Lionel Fourment
email: lionel.fourment@mines-paristech.fr

Antoine Potet
email: antoine.potet@mines-paristech.fr

Sabrina Gastebois

Katia Mocellin
email: katia.mocellin@mines-paristech.fr

Katia Mocellin Numerical

Numerical Modeling of Friction Stir (FSW) and Linear Friction (LFW) Welding Processes

Keywords: FSW, LFW, Numerical Simulation, Thermal Measurements, Calibration

Numerical simulation is regarded as a powerful tool to understand the physical mechanisms taking place during friction welding, provided that the numerical approach accurately models the complex interface phenomena and as long as the related model parameters are identified from experimental data acutely measured close to the interface. After describing the utilized finite element formulation, the friction and heat and friction interface models are calibrated from thermal measurements, both for the FSW welding of aluminum under a lap joint configuration and for the LFW of titanium alloys.

Numerical Simulation of Friction Welding processes

Numerical simulation provides essential information for understanding complex bonding mechanisms occurring in friction welding processes like FSW (Friction Stir Welding) and LFW (Linear Friction Welding). Obtaining sufficiently accurate 3D results requires developing specific numerical methods such as adaptive mesh refinement, treatment of unilateral contact conditions, and formulations dedicated to very large material deformations like the ALE (Arbitrary Lagrangian or Eulerian) [START_REF] Fourment | 3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration[END_REF] or Updated Lagrangian (including automatic 3D remeshing ) [START_REF] Potet | Numerical Simulation of Linear Friction Welding of Aeronautical alloys[END_REF] formulations. Such specific model allows predicting the actual tool print formed during FSW, which has significant influence on material local temperature [START_REF] Assidi | Accurate 3D friction stir welding simulation tool based on friction model calibration[END_REF].

A main difficulty is however encountered to model material and interface behaviors under widely ranging thermo-mechanical conditions. An inverse approach is consequently necessary to assess the selected JMatPro material model and to calibrate the still unknown friction parameters [START_REF] Assidi | Accurate 3D friction stir welding simulation tool based on friction model calibration[END_REF], based on instrumented experiments providing displacements, forces and especially temperature measurements. A special attention is paid to acutely monitor data as close as possible to the welding interface.

Application to FSW

This approach is first applied to FSW welding of aluminum under a lap joint configuration of 7175 on 2024 using a scrolled tool [START_REF] Fourment | Calibration of 3D ALE finite element model from experiments on friction stir welding of lap joints[END_REF] (see Figure 1(a)), which shows quite difficult to model. Tool temperature is consequently measured close to the surface (using an embarked recording system) under different welding configurations. Calibration of the physical coefficients of the numerical model, which is based on an ALE formulation with unilateral contact conditions [START_REF] Fourment | 3D numerical simulation of the three stages of Friction Stir Welding based on friction parameters calibration[END_REF][3], allows obtaining a rather adequate agreement with experimental measures. On the other hand, the sensitivity of worm hole formation to the advancing and rotating velocities is very satisfactorily simulated as presented in Figure 1(b).

Application to LFW

This approach is extended to the LFW of titanium alloys, a process presenting same numerical and modeling difficulties along with similar bonding mechanisms. The numerical simulation is based on an Updated Lagrangian formulation with frequent adaptive remeshing [START_REF] Potet | Numerical Simulation of Linear Friction Welding of Aeronautical alloys[END_REF]. It allows properly predicting the material consumption rate (downhill velocity of the upper tool under a constant forging force). As shown in Figure 2

Conclusions

Accurate numerical simulation of temperature evolution close to the friction welding interface shows delicate. It raises experimental and especially numerical difficulties, which are though essential to be tackled and solved, following the proposed approach based on adaptive meshing, unilateral contact and friction calibration. This approach aims at acutely understanding the metallurgical evolutions within the welded join.

  (a), temperatures are recorded at different locations of the Ti6242 sample, as close as possible to the welding interface. Although in good agreement, numerical results (see Figure2(b)) obtained before calibration of the friction coefficient do not properly follow the experimental curves. It shows the necessity of both enhancing the accuracy of the numerical model through mesh adaption based on error estimation, and of precisely identifying the friction coefficient, a very sensitive model parameter.

Figure 1 :

 1 Figure 1: (a) Conical tool with threaded pin and scrolled shoulder (b) Development of worn hole defect in the simulation that perfectly corresponds with macrographs from FSW trials [3].

Figure 2 :

 2 Figure 2: (a) Location of thermocouples on the sample (b) Time evolution of measured and computedthermocouples temperatures, using a symmetrized numerical model[START_REF] Potet | Numerical Simulation of Linear Friction Welding of Aeronautical alloys[END_REF]

Acknowledgements

Authors express their acknowledgments for the financial support provided by the aeronautical chair DAHER and by the OPTIMUM (ANR-14-CE27-0017) project, and to the OPTIMUM project partners who made the experimentations possible.


Laurence Cholvy

Opinion Diffusion and Influence: A logical Approach

This paper aims at modelling opinion diffusion among agents with a logical approach by assuming that opinions are propositional formulas. In a first part, we present a model in which an agent changes its opinion by merging the opinions of some influential agents, its influencers. More precisely, it merges these opinions, which may be contradictory, by taking into account an order of importance among its influencers. In a second part, we generalize this model so that influencers are ordered according to several orders of influence which depend on the topics of opinions.

Introduction

Understanding the dynamics of opinion among agents is an important question which has recently received considerable attention in the community of autonomous agents and multi-agent systems [START_REF] Crawford | Opposites repel: the effect of incorporating repulsion on opinion dynamics in the bounded confidence model[END_REF][START_REF] Chatterjee | Predicting migration and opinion adoption patterns in agent communities[END_REF][START_REF] Chau | Social judgment theory based model on opinion formation, polarization and evolution[END_REF][START_REF] Tsang | Opinion dynamics of skeptical agents[END_REF][START_REF] Christoff | A logic for diffusion in social networks[END_REF][START_REF] Grandi | Propositional opinion diffusion[END_REF]19,[START_REF] Schwind | Belief revision games[END_REF]. This question depends on several parameters.

The first important parameter is the population of agents. This population may be unstructured, in such a case, agents interact randomly [START_REF] Jager | A dynamical perspective on attitude change[END_REF][START_REF] Crawford | Opposites repel: the effect of incorporating repulsion on opinion dynamics in the bounded confidence model[END_REF]. But generally, some relations exist between agents. The population of agents may be divided into communities modelling neighborhoord relations between agents [START_REF] Feyza | Analysis of opinion spread through migration and adoption in agent communities[END_REF][START_REF] Chatterjee | Predicting migration and opinion adoption patterns in agent communities[END_REF][START_REF] Christoff | A logic for diffusion in social networks[END_REF]. The population may also be structured via an influence relation which relates two agents, the opinion of one of these agents being influenced by the opinion of the other [START_REF] Tsang | Opinion dynamics of skeptical agents[END_REF]. Graphs are widely used to model the structured population: nodes are agents and links are the relations between agents. Links are symmetrical or not, depending on the type of relations and they may also be labelled with probabilities [18].

The second parameter is the model of opinion. Here again, several options exist. Most of the works previoulsy cited consider only one opinion and model it as a real number in [0, 1]. For instance, if the question is to evaluate the opinion of people about the fact that Canada will host the Winter Olympics in 2026, then an opinion which is close to 1 means that the agent is quite confident in Canada candidature or that according to this agent, the probability that Canada will host the Olympics is high. An opinion which is close to 0 means that the agent thinks that Canada candidature will be rejected or that according to this agent, the probability that Canada will host the Olympics is low. Some other works are based on formal logic and model opinions as propositional formulas or, more precisely, as the sets of their models. In [START_REF] Grandi | Propositional opinion diffusion[END_REF], an opinion is a single interpretation, called a ballot. For instance, an agent whose opinion is CAN ∧ acroski, thinks that Canada will organize the Olympics in 2026 and that there will be acroski trials. Another agent whose opinion is ¬CAN ∧ acroski, thinks that 2026 Games will not be hosted by Canada but there will be acroski trials. More generally, [START_REF] Schwind | Belief revision games[END_REF] considers that an opinion is any propositional formula, thus modelled by a set of interpretations which is not necessarly a singleton. For instance, an agent opinion is (acroski ∨ skijoëring) ∧ CAN when it thinks that the 2026 Winter Olympics will be hosted by Canada and that there will be acroski trials or skijoëring trials.

The last parameter is the model of opinion dynamics. Many works in the field of opinion dynamics in multi-agent systems are based on a theory introduced in the field of Social Psychology called Social Judgment Theory (SJT). The basic idea of SJT is that individual opinion changing is a judgmental process: if an agent considers that a presented opinion is close to its current opinion, then it is likely to shift in the direction of this opinion (assimilation); if it considers that the presented opinion is distant to its current opinion, then it is likely to shift away from this opinion (contrast); otherwise, the agent does not change its opinion (non-commitment). This general idea has led to different formal models [START_REF] Jager | A dynamical perspective on attitude change[END_REF][START_REF] Crawford | Opposites repel: the effect of incorporating repulsion on opinion dynamics in the bounded confidence model[END_REF][START_REF] Chau | Social judgment theory based model on opinion formation, polarization and evolution[END_REF] in which the thresholds agents use to characterize what are close and distant opinions are identical or not, universal or agent dependent. Some other works, like [START_REF] Tsang | Opinion dynamics of skeptical agents[END_REF], are based on the theory of motivated cognition, defined in Cognitive Psychology, and which also says that agents are skeptical of another agent when their opinions diverge, but are more receptive to persuasion when their opinions better align. Some other works in the field of diffusion in multi agent systems claim to be based on models provided by the Network Science community. For instance, [START_REF] Christoff | A logic for diffusion in social networks[END_REF] is based on the SIR model which says that the value of an agent's feature evolves according to the values of its neighbors feature values. For instance an agent is infected if one of its neighbor is. Or an agent may say that it believes something if its neighbors said that they also do.

Simulating opinion diffusion may be applied in many contexts. In particular, simulation of opinion diffusion is important to prepare Pyschological Actions (PSYOPS) in the military context. These actions aim at changing the perception and the behavior of some people. They consist in elaborating and spreading out a message that must reach these individuals, directly or indirectly via their social networks [START_REF] Faucher | PSYOPS and CIMIC operations: from concepts to G-DEVS models[END_REF][START_REF] Faucher | Propagation of the effects of certain types of military psychological operations in a networked population[END_REF].

In the present paper, we extend a work recently presented in [START_REF] Cholvy | Diffusion of opinion and influence[END_REF] 1 . We model opinions by any kind of propositional formulas. As a consequence, opinions may be in disjunctive form and thus be incomplete. We also assume that the population of agents is structured by a binary relation of influence which relates two agents when one influences the other. In a first part, we assume that any agent orders its influencers (i.e. agents which influence it) according to the strength of the influence relation. Then, any agent changes its opinion by merging the opinions of its influencers from the most influential one to the least. In a second part, we consider that agents order their influencers according to the topics of opinions. For instance, you may be more influenced by your friend P aul than by your friend M ary about winter sport events while being more influenced by M ary than by P aul about litterature. In this case, any agent changes its opinion by merging the opinions of its influencers, topic by topic.

This paper is organized as follows. Section 2 presents the notion of Importance-Based Merging Operators and provides some original properties of these operators. Section 3 presents the notion of Influence-Based Opinion Diffusion Structures (IODS) to model opinion diffusion. Section 4 presents some properties of IODS. Section 5 extends IODS so that influencers may be ordered according to several orders of influence, depending on the topics of opinions. It also studies some of their properties. Finally, Section 6 presents some conclusions and discussions.

Importance-Based Merging Operators

In this section, we aim at defining a merging operator which takes into account the relative importance of the formulas to be merged for building the result. For doing this, we adopt the same kind of approach introduced in [14]: we adopt a semantical approach and we characterize the models of the result from the models of the initial formulas; we also consider a special formula, called integrity constraint, which expresses some law of nature and which restricts the possible models.

We consider a finite propositional language L i.e. a finite set of propositional letters. A literals is a propositional letter or the negation of a propositional letter. By convention, an interpretation of L is represented by a set of literals so that a propositional letter is positive iff it is satisfied in the interpretation, negative iff it is not satisfied in the interpretation. If ϕ is a formula of L, M od(ϕ) denotes the set of models of ϕ i.e., the set of interpretations in which ϕ is true. A multi-set of formulas {ϕ 1 , ..., ϕ n } equipped with a total order ≺ s.t. ϕ i ≺ ϕ i+1 (i = 1...n -1) is called an ordered multi-set of formulas and denoted ϕ 1 ≺ ... ≺ ϕ n .

Given a consistent formula µ and an ordered multi-set of formulas ϕ 1 ≺ ... ≺ ϕ n , an Importance-Based Merging Operator characterizes a formula ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) whose models are selected among the models of µ by taking into account the relative importance of the formulas ϕ i . More precisely, the operator selects the models of µ which first are the closest to the models of ϕ 1 , then the closest to the models of ϕ 2 etc. For doing this, we assume a pseudo-distance d between interpretations of L. Notice that ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) should be indexed by d but we write ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) to simplify the notation. The formal definition of ∆ is the following: Definition 1 (Importance-Based Merging Operator). An Importance-Based Merging Operator is a function ∆ which associates a formula µ and a non empty ordered multi-set of consistent formulas ϕ 1 ≺ ... ≺ ϕ n with a formula denoted

∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) so that: M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) = M in ≤ d,ϕ 1 ≺...≺ϕn M od(µ) with: • w ≤ d,ϕ1≺...≺ϕn w iff [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]
• [D(w, ϕ 1 ), ..., D(w, ϕ n )] is a vector whose k th element is D(w, ϕ k )

• D(w, ϕ) = min w ∈M od(ϕ) d(w, w ), w and w being two interpretations and d is a pseudo-distance2 between interpretations.

• ≤ lex is a lexicographic comparison of vectors of reals defined by:

[v 1 , ...v n ] ≤ lex [v 1 , ...v n ] iff (i) ∀k v k = v k or (ii) ∃k v k < v k and ∀j < k v j = v j
∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = M in ≤ d,a∨b≺¬a≺¬b∧c ({w 1 , ...w 8 }). With d = d H we get: M od(∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = {w 5 }, thus ∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = ¬a ∧ b ∧ c. With d = d D we get: M od(∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = {w 5 , w 6 }, thus ∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 ) = ¬a ∧ b. Suppose now that µ = a ∧ b i.e., M od(µ) = {w 1 , w 2 }. With d = d H we get: M od(∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = {w 1 }, thus ∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = a ∧ b ∧ c. With d = d D we get: M od(∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 )) = {w 1 , w 2 }, thus ∆ µ (ϕ 1 ≺ ϕ 2 ≺ ϕ 3 ) = a ∧ b.
Let's now examine some properties of Importance-Based Merging Operators. First we consider the nine postulates that merging operators should satisfy according to [START_REF] Konieczny | Merging information under constraints: A logical framework[END_REF]. We reformulate them within our context since here, formulas to be merged are ordered. Results are given in the following proposition 3 .

Proposition 1. • ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) |= µ • If µ is consistent then ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) is consistent • If n i=1 ϕ i ∧ µ is consistent then |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ↔ n i=1 ϕ i ∧ µ • It is not the case that if |= µ ↔ µ and if there is a permutation f over {1...n} such that ∀i = 1...n, |= ϕ i → ϕ f (i) then |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ↔ ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ). We only have: If |= µ ↔ µ and if ∀i = 1...n, |= ϕ i ↔ ϕ i then |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ↔ ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) • It is not necessarly the case that if ϕ 1 |= µ and ϕ 2 |= µ then ∆ µ (ϕ 1 ≺ ϕ 2 ) ∧ ϕ 1 is consistent if f ∆ µ (ϕ 1 ≺ ϕ 2 ) ∧ ϕ 2 is consistent • ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ∧ ∆ µ (ϕ n+1 ≺ ... ≺ ϕ m ) |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ m ) • If ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ∧ ∆ µ (ϕ n+1 ≺ ... ≺ ϕ m ) is consistent then ∆ µ (ϕ 1 ≺ ... ≺ ϕ m ) |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ∧ ∆ µ (ϕ n+1 ≺ ... ≺ ϕ m ) • ∆ µ1 (ϕ 1 ≺ ... ≺ ϕ n ) ∧ µ 2 |= ∆ µ1∧µ2 (ϕ 1 ≺ ... ≺ ϕ n ) • If ∆ µ1 (ϕ 1 ≺ ... ≺ ϕ n ) ∧ µ 2 is consistent then ∆ µ1∧µ2 (ϕ 1 ≺ ... ≺ ϕ n ) |= ∆ µ1 (ϕ 1 ≺ ... ≺ ϕ n )
Furthermore we can also prove the following propositions.

Proposition 2. If ϕ 1 |= µ then M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) = M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n ))
I.e., if the most important formula to be merged satisfies µ then replacing the integrity constraint µ by that formula does not change the result of merging 4 . As a consequence we have:

Proposition 3. If ϕ 1 |= µ then ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) |= ϕ 1 .
Proposition 4. Consider n formulas ϕ 1 , ..., ϕ n and ≺ an order on these formulas. Let k ∈ {1...n} and let ≺ k be the order obtained from ≺ by replacing ϕ k by ∆ µ (≺). Then,

M od(∆ µ (≺)) = M od(∆ µ (≺ k )).
I.e., the result of merging different formulas with ∆ µ does not change when one formula, whatever its importance, is replaced by the merged formula. This shows that ∆ µ satisfies the property called "ballot-monotonicity" in [START_REF] Grandi | Propositional opinion diffusion[END_REF] and which was there restricted to ballots.

Influence-Based Opinion Diffusion Structure (IODS)

In the following, we present the notions of Influence-Based Opinion Diffusion Structure and Influence-Based Opinion Sequence which are inspired by the notions of Belief Revision Games and Belief Sequences introduced in [START_REF] Schwind | Belief revision games[END_REF].

Definition 2 (Preliminary definition and notation).

• A totally-ordered set is a set S equipped with a total order . It is usually denoted (S, ).

• If S = {a 1 , ..., a n } is a finite set and is a total order on S so that ∀i = 1..(n -1) a i a i+1 , then by convention, we will write S = {a 1 ≺ ... ≺ a n } to compactly denote the totally ordered set (S, ). As usual, we will write e ∈ S iff e is one of the a i s.

Definition 3 (Influence-Based Opinion Diffusion Structure

). An Influence-Based Opinion Diffusion Structure (IODS) is a quadruplet S = (A, µ, B, Inf ) where:

• A = {1, ..., n} is a finite set of agents.

• µ is a consistent formula of L.

• B is a function which associates any agent i of A with a consistent formula of L denoted for short B i such that B i |= µ.

• Inf is a function which associates any agent i of A with a totally ordered set

Inf (i) = {i 1 ≺ i ... ≺ i i ni }, n i ≥ 1.
A is the finite set of agents. The formula µ represents the information which is true in the world. It is called integrity constraint. For any agent i, the formula B i represents its initial opinion. We assume that agents are rational and thus that B i is consistent and satisfies the integrity constraint µ. For any agent i, agents i 1 , ..., i ni are the influencers of i. With the total order i , i ranks them according to their degree of influence: for any agents j and k in Inf (i), j ≺ i k means that, according to i, it own opinion is (strictly) more influenced by j's opinion than by k's opinion. If

Inf (i) = {i 1 ≺ i ... ≺ i i ni }, i 1 is called the main influencer of i.
Notice that i may belong to Inf (i) i.e., we do not require that i's opinion is only influenced by other agents opinion: i may take it own opinion into account in the process of opinion changing. Consequently, i has to rank itself in Inf (i). For instance, i ≺ i j, for any j ∈ Inf (i) and j = i when i considers that, even if it is influenced by other influential agents, its own opinion will only be strenghened by their opinions. At the opposite, j ≺ i i, for any j ∈ Inf (i) and j = i when i is not confident in its own opinion and wants to take opinions of its influencers (but itself) first. As a particular case, Inf (i) = {i} is allowed and represents the fact that agent i is not influenced by some other agent but itself.

Notice also that i may not belong to Inf (i). This means that i is only influenced by other people and is not confident at all in its own opinion. Definition 4 (Influence-Based Opinion Sequence). Let S = (A, µ, B, Inf ) be an IODS and i ∈ A with

Inf (i) = {i 1 ≺ i ... ≺ i i ni }. The Influence-Based Opinion Sequence of i, denoted (B s i ) s∈N , is defined by: (i) B 0 i = B i (ii) ∀s > 0, B s i = ∆ µ (B s-1 i1 ≺ ... ≺ B s-1 in i )
The Influence-Based Opinion Sequence (or Opinion Sequence for short) of agent i, (B s i ) s∈N , represents the history of i's opinion evolution. This evolution is done according to the Importance-Based Merging Operator ∆ µ : i's opinion at step s is the result of ∆ µ applied to the ordered multi-set of opinions:

B s-1 i1 ≺ ... ≺ B s-1 in i .
As already written, the definition of IODS is inspired by the definition of Belief Revision Games (BRG), given in [START_REF] Schwind | Belief revision games[END_REF] according to which a BRG G is a 5-uple G = (A, Ac, L, B, R) 5 where:

• A is a finite set of agents, • Ac is a binary irreflexive relation on A. {j : (j, i) ∈ Ac} is the set of acquain- tances of i • L is a finite propositional language,
• B is a mapping from A to L.

• R = {R i : i ∈ A}. Each R i (called the revision policy of agent i) is a mapping from L × L in(i) to L with in(i) =| {j : (j, i) ∈ Ac} |, such that, 1. if ϕ 1 0 ≡ ϕ 2 0 and < ϕ 1 1 , ..., ϕ 1 in(i) >≡< ϕ 2 1 , ..., ϕ 2 in(i) > 6 then R i (ϕ 1 0 , ϕ 1 1 , ..., ϕ 1 in(i) ) ≡ R i (ϕ 2 0 , ϕ 2 1 , ..., ϕ 2 in(i) ) 2. if in(i) = 0 then R i is the identity function
At first glance, IODS could be seen as a particular case of BRG in which all the agents have the same revision policy: ∆ µ . But this is not true because ∆ µ is not a revision policy as defined previously. Indeed, the first constraint imposed on R i in BRG definition implies that acquaintances cannot be ordered: i's opinion will evolve the same, whatever a permutation of its acquaintance. This is obviously not satisfied by ∆ µ since, as proved by the fourth item of proposition 1, inputs of the Importance-Based Merging Operator ∆ µ cannot be permuted.

However, an interesting result proved in proposition 1 of [START_REF] Schwind | Belief revision games[END_REF] is still valid for IODS. This result shows that in a BRG, the opinion sequence of any agent is cyclic i.e., is characterized by an initial segment B 0 i ...B b-1 i and a cycle B b i ...B e i which will be repeated ad infinitum. The proof is based on the fact that since the language is finite, the number of possible opinions which are distinct up to logical equivalence is finite. Moreover, there is a finite number of agents and for each agent, the set of its acquaintances is fixed. So the number of possible n-uples BRG s = (B s 1 , ...., B s n ) (vectors of agent opinions at step s) is finite, up to equivalence. Consequently, the belief sequence of each agent s cyclic. The proof does not use the fact that R i is irrelevant to permutation of its arguments. So, we can re-use this proof for proving the following: Proposition 5. In an IODS, the opinion sequence of any agent is cyclic i.e., the opinion sequence of any agent i is characterized by an initial segment B 0 i ...B b-1 i and a cycle B b i ...B e i which will be repeated ad infinitum.

If the cycle of the opinion sequence of i is B b i ...B e i , then the size of the cycle is defined by: | Cyc(B i ) |= e -b + 1. 5 We change the initial notation in order to easily compare IODS and BRG 6

< ϕ 1 1 , ..., ϕ 1 in(i) >≡< ϕ 2 1 , ..., ϕ 2 in(i) > iff there is a permutation f over {1...in(i)} so that ∀j = 1..in(i) ϕ 1 j ≡ ϕ 2 f (j)
Example 2. Let S = (A, µ, B, Inf ) be an IODS with A = {1, 2, 3} (I.e., there are three agents named 1, 2 and 3), µ is a tautology, The following example shows a case where lengths of cycles are greater than 1. Finally, let us introduce some more definitions which will be useful for the next section. The following adapts the definition provided in [START_REF] Grandi | Propositional opinion diffusion[END_REF] and defines some type of IODS in which only some particular loops are permitted in the relation of influence.

B 1 = a ∨ b, B 2 = ¬a ∧ ¬b, B 3 = a ∨ ¬a, Inf (1) = {1}, Inf (2) = {2 ≺ 2 1}, Inf (3) = {1 ≺ 3 2}. I.e.,
s = 0 s ≥ 1 i = 1 a ∨ b a ∨ b i = 2 ¬a ∧ ¬b ¬a ∧ ¬b i = 3 a ∨ ¬a (a ∨ b) ∧ (¬a ∨ ¬b)
Definition 5 (DAG with self-loops). From S = (A, µ, B, Inf ), we can build a graph whose nodes are agents of A and edges are i → j iff i ∈ Inf (j). We say that S is a DAG with self-loops if this graph is a directed graph where the only permitted cycles are of type i → i.

The following introduces the notion of sphere of influence of an agent. Definition 6 (Sphere of Influence of an agent). Let S = (A, µ, B, Inf ) and i ∈ A. The sphere of influence of i is defined by: Sphere(i) = k≥1 Sphere k (i) with

Sphere

1 (i) = {j 1 : Inf (j 1 ) = {i ≺ ...}} Sphere k (i) = {j k : Inf (j k ) = {j k-1 ≺ ...} and j k-1 ∈ Sphere k-1 (i)}
The sphere of influence of an agent is thus the set of agents which are directly or indirectly mostly influenced by i. Let's now introduce some different types of agents.

Definition 7 (Dogmatic agent). Let S = (A, µ, B, Inf ) and i ∈ A. i is a dogmatic agent iff Inf (i) = {i}. Definition 8 (Self-confident agent). Let S = (A, µ, B, Inf ) and i ∈ A. i is self- confident iff i is its main influencer. Definition 9 (Diffident agent). Let S = (A, µ, B, Inf ) and i ∈ A. i is diffident iff i ∈ Inf (i) or Inf (i) = {i 1 ≺ i ... ≺ i i} with i 1 = i.
An agent is dogmatic when it is not influenced by other agents. As a consequence, a dogmatic agent i will never change its opinion i.e., ∀s ≥ 0 B s i = B 0 i . An agent is self-confident when it is its main influencer. Notice that dogmatic agents are selfconfident. Moreover, if i is self-confident then i ∈ Sphere(i). An agent is diffident when it is influenced only by others or it has at least two influencers and it is its least influential influencer. Finally, notice that if S is a DAG with self-loops then there is at least one dogmatic agent.

Definition 10 (Leader of a group, leader). Let S = (A, µ, B, Inf ) and i ∈ A.

• Let I ⊆ A. i is the leader of I iff i is dogmatic and I ⊆ Sphere(i).

• i is a leader iff there is a subset I of A st i is the leader of I.

In other words, for being a leader, an agent must be the leader of a group. And, for being the leader of a group of agents I, agent i must not be influenced by no other agent and any agent of I must be directly or indirectly mostly influenced by i.

Properties of IODS

In this section, we consider some properties, many of them being introduced in [START_REF] Schwind | Belief revision games[END_REF], and we check whether Importance-Based Opinion Structures satisfy them or not. Proposition 6. Let S = (A, µ, B, Inf ) be an IODS. Then, ∀i ∈ A ∀s ∈ N B s i is consistent.

This property corresponds to the property of BRG called Consistency Preservation (CP) defined in [START_REF] Schwind | Belief revision games[END_REF] as: ∀i ∈ A, if B i is consistent then ∀s ∈ N B s i is consistent. In the case of an IODS, the premisse is omitted because the initial opinions are consistent. This proposition shows that agents take integrity constraints into account to revise their opinion. Proposition 8. Let S = (A, µ, B, Inf ) be an IODS and ϕ be a consistent formula of

L. Then, if ∀i ∈ A, ϕ |= B i then ∀i ∈ A ∀s ∈ N, ϕ |= B s i .
This proves that IODS satisfy the property called Agreement Preservation (AP) defined in [START_REF] Schwind | Belief revision games[END_REF]: if all agents initially agree on some alternatives then they will not change their mind about them. Proposition 9. Let S = (A, µ, B, Inf ) be an IODS and let ϕ be a formula of L. Then,

if ∀i ∈ A, B i |= ϕ then ∀i ∈ A, ∀s ∈ N, B s i |= ϕ
This proposition proves that any Importance-Based Opinion Structures satisfy the property called Unanimity Preservation (UP) defined in [START_REF] Schwind | Belief revision games[END_REF]: every formula which is a logical consequence of the initial opinions remains so after opinion diffusion. I.e, any opinion initially shared by the agents remains so after opinion diffusion.

The property of Responsiveness (Resp) introduced in [START_REF] Schwind | Belief revision games[END_REF] states that an agent should take into account the opinions of the agents who influence it whenever (i) its opinions are inconsistent with the opinions of its acquaintances (but itself) and (ii) its acquaintances (but itself) agree on some alternatives. It is adapted here as follows:

Definition 11 (Resp). S = (A, µ, B, Inf ) satisfies (Resp) iff ∀i ∈ A ∀s ∈ N, if ∀j ∈ Inf (i) \ {i} B s j ∧ B s i is inconsistent and if ∧ j∈Inf (i)\{i} B s j is consistent, then B s+1 i |= B s i .
(Resp) is not necessarily satisfied here. As a counterexample, consider an IODS in which A = {1, 2, 3},

Inf (1) = {1}, Inf (2) = {2}, Inf (3) = {3 ≺ 1 ≺ 2}, B 1 = a ∧ b, B 2 = a, B 3 =
¬a and µ is a tautology. Notice that agents 1 and 2 influence agent 3 and that 1 and 2 agree on a. However, if the underlying distance is Hamming, we have ∀s ≥ 1,

B s 3 = ¬a ∧ b. Consequently, B s 3 |= B 3 .
Let us now focus on the property of Convergence (Conv) introduced in [START_REF] Schwind | Belief revision games[END_REF] and in [START_REF] Grandi | Propositional opinion diffusion[END_REF] as well, which states that there is a step in the evolution process when all opinions stop evolving.

Definition 12 (Conv). An IODS S = (A, µ, B, Inf ) satisfies (Conv) iff ∀i ∈ A | Cyc(B i ) |= 1.
Some IODS satisfy (Conv) and some don't. In example 2, S converges: the stable opinions of 1, 2 and 3 are respectively a ∨ b, ¬a ∧ ¬b and (a ∨ b) ∧ (¬a ∨ ¬b). In example 3, S does not converge.

In the case when opinions are single logical interpretations (i.e., ballots), [START_REF] Grandi | Propositional opinion diffusion[END_REF] proves that, if the aggregation operator satisfies the property of ballot-monotonicity, then a DAG with self-loops converges. In the present work, assumptions are different since opinions are sets of logical interpretations (i.e., sets of ballots). Fortunately, proposition 4 proves that ∆ µ satisfies ballot-monotonicity and the proof given in [START_REF] Grandi | Propositional opinion diffusion[END_REF] does not use the fact that opinions are supposed to be ballots. Thus we can re-use the proof of [START_REF] Grandi | Propositional opinion diffusion[END_REF] and claim: Proposition 10. IODS which are DAG with self-loops satisfy (Conv).

Let us finally consider the property of Acceptability given in [START_REF] Schwind | Belief revision games[END_REF]. We extend it for IODS as follows:. Definition 13 (Acceptability). Let S = (A, µ, B, Inf ) be an IODS and ϕ a formula of L. ϕ is accepted by agent i of A iff for all B s i ∈ Cyc(B i ), we have B s i |= ϕ. ϕ is unanimously accepted in S iff ϕ is accepted by all i in A. ϕ is majoritary accepted if the number of agents who accept it is strictly greater than the number of agents who do not.

In example 2, for instance, a ∨ b is majoritary accepted, as well as ¬a ∨ ¬b. The following proposition identifies cases in which an IODS satisfies Acceptability.

Proposition 11. Let S be an IODS and i the leader of a set of agents I. Then B i is accepted by any agent in I.

This means that the opinion of the leader of a group is accepted by any agent in this group.

Proposition 12. Let S be an IODS and i a dogmatic agent. Then:

(i) If Sphere(i) = A then B i is unanimously accepted.

(ii) If | Sphere(i) |> |A| 2 then B i is majoritary accepted. Obviously, if a dogmatic agent is the main infuencer of all agents in the population, then its opinion is unanimously accepted. If a dogmatic agent is the main influencer of more than a half population, then its opinion is majoritary accepted. Proposition 13. Let S be an IODS and i a self-confident agent. Then ∀s

B s i |= B s-1 i
This shows that the opinion of a self-confident agent becomes stronger and stronger. This is illustrated below.

Example 4. Assume µ is a taulology. Consider three agents 1, 2, 3 so that:

Inf (1) = {1}, Inf (2) = {1 ≺ 2}, Inf (3) = {3 ≺ 2 ≺ 1} and B 1 = a, B 2 = b, B 3 = a ∨ ¬a.
Agent 3 is a self confident who initially has no opinion. At step 1, its opinion becomes b, which is stronger. Finally, after step 2, its opinion is a ∧ b, which is even stronger. Proposition 14. Let S be an IODS. Let i be a dogmatic agent st | M od(B i ) |= 1. Then ∀j ∈ Sphere(i) ∃k ∀s > k | M od(B s j ) |= 1 We can consider that an opinion with only one model is a strong opinion since, the agent which has this opinion has a firm position (yes or no) towards each propositional letter. This proposition shows that any agent in the sphere of influence of a dogmatic agent whose opinion is strong converges to an opinion which is strong.

Taking topics into account

In this section, we extend IODS in order to take topics of opinions into account. Our motivation is to model the fact that an agent may influence another agent for opinions which belong to a given topic only and not for every opinion. As a consequence, topics will parametrize the ranking of influencers, i.e., for each agent, there will be as many rankings of influencers as topics.

Topics

Definition 14 (Topics). Topics T 1 , ..., T m of L are some sets of propositional literals of L so that: (i) any literal of L belongs to a topic; (ii) for any proposition letter p of L, for any topic T i in {T 1 , ...T m }, we have: p ∈ T i ⇐⇒ ¬p ∈ T i .

Example 5. Suppose that the letters of the language are a, b, c. We can define: T 1 = {a, ¬a} and T 2 = {b, ¬b, c, ¬c}. In this case these two topics are disjoint. We can also define: T 1 = {a, ¬a, b, ¬b}, T 2 = {b, ¬b, c, ¬c}, T 3 = {b, ¬b}. There,

T 3 = T 1 ∩ T 2 .
As written, each agent will rank its influencers according to several rankings, each of them depending on a topic. So the question of compatibility between the topics and the rankings they parametrize must be addressed: in particular, given a set of influencers, is there a relation between their ranking parametrized by the topic T and their ranking parametrized by a topic T ⊆ T ? More generally, is there a relation between their ranking parametrized by the topic T and their ranking parametrized by a topic T when T ∩ T = ∅ ? We answer this question by considering that if an agent i is strictly more influent than another agent j for opinions related to the topic T , then j cannot be strictly more influent than i for opinions related to topic T when T ∩T = ∅. Indeed, if it was not the case, then, for any opinion belonging to T ∩ T , i would be considered as strictly more influent than j because T ∩ T ⊆ T and j would be considered as strictly more influent than i because T ∩ T ⊆ T . Which is not possible. For instance, consider the two topics skiing and ballets (i.e choregraphic disciplines). They are not disjoint because acroski is a kind of skiing with a choregraphy. Consider now two agents P aul and M ary providing information related to these topics. Assume that P aul influences you strictly more than M ary regarding skiing and that M ary influences you strictly more than P aul regarding ballets. On a first hand, you can conclude that P aul influences you strictly more than M ary regarding acroski since acroski is a kind of skiing. On the other hand, you can conclude that M ary influences you strictly more than P aul regarding acroski since since acroski is a kind of ballets. But concluding both is impossible.

The following definition characterize rankins which are compatible with topics.

Definition 15 (Topic compatible orders). Consider m topics T 1 ...T m . Let ≺ T1 .... ≺ Tm be m total orders on a given set of formulas. We say that ≺ T1 .... ≺ Tm are topiccompatible iff for any i and j, if T i ∩ T j = ∅ then ≺ Ti and ≺ Tj are identical.

Notice that we could have followed a different solution and adopted a default reasoning by stating that: if an agent i is strictly more influent than another agent j for opinions related to the topic T , and if T ∩ T = ∅ then, typically, j cannot be strictly more influent than i for opinions related to topic T . This rule could be modeled in Default Logic [START_REF] Reiter | A logic for default reasoning[END_REF]. In the example, this would lead to two extensions: one in which Paul is considered as strictly more influent than Mary for opinions related to acroski; the other in which Mary is considered as strictly more influent than Paul for opinions related to acroski. This is not the solution we followed in the present paper.

Topic-dependent Importance-Based Merging Operators

We now extend the notion of Importance-Based Merging Operators for taking topics into account. Before, we introduce the following definition: Definition 16. Let ϕ a formula and T i a topic.

i ϕ is the formula defined by: M od( i ϕ) = {w ∩ T i : w ∈ M od(ϕ)} Definition 17 ( Topic-dependent Importance-Based Merging Operator). A Topicdependent Importance-Based Merging Operator, is a function Θ which, given a formula µ, given a multi-set of consistent formulas of L and m topic-compatible orders on this multi-set denoted ≺ T1 ... ≺ Tm , produces a formula denoted Θ µ (≺ T1 ... ≺ Tm ) so that:

M od(Θ µ (≺ T1 ... ≺ Tm )) = i=1...m M od(∆ i µ (≺ Ti ))
with

• ∆ an importance-based merging operator as defined in 1.

• i=1...m M i = {w : w ∈ M od(µ) and ∀i w ∩ T i ∈ M i } if not empty; i=1...m M i = M od(µ) else.
First, Θ merges formulas topic by topic and computes M od(∆ i µ (≺ Ti )). Then it agregates the results with operator . This agregation operator aims at selecting models of µ whose projections on all topics are results of independent mergings (i.e., belong to M od(∆ i µ (≺ Ti ))). But it may happen that no model of µ satisfies this condition, even if each initial formula satisfies µ as shown in example 6. In such a case, the operator selects all the models of µ i.e., the merging is vacuous and the resulting formula is nothing else than µ. Notice that, when µ is consistent, M od(∆ µ (≺ T1 ... ≺ Tm )) is not empty.

Example 6. Consider a language with two letters a, b and two topics: Notice that definition 1 is a special case of definition 17 i.e., if there is only one topic and only one order on a multi-set of formulas, then Θ µ (≺) as defined in definition 17 is identical to ∆ µ (≺) as defined in definition 1.

T 1 = {a, ¬a}, T 2 = {b, ¬b}. Consider ϕ 1 = a ∧ b, ϕ 2 = ¬a ∧ ¬b , µ = a → b. Notice that ϕ 1 |= µ and ϕ 2 |= µ. Consider two orders, {ϕ 1 ≺ T1 ϕ 2 , ϕ 2 ≺ T2 ϕ 1 }. If
Again, we consider the different postulates that merging operators should satisfy according to [START_REF] Konieczny | Merging information under constraints: A logical framework[END_REF]. Here, we focus on the only ones which will be necessary for proving properties about TIODS. We reformulate them within the context of several orders. Proposition 15.

• Θ µ (≺ T1 ... ≺ Tm ) |= µ • If µ is consistent then Θ µ (≺ T1 ... ≺ Tm ) is consistent • If n i=1 ϕ i ∧ µ is consistent then |= Θ µ (≺ T1 ... ≺ Tm ) ↔ n i=1 ϕ i ∧ µ
M od(Θ µ (≺ T1 ... ≺ Tm )) = M od(Θ µ (≺ k T1 ≺ ... ≺ k Tm ))
This shows that the result of merging different formulas ordered by several topicdependant orders does not change when one formula, whatever its importance in any order, is replaced by the merged formula. This is an extension of proposition 4.

Topic-dependent Influence-Based Opinion Diffusion Structure (TIODS)

We now extend the notion of Influence-Based Opinion Diffusion Structure to take topics into account.

Definition 18 (Topic-dependent Influence-Based Opinion Diffusion Structure). A Topic-dependent Influence-Based Opinion Diffusion Structure (TIODS) is a quadruplet S = (A, µ, B, Inf ) where:

• A = {1, ..., n} is a finite set of agents.

• µ is a consistent formula of L.

• B is a function which associates any agent i of A with a consistent formula of L denoted B i such that B i |= µ.

• For any agent i of A, Inf (i) = {≺ T1 , ..., ≺ Tm } where ≺ T1 ,... ≺ Tm are m total topic-compatible orders on a single set of agents {i i1 ...i ni }.

A, µ, B i are defined as before. Inf (i) is here defined by m orders on the set of i's influencers. Each order corresponds to a topic. Given a topic T k and ≺ T k ∈ Inf (i), i 1 ≺ T k i 2 means that i's opinion is more influenced by i 1 's opinion than by i 2 's opinion regarding the topic T k . Obviously, this definition extends definition 3.

Definition 19 (Topic-dependent Influence-Based Opinion Sequence). Consider a TIODS S = (A, µ, B, Inf ). Let i ∈ A with Inf (i) = {≺ T1 , ... ≺ Tm } m orders on the set {i i1 ...i ni }. The Topic-dependent Influence-Based Opinion Sequence of i, denoted (B s i ) s∈N , is defined as follows:

• B 0 i = B i • ∀s ∈ N, B s+1 i = Θ B s i 1 ∨...∨B s in i (≺ T1 ... ≺ Tm ) with B s i1 ≺ T k B s i2 iff i 1 ≺ T k i 2 for any k ∈ {1...m}.
As before, the Opinion Sequence of agent i, (B s i ) s∈N , represents the history of i's opinion evolution. Here, i's opinion evolves according to the Topic-Dependant Importance-Based Merging Operator Θ. More precisely, i's opinion at step s is the result of Θ B s Proposition 17. When there is only one topic and only one order on a multi-set of formulas, then the Topic-dependent Influence-Based Opinion Diffusion Sequence as defined in definition 19 is identical to the Influence-Based Opinion Diffusion Sequence as defined in definition 4.

Proposition 18. In a TIODS, the opinion sequence of any agent is cyclic i.e., the opinion sequence of any agent i is characterized by an initial segment B 0 i ...B b-1 i and a cycle B b i ...B e i which will be repeated ad infinitum.

Example 7. Consider a language whose topics are T 1 = {a, ¬a} and T 2 = {b, ¬b} and a TIODS S = (A, µ, B, Inf ) with:

A = {1, 2}, µ is a tautology, Inf (1) = {1}, Inf (2) = {1 ≺ T1 2, 2 ≺ T2 1}, B 1 = a, B 2 = ¬a ∧ b.
In other words, we consider two agents 1 and 2. 1 is its own and only influencer. Regarding topic T 1 , 2 is influenced by 1 and by itself, but more by 1 than by itself; regarding topic T 2 , 2 is influenced by 1 and by itself, but more by itself than by 1. Initially, 1's opinion is a and 2's opinion is ¬a ∧ b. Table 3 shows the evolution of opinions (for distance d H and d D as well).

s = 0 s ≥ 1 i = 1 a a i = 2 ¬a ∧ b a ∧ b

Some properties of TIODS

We present here some results about TIODS. First we prove that Consistency preservation, Agreement preservation and Unanimity preservation are still satisfied.

Proposition 19. Let S = (A, µ, B, Inf ) be a TIODS. Then ∀i ∈ A ∀s ∈ N B s i |= µ.
Proposition 20. Let S = (A, µ, B, Inf ) be a TIODS. Then ∀i ∈ A ∀s ∈ N B s i is consistent.

Proposition 21. Let S = (A, µ, B, Inf ) be a TIODS and ϕ a consistent formula of L.

If ∀i ∈ A, ϕ |= B i then ∀i ∈ A ∀s ∈ N, ϕ |= B s i .
Proposition 22. Let S = (A, µ, B, Inf ) be a TIODS and ϕ a consistent formula of L.

If ∀i ∈ A, B i |= ϕ then ∀i ∈ A, ∀s ∈ N, B s i |= ϕ.
Responsiveness is obviously not satisfied (see counter example given in section 4). Moreover, the definition of DAG with self-loops can obviously be extended to TIODS, as well as the definition of convergence. Then we can prove the following: Proposition 23. TIODS which are DAG with self-loops converge.

Conclusion and perspectives

The present paper adressed opinion diffusion in the case when opinions are modeled by propositional formulas. In a first part, we have presented a formal model which assumes that each agent changes its opinion by merging the opinions of its influencers, from the most influential to the least one. In a second part, we have extended this model in order to take the topics of opinions into account. This work is original since such models had never been studied before and properties that have been proved are rather encouraging.

However,this work is based on several assumptions which can be discussed and changing them opens up new perspectives. Let's present some of them.

The first main assumption on IODS is that each agent has a fixed set of influencers and the way it orders them is fixed also. This means that all their life long, agents are influenced by the same people, in the same way. This is of course very restrictive and a model in which, for each agent, the ordered set Inf (i) may change would be much more general than the one presented here. For sure, convergence will not be guaranteed. Notice that Inf (i) could change because the preorder ≺ changes or because the agents in Inf (i) change (for instance, i choses its influencers mong the ones whose opinions are close to its own).

The second assumption made here is that agents order their influencers according to a strict order (or according to several strict orders if many topics). Thus, two agents cannot be considered as equally influential for a given agent. Changing this assumption will lead us to consider preorders instead of orders. Defining a merging operator which takes into account preorders is an open question.

Moreover, in TIODS, an agent has a single set of influencers,which are ordered differently depending on the topics. Again, this is restrictive and it will be interesting to consider that the sets of influencers are themselves topic-dependent.

A generalization of this work will be to consider that there is no global integrity constraint as it is the case here, but only some local ones, shared by the agents who, in some way, belong the same community. Studying the acceptability property in such a context is challenging.

Another perspective we plan to work on in the near future is to inject in our model the Bounded-Confidence assumption. According to this assumption, agents are only influenced by people whose opinions are not absolutely different from their owns. In the present model, there is no such constraint on influencers but we plan to take it into account and to examine what become the properties of IODS then.

We would also like to extend our model in order to take the communication durations into account. Indeed, the present models assume that agent opinions change regularly at the same time. This comes to assume that communication between agents have the same duration. But this is a too strong assumption because in real applications, some interactions may be very short (electronic communications, emails, telephone..) and some others may be longer (mails, physical meeting ...). Taking this into account will lead ud to redefine the notion of Opinion Sequence and to wheck if the properties are still satisfied.

Finally, we would like to revisit our work in order to represent uncertain agents i.e., agents whose opinions are weighted propositional formulas, weights intending to represent the degrees of uncertainty of agents. Following [START_REF] Dabarera | Dynamics of belief theoretic agent opinions under bounded confidence[END_REF][START_REF] Cho | Dynamics of uncertain opinions in social networks[END_REF][START_REF] Crosscombe | A model of multi-agent consensus for vague and uncertain beliefs[END_REF] we think that the Belief Functions theory is a good candidate for doing so. But redefining IODS in such formal tool and studying their properties is an open question.

[18] Zhefeng Wang, Enhong Chen, Qi Liu, Yu Yang, Yong Ge, and Biao Chang.

Maximizing the coverage of information propagation in social networks Proof 1.

• The first item is obviously satisfied.

• The second item is satisfied. Indeed, if µ is consistent then

M od(µ) = ∅, thus M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) = ∅, i.e, ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) is consistent.
• The third item is satisfied. Indeed, assume that i=1..n ϕ i ∧ µ is consistent.

-

)] is minimal in M od(µ). Thus w |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ). • If |= µ ↔ µ and ∀i = 1...n, |= ϕ i ↔ ϕ i then M od(µ) = M od(µ ). Thus M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) = M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) i.e., |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ↔ ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) • Consider µ a tautology, ϕ 1 = a ∧ b and ϕ 2 = a ∧ ¬b. We have ϕ 1 |= µ, ϕ 2 |= µ and ∆ µ (ϕ 1 ≺ ϕ 2 ) = a ∧ b. ∆ µ (ϕ 1 ≺ ϕ 2 ) ∧ ϕ 1 is consistent but ∆(ϕ 1 ≺ ϕ 2 ) ∧ ϕ 2 is inconsistent.
• Consider w and w two models of µ. We can show that: If

.. ≺ ϕ n ) and w |= ∆(ϕ n+1 ≺ ... ≺ ϕ m ), i.e, ∆(ϕ 1 ≺ ... ≺ ϕ m ) |= ∆(ϕ 1 ≺ ... ≺ ϕ n ) ∧ ∆(ϕ n+1 ≺ ... ≺ ϕ m ) • If w |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ) ∧ µ then w |= µ and w |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ). Thus w ∈ M od(µ ∧ µ ) and ∀w ∈ M od(µ) [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. Let w ∈ M od(µ∧w ). Thus w ∈ M od(µ) thus [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. Thus w |= ∆ µ∧µ (ϕ 1 ≺ ... ≺ ϕ n ). • Assume ∆ µ1 (ϕ 1 ≺ ... ≺ ϕ n ) ∧ µ 2 is consistent. Then ∃w 0 ∈ M od(µ 1 ∧ µ 2 ) st [D(w 0 , ϕ 1 ), ..., D(w 0 , ϕ n )] is minimal in M od(µ 1 ). I.e., ∀w ∈ M od(µ 1 ) [D(w 0 , ϕ 1 ), ..., D(w 0 , ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. Consider w |= ∆ µ1∧µ2 (ϕ 1 ≺ ... ≺ ϕ n ). Then w ∈ M od(µ 1 ) and w ∈ M od(µ 2 ) and [D(w, ϕ 1 ), ..., D(w, ϕ n )] is minimal in M od(µ 1 ∧µ 2 ). I.e., ∀w ∈ M od(µ 1 ∧ µ 2 ) [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. In particular [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w 0 , ϕ 1 ), ..., D(w 0 , ϕ n )]. Thus ∀w ∈ M od(µ 1 ), [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. Conse- quently, [D(w, ϕ 1 ), ..., D(w, ϕ n )] is minimal in M od(µ 1 ). I.e., w |= ∆ µ1 (ϕ 1 ≺ ... ≺ ϕ n ). Proposition 2. If ϕ 1 |= µ then M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) = M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )) Proof 2. • Assume M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) ⊆ M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n ). Then ∃w O ∈ M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) st w 0 ∈ M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )).
On the first hand,

w O ∈ M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )) implies ∀w ∈ M od(µ), [D(w 0 , ϕ 1 ), ..., [D(w 0 , ϕ n )] ≤ [D(w, ϕ 1 ), ..., [D(w, ϕ n )]
On the other hand,

w 0 ∈ M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )) implies ∃w 1 ∈ M od(ϕ 1 ) st [D(w 1 , ϕ 1 ), ..., [D(w 1 , ϕ n )] < [D(w 0 , ϕ 1 ), ..., [D(w 0 , ϕ n )].
Since, ϕ 1 |= µ, we have M od(ϕ 1 ) ⊆ mod(µ). Thus the two previous assertions are contradictory. Proof 10. In the case when opinions are single logical interpretations (i.e., ballots), [START_REF] Grandi | Propositional opinion diffusion[END_REF] proves that, if the aggregation operator satisfies the property of "ballot-monotonicity" then a DAG with self-loops converges. Here, assumptions are different since opinions are sets of logical interpretations (i.e., sets of ballots). But fortunately, proposition 4 proves that ∆ µ satisfies ballot-monotonicity. Moreover, the proof given in [START_REF] Grandi | Propositional opinion diffusion[END_REF] does not use the fact that opinions are supposed to be ballots. Thus we can re-use the proof of [START_REF] Grandi | Propositional opinion diffusion[END_REF] in our case.

• Assume M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )) ⊆ M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )). Then ∃w 1 ∈ M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )) st w 1 ∈ M od(∆ µ (ϕ 1 ≺ ... ≺ ϕ n )). I.e, ∃w 2 ∈ M od(µ) st [D(w 2 , ϕ 1 ), ..., D(w 2 , ϕ n )] < [D(w 1 , ϕ 1 ), ..., D(w 2 , ϕ n )]. But, w 1 ∈ M od(∆ ϕ1 (ϕ 1 ≺ ... ≺ ϕ n )) implies D(w 1 , ϕ 1 ) = 0 (because
Proposition 11. Let S be an IODS and i the leader of a set of agents I. Then B i is accepted by any agent in I.

Proof 11. We first show that for i being dogmatic, ∀j ∈ Sphere(i), ∃s j ∀k ≥ s j B k j |= B i . I.e, B i is accepted by any agent in Sphere(i).

First, notice that since i is dogmatic, ∀k ≥ 0 B k i = B i . Now, consider j ∈ Sphere(i).

• Suppose that Inf (j) = {i ≺ ...}. Then ∀k ≥ 1, B k j = ∆ µ (B i ≺ ...). Thus, because B i |= µ, we have B k j |= B i , i.e, B i is accepted by j. • Suppose that there exists j 1 ...j k-1 so that Inf (j 1 ) = {i ≺ ...}, Inf (j 2 ) = {j 1 ≺ ...}, Inf (j k-1 ) = {j k-2 ≺ ...}, Inf (j) = {j k-1 ≺ ...}. Thus ∀r ≥ 1 B r j1 = B i , ∀r ≥ 2 B r j2 = B i ,... , ∀r ≥ k B r j = B i . I.e, B i is accepted by j. As a consequence, if i is the leader of a group I, then I ⊆ Sphere(i) thus B i is accepted by any agent in I. Proposition 18. In a TIODS, the opinion sequence of any agent is cyclic i.e., the opinion sequence of any agent i is characterized by an initial segment B 0 i ...B b-1 i and a cycle B b i ...B e i which will be repeated up as infinitum.

Proof 18. Here again, the propositional language is finite, the number of agents is finite, the integrity constraint and the graph of influences topic-by-topic do not change over the diffusion process.

Proposition 19. Let S = (A, µ, B, Inf ) a TIODS. Then ∀i ∈ A ∀s ∈ N B s i |= µ.

Proof 19. We prove this by induction on s.

• For s = 0 we have ∀i ∈ A B 0 i |= µ by definition of TIODS.

• Suppose that the property is true till step s -1.

Example 1 .

 1 Some simple distances d which can be used for instanciating the previous definition are: d D , the drastic distance, (d D (w, w ) = 0 iff w = w , 1 otherwise); d H , the Hamming distance (d H (w, w ) = m iff w and w differ on m variables). Let µ be a tautology. ϕ 1 = a ∨ b, ϕ 2 = ¬a, ϕ 3 = ¬b ∧ c. The eight interpretations are w 1 = {a, b, c}, w 2 = {a, b, ¬c}, w 3 = {a, ¬b, c}, w 4 = {a, ¬b, ¬c}, w 5 = {¬a, b, c}, w 6 = {¬a, b, ¬c}, w 7 {¬a, ¬b, c}, w 8 = (¬a, ¬b, ¬c} and M od(

Example 3 .

 3 Let S = (A, µ, B, Inf ) be an IODS with: A = {1, 2}, µ being a tautology, Inf (1) = {2}, Inf (2) = {1}, B 1 = a, B 2 = ¬a. S represents a network of two agents each one being influenced by the other one only. Agent 1 initially believes a and agent 2 initially believes ¬a. Assume that µ is a tautology.

Proposition 7 .

 7 Let S = (A, µ, B, Inf ) be an IODS. Then, ∀i ∈ A ∀s ∈ N B s i |= µ.

1 µ 2 µ

 12 we consider the first topic only we get: M od(∆ T (≺ T1 )) = {{a}}. If we consider the second topic only we get: M od((∆ T (≺ T2 ) = {{¬b}}. By definition 17 we get {{a}} ⊕ {{¬b}} = M od(µ) because {a, ¬b} ∈ M od(µ). Finally, we get M od(Θ µ (≺ T1 ≺ T2 ) = M od(µ) i.e., Θ µ (≺ T1 ≺ T2 ) = a → b.

Proposition 16 .

 16 Consider n formulas ϕ 1 , ...ϕ n and m orders on these formulas ≺ T1 ... ≺ Tm . Let k ∈ {1, ...n} and ≺ k T1 , ... ≺ k Tm be the orders obtained from ≺ T1 ... ≺ Tm by replacing ϕ k by Θ µ (≺ T1 ... ≺ Tm )). Then,

i 1 ∨

 1 ...∨B s in i applied to the multi-set of opinions: B s i1 ....B s in i ordered topic by topic by ≺ T1 ... ≺ Tm . The following proposition shows that definition 19 extends definition 4.

  Consider w |= ∆ µ (ϕ 1 ≺ ... ≺ ϕ n ). This implies that [D(w, ϕ 1 ), ...D(w, ϕ n )] is minimal in M od(µ). But, since i=1..n ϕ i ∧ µ is consistent, there exists w 0 st w 0 |= i=1..n ϕ i ∧ µ. Thus [D(w 0 , ϕ 1 ), ...D(w 0 , ϕ n )] = [0, ..., 0], i.e, [D(w 0 , ϕ 1 ), ...D(w 0 , ϕ n )] is minimal. Consequently, ∀i D(w, ϕ i ) = 0 i.e, ∀i w |= ϕ i . Thus w |= i=1..n ϕ i ∧ µ. -Now consider w |= i=1..n ϕ i ∧ µ. (such a model exists since i=1..n ϕ i ∧ µ is consistent). Thus by definition, w |= µ and ∀i = 1...n D(w, ϕ i ) = 0. Thus [D(w, ϕ 1 ), ...D(w, ϕ n

  [D(w, ϕ1), ..., D(w, ϕn] ≤ lex [D(w , ϕ1), ..., D(w , ϕn] and [D(w, ϕn+1), ..., D(w, ϕm] ≤ lex [D(w , ϕn+1), ..., D(w , ϕm] then [D(w, ϕ1), ..., D(w, ϕm] ≤ lex [D(w , ϕ1), ..., D(w , ϕm]. Thus, if w|= ∆(ϕ 1 ≺ ... ≺ ϕ n ) and w |= ∆(ϕ n+1 ≺ ... ≺ ϕ m ) then w |= ∆(ϕ 1 ≺ ... ≺ ϕ m ) • Assume ∆(ϕ 1 ≺ ... ≺ ϕ n ) ∧ ∆(ϕ n+1 ≺ ... ≺ ϕ m ) consistent.Thus ∃w 0 st [D(w 0 , ϕ 1 ), ..., D(w 0 , ϕ n )] and [D(w 0 , ϕ n+1 ), ..., D(w 0 , ϕ m )] are minimal in M od(µ). Thus [D(w 0 , ϕ 1 ), ..., D(w 0 , ϕ m )] is minimal in M od(µ). Consider w |= ∆(ϕ 1 ≺ ... ≺ ϕ m ). I.e, [D(w, ϕ 1 ), ..., D(w, ϕ m )] is minimal in M od(µ). Thus ∀i, D(w, ϕ i ) = D(w 0 , ϕ i ). Thus both [D(w, ϕ 1 ), ..., D(w, ϕ n )] and [D(w, ϕ n+1 ), ..., D(w, ϕ m )] are minimal in M od(µ). Thus, w |= ∆(ϕ 1 ≺ .

Proposition 12 . 1 i--

 121 Let S be an IODS and i a dogmatic agent. Then: (i) If Sphere(i) = A then B i is unanimously accepted. (ii) If | Sphere(i) |>|A| 2 then B i is majoritary accepted. Proof 12. This is a corollary of proposition 11 Proposition 13. Let S be an IODS and i a self-confident agent. Then∀s B s i |= B s-Proof 13. i being self-confident, we can write Inf (i) = {i ≺ i 1 ≺ ...i ni } (with n i ≥ 0). Because B i |= µ,we can apply proposition 2 and conclude that ∀s∆ µ (B sAssume that there is only one topic. We haveM od(Θ µ (≺)) = M od(∆ µ (≺ )) and M od(Θ µ (≺ k )) = M od(∆ µ (≺ k )). Moreover proposition 4 implies that M od(∆ µ (≺)) = M od(∆ µ (≺ k )). Thus finally, M od(Θ µ (≺)) = M od(Θ µ (≺ k ))Assume that the property is true for t -1 topics. Consider t topics.By definition 18, we can write M od(Θ µ (≺ T1 ...≺ t )) = M od(Θ µ (≺ T1 ... ≺ Tt-1 )) ∩ M od(Θ µ (≺ Tt )).We can then apply the induction hypothesis on both subsets and write: M od(Θ µ (≺ T1 ...≺ Tt )) = M od(Θ µ (≺ k T1 ... ≺ k Tt-1 ))∩M od(Θ µ (≺ k Tt )). Finally M od(Θ µ (≺ T1 ... ≺ Tt )) = M od(Θ µ (≺ k T1 ... ≺ k Tt )) Proposition 17.When there is only one topic and only one order on a multi-set of formulas, then the Topic-dependant Influence-Based Opinion Diffusion Sequence, as defined in definition 19, is identical to the Influence-Based Opinion Diffusion Sequence as defined in definition 4.Proof 17. We proveϕ 1 |= µ =⇒ M od(Θ ϕ1∨..∨ϕn (ϕ 1 ≺ .. ≺ ϕ n )) = M od(∆ µ (ϕ 1 ≺ .. ≺ ϕ n )) Definition 18 implies M od(Θ ϕ1∨..∨ϕn (ϕ 1 ≺ .. ≺ ϕ n )) = M od(∆ ϕ1∨..∨ϕn (ϕ 1 ≺ .. ≺ ϕ n )). Proposition 2, allows us to conclude M od(∆ µ (ϕ 1 ≺ .. ≺ ϕ n )) = M od(∆ ϕ1 (ϕ 1 ≺ .. ≺ ϕ n )) (since ϕ 1 |= µ). Proposition 2 also allows us to conclude M od(∆ ϕ1∨..∨ϕn (ϕ 1 ≺ .. ≺ ϕ n )) = M od(∆ ϕ1 (ϕ 1 ≺ .. ≺ ϕ n )) (since ϕ 1 |= ϕ 1 ∨ ... ∨ ϕ n ).From these three equalities, we conclude M od(Θ ϕ1∨..∨ϕn (ϕ 1 ≺ .. ≺ ϕ n ) = M od(∆ µ (ϕ 1 ≺ .. ≺ ϕ n )).

  ... ≺ Tm ). Because Θ satisfies proposition 15 (second item), we haveB s i |= B s-1 i1 ∨..∨B s-1 in i. Thus, by induction hypothesis, we have B s i |= µ.Proposition 20. Let S = (A, µ, B, Inf ) a TIODS. Then ∀i ∈ A ∀s ∈ N B s i is consistent.

  agent 1 is its own and only influential agent. Agent 2 is influenced by 1 and by itself, but more influenced by itself than by 1. Agent 3 is influenced by 1 and by 2, 1 being the agent who influences it the most. 1 initially believes a ∨ b, 2 initially believes ¬a ∧ ¬b, and 3 has initially no opinion. Table1shows the evolution of agent opinions when the distance used is d H . Agent 1 is not influenced by anyone except itself, so its opinion remains a ∨ b. Agent 2 is keen to change its opinion by integrating opinion of 1 if possible. But here, 2 has a complete opinion which is inconsistent with 1's opinion. Consequently, 2's opinion remains ¬a ∧ ¬b. Finally, 3 who had intially no opinion, is keen to form its own opinion by integrating 1's opinion and 2's opinion by giving preference to the first

one. Here it gets: (a∧¬b)∨(b∧¬a) Here, | Cyc(B 1 ) |=| Cyc(B 2 ) |=| Cyc(B 3 ) |= 1.

Table 1 :

 1 Opinion evolution in example 2

Table 2

 2

		s = 0 s = 1 s=0 mod 2 s=1 mod 2
	i = 1	a	¬a	a	¬a
	i = 2	¬a	a	¬a	a

shows the evolution of opinions (for distance d H and d D as well). By definition, agent 1 adopts 2's opinion and agent 2 adopts 1's opinion at the same time. Since their initial opinions contradict, the agents change opinion recursively. Here, | Cyc(B 1 ) |=| Cyc(B 2 ) |= 2

Table 2 :

 2 Opinion evolution in example 3

Table 3 :

 3 Opinion evolution in example 7

  of the first point of proposition 1). Thus [D(w 2 , ϕ 1 ), ...] < [D(w 1 , ϕ 1 ), ...] is impossible.Proof 9. We prove it by induction. Assume ∀i ∈ A, B i |= ϕ.• For s = 0. If ∀i ∈ A, B i |= ϕ, then obviously ∀i ∈ A, B 0 i |= ϕ • Suppose the property true for any t ≤ s-1 and let's prove it for s. By hypothesis we have ∀i ∈ A B s-1 Consider an agent i so that Inf (i) = {i 1 ≺ ...i n }. Proposition 10. IODS which are DAG with self-loops satisfy (Conv).

	We have in particular, B s-1 i1 B s i |= ϕ	|= ϕ. By proposition 3, we have B s i |= B s-1 i1 . Thus

i |= ϕ.

Preprint submitted to ElsevierJuly 18, 2017

In this present paper, we prove many more properties on IODS and TIODS that we did in the previous paper. But the main difference between the two papers is that here, we changed the definition for TIODS so that Unanimity Preservation is now satisfied while it was not in the previous paper. It is also proved that TIODS extend IODS, which was not the case in the previous paper. Moreover, proofs are given in extension

∀w∀w d(w, w ) = d(w , w) and d(w, w ) = 0 =⇒ w = w

Proofs are given in Appendix.

We can prove the equality if ϕ 1 ∧ µ is consistent, which is a less strong condition. But we need the stronger condition in the rest of the paper

Acknowledgements: This work was supported by ONERA under grant number 25348.01F (ROSARIO project).

Proof 3. This is a consequence of proposition 1 and proposition 2.

Proposition 4. Consider n formulas ϕ 1 , ..., ϕ n and ≺ an order on these formulas. Let k ∈ {1...n} and let ≺ k be the order obtained from ≺ by replacing ϕ k by ∆ µ (≺). Then, M od(∆ µ (≺)) = M od(∆ µ (≺ k )).

Proof 4.

Let us assume that ≺ is defined by ϕ 1 ≺ ... ≺ ϕ n . Let k ∈ {1...n}.

(⊆) Let w |= ∆ µ (≺). Thus (1) w ∈ M od(µ), [START_REF] Chau | Social judgment theory based model on opinion formation, polarization and evolution[END_REF] [D(w, ϕ 1 ), ..., D(w, ϕ n )] is minimal in M od(µ) and (3) D(w, ∆ µ (≺)) = 0.

(2) implies, ∀w ∈ M od(µ) [D(w, ϕ 1 ), ..., D(w, ϕ n )] ≤ lex [D(w , ϕ 1 ), ..., D(w , ϕ n )]. I.e., ∀j = 1..n D(w , ϕ j ) = D(w, ϕ j ) or ∃k ∈ {1, ..., n} st ∀j < k D(w, ϕ j ) = D(w , ϕ j ) and D(w, ϕ k ) < D(w , ϕ k ) Thus finally, in any case we have:

(4) and ( 5) implies [D(w , ϕ 1 ), ..., D(w , ϕ k ), ..., D(w , ϕ n )] < [D(w, ϕ 1 ), ..., D(w, ϕ k ), ..., D(w, ϕ n )] Thus, (6

Besides, ( 5) and ( 7) implies

. This contradicts (4).

Proposition 5. In an IODS, the opinion sequence of any agent is cyclic i.e., the opinion sequence of any agent i is characterized by an initial segment B 0 i ...B b-1 i and a cycle B b i ...B e i which will be repeated up as infinitum.

Proof 5. This result has been proved in proposition 1 of [START_REF] Schwind | Belief revision games[END_REF]. The proof is based on the fact that since the language is finite, the number of possible opinions which are distinct up to logical equivalence is finite. Moreover, there is a finite number of agents and for each agent, th set of its acquaintances is fixed. So the number of possible n-uples BRG s = (B s 1 , ...., B s n ) (vectors of agent opinions at step s) is finite, up to logical equivalence. Consequently, the belief sequence of each agent s cyclic. The proof does not use the fact that revision policies have to be irrelevant to permutation of their arguments. So we can transpose this proof for IODS. Proposition 6. Let S = (A, µ, B, Inf ) be an IODS. Then, ∀i ∈ A ∀s ∈ N B s i is consistent.

Proof 6. By definition, the initial opinion of agent i is consistent and µ is consistent. Moreover, Importance-based Merging Operators satisfy proposition 1 (second item). So i's opinion remains consistent at any step of the sequence.

Because Importance-based Merging Operators satisfy proposition 1 (third item), we conclude that Proof 14. This is a consequence of propositions 11 and 13.

Proposition 15.

• The first item is proved by definition of .

• The second item is satisfied because

Proof 16. Let k ∈ {1...n}. We can write {w : w ∈ M od(µ) and ∀i ∈ {1...

• Suppose {w : w ∈ M od(µ) and ∀i ∈ {1...m} i w ∈ M od(∆ i µ (≺ Ti ))} = ∅. Then, by the previous remark, we also have {w : w ∈ M od(µ) and ∀i ∈ {1...m} i w ∈ M od(∆ i µ (≺ k Ti ))} = ∅. Applying definition 18 twice, we conclude M od(Θ µ (≺ T1 ... ≺ Tm )) = M od(µ) and M od(Θ µ (≺ k T1 ... ≺ k Tm )) = M od(µ). Finally, M od(Θ µ (≺ T1 ... ≺ Tm )) = M od(Θ µ (≺ k T1 ... ≺ k Tm )).

• Assume now that {w : w ∈ M od(µ) and ∀i ∈ {1...m} i w ∈ M od(∆ i µ (≺ Ti ))} = ∅ We prove the result by induction on the number of topics.

Proof 20. We prove this by induction on s.

• For s = 0 we have ∀i ∈ A B 0 i is consistent by definition of TIODS.

• Suppose that the property is true till step s -1. Proof 22. We prove this by induction.

• For s = 0 the property holds.

• Assume that the property holds till s -


Antoine Potet

Katia Mocellin

Lionel Fourment

Lionel Fourment Linear

Linear Friction Welding of Aeronautical alloys Modeling and Numerical Simulation

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

•

  full 3D, entire process simulation Forge® solver (metal forming framework) • Implicit formulation • Updated lagrangian • (v-p) based formulation • Remeshing Modeling challenge • Friction model is critical but unknown Proposed solution : inverse analysis from recorded real-process data Results presented here are based on a Coulomb model Assumption of a perfect perfect weld • Can simulate the end of the process • No friction / perfect contact • Requires initial temperature distribution