of n a,b (ψ). Let us first denote by a 0 , a 1 and b 0 , b 1 the corresponding endpoints of the edges a and b, and by ψ(p a,b ) ∈ ψ(a) and ψ(p b,a ) ∈ ψ(b) the set of points that minimize the distance between ψ(a) and ψ(b) (Figure 5):

ψ(p a,b ) = αψ(a 0 ) + (1 -α)ψ(a 1 ), ψ(p b,a ) = βψ(b 0 ) + (1 -β)ψ(b 1 ), with α, β ∈ R, 0 ≤ α, β ≤ 1. ψ (a0) ψ(a1) ψ(b0
Note that problem (8) admits always a unique solution, except when ψ(a) and ψ(b) are colinear, that is when

a 2 b 2 -(a 1 -a 0 , b 1 -b 0 ) 2 = 0.
From a practical point of view it is convenient to first solve the corresponding unconstrained minimization problem and then evaluate the fulfillment of the constraints. If one of the constraints is not satisfied or if the edges are colinear, the solution of problem ( 8) is equivalent to Let us now consider the definition of the normal vector n T,x (ψ) . In a similar way to the edge/edge contact case, we introduce the vertices (t 0 , t 1 , t 2 ) of the triangle T and the barycentric coordinates for the point ψ(p T,x ) ∈ ψ(T ) that minimizes the distance from the vertex ψ(x) (Figure 7):

ψ(p T,x ) = αψ(t 0 ) + βψ(t 1 ) + (1 -α -β)ψ(t 2 ), with α, β, γ ∈ R, 0 ≤ α, β, γ ≤ 1 and γ = 1 -α -β.
The coefficients α, β and γ are determined by

(α, β) = argmin 0≤α,β,γ≤1 γ=1-α-β g(α, β) = argmin 0≤α,β,γ≤1 γ=1-α-β ψ(p T,x ) -ψ(x) , (9)
and n T,x (ψ) is obtained from:

n T,x (ψ) = ψ(p T,x ) -ψ(x) ψ(p T,x ) -ψ(x) .
From a computational point of view, instead of solving directly the constrained problem [START_REF] Hart | Evaluation of a fictitious domain method for predicting dynamic response of mechanical heart valves[END_REF], it is easier to determine first the relative position of the point ψ(x) with respect to ψ(T ) and then solve a subproblem to calculate ψ(p T,x ).

M1 M2 ψ(pT,x) ψ(x) nT,x(ψ) ψ(t0) ψ(t1) ψ(t2)
In particular, considering the triangle in Figure 8.a, ψ(p T,x ) will be computed as the projection of ψ(x) on the edge ψ(e i ) (∀i ∈ {0, 1, 2}) if ψ(x) is in the part of the space opposed to ψ(t i ) with respect to the plane perpendicular to ψ(T ) that contains ψ(e i ). Therefore, in this case, the triangle/vertex problem is equivalent to an edge/vertex problem in three-dimensions. On the other hand (Figure 8.b), if ψ(x) belongs to the cylinder determined by the intersection of the three perpendicular planes previously defined, ψ(p T,x ) will be determined by the analytical solution of the unconstrained problem

(α, β) = argmin g(α, β) = argmin ψ(p T,x ) -ψ(x) ,
which is equivalent to the projection of the vertex ψ(x) on the infinite plane defined by (t

0 , t 1 , t 2 ). b) a) ψ(pT,x) ψ(pT,x) ψ(x) ψ(x) nT,x(ψ) nT,x(ψ) ψ(t0) ψ(t0) ψ(t1) ψ(t1) ψ(t2) ψ(t2) ψ(e0) ψ(e0) ψ(e1) ψ(e1) ψ(e2) ψ(e2)
Figure 8: Possible configurations of the unit vector n T,x (ψ). a) ψ(x) is in the part of the space opposed to ψ(t 1 ) with respect to the plane that contains ψ(e 1 ). b) ψ(x) is inside the cylinder determined by the intersection of the three perpendicular planes to ψ(T ).

As in the bidimensional case, we can note that the convex neighborhood can also be rewritten as:

C(ψ) = ϕ ∈ X, F j,k a,b (ϕ) ≤ 0 ∀j, k ∈ {0, 1}
, for all edges a and b of the mesh T such that a ∩ b = ∅ and F k T,xi (ϕ) ≤ 0 ∀k ∈ {0, 1, 2}, for all triangle T and all vertex x of the mesh T such that x ∈ T , [START_REF] Diniz Dos Santos | A partitioned fluidstructure algorithm for elastic thin valves with contact[END_REF] where

F j,k a,b,ψ (ϕ) = ε g -n a,b (ψ) • (ϕ(a j ) -ϕ(b k )), and F k T,xi,ψ (ϕ) = ε g -n T,xi (ψ) • (ϕ(t k ) -ϕ(x i )).

INRIA

If loop 2 converges, the limit satisfies the following optimality system:

                         J ′ (ϕ), ξ - a b a∩b=∅ n T,xi • 2 j=0 2 k=0 λ j,k a,b ξ(a j ) -λ j,k a,b ξ(b k ) - T xi ∈T n T,xi • 3 k=0 λ k T,xi ξ(x i ) -λ k T,xi ξ(t k ) = 0 ∀ξ ∈ X, λ j,k a,b F j,k a,b,ψ (ϕ) = 0 ∀j, k ∈ {0, 1}, λ k T,xi F k T,xi,ψ (ϕ) = 0 ∀k ∈ {0, 1, 2}. (11)
where λ j,k a,b ≥ 0 and λ k T,x ≥ 0 represent respectively the four Lagrange multipliers associated to the (a, b) edge/edge problem and the three ones associated to the (T, x i ) triangle/vertex problem.

Minimization with convex constraints (loop 3)

The most inner loop aims at solving an optimization problem with convex constraints: given an hydrodynamic force σ j f (loop 1), given a convex neighborhood C(ψ) of the current solid deformation ψ = ϕ j,k (loop 2), we have to solve inf ϕ∈C(ψ)

J(ϕ), (12)
The convex set C(ψ) being defined by ( 6) in 2D and by [START_REF] Diniz Dos Santos | A partitioned fluidstructure algorithm for elastic thin valves with contact[END_REF] in 3D.

To solve problem [START_REF] Laursen | Computational contact and impact mechanics. Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis[END_REF], various methods -like penalization or relaxation with projection -may yield substantial changes of the structure solver. Here we adopt a method which consists in maximizing a dual energy. We present it in 2D, the extension in 3D being obtained mutatis mutandis.

Denoting by µ the vector (µ ± e,xi ), where (e, x i ) describes all the couples edge/nodes such that x i ∈ e, we look for the maximum of the dual energy

G(µ) = inf ϕ∈X   J(ϕ) + e xi ∈e µ - e,xi F - e,xi,ψ (ϕ) + µ + e,xi F + e,xi,ψ (ϕ)   ,
under the constraint µ ± e,xi ≥ 0. In a gradient method with projection, these constraints are very easy to implement, whereas the original one, namely ϕ ∈ C(ψ), is complicated. This is the usual motivation of the dual approach. In our specific framework, this method has another advantage: during the resolution by a gradient method of the dual problem, the structure solver exchanges the same kind of information as for the coupling with the fluid (it receives loads, it sends displacements, see Figure 9). The contact treatment can therefore be easily included as an inner-loop in the global algorithm without any change in the structure solvers. Even if other optimization methods are known to perform better than the gradient method, the possibility to use the structure solver as a "black-box" is a strong motivation for the proposed approach.

We can sum up the loop 3 as follows:

(i) Initial guess: λ 0 .

(ii) Solve the structure problem: find ϕ l ∈ X such that for all ξ ∈ X,

J ′ (ϕ l ), ξ = - e xi ∈e λ - e,xi (F - e,xi,ψ ) ′ (ϕ l ), ξ + λ + e,xi (F + e,xi,ψ ) ′ (ϕ l ), ξ = e xi ∈e n e,xi (ψ) • (λ - e,xi + λ + e,xi )ξ(x i ) -λ - e,xi ξ(e -) -λ + e,xi ξ(e + ) .
(iii) Gradient iteration with projection:

λ l+1,± e,xi = P R + λ l,± e,xi + α l F ± e,xi (ϕ l ) .
(iv) Go to (ii) until convergence.

The projection operator introduced in step (iii) is defined by:

P R + (x) = x if x > 0 0 if x ≤ 0
Remark 1 For efficiency, it is of course recommended to restrict the sets edge/vertex in 2D or edge/edge, triangle/vertex in 3D to those elements which can actually experience contact.

Remark 2 [chordae tendineae] Contacts are not the only relevant constraints in the applications to cardiac valves. For example, the chordae tendineae prevent the leaflet of the mitral valves from everting into the atrium. We have also implemented the capability to deal with such constraints in our framework. More precisely, let C be a point on the ventricular wall, and let M be the point of the valve to which a chorda (length L) is attached. It is straightforward to adapt the dual algorithm presented above to the constraint:

dist(C, M ) ≤ L.
The Lagrange multiplier corresponds in this case to the tension applied on the valve by the string. Once again, the structure codes have not been modified which is an additional illustration of the versatility of the method.

Remarks on implementation

The independent solvers are coupled by exchanging "messages" (through PVM or MPI). The organization is sketched in Figure 9: a "fluid-structure master" manages the FSI coupling algorithm (loop 1), while a "structure master" manages the contact (loop 2 and 3). Whatever the coupling algorithm (loosely coupled, strongly coupled, etc.), whatever the fluid formulation (ALE, fictitious domains, or both), whatever the number and the kind of structures (valves, walls), in presence of contact or not, the only modification to perform in existing solvers are as limited as possible: for the fluid, it only consists in sending a load and receiving displacements whereas, for the structure it only consists in sending displacements and receiving a load.

INRIA f σ f σ f σ c σ + f σ c σ + f σ c σ +

Numerical experiments

In this section we present some numerical results obtained on a realistic geometry of an aortic valve with the aim of testing the proposed algorithm. The aortic valve lets the blood flowing in the ascending aorta, and prevents its back flow to the heart. It is composed of three semilunar leaflets attached to the aortic root. Behind them, three anatomic dilatations define the Valsalva sinuses in which the two coronary arteries are attached. A bidimensional sketch of the valve is presented in Figure 10. Some anatomical characteristics and mechanical properties of the valve can be found in [START_REF] Hart | A three-dimensional computational analysis of fluid-structure interaction in the aortic valve[END_REF][START_REF] Thiriet | Biology and Mechanics of Blood Flows, part II: Mechanics and Medical Aspects of Blood Flows[END_REF]. Since the target of the numerical simulation is to test the multi-body contact algorithm, only the leaflet of the valve are considered flexible (red colored in Figure 11); the remaining part, blue colored, is the fluid domain boundary, Γ wall , which is fixed. For a more realistic simulation also the fluid-structure interaction between the aorta wall and the blood has to be taken into account. The discretized domain contains approximately 80000 tetrahedra for the fluid (Figure 12) and 2500 shell elements for the solid (Figure 13). From the mathematical viewpoint, the issue of contacts between bodies immersed in a viscous fluid is complicated and can lead to paradoxical results. For example, it is proved in [START_REF] Hillairet | Lack of collision between solid bodies in a 2D incompressible viscous flow[END_REF] that an immersed body cannot reach in finite time the boundary of the cavity surrounding the fluid. This interesting problem is INRIA beyond the scope of this paper. From the computational viewpoint, it clearly appears that contacts do occur and have to be handled. To illustrate this point, we propose two simulations: the first one without handling the contact, the second one handling it with the contact algorithm proposed above.

We provide here the details of the test case. A periodic pressure difference is applied between the inlet and outlet of the fluid domain. On Γ in the following pressure function is imposed:

p in = A if 0 ≤ t ≤ 13T 28 -A if 13T 28 ≤ t ≤ T ,
with amplitude A = 130 dyne cm -2 and period T = 0.28 s. Note that the duration of the systolic phase roughly corresponds to the physiological one. But, in order to test the robustness of the algorithm over several opening and closure cycles, we have artificially reduced the duration of the diastole. On Γ out free boundary conditions are set: p out = 0; while on Γ wall and Γ s no slip boundary conditions are imposed. For the fluid, the density ρ f and the dynamic viscosity µ f are respectively 1.0 g cm -2 and 0.03 poise. For the solid, a valve thickness of 0.65 mm has been considered. The structure density ρ s and the Young modulus E are respectively 1.2 g cm -2 and 10000 dyne/cm 2 . Both the simulations run with a timestep ∆t of 10 -3 s for a total time of 3T to attend the periodic condition; the fluid-structure tolerance ε f si is finally set to 10 -4 .

For the contact algorithm a gap of 10 -3 cm is imposed and the tolerances ε C and ε cvx are respectively fixed to 5. 10 -6 and 5. 10 -7 . At the closure of the valve, if the contact is not handled, a non-physical overlap of the leaflets is observed, as illustrated in Figure 14. This of course results in a dramatic change of the flow and the structure displacements. Moreover, in this case, locking phenomena among the leaflets or numerical instabilities can also happen, as noticed in [START_REF] Diniz | Numerical methods for fluid-structure interaction problems with valves[END_REF]. These observations confirm the importance of correctly manage the contact. In Figures 15 and16, the valve displacements, the blood velocity and pressure are reported for different timesteps in the case of contact handling. A maximum velocity of approximatively 20 cm/s has been obtained during the simulation. At timesteps 0.212 and 0.268, the velocity vectors show in partic-ular the blood recirculations that happen behind the aortic valve. Moreover, a pressure jump across the valve could be observed during the closure period.

Conclusion

We have presented a partitioned strategy to solve fluid-structure interaction problems in which contacts among different deformable bodies can occur. A possible application of this partitioned strategy is the movements of thin valves immersed in an incompressible viscous fluid.

The proposed scheme is conceived to address the problem with existing structure solvers that are not supposed to handle contact by themselves. The non-convex constraint optimization problem defined by the hypothesis of nonpenetration among solid objects is solved with an iterative algorithm which solves a series of convex constraint optimization problem. The bidimensional and tridimensional formulations are introduced and some details of the implementation are given.

A simulation of an idealized aortic valve is presented with the purpose of illustrating the algorithm in a non trivial configuration. In the proposed test case, several simplifications have been done, in particular the aortic root has been assumed to be fixed. More realistic physiological conditions will be investigated in future works.

Figure 1 :

 1 Figure 1: General algorithm

Figure 2 :Figure 3 :

 23 Figure 2: Definition of n e,x in two configurations.

1 .Figure 4 :

 14 Figure 4: Possible contacts between two solids M 1 and M 2 in three-dimensions. a) Contact between two edges. b) Contact between a vertex and a triangle.

Figure 5 :

 5 Figure 5: Computation of the unit vector n a,b (ψ) for the edge/edge couple (a, b). The coefficients α and β represent respectively the barycentric coordinates for ψ(p a,b ) and ψ(p b,a ) and are evaluated analytically by solving the following minimization problem: (α, β) = argmin 0≤α,β≤1

  b,a )ψ(p a,b ) . Once the couple (ψ(p a,b ), ψ(p b,a )) is computed, the normal vector n a,b (ψ) is finally obtained by n a,b (ψ) = ψ(p b,a )ψ(p a,b ) ψ(p b,a )ψ(p a,b ) . INRIA Some possible configurations of n a,b (ψ) for two generic edges a and b are presented in Figure 6. a,b ) ψ(p a,b ) ψ(p a,b ) ψ(p b,a ) ψ(p b,a ) ψ(p b,a ) n a,b (ψ) n a,b (ψ) n a,b (ψ)

Figure 6 :

 6 Figure 6: Possible configurations of the unit vector n a,b (ψ).

Figure 7 :

 7 Figure 7: Computation of the unit vector n T,x (ψ) for the triangle/vertex couple (T, x).

Figure 9 :

 9 Figure 9: FSI with multi-body contacts: σ f stands for hydrodynamic force, and σ c for the contact force.

Figure 10 :

 10 Figure 10: Bidimensional representation of the aortic valve.

Figure 11 :

 11 Figure 11: Computational domain.

Figure 12 :Figure 13 :

 1213 Figure 12: Fluid computational domain. On the left: surface mesh. On the right: inside clipping to show the spatial discretization step.

Figure 14 :

 14 Figure 14: Comparison between the two simulations at t = 0.259 s. On the left side, the contact among leaflets is handled with the contact algorithm, on the right side, it isn't.

Figure 15 :

 15 Figure 15: Valve displacements for five different timesteps in the simulations with (left) and without (right) contact handling. Note the dramatic difference between them at the reopening of the valve.

Figure 16 :

 16 Figure 16: Blood velocity vectors (left) and pressure field (right) for 5 different timesteps.

INRIA

RR n°6583

Acknowledgments

This work has been partially supported by the CardioSense3D action (INRIA) and by a grant from Région Ile-de-France.

We are debtful to Marc Thiriet and Tran Tan for the mesh of the aortic root.