Pierre-Emmanuel Arduin
email: pierre-emmanuel.arduin@dauphine.fr

Michel Grundstein
email: mgrundstein@mgconseil.fr

Camille Rosenthal-Sabroux
email: camille.rosenthal-sabroux@dauphine.fr

From Knowledge Sharing to Collaborative Decision Making

Keywords: Knowledge Management, Knowledge Retention, Collaborative Decision Making, Ba, Semi-opened Infrastructure Model (SoPIM)

This paper introduces an approach developed at AXA, a main French insurance company, in order to retain the knowledge acquired by outsourcing teams. This approach is based on the assumption that sharing and disseminating knowledge are two key factors for actually getting knowledge retention. It has been used, tested and improved within AXA France's industrial environment.

By presenting the problem raised by knowledge retention within organizations and the different ways envisaged to improve it, we propose a discussion about their efficiency, and describe our approach in order to retain knowledge in a specific case. This approach, based on our approach of Knowledge Management, has implied Collaborative Decision Making, which will be highlighted in this paper.

Bridging Knowledge Management and Collaborative Decision Making has been a means to reinforce the ability of the stakeholders to actually retain knowledge and to easily elaborate a consensus, the starting point of an efficient Collaborative Decision Making.

INTRODUCTION

In Knowledge society, as presented by [START_REF] Sharma | A Policy Framework for Developing Knowledge Societies[END_REF], knowledge must be considered as a core resource for organizations. So, beyond to be a medium to progress and to innovate, knowledge is one of our most important resources: something necessary to decide.

As emphasized by [START_REF] Liebowitz | Knowledge retention: strategies and solutions[END_REF], organizations that are embracing knowledge retention activities are gaining a competitive advantage. Organizational rearrangements from companies, notably outsourcing, increase a possible lost of knowledge, making knowledge retention sounds as an essential need for them.

When Knowledge is less shared, collaborative decision making seems harder to obtain insofar as a "communication breakdown" characterizes participants' discourse [START_REF] Kuhn | Reflections on my critics[END_REF]. At best stakeholders have to found a consensus according to their knowledge. Sharing knowledge ensures its retention and catalyzes the construction of this consensus.

After presenting the industrial context of the knowledge retention problems we have met, we present our visions of collaborative decision making, knowledge in the organization and Knowledge Management. We highlight the requirements and characteristics, but also the strengths and weaknesses of the two main approaches of Knowledge Management: (i) the Technological approach, and (ii) the Managerial and Sociotechnical approach. This finally allows us to see how the elaboration of an appropriated environment has efficiently shared knowledge, supporting several Knowledge Management theories, and has additionally aided people to collaborative decision making.

INDUSTRIAL CONTEXT

Knowledge retention: a true need AXA was created by the fusion between major French insurance groups. Every fusion pulled organizational and technological rearrangements. As a consequence, today the computer equipment of AXA counts more than 32,000 applications. The technology used is the "AXAPAC mainframe" technology. Many updates of a system using this old but efficient technology are a source of errors due to young programmers. Those errors can only be resolved by an expert of the AXAPAC mainframe technology. The problem is that these experts become more and more rare.

The teams working with the AXAPAC mainframe technology are always composed of an expert and a three-year-contract supplier. This organization seems dangerous to the extent that the absence of one expert leads to the loss of an incredible volume of knowledge, notably crucial knowledge: the expertise required to maintain AXAPAC mainframe technology and thus to enable the business activity. Furthermore, the supplier, which is present for three years, cannot acquire fast enough all the knowledge needed to become an AXAPAC mainframe technology expert.

The problems highlighted here are the problem of retaining expert's knowledge and the problem of aggregating experts' and suppliers' decisions. First of all, experts have to share their knowledge in order to enable AXA to retain it. Could this sharing make collaborative decision making easier? How can it be introduced in practice? Reliying on [START_REF] Wilson | The nonsense of 'knowledge management[END_REF], [START_REF] Bafoutsou | Review and functional classification of collaborative systems[END_REF], those are our research questions and they will be answered in this paper.

Two teams with two different purposes

Two teams work with the AXAPAC mainframe technology (see Fig. 1):

-The support team: it does a support activity. It is worrying that in spite of coming with changing volume, instant availability is expected from it. It is polarized around a file which simultaneously is a medium to obtain statistics, a pedagogic tool, and a documentary base. This file is called the "incidents' file". When this team has to deal with an incident, the incidents' file is updated. The update is done individually, directly at the end of the process, including information such as the person who investigated it, the date, a description of the demand, a description of the answer, etc and increase the quality of the AXAPAC mainframe's programs. This activity is worrying less than the support activity: knowledge is considered easier to transfer by the workers.

These teams are both composed of an expert and a three-year-contract supplier. The experts form the newcomer suppliers, and the suppliers assist the experts. Nevertheless, in practice there was so much incidents that the experts and the suppliers were working both on deciding the best way to resolve the incidents. Having no time to discuss, they only updated the incidents' file, without checking the acceptance of what they wrote inside.

Furthermore, this divided organization reinforces the feeling of working alone for the workers. Considering Barratt (2002) in the case of the support center, every worker has its knowledge and its methodology to resolve the incidents, and can thus be viewed as a decisionmaker. Although they are working on the same subject: to resolve incidents on AXAPAC mainframe's programs, they are taking decisions on their own. Knowledge is poorly shared, and collaborative decision making is not envisaged.

BACKGROUND THEORY

Our vision of collaborative decision making

As said by [START_REF] Ball | Collaborative Decision Making in Air Traffic Flow Management[END_REF], one of the maxims of collaborative decision making is that more information is better. Although it has sometimes been extended, the classic decision making process stays the same as its first description by [START_REF] Simon | The new science of management decision[END_REF]: intelligence, design and choice.

Our vision of collaborative decision making aims not only at increasing the quality of the first parts of the decision making process: intelligence and design, but also at increasing the 4 acceptance of the choice. Intelligence and design will be done by more than one individual, and constructed together the decision is more easily accepted. The decided choice will be then a shared choice. Thereby where decision making could be seen as a constructed model, collaborative decision making, for us, is seen as the use of socio-technical media in order to improve the performance and the acceptability of decision making. The location of knowledge in the realization of collaborative decision making has to be regarded insofar as, according to Gzara-Yesilbas and Lombard (2004, p. 348), knowledge sharing leads to improve collaborative decision making: a "static view" has to be structured and constitutes the "collaborative knowledge". Knowledge having an important role in individual decision making, we consider that for collaborative decision making, knowledge has to be shared. As said by [START_REF] Olson | Computer-supported co-operative work: research issues for the 90s[END_REF]: "What is required is a better understanding of the nature of group work". Knowledge has to be shared, but how do we share knowledge? Knowledge in the organization [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF], considering tacit knowledge structuring, presents the concepts of "sensegiving" and "sense-reading", which lead us to the following observation: we continuously appropriate information which is not ours. Information is transmitted by speaking, writing, or acting, during a sense-giving process. We perceive some data from that information by listening, reading, or watching, during a sense-reading process. This process activates relevant interpretative frameworks and previous knowledge, depending of the context, our situation and our intention. Eventually, new tacit knowledge is created.

As the authors of this paper, we have got tacit knowledge that we have structured into information during a process of sense-giving. As the readers of this paper, you have interpreted this information perceiving forms and colors, integrated words, data, during a process of sense-reading possibly creating new tacit knowledge (see Fig. 2).

When a person P1 structures its tacit knowledge and externalizes it, he creates information. A person P2 perceiving some data from this information and internalizing it, possibly creates new tacit knowledge. Thus knowledge is the result of the interpretation of information by someone. This interpretation is done through an interpretative framework that filters data contained in the information and the use of previous tacit knowledge as presented by [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF].

How can we ensure that the information, regardless who is receiving it, will give the same meaning? How can we avoid the creation of non pertinent knowledge issue from wrongly interpreted information? And more than everything, how can we ensure an efficient knowledge transfer?

The efficiency of knowledge transfer is critical insofar as knowledge cannot exist without people. Retaining knowledge is not reduced to maintaining it. Knowledge continues existing through the actors in presence. If we only store it, for example keeping externalized 5

S e n s e -g i v i n g

Based on Shigehisa Tsuchiya's works, 1993

P 1 P 2
information into a computer, we need someone to interpret this information, someone who has an interpretative framework which allows him to give it the same meaning, and this only, correctly.

If the probability that two people will give the same meaning to the same information is high, it is said that their interpretative frameworks have a strong commensurability. On the contrary, if this probability is low, it is said that their interpretative frameworks have a low commensurability.

Thus it is now clear that the efficiency of a knowledge transfer can be assured if the interpretative frameworks of the actors of this transfer have a strong commensurability. In the same way, in our specific case, collaborative decision making can be considered efficient. The interpretative frameworks of the stakeholders have a strong commensurability, doing a consensus being more evident. Many of our studies aim at setting a mean to measure this commensurability.

Increasing the commensurability of interpretative frameworks allows to consider knowledge as an object. Indeed, it is more and more likely that the same information will have the same meaning for the sender and the receiver. To handle information, for these two people, becomes just the same as to handle knowledge. However how can we handle knowledge?

The two approaches of Knowledge Management

Most of the time thinking Knowledge Management is thinking in knowledge representation, collaborative or semantic technologies, e-learning, etc. Yet Knowledge Management is not limited to its technological approach, which rests on the codification of explicit knowledge. Knowledge Management includes a managerial and sociotechnical approach which considers knowledge as a resource participating in companies' performance, as presented by [START_REF] Grundstein | MGKME: A Model for Global Knowledge Management within the Enterprise[END_REF] and according to [START_REF] Daft | Toward a model of organizations as interpretation systems[END_REF].

The technological approach: here knowledge is considered as an object independent of the person who is handling it. Thus some "knowledge" can be found on collaborative tools or be stored in "knowledge-bases". It is notably the point of view of [START_REF] Grangel | A Modelling Framework for Sharing Knowledge[END_REF]. This approach is suitable if it is used by people for whom knowledge can be considered as an object: experts, specialized staff, business community, etc. Actually, people whom interpretative frameworks have an extremely strong commensurability. And this is the most important limitation of this approach: knowledge is regarded as an object. An object which can be handled by people who give it a meaning, and the same meaning. Otherwise we will conclude that the knowledge transfer is not efficient.

The managerial and sociotechnical approach: this approach takes into consideration the variety of the situations. It places the constraints of the social system and the specificities of the actors at the heart of its analysis. More supported by [START_REF] Miller | I = 0 (Information has no intrinsic meaning)[END_REF] among other, this apporach is centered on the processes but it does not center them on technological solutions conceivable a priori. The managerial and sociotechnical approach is articulated around two purposes : a) Patrimony purpose aims at knowledge conservation, for its retention and its transfer. It aims also at uncovering tacit knowledge to retain it. b) Sustainable innovation purpose is complementary to patrimony purpose. It is about encouraging individual knowledge creation, without neglecting its appropriation by the organization. It stimulates knowledge creation from an individual point of view and its retention from the collectivity point of view. This purpose can be regarded as a medium to encourage original decision elaborating and collaborative decision making.

These two approaches have requirements and characteristics, which make them more or less suitable depending on the expectations and the characteristics from the actors in presence (see Fig. 3).

In the case of people having the same specialty for instance, people of whom their interpretative frameworks have a strong commensurability, the technological approach is completely satisfactory to share knowledge. As long as there is a user to correctly interpret information.

On the contrary, if all the people are not experts, if their interpretative frameworks have a poor commensurability, it is likely that the same information will have different meanings. It is then extremely risky to use the technological approach, which handles knowledge as an object; object which here, does not have the same meaning for everybody. The managerial and sociotechnical approach, as it takes in consideration the specificities of the individuals, ensures enhancing their skills and giving them the means to give the same meaning to the same information.

As much as the commensurability of the interpretative frameworks of the actors will increase, we consider that it will be easier to elaborate collaborative decision making. Sharing the same knowledge and abilities to interpret, they will share more affinities. They will find a consensus easily, starting point of collaborative decision making.

If the technological approach is operational from the beginning, the managerial and sociotechnical approach needs a certain time to ensure an efficient knowledge retention (see Fig. 3 and4). On the base of our observations, we propose a representation from the evolution of the commensurability in both cases (Fig. 4).

As shown in the Fig. 3, the technological approach, applied in an environment where the actors have strong commensurabilities between their interpretative frameworks is effective. On condition that there is always a user of the system able to appropriate correctly and entirely the knowledge.

But then how can we retain knowledge in the case of actors having poor commensurabilities between their interpretative frameworks? How can we prevent the risk of generating incorrect knowledge?

The technology is not enough here. The managerial and sociotechnical approach comes in and reminds the Man that knowledge is not exchanged, it is transferred.

The SECI model and the concept of Ba

The postulate of the existence of explicit and tacit knowledge [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF] leads to ask how we can handle tacit knowledge. Relaying on conversions for example from tacit to tacit or tacit to explicit, [START_REF] Nonaka | The Knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF] proposed a model of knowledge conversion.

1. From tacit to tacit: these authors called it "socialization". A transfer of tacit knowledge is done by watching and practical experience. It is the case for example between a master and his apprentice. 2. From tacit to explicit: it is called the "externalization". An individual shapes his tacit knowledge to share it with a group. 3. From explicit to explicit: it is called the "combination". Several instances of explicit knowledge are combined to constitute new explicit knowledge. 4. From explicit to tacit: it is called the "internalization". By playing and repeating, the individual appropriates the explicit knowledge disseminated in the organization. It becomes, for him, tacit knowledge. With the concept of "Ba", Nonaka and Konno (1998) proposed a place to let emerge and / or share tacit knowledge. Concretely, the Ba can be physical (a meeting room for example) or virtual (a collaborative work space for example). It is a place to make progress individual and collective knowledge. Share it to retain it, but also ensure innovation, knowledge creation. That is why the Ba suits perfectly the two purposes of the Knowledge Management in its managerial and sociotechnical approach. It ensures knowledge retention and knowledge appropriation by a group, starting collaborative decision making. The "semi-opened infrastructure model", introduced by Grundstein ( 2005) is rather close to the Ba to the extent that it sets up conditions leading to the establishment of exchange relations between people. That kind of exchange relations are considered as a base for an efficient knowledge transfer, and consequently, an efficient knowledge retention.

It aims at catalyzing individual learning and collective dissemination of the acquired knowledge.

A "multidisciplinary group" ensures the "evolution and progress space" the capacity of making building up the skills of the actors of the working space. Actors who thereafter disseminate these skills into their working space (see Fig. 6).

The SECI model as well as the SopIM, insist on the existence of a space where people can discuss, let emerge some tacit knowledge to share it, to disseminate it, and thus, to retain it, by the building-up skills that these actions have involved. The commensurability of their interpretative frameworks tends to increase, doing, according to our observations, collaborative decision making easier to elaborate.

When the actors are not all experts, when it is impossible to ensure that the same information will take the same meaning for the sender and the receiver, it is now clear that in order to share knowledge, we must create specific spaces where people can interact together. These places will also ensure collaborative decision making: knowledge is discussed, knowledge is shared, and knowledge is appropriated. It is transferred in the sense of Davenport and Prusak that is: "Transfer = Transmission + Absorption (and Use)" (1998, page 101).

THE IMPLEMENTED COLLABORATIVE APPROACH

In this part we will present the approach developed at AXA France in order to retain the knowledge of the teams working with the AXAPAC mainframe technology. This approach has been used, tested and improved within AXA France's industrial environment. Relying on the elaboration of a collaborative space where the actors can share their knowledge, we observed that this space facilitates collaborative decision making: they now decide together on the best way to solve the incidents. 9

A planned rearrangement: the origin of disruptions

As presented in the first part of this paper (Fig. 1), the teams working with the AXAPAC mainframe technology were divided, what reinforced the feeling of "working alone" for the decision-makers. In order to catalyze knowledge sharing, we suggested merging the knowledge of the teams. We created a "competence center" (Fig. 7). Suppliers will work there on support and quality together. Not only it will ensures knowledge retention, but also it will aid collaborative decision making. We will see now how the disruptions inferred by this rearrangement will imply structural changes instigators of our approach to retain knowledge and to promote collaborative decision making.

The idea of this rearrangement is to merge the knowledge of the suppliers so that when a new supplier arrives, the elder is able to assist and form him as in the "socialization" of the SECI model from [START_REF] Nonaka | The Knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF]. The competence center created is the one to work with the incidents' file. This comment is what brings us to the following proposition: "to update together the incidents' file".

Indeed, the update was done individually, directly at the end of an incident, what implied no dissemination of acquired knowledge and impeded collaborative decision making. Why don't make the actors of the competence center met in order to discuss the incidents, the ways they resolved them, share their knowledge, and then update the file together? There is consequently a dissemination and a sharing of knowledge from where will emerge a consensus on the update of the incidents' file. An update on which everybody will have taken part, resulting of collaborative decision making. An update, which is done by the group and which thus is profitable for the group.

That kind of meetings, which can be weekly, make all the actors of the competence center meet so that they express their tacit knowledge, they share their experiences, their knowledge and they finally disseminate them as in the "externalization" of the SECI model from [START_REF] Nonaka | The Knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF].

The question of the use of the incidents' file does not raise problems any more: the update resulting of collaborative decision making, every member of the group knows what the file contains, and thus knows what he can search in it. Furthermore, the impregnation of the actors by the contents of the file will be much greater than in the context of an individual update: they talked, exchanged, and all participated in the formalization of their knowledge.

These meetings remind us of the "evolution and progress space" of the semi-opened infrastructure model from Grundstein (2005, see Fig. 6). During the time of a meeting, the actors are as in this "evolution and progress space": knowledge is shared by the interactions between them and a "multidisciplinary group". In the case of these meetings which have been called "incidents' meetings", the evolution and progress space, the multidisciplinary group, and the meeting as such are the same. The actual rearrangement: the implementation of a Ba

For practical purposes, when a new supplier arrives we observe that the "socialization" is effectively made between the two suppliers. However, the "socialization" is also effective between the new supplier and every member of the group, what was less predictable. A kind of generalized tacit knowledge sharing was set up at the arrival of the new supplier.

We began planning one incidents' meeting a week. Always in a small meeting room, no more than five persons were present. The duration was generally between one and a half hours and two hours. They were sometimes dangerously too technical at the beginning (only list incidents and solutions, not necessarily understandable by all the stakeholders), but they rapidly became more interactive and make the actors design new axes to optimize knowledge disseminating and sharing. We remind that we consider these two vectors as necessary to retain knowledge in the case of actors having poor commensurabily between their interpretative frameworks, like in such case. The concern is the appropriation of the shared and disseminated knowledge by the actors. From this knowledge sharing they began to think about elaborating collaborative decision making.

If the "socialization" is made when a new supplier arrives and the "externalization" during the incidents' meetings, the "combination" (combination of several instances of explicit knowledge in order to constitute new explicit knowledge) has been observed during these incidents' meetings: tacit knowledge is first externalized and then combined, for instance to elaborate decisions with collaborative decision making. This is a real quality of these incidents' meetings: they not only ensure knowledge retention but also catalyze innovation, knowledge creation, and collaborative decision making. How to optimize knowledge sharing? How to update the incidents' file? These questions and others have been discussed and decided during incidents' meetings.

Practice, repetition and an active participation to the incidents' meetings enable the "internalization": actors appropriate the shared and disseminated knowledge and finally accept the discussed and taken decisions. We are here in a case of application of the SECI model from [START_REF] Nonaka | The Knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF]: knowledge is shared, knowledge is appropriated, knowledge is actually retained, and finally, a place where collaborative decision making can be done has been constructed (see Fig. 8), this place is a Ba, as defined by [START_REF] Nonaka | The concept of « Ba » : building a foundation for knowledge creation[END_REF].

A major quality of the incidents' meetings is that they allow their self-amelioration. Thereafter a slight generalization of this self-amelioration has been observed, as if the actors of the competence center tend to spread this passion for knowledge manipulation and collaborative decision making. 11 In this way the use of a vocabulary for updating the incidents' file has been suggested. This idea has been immediately and unanimously decided by the group, showing the necessity and the will to normalize the way of updating the incidents' file. Until then the file was updated by the person who dealt with an incident specifying some precise information (processing time, concerned domain, etc.) but also more subjective information (description of the incident, of the decided solution, etc.). Thus, when a similar incident occurred to another person, it was not able to find the first incident in the incidents' file due to the variety of technical vocabulary and to the different expressions used by each author to describe the same incident. Collaborative decision making was difficult to obtain.

A Ba having been developed in the incidents' meetings and the update being now done together, the problem of seeking and finding in the incidents' file will decrease, allowing the actors to discuss and decide together on the best way to solve the incidents discussed during the incidents' meetings.

The idea of the vocabulary restricts the expressiveness of the actors and above all makes them think about what they are about to insert in the incidents' file. The incidents' meetings became thus a place where the actors will discuss the incidents that had been met and the taken decisions to solve them, and now where they will decide about what they are going to insert into the incidents' file: they formalize their knowledge, they synthesize it, and they do it together, ensuring all the actors to appropriate it and to decide it.

So the approach to retain knowledge that we put forward and which facilitates collaborative decision making is articulated around these three dimensions (see Fig. 9):

-The incidents' file: indisputable, it collects the information source of the knowledge necessary for the realization of the support activity. -The organisation: indispensable, it is more about the rearrangement which was at the origin of the incidents' meetings: to conceive a "competence center", and organize meetings there. -The vocabulary: unnecessary, but requested so much by the team, the creation of a vocabulary ensures the actors that they understand each other and especially that they can easily reuse the information entered in the incidents' file. Nevertheless the complexity of this task has to be taken into consideration: before every entry in the incidents' file, a heavy collaborative decision making must be done to find the most significant and the most important keywords.

Gradually, the incidents' meetings became more and more generative of discussions concerning the core of the support activity. Discussions in which the various actors of the competence center actively took part, not only the expert, but also the supplier and the newcoming supplier. Working together in the dealing of the incidents and in deciding the formalization of the knowledge necessary to these interventions, each actor appropriated the 12 knowledge. This appropriation retain knowledge and makes easier the elaboration of the next decision with collaborative decision making. By involving people in the implementation of an approach and in the improvement of a process, we make them appropriate the knowledge necessary for the realization of this process.

Within AXA France's industrial environment, our approach has been used, tested and improved. One year after the beginning of this work, the activities of the concerned teams integrate this approach. Knowledge retention became a key step of their activities, and the knowledge retention process continues to give them a place where they can efficiently discuss and initiate collaborative decision making.

RELATED WORKS

Several works present methods to retain knowledge close to our approach, which is centered on sharing and disseminating knowledge in order to retain it. We think for example of the concept of "thinkLets" introduced by De Vreede ( 2003) in order to improve collaborative work. It has been observed that improving the quality of collaborative working leaded to improve the sharing of knowledge.

A parallel should be drawn between our vision of collaborative decision making and Raiffa (2002, p. 389) notably, who not only proposes to "look at traditional negotiation advice, organizational behavior, and sociology to illuminate the problem of how to use quantitative methods.", but also to "use quantitative methods and game-theoretic analysis to illuminate negotiation problems". In this paper, we suggested to retain knowledge by sharing it. This sharing increased the commensurability of interpretative frameworks, what, for us, facilitated collaborative decision making. We observed and participated in the collaborative decision making, we observed and participated in the processes.

Considering face-to-face meetings as being the main tool for collaborative decision making, [START_REF] Camilleri | Towards a dynamic estimation of collective decisionmaking meetings efficiency[END_REF] outline an approach to dynamically estimate the efficiency of collective decision making meetings. This can constitute a future for our works: evaluate the commensurability of interpretative frameworks in order to identify people more able to easily work together with collaborative decision making. [START_REF] Schmidt | Taking CSCW seriously: Supporting articulation work[END_REF] precise that the term of Computer Supported Cooperative Work (CSCW) also covers the study of the organizational effects of using a collaborative working tool. Our approach can be viewed as an echo to it. Yet it have to be noticed that CSCW is often confused with groupware: computer software designed to cooperative work. Contrarily to [START_REF] Carstensen | Computer Supported Cooperative Work: New Challenges to Systems Design[END_REF], our approach does not gives to the workers a technological system in order to share their knowledge and elaborate collaborative decision making. However it gives them the opportunity to decide to design it together, for example deciding to change the structure of the incidents' file. [START_REF] Jordan | Ethnographic Workplace Studies and Computer Supported Cooperative Work[END_REF] applied a mix of ethnographic methods studying and participating in collaborative decision making processes. She considered Communities of Practice, focusing on "how work is accomplished as a collaborative enterprise" (p. 20). Knowledge is seen as a meaning, what strengthen our approach: knowledge is a meaning, we have to ensure that it is the same for every stakeholder. She insisted on the ethnographic field methods: participant observation, "in-situ question asking" (p. 25) and video review sessions. All of these methods can help to explicit tacit knowledge and share it.

We consider that knowledge has to be shared to ensure its retention. This sharing increases the commensurability of the interpretative frameworks of the stakeholders what, in our specific case, facilitated collaborative decision making. Finding a way to measure the commensurability of interpretative frameworks, would allows us to present quantified data supporting this observation. A real strength of our approach is the accent given to the tacit dimension. Several related works presented in this section can complete our approach by giving it a more quantifiable dimension, which is currently one of its weaknesses.

CONCLUSION

In this paper we began presenting the industrial context of the knowledge retention problems that we have met, and then we introduced background theories, notably our visions of collaborative decision making and of knowledge in the organization. We finally explained our collaborative approach based on these background theories. This approach has been implemented at AXA France, where it has promoted collaborative decision making.

Whereas collaborative decision making was unthinkable because they worked alone, on their own, we have set up an original approach, sometimes viewed as a "strange idea" by the workers: to continue working alone, but discussing once a week how to do the work. These discussions rapidly shared knowledge and this sharing rapidly increased their abilities for collaborative decision making. The implemented Ba gives them a place where they can efficiently share their knowledge, but also a place where they can create knowledge together, externalizing and combining it. This creation is the basis of collaborative decision making.

Considered as the result of the interpretation of information by someone [START_REF] Tsuchiya | Improving Knowledge Creation Ability through Organizational Learning[END_REF], knowledge has to be actually shared in order to be retained. By creating an appropriated environment for the actors, giving them a place where they can met, exchange and share, the Knowledge Management, in its managerial and sociotechnical approach considers that an efficient knowledge sharing will be envisaged under these circumstances. This efficient knowledge sharing will be a stiff support to its retention.

A knowledge sharing can be done if we give to the actors a place to do it. This place is a Ba. It is impressive because it simultaneously catalyzes the transfer of knowledge, proposes a support to the updating and the sharing of tacit knowledge, and it has been observed that it promotes collaborative decision making. The actors of the knowledge sharing process have become more and more efficients in terms of performance and acceptability of the decisions elaborated.

Sharing knowledge, thus, increased the commensurability of the stakeholders' interpretative frameworks. This increase not only made knowledge transfer efficient (Davenport and Prusak, 1998, page 101), knowledge is thus retained, but also it facilitates collaborative decision making: a consensus become more evident, as said by Gzara-Yeslbas and Lombard ( 2004) "collaborative knowledge" is generated, and the risk of "communication breakdown" as said by [START_REF] Kuhn | Reflections on my critics[END_REF] is reduced.

Many of our current studies aim at setting a mean to measure the commensurability of interpretative frameworks. A perspective to the work presented in this paper is to try to use a measure of the commensurability in order to evaluate the efficiency of knowledge sharing and thus, the predisposition of people to easily work together with collaborative decision making.

Bridging Knowledge Management and Collaborative Decision Making has been here a medium to obtain surprising results, which have highlighted critical aspects of knowledge retention and collaborative decision making: if it is not shared, knowledge is not retained, and if there is no shared knowledge, collaborative decision making is harder to obtain.

Figure 1 :

 1 Figure 1: Initial configuration of the teams working with the AXAPAC mainframe technology

Figure

  Figure 2: Tacit Knowledge transfert

  7

Figure 3 :Figure 4 :

 34 Figure 3: Both approaches, their requirements and characteristics

Fig. 5

 5 Fig. 5 shows how these conversions follow each other in the style of a spiral: Socialization, Externalization, Combination, and Internalization. From there comes the name of their model: the SECI model.

Figure 5 :

 5 Figure 5: The SECI model

Figure 6 :

 6 Figure 6: The semi-opened infrastructure model (SopIM)

  10

Figure 7 :

 7 Figure 7: Planned configuration of the teams working with the AXAPAC mainframe technology

Figure 8 :

 8 Figure 8: The SECI model in the case of the competence center

Figure 9 :

 9 Figure 9: The principle of our approach to retain knowledge

  . What is entered into the incidents' file is quite clear. What raises more concerns is what comes out from it: this file is difficult to use because it is updated individually. People does not know what it contains, doing collaborative decision making harder to get. -The quality team: it does a quality activity. It relies on formalized rules which ensure


Jean-Daniel Boissonnat

Mael Rouxel-Labbé

Mathijs Wintraecken

Anisotropic triangulations via discrete Riemannian Voronoi diagrams

Keywords: 1998 ACM Subject Classification Computational Geometry and Object Modeling Riemannian Geometry, Voronoi diagram, Delaunay triangulation, Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.19

The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics.

To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R 2 and on surfaces embedded in R 3 as detailed in our experimental companion paper.

In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Ω under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.

Introduction

Anisotropic triangulations are triangulations whose elements are elongated along prescribed directions. Anisotropic triangulations are known to be well suited when solving PDE's [START_REF] Simpson | On optimal interpolation triangle incidences[END_REF][START_REF] Mirebeau | Optimal meshes for finite elements of arbitrary order[END_REF][START_REF] Shewchuk | What is a good linear finite element? Interpolation, conditioning, anisotropy, and quality measures[END_REF]. They can also significantly enhance the accuracy of a surface representation if the anisotropy of the triangulation conforms to the curvature of the surface [START_REF] Garland | Surface simplification using quadric error metrics[END_REF].

Many methods to generate anisotropic triangulations are based on the notion of Riemannian metric and create triangulations whose elements adapt locally to the size and anisotropy prescribed by the local geometry. The numerous theoretical and practical results [START_REF] Aurenhammer | Voronoi diagrams[END_REF] of the Euclidean Voronoi diagram and its dual structure, the Delaunay triangulation, have pushed authors to try and extend these well-established concepts to the anisotropic setting. Labelle and Shewchuk [START_REF] Labelle | Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF] and Du and Wang [START_REF] Du | Anisotropic centroidal Voronoi tessellations and their applications[END_REF] independently introduced two anisotropic Voronoi diagrams whose anisotropic distances are based on a discrete approximation of the Riemannian metric field. Contrary to their Euclidean counterpart, the fact that the dual of these anisotropic Voronoi diagrams is an embedded triangulation is not immediate, and, despite their strong theoretical foundations, the anisotropic Voronoi diagrams of Labelle and

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Shewchuk and Du and Wang have only been proven to yield, under certain conditions, a good triangulation in a two-dimensional setting [START_REF] Cañas | Orphan-free anisotropic Voronoi diagrams[END_REF][START_REF] Cañas | Duals of orphan-free anisotropic Voronoi diagrams are embedded meshes[END_REF][START_REF] Cheng | Anisotropic surface meshing[END_REF][START_REF] Du | Anisotropic centroidal Voronoi tessellations and their applications[END_REF][START_REF] Labelle | Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF].

Both these anisotropic Voronoi diagrams can be considered as an approximation of the exact Riemannian Voronoi diagram, whose cells are defined as V g (p i ) = {x ∈ Ω | d g (p i , x) ≤ d g (p j , x), ∀p j ∈ P\p i }, where d g (p, q) denotes the geodesic distance. Their main advantage is to ease the computation of the anisotropic diagrams. However, their theoretical and practical results are rather limited. The exact Riemannian Voronoi diagram comes with the benefit of providing a more favorable theoretical framework and recent works have provided sufficient conditions for a point set to be an embedded Riemannian Delaunay complex [START_REF] Boissonnat | Delaunay triangulation of manifolds[END_REF][START_REF] Dyer | Surface sampling and the intrinsic Voronoi diagram[END_REF][START_REF] Leibon | Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization[END_REF]. We approach the Riemannian Voronoi diagram and its dual Riemannian Delaunay complex with a focus on both practicality and theoretical robustness. We introduce the discrete Riemannian Voronoi diagram, a discrete approximation of the (exact) Riemannian Voronoi diagram. Experimental results, presented in our companion paper [START_REF] Rouxel-Labbé | Discretized Riemannian Delaunay triangulations[END_REF], have shown that this approach leads to good anisotropic triangulations for two-dimensional domains and surfaces, see Figure 1.

Figure 1 Left, the discrete Riemannian Voronoi diagram (colored cells with bisectors in white) and its dual complex (in black) realized with straight simplices of a two-dimensional domain endowed with a hyperbolic shock-based metric field. Right, the discrete Riemannian Voronoi diagram and the dual complex realized with curved simplices of the "chair" surface endowed with a curvature-based metric field [START_REF] Rouxel-Labbé | Discretized Riemannian Delaunay triangulations[END_REF].

We introduce in this paper the theoretical side of this work, showing that our approach is theoretically sound in all dimensions. We prove that, under sufficient conditions, the discrete Riemannian Voronoi diagram has the same combinatorial structure as the (exact) Riemannian Voronoi diagram and that the dual discrete Riemannian Delaunay complex can be embedded as a triangulation of the point set, with either curved or straight simplices. Discrete Voronoi diagrams have been independently studied, although in a two-dimensional isotropic setting by Cao et al. [START_REF] Cao | Proof of correctness of the digital Delaunay triangulation algorithm[END_REF].

Riemannian geometry

In the main part of the text we consider an (open) domain Ω in R n endowed with a Riemannian metric g, which we shall discuss below. We assume that the metric g is Lipschitz continuous. The structures of interest will be built from a finite set of points P, which we call sites.

Riemannian metric

A Riemannian metric field g, defined over Ω, associates a metric g(p) = G p to any point p of the domain. This means that for any v, w ∈ R n we associate an inner product v, w g = v t g(p)w, in a way that smoothly depends on p. Using a Riemannian metric, we can associate lengths to curves and define the geodesic distance d g as the minimizer of the lengths of all curves between two points. When the map g : p → G is constant, the metric field is said to be uniform. In this case, the distance between two points x and y in Ω is

d G (x, y) = x -y G = (x -y) t G(x -y).
Most traditional geometrical objects can be generalized using the geodesic distance. For example, the geodesic (closed) ball centered on p ∈ Ω and of radius r is given by

B g (p, r) = {x ∈ Ω | d g (p, x) ≤ r}.
In the following, we assume that Ω ⊂ R n is endowed with a Lipschitz continuous metric field g.

We define the metric distortion between two distance functions d g (x, y) and d g (x, y) to be the function ψ(g, g ) such that for all x, y in a small-enough neighborhood we have: 1/ψ(g, g ) d g (x, y) ≤ d g (x, y) ≤ ψ(g, g ) d g (x, y). Observe that ψ(g, g ) ≥ 1 and ψ(g, g ) = 1 when g = g . Our definition generalizes the concept of distortion between two metrics g(p) and g(q), as defined by Labelle and Shewchuk [START_REF] Labelle | Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation[END_REF] (see Appendix B of the full version of this paper, [START_REF] Boissonnat | Anisotropic triangulations via discrete Riemannian Voronoi diagrams[END_REF]).

Geodesy

Let v ∈ R n . From the unique geodesic γ satisfying γ(0) = p with initial tangent vector γ = v, one defines the exponential map through exp(v) = γ [START_REF] Aurenhammer | Voronoi diagrams[END_REF]. The injectivity radius at a point p of Ω is the largest radius for which the exponential map at p restricted to a ball of that radius is a diffeomorphism. The injectivity radius ι Ω of Ω is defined as the infimum of the injectivity radii at all points. For any p ∈ Ω and for a two-dimensional linear subspace H of the tangent space at p, we define the sectional curvature K at p for H as the Gaussian curvature at p of the surface exp p (H).

In the theoretical studies of our algorithm, we will assume that the injectivity radius of Ω is strictly positive and its sectional curvatures are bounded.

Power protected nets

Controlling the quality of the Delaunay and Voronoi structures will be essential in our proofs. For this purpose, we use the notions of net and of power protection.

Power protection of point sets

Power protection of simplices is a concept formally introduced by Boissonnat, Dyer and Ghosh [START_REF] Boissonnat | Delaunay triangulation of manifolds[END_REF]. Let σ be a simplex whose vertices belong to P, and let B g (σ) = B g (c, r) denote a circumscribing ball of σ where r = d g (c, p) for any vertex p of σ. We call c the circumcenter of σ and r its circumradius.

For 0 ≤ δ ≤ r, we associate to B g (σ) the dilated ball

B +δ g (σ) = B(c, √ r 2 + δ 2
). We say that σ is δ-power protected if B +δ g (σ) does not contain any point of P \ Vert(σ) where Vert(σ) denotes the vertex set of σ. The ball B +δ g is the power protected ball of σ. Finally, a point set P is δ-power protected if the Delaunay ball of its simplices are δ-power protected.

Nets To ensure that the simplices of the structures that we shall consider are well shaped, we will need to control the density and the sparsity of the point set. The concept of net conveys these requirements through sampling and separation parameters.

S o C G 2 0 1 7 19:4

The sampling parameter is used to control the density of a point set: if Ω is a bounded domain, P is said to be an ε-sample set for Ω with respect to a metric field g if d g (x, P) < ε, for all x ∈ Ω. The sparsity of a point set is controlled by the separation parameter: the set P is said to be µ-separated with respect to a metric field g if d g (p, q) ≥ µ for all p, q ∈ P. If P is an ε-sample that is µ-separated, we say that P is an (ε, µ)-net.

Riemannian Delaunay triangulations

Given a metric field g, the Riemannian Voronoi diagram of a point set P, denoted by Vor g (P), is the Voronoi diagram built using the geodesic distance d g . Formally, it is a partition of the domain in Riemannian Voronoi cells

{V g (p i )}, where V g (p i ) = {x ∈ Ω | d g (p i , x) ≤ d g (p j , x), ∀p j ∈ P \ p i }.
The Riemannian Delaunay complex of P is an abstract simplicial complex, defined as the nerve of the Riemannian Voronoi diagram, that is the set of simplices Del g (P) = {σ | Vert(σ) ∈ P, ∩ p∈σ V g (p) = 0}. There is a straightforward duality between the diagram and the complex, and between their respective elements.

In this paper, we will consider both abstract simplices and complexes, as well as their geometric realization in R n with vertex set P. We now introduce two realizations of a simplex that will be useful, one curved and the other one straight.

The straight realization of a n-simplex σ with vertices in P is the convex hull of its vertices. We denote it by σ. In other words,

σ = {x ∈ Ω ⊂ R n | x = p∈σ λ p (x) p, λ p (x) ≥ 0, p∈σ λ p (x) = 1}. (1)
The curved realization, noted σ is based on the notion of Riemannian center of mass [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF][START_REF] Dyer | Riemannian simplices and triangulations[END_REF]. Let y be a point of σ with barycentric coordinate λ p (y), p ∈ σ. We can associate the energy functional E y (x) = 1 2 p∈σ λ p (y)d g (x, p) 2 . We then define the curved realization of σ as

σ = {x ∈ Ω ⊂ R n | x = argmin E x(x), x ∈ σ}. ( 2
)
The edges of σ are geodesic arcs between the vertices. Such a curved realization is well defined provided that the vertices of σ lie in a sufficiently small ball according to the following theorem of Karcher [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF].

Theorem 1 (Karcher). Let the sectional curvatures K of Ω be bounded, that is Λ -≤ K ≤ Λ + . Let us consider the function E y on B ρ , a geodesic ball of radius ρ that contains the set {p i }. Assume that ρ ∈ R + is less than half the injectivity radius and less than π/4 Λ + if Λ + > 0. Then E y has a unique minimum point in B ρ , which is called the center of mass.

Given an (abstract) simplicial complex K with vertices in P, we define the straight (resp., curved) realization of K as the collection of straight (resp., curved) realizations of its simplices, and we write K = {σ, σ ∈ K} and K = {σ, σ ∈ K}.

We will consider the case where K is Del g (P). A simplex of Del g (P) will simply be called a straight Riemannian Delaunay simplex and a simplex of Del g (P) will be called a curved Riemannian Delaunay simplex, omitting "realization of". In the next two sections, we give sufficient conditions for Del g (P) and Del g (P) to be embedded in Ω, in which case we will call them the straight and the curved Riemannian triangulations of P.

Sufficient conditions for Del g (P) to be a triangulation of P

It is known that Del g (P) is embedded in Ω under sufficient conditions. We give a short overview of these results. As in Dyer et al. [START_REF] Dyer | Riemannian simplices and triangulations[END_REF], we define the non-degeneracy of a simplex σ of Del g (P). Definition 2. The curved realization σ of a Riemannian Delaunay simplex σ is said to be non-degenerate if and only if it is homeomorphic to the standard simplex.

Sufficient conditions for the complex Del g (P) to be embedded in Ω were given in [START_REF] Dyer | Riemannian simplices and triangulations[END_REF]: a curved simplex is known to be non-degenerate if the Euclidean simplex obtained by lifting the vertices to the tangent space at one of the vertices via the exponential map has sufficient quality compared to the bounds on sectional curvature. Here, good quality means that the simplex is well shaped, which may be expressed either through its fatness (volume compared to longest edge length) or its thickness (smallest height compared to longest edge length).

Let us assume that, for each vertex p of Del g (P), all the curved Delaunay simplices in a neighborhood of p are non-degenerate and patch together well. Under these conditions, Del g (P) is embedded in Ω. We call Del g (P) the curved Riemannian Delaunay triangulation of P.

Sufficient conditions for Del g (P) to be a triangulation of P

Assuming that the conditions for Del g (P) to be embedded in Ω are satisfied, we now give conditions such that Del g (P) is also embedded in Ω. The key ingredient will be a bound on the distance between a point of a simplex σ and the corresponding point on the associated straight simplex σ (Lemma 3). This bound depends on the properties of the set of sites and on the local distortion of the metric field. When this bound is sufficiently small, Del g (P) is embedded in Ω as stated in Theorem 4.

Lemma 3.

Let σ be an n-simplex of Del g (P). Let x be a point of σ and x the associated point on σ (as defined in Equation 1). If the geodesic distance d g is close to the Euclidean distance d E , i.e. the distortion ψ(g, g E ) is bounded by

ψ 0 , then | x -x| ≤ 2 • 4 3 (ψ 0 -1)ε 2 .
We now apply Lemma 3 to the facets of the simplices of Del g (P). The altitude of the vertex p in a simplex τ is noted D(p, τ ). Theorem 4. Let P be a δ-power protected (ε, µ)-net with respect to g on Ω. Let σ be any n-simplex of Del g (P) and p be any vertex of σ. Let τ be a facet of σ opposite of vertex p.

If, for all x ∈ τ , we have | x -x| ≤ D(p i , σ) (x is defined in Equation 1), then Del d (P) is embedded in Ω. The condition | x -x| ≤ D(p i , σ
) is achieved for a sufficiently dense sampling according to Lemma 3 and the fact that the distortion ψ 0 = ψ(g, g E ) goes to 1 when the density increases. The complete proofs of Lemma 3 and Theorem 4 can be found in[4, Appendix F].

Discrete Riemannian structures

Although Riemannian Voronoi diagrams and Delaunay triangulations are appealing from a theoretical point of view, they are very difficult to compute in practice despite many studies [START_REF] Peyré | Geodesic methods in computer vision and graphics[END_REF]. To circumvent this difficulty, we introduce the discrete Riemannian Voronoi diagram. This discrete structure is easy to compute (see our companion paper [START_REF] Rouxel-Labbé | Discretized Riemannian Delaunay triangulations[END_REF] for details) and, as will be shown in the following sections, it is a good approximation of the exact S o C G 2 0 1 7

19:6

Riemannian Voronoi diagram. In particular, their dual Delaunay structures are identical under appropriate conditions.

We assume that we are given a dense triangulation of the domain Ω we call the canvas and denote by C. The canvas will be used to approximate geodesic distances between points of Ω and to construct the discrete Riemannian Voronoi diagram of P. This bears some resemblance to the graph-induced complex of Dey et al. [START_REF] Dey | Graph induced complex on point data[END_REF]. Notions related to the canvas will explicitly carry canvas in the name (for example, an edge of C is a canvas edge). In our analysis, we shall assume that the canvas is a dense triangulation, although weaker and more efficient structures can be used (see Section 9 and [23]).

The discrete Riemannian Voronoi Diagram

To define the discrete Riemannian Voronoi diagram of P, we need to give a unique color to each site of P and to color the vertices of the canvas accordingly. Specifically, each canvas vertex is colored with the color of its closest site.

Definition 5 (Discrete Riemannian Voronoi diagram). Given a metric field g, we associate to each site p i its discrete cell V d g (p i ) defined as the union of all canvas simplices with at least one vertex of the color of p i . We call the set of these cells the discrete Riemannian Voronoi diagram of P, and denote it by Vor d g (P).

Observe that contrary to typical Voronoi diagrams, our discrete Riemannian Voronoi diagram is not a partition of the canvas. Indeed, there is a one canvas simplex-thick overlapping since each canvas simplex σ C belongs to all the Voronoi cells whose sites' colors appear in the vertices of σ C . This is intentional and allows for a straightforward definition of the complex induced by this diagram, as shown below.

The discrete Riemannian Delaunay complex

We define the discrete Riemannian Delaunay complex as the set of simplices Del d g (P) = {σ | Vert(σ) ∈ P, ∩ p∈σ V d g (p) = 0}. Using a triangulation as canvas offers a very intuitive way to construct the discrete complex since each canvas k-simplex σ of C has k + 1 vertices {v 0 , . . . , v k } with respective colors {c 0 , . . . , c k } corresponding to the sites {p c0 , . . . , p c k } ∈ P. Due to the way discrete Voronoi cells overlap, a canvas simplex σ C belongs to each discrete Voronoi cell whose color appears in the vertices of σ. Therefore, the intersection of the discrete Voronoi cells {V d g (p i )} whose colors appear in the vertices of σ is non-empty and the simplex σ with vertices {p i } thus belongs to the discrete Riemannian Delaunay complex. In that case, we say that the canvas simplex σ C witnesses (or is a witness of) σ. For example, if the vertices of a canvas 3-simplex τ C have colors yellow-blue-blue-yellow, then the intersection of the discrete Voronoi cells of the sites p yellow and p blue is non-empty and the one-simplex σ with vertices p yellow and p blue belongs to the discrete Riemannian Delaunay complex. The canvas simplex τ C thus witnesses the (abstract, for now) edge between p yellow and p blue .

Figure 2 illustrates a canvas painted with discrete Voronoi cells, and the witnesses of the discrete Riemannian Delaunay complex.

Remark. If the intersection i=0...k V d g (p ci
) is non-empty, then the intersection of any subset of {V d g (p ci )} i=0...k is non-empty. In other words, if a canvas simplex σ C witnesses a simplex σ, then for each face τ of σ, there exists a face τ C of σ C that witnesses τ . As we assume that there is no boundary, the complex is pure and it is sufficient to only consider canvas n-simplices whose vertices have all different colors to build Del d g (P). Similarly to the definition of curved and straight Riemannian Delaunay complexes, we can define their discrete counterparts we respectively denote by Del d g (P) and Del d g (P). We will now exhibit conditions such that these complexes are well-defined and embedded in Ω.

Equivalence between the discrete and the exact structures

We first give conditions such that Vor d g (P) and Vor g (P) have the same combinatorial structure, or, equivalently, that the dual Delaunay complexes Del g (P) and Del d g (P) are identical. Under these conditions, the fact that Del d g (P) is embedded in Ω will immediately follow from the fact that the exact Riemannian Delaunay complex Del g (P) is embedded (see Sections 3.1 and 3.2). It thus remains to exhibit conditions under which Del d g (P) and Del g (P) are identical.

Requirements will be needed on both the set of sites in terms of density, sparsity and protection, and on the density of the canvas. The central idea in our analysis is that power protection of P will imply a lower bound on the distance separating two non-adjacent Voronoi objects (and in particular two Voronoi vertices). From this lower bound, we will obtain an upper bound on the size on the cells of the canvas so that the combinatorial structure of the discrete diagram is the same as that of the exact one. The density of the canvas is expressed by e C , the length of its longest edge.

The main result of this paper is the following theorem.

Theorem 6. Assume that P is a δ-power protected (ε, µ)-net in Ω with respect to g. Assume further that ε is sufficiently small and δ is sufficiently large compared to the distortion between g(p) and g in an ε-neighborhood of p. Let {λ i } be the eigenvalues of g(p) and 0 a value that depends on ε and δ (Precise bounds for ε, δ and l 0 are given in the proof). Then,

if e C < min p∈P min i √ λ i min {µ/3, 0 /2} , Del d g (P) = Del g (P).
The rest of the paper will be devoted to the proof of this theorem. Our analysis is divided into two parts. We first consider in Section 6 the most basic case of a domain of R n endowed with the Euclidean metric field. The result is given by Theorem 7. The assumptions are then relaxed and we consider the case of an arbitrary metric field over Ω in Section 7. As we shall see, the Euclidean case already contains most of the difficulties that arise during the proof and the extension to more complex settings will be deduced from the Euclidean case by bounding the distortion.

Equality of the Riemannian Delaunay complexes in the Euclidean setting

In this section, we restrict ourselves to the case where the metric field is the Euclidean metric g E . To simplify matters, we initially assume that geodesic distances are computed exactly on the canvas. The following theorem gives sufficient conditions to have equality of the complexes. Theorem 7. Assume that P is a δ-power protected (ε, µ)-net of Ω with respect to the Euclidean metric field g E . Denote by C the canvas, a triangulation with maximal edge length e C . If e C < min µ/16, δ 2 /64ε , then Del d E (P) = Del E (P). We shall now prove Theorem 7 by enforcing the two following conditions which, combined, give the equality between the discrete Riemannian Delaunay complex and the Riemannian Delaunay complex:

(1) for every Voronoi vertex in the Riemannian Voronoi diagram v = ∩ {pi} V g (p i ), there exists at least one canvas simplex with the corresponding colors {c pi }; (2) no canvas simplex witnesses a simplex that does not belong to the Riemannian Delaunay complex (equivalently, no canvas simplex has vertices whose colors are those of nonadjacent Riemannian Voronoi cells).

Condition ( 2) is a consequence of the separation of Voronoi objects, which in turn follows from power protection. The separation of Voronoi objects has previously been studied, for example by Boissonnat et al. [START_REF] Boissonnat | Delaunay triangulation of manifolds[END_REF]. Although the philosophy is the same, our setting is slightly more difficult and the results using power protection are new and use a more geometrical approach (see [4, Appendix C]).

Sperner's lemma

Rephrasing Condition (1), we seek requirements on the density of the canvas C and on the nature of the point set P such that there exists at least one canvas n-simplex of C that has exactly the colors c 0 , . . . , c d of the vertices p 0 , . . . , p d of a simplex σ, for all σ ∈ Del g (P). To prove the existence of such a canvas simplex, we employ Sperner's lemma [START_REF] Sperner | Fifty years of further development of a combinatorial lemma[END_REF], which is a discrete analog of Brouwer's fixed point theorem. We recall this result in Theorem 8 and illustrate it in a two-dimensional setting (inset).

Theorem 8 (Sperner's lemma).

Let σ = (p 0 , . . . , p n ) be an n-simplex and let T σ denote a triangulation of the simplex. Let each vertex v ∈ T σ be colored such that the following conditions are satisfied:

The vertices p i of σ all have different colors. If a vertex p lies on a k-face (p i0 , . . . p i k ) of σ, then p has the same color as one of the vertices of the face, that is p ij .

Then, there exists an odd number of simplices in T σ whose vertices are colored with all n + 1 colors. In particular, there must be at least one.

We shall apply Sperner's lemma to the canvas C and show that for every Voronoi vertex v in the Riemannian Voronoi diagram, we can find a subset C v of the canvas that fulfills the assumptions of Sperner's lemma, hence obtaining the existence of a canvas simplex in C v (and therefore in C) that witnesses σ v . Concretely, the subset C v is obtained in two steps:

-We first apply a barycentric subdivision of the Riemannian Voronoi cells incident to v.

From the resulting set of simplices, we extract a triangulation T v composed of the simplices incident to v (Section 6.2). -We then construct the subset C v by overlaying the border of T v and the canvas (Section 6.3).

We then show that if the canvas simplices are small enough -in terms of edge lengththen C v is the triangulation of a simplex that satisfies the assumptions of Sperner's lemma. The construction of C v is detailed in the following sections and illustrated in Figure 3: starting from a colored canvas (left), we subdivide the incident Voronoi cells of v to obtain T v (middle), and deduce the set of canvas simplices C v which forms a triangulation that satisfies the hypotheses of Sperner's lemma, thus giving the existence of a canvas simplex (in green, right) that witnesses the Voronoi vertex within the union of the simplices, and therefore in the canvas.

v v v T v C v Figure 3
Illustration of the construction of Cv. The Riemannian Voronoi diagram is drawn with thick orange edges and the sites are colored squares. The canvas is drawn with thin gray edges and colored circular vertices. The middle frame shows the subdivision of the incident Voronoi cells with think black edges and the triangulation Tv is drawn in yellow. On the right frame, the set of simplices Cv is colored in purple (simplices that do not belong to C) and in dark yellow (simplices that belong to C).

The triangulation T v

For a given Voronoi vertex v in the Euclidean Voronoi diagram Vor E (P) of the domain Ω, the initial triangulation T v is obtained by applying a combinatorial barycentric subdivision of the Voronoi cells of Vor E (P) that are incident to v: to each Voronoi cell V incident to v, we associate to each face F of V a point c F in F which is not necessarily the geometric barycenter. We randomly associate to c F the color of any of the sites whose Voronoi cells intersect to give F . For example, in a two-dimensional setting, if the face F is a Voronoi edge that is the intersection of V red and V blue , then c F is colored either red or blue. Then, the subdivision of V is computed by associating to all possible sequences of faces Denote by Σ V the set of simplices obtained by barycentric subdivision of V and

{F 0 , F 1 , . . . F n-1 , F n } such that F 0 ⊂ F 1 • • • ⊂ F n = V and dim(F i+1 ) = dim(F i ) +
Σ v = {∪Σ V | v ∈ V }.
The triangulation T v is defined as the star of v in Σ v , that is the set of simplices in Σ v that are incident to v. T v is illustrated in Figure 4 in dimension 3. As shall be proven in Lemma 9, T v can be used to define a combinatorial simplex that satisfies the assumptions of Sperner's lemma. By construction, the triangulation T v is a triangulation of the (Euclidean) Delaunay simplex σ v dual of v as follows. We first perform the standard barycentric subdivision on this Delaunay simplex σ v . We then map the barycenter of a k-face τ of σ v to the point c Fi on the Voronoi face F i , where F i is the Voronoi dual of the kface τ . This gives a piecewise linear homeomorphism from the Delaunay simplex σ v to the triangulation T v . We call the image of this map the simplex σ S and refer to the images of the faces of the Delaunay simplex as the faces of σ S . We can now apply Sperner's lemma.

Lemma 9.

Let P be a δ-power protected (ε, µ)-net. Let v be a Voronoi vertex in the Euclidean Voronoi diagram, Vor E (P), and let Σ v be defined as above. The simplex σ S and the triangulation T v satisfy the assumptions of Sperner's lemma in dimension n.

Proof. By the piecewise linear map that we have described above, T v is a triangulation of the simplex σ S . Because by construction the vertices c Fi lie on the Voronoi duals F i of the corresponding Delaunay face τ , c Fi has the one of the colors of of the Delaunay vertices of τ . Therefore, σ S satisfies the assumptions of Sperner's lemma and there exists an n-simplex in T v that witnesses v and its corresponding simplex σ v in Del g (P).

Building the triangulation C v

Let p i be the vertices of the k-face τ S of σ S . In this section we shall assume not only that τ S is contained in the union of the Voronoi cells of V (p i ), but in fact that τ S is a distance 8e C removed from the boundary of ∪V (p i ), where e C is the longest edge length of a simplex in the canvas. We will now construct a triangulation C v of σ S such that: σ S and its triangulation C v satisfy the conditions of Sperner's lemma, the simplices of C v that have no vertex that lies on the boundary ∂σ S are simplices of the canvas C.

The construction goes as follows. We first intersect the canvas C with σ S and consider the canvas simplices σ C,i such that the intersection of σ S and σ C,i is non-empty. These simplices σ C,i can be subdivided into two sets, namely those that lie entirely in the interior of σ S , which we denote by σ int C,i , and those that intersect the boundary, denoted by σ ∂ C,i . The simplices σ int C,i are added to the set C v . We intersect the simplices σ ∂ C,i with σ S and triangulate the intersection. Note that σ ∂ C,i ∩ σ S is a convex polyhedron and thus triangulating it is not a difficult task. The vertices of the simplices in the triangulation of σ ∂ C,i ∩ σ S are colored according to which Voronoi cell they belong to. Finally, the simplices in the triangulation of σ ∂ C,i ∩ σ S are added to the set C v . Since T v is a triangulation of σ S , the set C v is by construction also a triangulation of σ S . This triangulation trivially gives a triangulation of the faces τ S . Because we assume that τ S is contained in the union of its Voronoi cells, with a margin of 8e C we now can draw two important conclusions:

The vertices of the triangulation of each face τ S have the colors of the vertices p i of τ S . None of the simplices in the triangulation of σ ∂ C,i ∩ σ S can have n + 1 colors, because every such simplex must be close to one face τ S , which means that it must be contained in the union of the Voronoi cells V (p i ) of the vertices of τ S .

We can now invoke Sperner's lemma; C v is a triangulation of the simplex σ S whose every face has been colored with the appropriate colors (since σ S triangulated by T v satisfies the assumptions of Sperner's lemma, see Lemma 9). This means that there is a simplex C v that is colored with n + 1 colors. Because of our second observation above, the simplex with these n + 1 colors must lie in the interior of σ S and is thus a canvas simplex.

We summarize by the following lemma:

Lemma 10. If every face τ S of σ S with vertices p i is at distance 8e C from the boundary of the union of its Voronoi cells ∂(∪V (p i )), then there exists a canvas simplex in C v such that it is colored with the same vertices as the vertices of σ S .

The key task that we now face is to guarantee that faces τ S indeed lie well inside of the union of the appropriate Voronoi regions. This requires first and foremost power protection. Indeed, if a point set is power protected, the distance between a Voronoi vertex c and the Voronoi faces that are not incident to c, which we will refer to from now on as foreign Voronoi faces, can be bounded, as shown in the following Lemma: Lemma 11. Suppose that c is the circumcenter of a δ-power protected simplex σ of a Delaunay triangulation built from an ε-sample, then all foreign Voronoi faces are at least δ 2 /8ε far from c.

The proof of this Lemma is given in the full version of this paper (see [4, Section C.2]).

In almost all cases, this result gives us the distance bound we require: we can assume that vertices {c F0 , c F1 , . . . , c Fn-1 , c Fn } which we used to construct T v , are well placed, meaning that there is a minimum distance between these vertices and foreign Voronoi objects. However it can still occur that foreign Voronoi objects are close to a face τ S of σ S . This occurs even in two dimensions, where a Voronoi vertex v can be very close to a face τ S because of obtuse angles, as illustrated in Figure 5.

v v T v v v
Figure 5 The point v can be arbitrarily close to Tv, as shown by the red segments (left and center). After piecewise linear deformation, this issue is resolved, as seen by the green segments (right).

Thanks to power protection, we know that v is removed from foreign Voronoi objects. This means that we can deform σ S (in a piecewise linear manner) in a neighborhood of v such that the distance between v and all the faces of the deformed σ S is lower bounded.

S o C G 2 0 1 7 19:12

In general the deformation of σ S is performed by "radially pushing" simplices away from the foreign Voronoi faces of v with a ball of radius r = min µ/16, δ 2 /64ε . The value µ/16 is chosen so that we do not move any vertex of σ v (the dual of v): indeed, P is µ-separated and thus d E (p i , p j ) > µ. The value δ 2 /64ε is chosen so that σ S and its deformation stay isotopic (no "pinching" can happen), using Lemma 11. In fact it is advisable to use a piecewise linear version of "radial pushing", to ensure that the deformation of σ S is a polyhedron. This guarantees that we can triangulate the intersection, see Chapter 2 of Rourke and Sanderson [START_REF] Rourke | Introduction to piecewise-linear topology[END_REF]. After this deformation we can follow the steps we have given above to arrive at a well-colored simplex.

Lemma 12. Let P be a δ-power protected (ε, µ)-net. Let v be a Voronoi vertex of the Euclidean Voronoi diagram Vor E (P), and T v as defined above. If the length e C of the longest canvas edge is bounded as follows: e C < r = min µ/16, δ 2 /64ε , then there exists a canvas simplex that witnesses v and the corresponding simplex σ v in Del E (P).

Conclusion

So far, we have only proven that Del g (P) ⊆ Del d g (P). The other inclusion, which corresponds to Condition (2) mentioned above, is much simpler: as long as a canvas edge is shorter than the smallest distance between a Voronoi vertex and a foreign face of the Riemannian Voronoi diagram, then no canvas simplex can witness a simplex that is not in Del g (P). Such a bound is already given by Lemma 11 and thus, if e C < δ 2 /8ε then Del d g (P) ⊆ Del g (P). Observe that this requirement is weaker than the condition imposed in Lemma 12 and it was thus already satisfied. It follows that Del d g (P) = Del g (P) if e C < min µ/16, δ 2 /64ε , which concludes the proof of Theorem 7.

Remark. Assuming that the point set is a δ-power protected (ε, µ)-net might seem like a strong assumption. However, it should be observed that any non-degenerate point set can be seen as a δ-power protected (ε, µ)-net, for a sufficiently large value of ε and sufficiently small values of δ and µ. Our results are therefore always applicable but the necessary canvas density increases as the quality of the point set worsens (Lemma 12). In our practical companion paper [START_REF] Rouxel-Labbé | Discretized Riemannian Delaunay triangulations[END_REF]Section 7], we showed how to generate δ-power protected (ε, µ)-nets for given values of ε, µ and δ.

7

Extension to more complex settings

In the previous section, we have placed ourselves in the setting of an (open) domain endowed with the Euclidean metric field. To prove Theorem 6, we need to generalize Theorem 7 to more general metrics, which will be done in the two following subsections.

The common path to prove Del d g (P) = Del g (P) in all settings is to assume that P is a power protected net with respect to the metric field. We then use the stability of entities under small metric perturbations to take us back to the now solved case of the domain Ω endowed with an Euclidean metric field. Separation and stability of Delaunay and Voronoi objects has previously been studied by Boissonnat et al. [START_REF] Boissonnat | Delaunay triangulation of manifolds[END_REF][START_REF] Boissonnat | Only distances are required to reconstruct submanifolds[END_REF], but our work lives in a slightly more complicated setting. Moreover, our proofs are generally more geometrical and sometimes simpler. For completeness, the extensions of these results to our context are detailed in the full version of this paper [START_REF] Boissonnat | Anisotropic triangulations via discrete Riemannian Voronoi diagrams[END_REF]Appendices C and E].

We now detail the different intermediary settings. For completeness, the full proofs are included in the appendices.

Uniform metric field

We first consider the rather easy case of a non-Euclidean but uniform (constant) metric field over an (open) domain. The square root of a metric gives a linear transformation between the base space where distances are considered in the metric and a metric space where the Euclidean distance is used (see [START_REF] Boissonnat | Anisotropic triangulations via discrete Riemannian Voronoi diagrams[END_REF]Appendix B.1]). Additionally, we show that a δ-power protected (ε, µ)-net with respect to the uniform metric is, after transformation, still a δ-power protected (ε, µ)-net but with respect to the Euclidean setting [4, Lemma 26], bringing us back to the setting we have solved in Section 6. Bounds on the power protection, sampling and separation coefficients, and on the canvas edge length can then be obtained from the result for the Euclidean setting, using Theorem 12. These bounds can be transported back to the case of uniform metric fields by scaling these values according to the smallest eigenvalue of the metric [4, Theorem 40].

Arbitrary metric field

The case of an arbitrary metric field over Ω is handled by observing that an arbitrary metric field is locally well-approximated by a uniform metric field. It is then a matter of controlling the distortion.

We first show that, for any point p ∈ Ω, density separation and power protection are locally preserved in a neighborhood U p around p when the metric field g is approximated by the constant metric field g = g(p) [4, Lemmas 27 and 39]: if P is a δ-power protected (ε, µ)-net with respect to g, then P is a δ -power protected (ε , µ )-net with respect to g . Previous results can now be applied to obtain conditions on δ , ε , µ and on the (local) maximal length of the canvas such that Del d g (P) = Del g (P) (see [START_REF] Boissonnat | Anisotropic triangulations via discrete Riemannian Voronoi diagrams[END_REF]Lemma 41]). These local triangulations can then be stitched together to form a triangulation embedded in Ω. The (global) bound on the maximal canvas edge length is given by the minimum of the local bounds, each computed through the results of the previous sections. This ends the proof of Theorem 6.

Once the equality between the complexes is obtained, conditions giving the embeddability of the discrete Karcher Delaunay triangulation and the discrete straight Delaunay triangulation are given by previous results that we have established in Sections 3.1 and 3.2 respectively.

Extensions of the main result

Approximate geodesic computations Approximate geodesic distance computations can be incorporated in the analysis of the previous section by observing that computing inaccurately geodesic distances in a domain Ω endowed with a metric field g can be seen as computing exactly geodesic distances in Ω with respect to a metric field g that is close to g [4, Section H.3].

General manifolds

The previous section may also be generalized to an arbitrary smooth n-manifold M embedded in R m . We shall assume that, apart from the metric induced by the embedding of the domain in Euclidean space, there is a second metric g defined on M. Let π p : M → T p M be the orthogonal projection of points of M on the tangent space T p M at p. For a sufficiently small neighborhood U p ⊂ T p M, π p is a local diffeomorphism (see Niyogi [START_REF] Niyogi | Finding the homology of submanifolds with high confidence from random samples[END_REF]).

S o C G 2 0 1 7 19:14

Denote by P Tp the point set {π p (p i ), p i ∈ P} and P Up the restriction of P Tp to U p . Assuming that the conditions of Niyogi et al. [START_REF] Niyogi | Finding the homology of submanifolds with high confidence from random samples[END_REF] are satisfied (which are simple density constraints on ε compared to the reach of the manifold), the pullback of the metric with the inverse projection (π -1 p ) * g defines a metric g p on U p such that for all q, r ∈ U p , d gp (q, r) = d g (π -1 p (q), π -1 p (r)). This implies immediately that if P is a δ-power protected (ε, µ)-net on M with respect to g then P Up is a δ-power protected (ε, µ)-net on U p . We have thus a metric on a subset of a n-dimensional space, in this case the tangent space, giving us a setting that we have already solved. It is left to translate the sizing field requirement from the tangent plane to the manifold M itself. Note that the transformation π p is completely independent of g. Boissonnat et al. [START_REF] Boissonnat | Delaunay triangulation of manifolds[END_REF]Lemma 3.7] give bounds on the metric distortion of the projection on the tangent space. This result allows to carry the canvas sizing field requirement from the tangent space to M.

Implementation

The construction of the discrete Riemannian Voronoi diagram and of the discrete Riemannian Delaunay complex has been implemented for n = 2, 3 and for surfaces of R 3 . An in-depth description of our structure and its construction as well as an empirical study can be found in our practical paper [START_REF] Rouxel-Labbé | Discretized Riemannian Delaunay triangulations[END_REF]. We simply make a few observations here.

The theoretical bounds on the canvas edge length provided by Theorems 6 and 7 are far from tight and thankfully do not need to be honored in practice. A canvas whose edge length are about a tenth of the distance between two seeds suffices. This creates nevertheless unnecessarily dense canvasses since the density does not in fact need to be equal everywhere at all points and even in all directions. This issue is resolved by the use of anisotropic canvasses.

Our analysis was based on the assumption that all canvas vertices are painted with the color of the closest site. In our implementation, we color the canvas using a multiple-front vector Dijkstra algorithm [START_REF] Campen | Practical anisotropic geodesy[END_REF], which empirically does not suffer from the same convergence issues as the traditional Dijkstra algorithm, starting from all the sites. It should be noted that any geodesic distance computation method can be used, as long as it converges to the exact geodesic distance when the canvas becomes denser. The Riemannian Delaunay complex is built on the fly during the construction of the discrete Riemannian Voronoi diagram: when a canvas simplex is first fully colored, its combinatorial information is extracted and the corresponding simplex is added to Del g (P).

Figure 2 A

 2 Figure 2 A canvas (black edges) and a discrete Riemannian Voronoi diagram drawn on it. The canvas simplices colored in red are witnesses of Voronoi vertices. The canvas simplices colored in grey are witnesses of Voronoi edges. Canvas simplices whose vertices all have the same color are colored with that color.

  1 the simplex with vertices {c F0 , c F1 , . . . , c Fn-1 , c Fn }. These barycentric subdivisions are allowed since Voronoi cells are convex polytopes.

T v as a triangulation of an nFigure 4

 4 Figure 4The triangulation Tv in 3D. A face (in green) and an edge (in red) of σS .

Acknowledgments

We thank Ramsay Dyer for enlightening discussions. The first and third authors have received funding from the European Research Council under the European Union's ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions).


V Trejo

M Shakoor

M Bernacki

P O Bouchard

A study of the effect of microstructural heterogeneities on ductile damage

Accurate prediction of ductile fracture of metallic materials requires sufficient understanding of the underlying micromechanical phenomena, especially if complex multiaxial and non-proportional loadings are to be considered. In this micromechanical study, real microstructures are submitted to realistic boundary conditions up to large plastic strains thanks to the combination of Laminography, Digital Volume Correlation and Finite Element Modelling. The obtained results evidence the effect of the microstructural heterogeneities on the ductile fracture process.

Introduction

A methodology developed for the study of the micromechanisms of ductile damage (void nucleation, growth and coalescence) for complex and heterogeneous microstructures up to large plastic strains is employed in this study. In this methodology, in-situ Synchrotron Radiation Computed Laminography (SRCL) (Centre des Matériaux, MINES ParisTech), Digital Volume Correlation (DVC) (Laboratoire de Mécanique et Technologie -École Normale Supérieure -Paris-Saclay) and Finite Element (FE) modelling (Centre de Mise en Forme de Matériaux, MINES ParisTech) are seamlessly coupled. In-situ SRCL tests provide three-dimensional images of the microstructure, and kinematic information is then obtained via DVC. A FE framework with advanced meshing capabilities is subsequently used in order to carry out micromechanical simulations thanks to the topological and kinematic information obtained by SRCL and DVC, respectively with the objective of studying the effect of the microstructural heterogeneities on the ductile fracture process.

Results

The described framework (SRCL-DVC-FE) was applied to a nodular cast iron specimen. This specimen has two laser-drilled holes of 1mm of diameter. Nodular cast iron was chosen because it provides strong contrast between the ferritic matrix and the graphite inclusions when observed via SRCL. This material also presents low levels of cohesive forces in the particle-matrix interface, which makes it possible to neglect them in the simulations. Since graphite inclusions are very soft with respect to the matrix, their mechanical contribution is not considered in the simulations, i.e., the particles are considered as voids. The mechanical behavior of the matrix is described with von Mises plasticity and isotropic hardening.

A section of the specimen along with the obtained plastic strain field is shown in Figure 1. Very good correspondence in the position of the deformed microstructural features is observed between the SRCL image and the simulation. Different localization bands with different orientations are predicted.

In Figure 2, the individual normalized void size evolution is presented for 53 voids in the domain. The line width of each curve in the figure is proportional to the corresponding initial void size. The absence of a clear trend between the initial void size and the final void growth hints at the complexity of the void growth due to the intervention of several microstructural variables.

Conclusions

The micromechanisms of ductile damage were assessed via a SRCL-DVC-FE combined approach, with emphasis on the effect of the microstructural heterogeneities on the fracture process of a nodular cast iron specimen. Analyzing the effect of these heterogeneities can provide insight into the importance of usually simplified microstructural variables such as void shape and distribution, and help recognize limitations of void nucleation, growth and coalescence models, as well as behavior laws.

Figure 1 .Figure 2 .

 12 Figure 1.(a) Section of the deformed microstructure of the and strain field obtained from FE simulation and (b) same section at the onset of coalescence

Conclusions

The authors would like to gratefulle thank the French ANR for funding the COMINSIDE project.