Nadia Jacoby
email: jacoby@univ-paris1.fr

Exploration and exploitation strategies. What kind of analytical models?

Keywords: Evolutionary approaches of firms, Exploration and Exploitation behaviors, NK models Stratégies d'exploration Approches évolutionnistes de la firme, Exploration, Exploitation, Modèles NK JEL classification: C63, L21

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Evolution is a method of search among a great number of possible "solutions". If, in biology, this set of possible solutions is the set of possible genetic sequences, evolution can also be seen as a method for designing innovative solutions to complex problems. The environment changes and thus, it is necessary to search continually new sets of possible solutions. Even if the evolution rules seem to be very complex, in particular because they are responsible for variety and complexity of species, in fact, there are rather simple. "Species evolve by means of random variation, followed by natural selection in which the fittest tend to survive and reproduce, thus propagating their genetic material to future generations", [START_REF] Mitchell | An introduction to genetic algorithm[END_REF].

In the evolutionary perspective, a central point in the study of the firms' evolution concerns adaptation and selection behaviors. Changes in the economic environment incite firms to adapt themselves. But, to analyze these adaptive processes, the relation between the exploration of new possibilities and the exploitation of old certainties1 should be observed. This paper focuses on adaptive systems engaged in exploration and exploitation processes.

Exploration strategy of firms includes search, variation, innovation, ..., whereas exploitation behavior concerns choice, efficiency, selection, ...2 But how can the firm combine at once exploration and exploitation? What is the right trade-off between exploration and exploitation? [START_REF] March | Exploration and exploitation in organizational learning[END_REF].

An useful starting point to analyze adaptation processes is to consider a mapping representation of the population. The concept of ″fitness landscape″ developed by [START_REF] Wright | Evolution in Mendelian populations[END_REF] is defined as a representation of the space of all different possible configurations of the population according to their fitness. Each individual of the population is represented by a dimension of the space associated to a fitness value, which depends on how well that individual solves the problem at hand. However as the economic rationality of agents is bounded and because they cannot observe all the possible positions over the landscape, an accurate representation has to consider a limited space within which components can move. Thus, the search process has to be local, as suggested by [START_REF] Wright | Evolution in Mendelian populations[END_REF].

Among several kinds of analytical models available, "NK model" [START_REF] Kauffman | The Origins of Order[END_REF] is a simple formal model of rugged fitness landscape that demonstrates the fitness landscape topology is determined by the interdependence degree of the fitness contribution of the various attributes of the organism. These interactions refer to epistatic effects3 [START_REF] Smith | Evolutionary Genetics[END_REF]. N represents the number of parts of the system, i.e. the number of individuals in the population, whereas K is the number of other attributes with which each individual interacts. Thus, the contribution of each element to the fitness landscape depends upon its own fitness level and upon the K other elements among N. Interdependency models like NK make easier the understanding of exploration/exploitation behaviors.

I -Exploration and exploitation strategies

The firm evolution refers to exploration and exploitation strategies and then to innovation processes. In the economic literature, organizational learning is understood at first and second order. The first order learning allows to do existing things better, whereas the second concerns the capability to do news things. This notion is clearly linked with those of parametric [START_REF] Langlois | Firms, Markets and Economic Change[END_REF] and architectural changes [START_REF] Abernathy | Innovation: Mapping the winds of creative destruction[END_REF][START_REF] Henderson | Architectural innovation: The reconstruction of existing product technologies and the failure of established firms[END_REF]. The way [START_REF] Holland | Adaptation in natural and artificial systems[END_REF] but above all [START_REF] March | Exploration and exploitation in organizational learning[END_REF] pose the dilemma between exploration of new possibilities and exploitation of old certainties sums up very well this question. This section will be devoted to the exploration/exploitation problem and its study proposed by the economic literature.

1°) Exploration, exploitation behaviors in the economic literature

To survive in the short run, firms need to exploit in an efficient way the existing resources. In the long run, it is necessary for the firm to explore new possibilities and to develop new competencies. The firm competitiveness can be guaranteed only thanks to its innovation process, which allows surviving to the market competition.

Exploration strategy

As March describes, "exploration includes things captured by terms such as search, variation, risk taking, experimentation, play, flexibility, discovery, innovation" [START_REF] March | Exploration and exploitation in organizational learning[END_REF]. Exploration is an adaptable and flexible process, which has to adapt itself to the new configuration the firm can discover and arises from individual deviance as a source of innovation [START_REF] Nooteboom | The Combination of Exploitation and Exploration: How does it work?[END_REF].

The study of the innovation process combining the theory of innovation cycle with the theory of product life cycle [START_REF] Vernon | ″International investment and international trade in the product cycle″[END_REF] leads to the identification of an innovation process in two stages. The first stage of innovation, characterized by volatility, concerns the creation of "Schumpeterian novel combinations", whereas in the second, "dominant designs", characterized by economies of scale and routinization, can emerge. At this second stage, the firm exploitation behavior is essential. The innovation developed in the first stage is consolidated in the second one. The degree of coordination between actors and the limitation of their autonomy depend on the innovation cycle stage. The best firm organizational structure for the exploration strategy is a disintegrated one. Members of organization should be able to operate independently without a too tight coordination process. The firm should be able to create some discontinuities in order to discover new possibilities. exploitation and existing resources? So, two underlying questions are raised. The first concerns creation with minimum of destruction, the second asks for the optimal discovery process. The model presented in the second section will try to answer those questions. The use of current competencies in novel contexts allows preserving exploitation while exploring new possibilities.

Existing routines are necessary for the exploration strategy of the firm. [START_REF] Abernathy | Innovation: Mapping the winds of creative destruction[END_REF] define two kinds of innovation. Innovation which "enhances the value or applicability of the firm's existing competence", and innovation that disrupt and destroy the firm's existing competence. The latest "changes the technology of process or product in a way that imposes requirements that existing resources, skills and knowledge satisfy poorly or not at all" [START_REF] Abernathy | Innovation: Mapping the winds of creative destruction[END_REF]. Therefore, the value of existing routines is reduced. As Schumpeter said, the "creative destruction" is the vehicle of growth.

But we focus on the first kind of innovation for which the changes don't need to be destructive.

"Innovation in process technology may require new procedures in handling information, but utilize existing labour skills in a more effective way. Such changes conserve the established competence of the firm" [START_REF] Abernathy | Innovation: Mapping the winds of creative destruction[END_REF]. Thus, the question of internal coherence of the firm described in particular by Dosi, Teece and [START_REF] Teece | Les frontières des entreprises: vers une théorie de la cohérence de la grande entreprise[END_REF] emerges. New routines created or discovered by the exploration process have to be consistent with the firm global activity. The firm should be able to manage all its routines, new and old, if it wants to survive in the long run. The compatibility between routines is fundamental for the firm durability4 . The firm diversification strategy is not randomly determined; at the opposite, it enforces some implicit rules such as those concerning its core activity.

Exploitation strategy

As [START_REF] March | Exploration and exploitation in organizational learning[END_REF] describes, "exploitation includes such things as refinement, choice, production, efficiency, selection, implementation, execution". Exploitation consists in a refinement of existing technology, requiring individual coordination [START_REF] Nooteboom | The Combination of Exploitation and Exploration: How does it work?[END_REF]. Indeed, the internal working of the firm refers to the coordination between members but also to the coherence among individual and collective knowledge. However, exploitation needs also fixed rules and routinization for an efficient management of the activities. The firm organizational structure should be sufficiently integrated to allow a large volume production, the emergence of economies of scale and the distribution of products in wider markets. This structure permits the emergence of a dominant design and a novel techno-economic paradigm (Freeman and Perez, 1988). The firm sustains a price competition; with an high level profit objective.

Compared to the returns from exploration and the high uncertainty degree associated, exploitation is safer for the firm. Returns from exploitation are less remote in time, less distant from the initial position. The exploitation results are more certain than those of exploration, there are quicker and more precise. Exploitation constitutes the main source of benefits for the firm; it allows expenses for exploration necessary for market competition. As [START_REF] March | Exploration and exploitation in organizational learning[END_REF] remarks, "basic research has less certain outcomes, longer time horizons, and more diffuse effects than does product development".

2°) Exploration, exploitation dilemma: a theoretical overview

Exploration as exploitation competes for scarce resources and the firm makes choices between them. So, the firm faces the dilemma between exploration and exploitation while knowing that it is necessary to make a trade-off between cognitive distance and proximity. The paradox the firm faces with, relies on the necessity to mix exploration and exploitation in a specific way, which guarantees its survival. The search horizon have to be sufficiently close to maintain exploitation and at the same time sufficiently distant to discover significant novelty. In evolutionary models, the discussion around exploration and exploitation is framed in terms of balancing variation and selection processes. In this theoretical perspective, the firm is a set of routines and core competencies evolving under the influence of variation and selection mechanisms. The evolutionary model of the firm is constructed at the intersection of the Darwinian triptych of "heredity, variety, selection" and the Lamarckian concept of "inheritability of acquired characteristics".

During the evolution process, the firm mutates, changes and those changes need to be selected in order to preserve the internal coherence. New routines emerging from the variation process are not necessarily consistent with the other routines of the firm. It is therefore essential those routines would be selected in order to preserve the internal coherence and then the firm durability.

Variation process applies to firm's exploration behavior whereas internal selection process refers to its exploitation behavior. The dilemma March wants to solve is reduced to the question of firm's internal equilibrium between variation and selection. What is the appropriate trade-off between changes and stability? What is the share of mutation the firm needs compared to the share of stability? In an evolutionary model, the firm exploratory behavior can be measured by the practice and the rate of change in the environment [START_REF] March | Exploration and exploitation in organizational learning[END_REF]. The higher the rate of change, the more the firm explores. Therefore, 1-p 1 reflects the exploration behavior, that is the development by individuals of visions different from those of the organization. Adaptation process acts according to the suitability to the reality. "The organizational code adapts to the beliefs of those individuals whose beliefs correspond with reality on more dimensions than does the code" [START_REF] March | Exploration and exploitation in organizational learning[END_REF].

The model measures the knowledge state at the individual and organizational levels. Individual beliefs and organizational code converge up to the stable equilibrium at which they share the same belief with respect to each dimension.

The model can be a closed system with neither entry nor exit, or an open system. In the closed system, March analyzes the effect of learning rates and of learning rates heterogeneity. When p 1 is high, we can observe a positive effect on the individual knowledge but an adverse effect on the improvement of organizational knowledge and then on the improvement on individual knowledge in the long run. Therefore, the organizational population has to be composed of slow learners and fast learners. In this case, the model shows that the heterogeneous population has always-higher knowledge equilibrium than the homogeneous population. Finally, the higher knowledge equilibrium can be obtained thanks to the mixed organizational population, that is with exploiters and explorers. In the open system, March is interested in the effect of environmental turbulence and personnel turnover, which is another way to introduce variability in the model. The first-order effect of turnover on individual knowledge is negative, but we observe there is a positive relation between the length of service and the individual knowledge level. On the whole, the turnover effect on individual knowledge is strong and slightly more complicated on the organizational knowledge where it reflects a trade-off between the learning and the turnover rates. Firstly, when the exploitation level is moderate (p 1 =0.1, for example) and the turnover relatively high, organizational knowledge level decreases. However, when exploitation is high (p 1 =0.9, for example), organizational knowledge increases till turnover achieves a particular threshold and then decreases. The environmental turbulence effect on the level of knowledge is negative. But this effect can be made up for the existence of turnover, even if it is moderate, especially on the organizational knowledge. However, the closer the new entrants to the current organizational code, the less efficient the turnover as source of exploration.

Finally, we observe that diversity increases level of knowledge. New entrants know less on average than incumbents but what they know is less redundant with the organizational code. In probability, they certainly contribute more to improve organizational knowledge.

II -Model

Interdependency models allow representing the relationship between members of organization. Adaptation processes are complex; variation and selection mechanisms, which refer to the exploration, exploitation dilemma in the evolutionary approach of the firm, are interdependent.

As [START_REF] Levinthal | Adaptation on rugged landscapes[END_REF] remarks, their complexity and interrelationship encourage the use of simulation models.

1°) Interdependency models

A useful starting point to analyze these processes is to consider a mapping representation of the population. The concept of ″fitness landscape″ developed by [START_REF] Wright | Evolution in Mendelian populations[END_REF] is a multidimensional space in which each attribute is represented by a dimension of the space according to its fitness value. The genotype represents different possible configurations of the population. Each genotype has fitness and the distribution of fitness values over the space of genotypes constitutes the ″fitness landscape″5 . Each individual of the population is represented by a dimension of the space associated to a fitness value, which depends on how well that individual solves the problem at hand. As described above, the overall distribution of the individual's fitness values over the population constitutes the fitness landscape. For simplicity, we consider that each population is a bit string of length l and that the distance between two populations is called ″hamming distance″. It represents the number of locations at which corresponding bits differ. So the fitness landscape can be represented in a (l+1)-dimensional plot. But the main interest of this kind of representation is to show how individuals can move all over the landscape, i.e. how to represent evolution process.

According to Wright, evolution causes populations to move along landscapes in particular ways, especially local search processes. Each population component searches a better position on the landscape, i.e. a position that gives it a higher fitness value. However as the economic rationality of agents is bounded and because they cannot observe all the possible position over the landscape, an accurate representation has to consider a limited space within which components can move. That is the reason why the search process is local. In this context, the concept of ″neighborhood search″, developed by [START_REF] March | Organizations[END_REF] can be compared to the ″local hill climbing″ [START_REF] Holland | Adaptation in natural and artificial systems[END_REF] notion. Adaptation can be seen as a moving of population towards local peaks [START_REF] Wright | Evolution in Mendelian populations[END_REF].

The ultimate goal for the population is to enhance the global fitness value through the increase of the component fitness value. So the population shape will be modified.

2°) NK model

"NK model" is a simple formal model of rugged fitness landscape that demonstrates the topology of the fitness landscape is determined by the degree of interdependence of the fitness contribution of various attributes of the organism. N represents the number of parts of the system, i.e. the number of genes of the genotype. An entity is composed of N attributes where each one can take on two possible values, 0 or 16 . Thus, the fitness space consists of 2 N possible configurations. Furthermore, the contribution of each element to the fitness landscape depends upon its own fitness level and upon K other elements among N. Thus, each attribute can take on 2 K+1 different values depending on the value of the attribute itself and on the value of the K other attributes with which it interacts. K measures the richness of epistatic interactions7 among the components of the system and influences the relative smoothness or ruggedness of the landscape. K is necessarily inferior to N but can assume all the values between 0 and N-18 .

Here the question is to discover an attractive peak on the rugged fitness landscape i.e. a peak with a higher fitness value. The search should be sensitive to the fitness value of alternative locations in the space of possible solutions. It is a local search process which examines alternatives in the immediate neighborhood of the current position. Search is a step-by-step process, which implies a moving only if the new fitness value is higher. At the end, the search mechanism stops when the local optimum is reached i.e. when there is no more possible position with a higher fitness value. During the search process and after a few iterations, the number of solutions, yet in touch, declines radically [START_REF] Kauffman | The Origins of Order[END_REF][START_REF] Levinthal | Adaptation on rugged landscapes[END_REF]. It clearly reflects the fact that many initial starting points share the same local optimum. The more rugged the landscape, the higher the number of local optima. The local search process results in a ″walk″ to the optimum from all the starting points. However, search efforts can be ″trapped″ on a suboptimal local peak. Indeed, search process is path-dependent and reveals here its limited nature.

However we can also identify another kind of search process focused on the adoption of alternatives far removed from the organization's current mode of operation. [START_REF] Kauffman | The Origins of Order[END_REF] calls, these kind of radical changes, ″long jump″. In practical terms, the "long jump" corresponds to the random specification of each of the N attributes of the organization.

Consequently, the organization adopts this new alternative if its fitness value is higher. Thus the likelihood of a radical organizational change is sensitive to the organization's current performance. The more efficient the organization, the less the likelihood for a radical change.

Local search and adaptation processes modeled in NK represent exploration behavior whereas selection refers to exploitation. We are interested in the impact exploration and exploitation have on the global performance, so we focus on their influence on individual fitness values. The results presented here, take into account the following initialization. The number of organizations is kept constant over time and, it is set at 100. We set N=10, number of organizations=100, number of runs=100 and runs the simulation for K=0, K=1, K=5. The results reflect the average behavior.

The first set of runs rests only on local adaptation -without ″long jump″ or selection -and is characterized by a high diversity degree. Organizations have only an exploration strategy.

Quite rapidly, the number of organizational forms decreases whatever the value of K. The search process continues till all the organizations reach a local optimum. While the organizations are initially randomly distributed on the fitness landscape, these results show that many of them share the same local optimum.

In a second set of simulation, there is no diversity and all the organizations are assigned to the same organizational form. It is assumed that organizations are engaged in local and distant search processes. The organizations' strategy is always only based on exploration. In the first few periods, roughly half of the organizations can identify attractive forms and adopt them. But quite rapidly, local search process leads to the decline in the number of distinct organizational forms in the population.

A third set of simulation postulates a pure selection process without adaptation; the strategy adopted by firms is only exploitation. Changes occur only by birth and death of organizational forms, which are not sensitive to the value of K. The landscape is driven towards the fittest organizational form. Contrary to adaptation processes, the rate of organizational forms differentiation is slower. However, selection process drives the population towards the existence of a unique form while adaptation leads to a set of organizational forms over which selection occurs.

But what happens if, now, we consider changing environments9 . What is the challenge for incumbents in a Schumpeterian environment? How do adaptation and selection interact in a changing external environment?

To explore these issues, after 25 periods of time, the simulation process specifies a new the fitness landscape by changing one dimension in the fitness contribution. When the degree of epistatic interaction equals zero, all the incumbents survive. But the higher the value of K, the less the share of surviving incumbents. With high levels of interaction, the local adaptation is not an efficient response to changes in the fitness landscape. As a result, in a changing environment, survivals are more dependent of a long-jump process result.

The diversity of organizational forms seems to be explained thanks to the diversity of environments in which organizations act. But this kind of argumentation requires the existence of a well-defined scheme of interrelationship between environment and organizations. The degree of epistatic interactions appears as an interesting factor to explain both organizational diversity and persistence of organizations in changing environment, [START_REF] Levinthal | Adaptation on rugged landscapes[END_REF]. [START_REF] Volberda | Building the flexible firm[END_REF] suggests possible solutions to solve the paradox of exploration/exploitation. First, the firm can be engaged in both exploration and exploitation mechanisms because one part of the organization is devoted to exploration while another part deals with exploitation. Thus, there is horizontal or vertical separation within the firm. In the first case, some divisions are engaged in exploration, typically R&D department, while others are engaged in exploitation, typically production department. But, in this case, the interface problem persists. Conversely, the vertical separation can go two ways. Either the management delegates the exploration activity and controls the upholding of internal coherence; or he determines the search direction and coordinates the staff.

Conclusion

In the horizontal separation, all the department interacting with the market and sources of technology can grip new opportunities they meet. There is no path pre-determined by the management. If the firm is vertically separated, the management claims specific paths the divisions have to respect in their exploration activity. Advantages and drawbacks are more or less the same [START_REF] Aoki | Horizontal vs. vertical information structure of the firm[END_REF] identifies for the horizontal structure of firms J and the vertical one of firms A.

The second Volberda's proposition concerns the separation between different specialized firms.

One firm focuses on a specific stage of exploration while another one offers complementary stages in exploitation. In the pharmaceutical industry, "large companies provide efficient production, marketing and distribution whereas biotechnological firms explore novel product forms", [START_REF] Nooteboom | The Combination of Exploitation and Exploration: How does it work?[END_REF].

Finally, the last proposition explains that separation can occur in time. The firm can explore during a certain period of time and then exploit during another one. Nooteboom refers to [START_REF] Burns | The management of innovation[END_REF] "oscillating" concept, specifying it's very difficult to achieve. Indeed, it requires the constant job redefinition; people periodically change their function in the firm, which changes itself its activity, oscillating between exploration and exploitation.

But, NK models are above all used to illustrate the temporal separation. So, the two first propositions remain research paths to explore, in particular thanks to those analytical models.

  the relation between exploration and exploitation, and exploring some implications of the relation for the accumulation and utilization of knowledge in organizations". In this mutual learning model, the dilemma between exploration and exploitation involves conflicts between short run and long run and between gains to individual knowledge and gains to collective knowledge. It's a simple model of the development and diffusion of organizational knowledge.The environment or external reality has m dimensions taking the value 1 or -1 with the independent probability of 0.5. The organization is composed by n individuals and is represented by the organizational code. Individuals and organizational code have m dimensions taking the value -1, 0 or 1, which can change over time according to the adaptation process. The individual beliefs change continuously as a consequence of learning from the code. P 1 represents the effectiveness of socialization, the learning from the code and then the exploitation behavior.

[START_REF] Schumpeter | The Theory of Economic Development[END_REF],[START_REF] Holland | Adaptation in natural and artificial systems[END_REF],[START_REF] March | Exploration and exploitation in organizational learning[END_REF].

[START_REF] March | Exploration and exploitation in organizational learning[END_REF].

Epistatic effects refer to the action of a specific gene on another one non-allele.

[START_REF] Chakir | How internal selection process leads to a cooperation agreement? An application to car industry[END_REF] proposes two case studies of cooperation agreements highlighting, in particular, the problem of internal coherence.

[START_REF] Kauffman | The Origins of Order[END_REF] notes that "often,[START_REF] Wright | Evolution in Mendelian populations[END_REF] thought of the fitness of a given gene or genotype as a function of its frequency in the population". Like Kauffman in his book (1993), we consider, here, the simpler idea that each genotype can be assigned fitness.

Describing this kind of model,[START_REF] Levinthal | Adaptation on rugged landscapes[END_REF] notes, "the model can be extended to an arbitrary finite number of possible values of an attribute".

Epistatic effects refer to the action of a specific gene on another one non-allele.

For a precise description of NK model, refer to[START_REF] Jacoby | Evolution of Firms in Complex Worlds: Generalized NK models[END_REF].

This option corresponds to the environmental turbulence proposed by[START_REF] March | Exploration and exploitation in organizational learning[END_REF].


Guillaume Charrondiere

Iheb Cherif

Gérard Poulachon

Dominique Cotton

José Outeiro

INFLUENCE OF TOOLPATH AND CLAMPING STRATEGIES ON STAINLESS STEEL PLATES DISTORSION AFTER MACHINING

Keywords: Residual Stresses, Toolpath Strategy, Clamping Strategy

Heat exchangers in nuclear power generation plants are made of thin stainless-steel plates assembled together in order to improve their efficiency and compactness. To ensure the assembly, the global distortion of those plates must be mastered and minimized, mainly by predicting the evolution of the residual stress field during their manufacturing process chain. The residual stresses generated during rolling are removed by heat treatments process that also induce another stress field while cooling. During machining, those residual stresses are redistributed to reach another equilibrium state, leading to a macroscopic part distortion. The objective of this work is to study the influence of the machining toolpath and clamping strategies on the global part distortion so as to optimize the manufacturing process chain.

Introduction

New generation of nuclear exchangers includes the use of thin plates sized 2000*1000*5 mm 3 . In order to improve their corrosion resistance, they are made of stainless steel 316 L, a refractory material which is difficult to machine. At the end of the manufacturing process, the plates are grooved by milling technique. The residual stress field is then redistributed and new stresses are introduced at the surface of the material.

The factors having an influence on this distortion are numerous and some authors already studied them. [L. Liangbao 2015], using the layer removal method coupled with numerical simulations, highlights the existence of a rate of material removal beyond which the distortion stabilizes. According to him, the first passes are those that induce the main distortion.

[K. H Fuh and C.F Wu 1995] were the first to study the influence of cutting parameters on the introduction of residual stresses while machining. They concluded that strong cutting conditions induce high residual stresses on the surface of the material. The influence of cutting conditions will not be studied in this document.

[S. Hassini 2015] and [X. Cerutti 2014] studied the influence of clamping strategies. They showed that the number of clamps has a negligible influence on the global distortion, but has an impact on dimensional tolerances. Their conclusions are similar regarding the influence of toolpath strategy.

Experimental methodology

Samples geometry

The geometry of samples was defined by applying a 5-ratio homothetic transformation on the dimensions of industrial parts. Figure 1 shows the comparison between industrial parts and academic samples. It also presents the rolling and machining directions that have been kept normal.

Test plan

This work only focuses on the influence of a few parameters of clamping and machining strategies. The influence of cutting conditions such as the cutting speed Vc (130 m/min), radial depth of cut ae (30 mm) or lubricant (central emulsion lubrication) has not been studied and thus were kept constant for the whole test campaign. Prior tests were realized according to "Tool Material Couple" method to validate cutting parameters values and to ensure cutting stability. 3a, the order corresponding to the machining order.

3 Experimental means The positioning is ensured by the dynamometric platform, a grinded wedge and a punctual contact.

Instrumented clamping system

Figure 2 presents the various components of the clamping system. Different force sensors are used: the 9255A piezoelectric Kistler dynamometric platform allows to measure the forces generated during machining in the three spatial directions. It is then possible to determine the cutting force and to control the tool wear. The Kistler load washers measure the instantaneous clamping force in each screw. These values give information about the part behavior during and after machining. The measurement error of load washers has been determined during their calibration. The value depends on the range of efforts applied and represents in this case 0.5%. To overcome this issue, an acquisition reset is made every pass (for fz=0.02 mm/rev/tooth) or every five passes (for fz = 0.12 mm/rev/tooth) and the gain or loss of efforts is calculated relatively. It also allows to reduce the electrical drift which reaches 1.5 N for 100 seconds which is the duration between two resets.

Geometrical control of parts

The geometrical control of parts is done with a DEA Coordinate Measuring Machine (CMM) for which the uncertainty of measurement is ± 2 µm. Each part is measured in three different steps: before machining, after machining clamped, after machining unclamped. The positioning as well as the measurement program are the same for all parts in order to ensure the repeatability.

Preliminary tests

To ensure the repeatability of the experiments, three parts were machined in similar conditions. The distortion profile is studied in a plane (XZ) located at y = 100 mm as shown in Figure 3a.

The maximum difference between the flatness defects of superior faces is about 8% which is negligible comparing to the next results. The repeatability being ensured, each test presented in Table 1 was realized once.

A temperature instrumented test has been carried out in order to ensure that the values measured by the load washers are due to the redistribution of residual stress within the material and not due to the dilatation of some components. The part as well as two clamping screws have been instrumented with K-type thermocouples. A 2°C increase in temperature is observed into the part leading to an expansion in the thickness of 0.7 µm. This dimensional variation can be neglected because it is offset by the different components and particularly by the spherical housing washers.

Results and discussion

Influence of initial clamping force

Three levels of initial effort have been tested (5, 10 and 20 kN per screw). A range of mounting and clamping have been implemented. The profiles of distortion obtained are presented in Figure 3b.

The distortion profile is similar for an initial clamping force of 5 or 10 kN per screw. By increasing it up to 20 kN, the global distortion is lower. The clamping forces create a stress field into the part which superimposes to the residual stresses field. While machining, the residual stresses are reorganized and the superposition of both stress fields can locally reach the elastic limit and induce plasticity. This hypothesis will have to be confirmed with numerical simulations.

Influence of axial depth of cut

As shown in Figure 3c, the depth of cut has a great influence on global distortion. This could be explained by the residual stress distribution into the part. This one has been determined with the X-ray diffraction method. The thickness of the compression layer is 0.5 mm. Between 0.5 mm and 2 mm, the stress is slightly positive. When removing a 1-mm layer, the traction-compression ratio in this layer is unbalanced, leading to a great distortion. On the contrary, a 3-mm layer is more balanced causing less distortion after machining.

The experiments also prove that the depth of cut has a great influence on the sequence effect. Figure 4 describes this effect: as the part deforms while machining, the sides of stiffener progressively lose their parallelism and some areas are machined at several passes.

Figure 5 shows the cartographies of the dimension of the left groove for depths of cut of 1 and 3 mm. v For test #2, during the first 10 passes, the tool successively machines the sides of the groove, countering the effect of distortion. Furthermore, during the 5 last passes, the 10 mm-height of the tool does not allow to machine the sides in the first 5 mm: the distortion can be seen in this area, the dimension is lower than the theoretical 30 mm. In the case of test #4, only the first 2 mm are subject to the distortion.

Influence of toolpath strategy

The distortion profile, available in Figure 3d, shows that toolpath has a great influence on it. This observation is in contradiction with the ones of [X. Cerutti 2014] and [S. Hassini 2015]. The asymmetry induced by the LMR strategy leads to a greater distortion.

Load washers highlight a heterogeneity of the redistribution of residual stresses while using the asymmetric strategy LMR. As shown in Figure 6, the gain of clamping force between the beginning and the end of the test is unbalanced: the washers on the right gain 257 N whereas the left ones lose 54 N. Cerutti 2014], toolpath strategy also seems to influence the part behavior and the redistribution of residual stresses. An asymmetric strategy increases the distortion and unbalances the redistribution of residual stress field.

These experimental results will be used to build and fit numerical models. These will be determinant to understand the phenomena highlighted experimentally and to validate the different hypothesizes presented in this paper.

Acknowledgement

The authors gratefully acknowledge FRAMATOME for providing the 316L samples used in this work.

7 Bibliography

Figure 1 :

 1 Figure 1: Geometry of the samples

  Figure 2: Instrumented clamping system

Figure 3 :

 3 Figure 3: Distortion profiles according different cutting parameters a) Notations Distortion profile according to: b) Initial clamping force -c) Axial depth of cut -d) Toolpath

TestFigure 4 :

 4 Figure 4: Sequence effect on stiffener sides

[Figure 6 :

 6 Figure 6: Results from load washers

  Table 1 presents the tests that have been carried out.

			Toolpath	Feed per tooth (mm/rev/tooth)	Axial depth of cut (mm)	Initial clamping force per screw (kN)
	Influence of	Test #1	LMR	0.12	1	5
	initial clamping	Test #2	LMR	0.12	1	10
	force	Test #3	LMR	0.12	1	20
	Influence of	Test #2	LMR	0.12	1	10
	depth of cut	Test #4	LMR	0.03	3	10
	Influence of	Test #2	LMR	0.12	1	10
	toolpath	Test #5	MLR	0.12	1	10
			Table 1: Test plan
	L, M and R stands for the designation of the grooves (Left, Middle, Right) as shown in Figure


R Coelho
email: rcoelho@ipfn.tecnico.ulisboa.pt

W Zwingmann

E Giovannozzi

P Mccarthy

E P Suchkov

F S Zaitsev

J Hollocombe

N Hawkes

G Szepesi

L Appel

S Silburn

G Poulipoulis

PLASMA EQUILIBRIUM RECONSTRUCTION OF JET DISCHARGES USING THE IMAS MODELLING INFRASTRUCTURE

The reconstruction of Tokamak plasma equilibrium is a fundamental step in the understanding of fusion plasma physics since it sets the starting point for all subsequent plasma modelling applications and experimental data interpretation. The verification and validation of the numerical codes used to reconstruct plasma equilibrium, using as many available input experimental data e.g. magnetic field or flux measurements, density and temperature diagnostics and polarimetry diagnostics, is essential both for physics model interpretation and when qualifying and extrapolating for ITER. In the framework of the EUROfusion Work Package on Code Development for Integrated Modelling, a scientific Kepler workflow for the reconstruction of Tokamak plasma equilibrium was prototyped, using the ITER Integrated Modelling and Analysis Suite (IMAS). The workflow can seamlessly use any sort of data from Tokamak experiments and call equilibrium reconstruction codes such as EQUAL, EQUINOX, NICE, EFIT++ and SDSS, all using the same physics and engineering data ontology and methods for accessing the data. In the paper, plasma equilibrium reconstructions on dedicated JET plasma ___________________________________________________________________________

INTRODUCTION

The reconstruction of Tokamak plasma equilibrium is a fundamental step in the understanding of fusion plasma physics since it sets the starting point for all subsequent plasma modelling applications and experimental data interpretation. Indeed, reliable and robust reconstructions of the plasma shape and position are crucial when steering plasma control, ensuring that the target scenario requested is met and that no machine operational limits are reached with the consequent termination of the discharge. In addition, interpretative plasma scenario modelling including additional heating and current drive physics modules as well as plasma stability strongly rely on the reference plasma equilibrium reconstruction being considered. The verification and validation of the numerical codes used to reconstruct plasma equilibrium, using all many available input experimental data -e.g. magnetic flux loops and magnetic field measurements, density and temperature diagnostics, polarimetry diagnostics and, when available, MHD markers as proxies for the safety factor profile -is an ambitious milestone that should ultimately provide full consistency between diagnostic data and physics model interpretation. For such an ambitious goal, it is easily recognised that multi code verification and validation is substantially facilitated when all codes share a common data model, physics data conventions, input data provenance and software infrastructure. This is the case of the integrated modelling framework developed by the EUROfusion Work Package on Code Development for Integrated Modelling (EU-IM) [START_REF] Falchetto | The European Integrated Tokamak Modelling (ITM) effort: achievements and first physics results[END_REF] and also more recently of the Integrated Modelling and Analysis Suite (IMAS) framework [START_REF] Pinches | Progress in the ITER Integrated Modelling Programme and the Use and Validation of IMAS within the ITER Members[END_REF][START_REF] Imbeaux | Design and first applications of the ITER integrated modelling & analysis suite[END_REF] developed by ITER. These frameworks are ideal testbed facilities for the verification of physics codes on arbitrary devices since the inherent complexity of adhering and interfacing to different datastructures, physics units or data storage I/O is avoided. In addition, one can easily adopt a device independent approach while designing and developing the scientific workflow, focusing on the device only when fine tuning the relevant set of code parameters.

In this work, results from a scientific workflow developed for the plasma equilibrium reconstruction on arbitrary Tokamak devices are shown, addressing dedicated plasma discharges on JET Tokamak. The workflow prototype is orchestrated in Kepler [4] and is being tested simultaneously on an IMAS modelling infrastructure installed on the EUROfusion Gateway cluster on MARCONI-HPC [START_REF] Voitsekhovitch | Recent EUROfusion Achievements in Support of Computationally Demanding Multiscale Fusion Physics Simulations and Integrated Modeling[END_REF] and also on the FREIA cluster at JET. In Section 2, a brief description of the IMAS software infrastructure is made together with an overview of the workflow developed and some of the functionalities included. In Section 3 some results from two different JET discharges are shown with different sets of codes and equilibrium reconstruction constrains. Lastly in the Conclusions a summary of the work is presented and a discussion on the future improvements of the software suite included in the workflow.

THE IMAS MODELLING INFRASTRUCTURE AND EQUILIBRIUM RECONSTRUCTION WORKFLOW

The ITER Integrated Modelling and Analysis Suite (IMAS), from a software perspective, is easily portrayed as a modular set of software components that enable the development and execution of coupled modeling applications and data processing. Pivotal to its design, the concept of a "data model" that encompasses in detail all possible aspects of an integrated Tokamak modeling simulator i.e. from actuators used for plasma operation to transport solver numerics, facilitates enormously the communication between different physics actors. From an interfacing point of view, physics or signal processing components in a workflow communicate primarily by exchanging objects from a data dictionary using multi-language (Fortran, C++, Python, Java, among others) implementations of input and output methods. Similarly to the concept of Consistent Physical Objects (CPOs) developed and used in EU-IM [6], in IMAS there are Interface Data Structures (IDSs) e.g. equilibrium, magnetics, interferometry, core_profiles, core_sources, that are exchanged by the various components in the scientific workflow [START_REF] Imbeaux | Design and first applications of the ITER integrated modelling & analysis suite[END_REF]. The data dictionary is meant to be fully independent of any particular data structure in use at the various EU Tokamak device facilities and sufficiently flexible in design to be extended according to particular needs and to ensure that a one to one correspondence between the different data models is possible. Accessing experimental data from the devices' databases and populating and IDS database is easily doable, including by coding the data mapping and data exchange directly into the platform's data access layer, transparently to the user. This is done using a Universal Data Access (UDA) plugin developed to parse the data access requests from the IMAS data access framework and map these to the local, experiment specific, data infrastructure and access APIs. The data obtained, as mapped from the IDS request, is then transformed as required (data-scaling, reshaping, etc.) before being passed back to the IMAS data access framework where it is returned as an IDS data object. The user only sees the IDS data object and is isolated from the source of the data, whether it has come from a local IDS database or remotely via the UDA plugin.

The layout of the workflow developed in IMAS, as observed from the KEPLER graphical user interface, is shown in Fig. 1. Provisioning of the experimental data is obtained using a dedicated script using the UDA plugin to download and assign automatically the experimental data IDSs and that can be easily embedded as an actor in the workflow. The workflow offers the possibility to perform single time or multi time calculations using a pool of reconstruction codes that currently includes EQUAL [START_REF] Zwingmann | Equilibrium analysis of steady state tokamak discharges[END_REF], NICE [START_REF] Faugeras | Equilibrium reconstruction at JET using Stokes model for polarimetry[END_REF] and EQUINOX [START_REF] Faugeras | 2D interpolation and extrapolation of discrete magnetic measurements with toroidal harmonics for equilibrium reconstruction in a tokamak[END_REF] codes but is easily extendable to other codes such as EFIT++ [START_REF] Appel | Equilibrium reconstruction in an iron core tokamak using a deterministic magnetisation model[END_REF] or CLISTE [START_REF] Mccarthy | Analytical solutions to the Grad-Shafranov equation for tokamak equilibrium with dissimilar source functions[END_REF] once IMAS compliant actors for these codes become available. The post processing, included in the actor composite shown in Fig. 1, includes for the moment the automatic calculation of a high resolution equilibrium obtained from the reconstructed equilibrium applying a cut-off to the boundary at a prescribed normalized poloidal flux. Post-processing may also include calls to the SDSS [START_REF] Coelho | Evaluation of Epsilon-Net Calculated Equilibrium Reconstruction Error Bars in the European Integrated Modeling Platform[END_REF] code for the calculation of confidence bands on the profiles in a nonprobabilistic approach.

FIG. 1 -Layout of the equilibrium reconstruction workflow in Kepler. Workflow settings are available either on the canvas or on the appropriate composite actors (in green).

All codes featured in the workflow use the same categorization of physics and engineering data and the same methods for accessing the data, thus reducing one of the possible origins of mismatches on the reconstructed equilibria. The workflow orchestration and design can easily accommodate different types of use, e.g. reconstruction with magnetics data only or including additional diagnostic constrains. Simple switches with conditional logic are implemented that allow for using different codes in the pool set as shown in Fig. 2.

FIG.2 -Workflow selection of the currently available equilibrium reconstruction actors (set by parameter FBEcode).

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EQUILIBRIUM RECONSTRUCTION RESULTS

As mentioned in Section 1, the underlying design of the modelling platform is flexible enough to allow for the same physics code to comply and run using different versions of the data model or workflow orchestration engine. In fact, different wrappers to the physics codes can easily be built with interfaces to the different type of hierarchical data models being used on input and output. Therefore, when comparing the physical results obtained by several codes, it is essential that all use the same underlying data model and that the data provenance is precisely the same. Taking this into consideration, two different JET plasma discharges were considered: #89140 and #84600. The former was analysed using both IMAS and EU-IM software infrastructures and the results shown correspond to the EU-IM case. The latter was analysed solely within IMAS, running either on the EUROfusion Gateway cluster installed in MARCONI HPC centre or on the FREIA cluster installed at JET. For #89140, EQUAL and EQUINOX codes were used and standalone EFIT results (data mapped into CPOs) ran on JET cluster are used as reference. For the #84600, EQUAL and NICE codes were used and an IMAS compliant EFIT++ version running on JET cluster is used as reference.

The JET hybrid discharge (#89140) was considered in order to highlight the influence of internal MSE data on the equilibrium reconstructions. As shown in Fig. 3, it has a plasma current and toroidal magnetic field of 1.4MA and 1.75T respectively and 12.5MW of NBI and 1.9MW of ICRH ([49-52s]). The discharge showed frequent (~200ms period) sawtooth activity during the NBI heated stage, as evidenced by the bursty behavior in the magnetic field perturbation with odd toroidal mode number (n-odd signal in Fig. 3), a clear indication that the central safety factor q(0) should periodically drop below unity in between sawtooth crashes.

FIG. 3 -Details on JET plasma discharge #89140 (left), odd (red) and even (blue) toroidal mode number perturbed magnetic field amplitudes (middle) and spectrogram of Mirnov magnetic coil (right). Units and scale of plasma current (Ip) are in MA, toroidal field (B vac ) in Tesla, NBI power (P NBI ) in 10MW and ICRH power (P ICRH ) in MW.

The time chosen for the equilibrium reconstruction was t=47.45s since MSE measurements were available at that time and, coincidentally, a n=1 sawtooth precursor was also present, indicating that a magnetic surface with safety factor q=1 ought to be present and reconstructed by the codes. When using magnetics only, standard intra-shot EFIT fails to recover the q=1 surface since q>1 in the entire plasma volume. On the other hand, by properly tuning the codes with less aggressive profile regularizations (as done for EFIT and EQUAL but not on EQUINOX for comparison), one can recover q<1 in the deep core although, as anticipated, the sawtooth crash cycle oscillations remain unresolved. The comparison between EFIT (labeled 'EFTF' to distinguish for magnetic only), EQUAL and EQUINOX is shown in Fig. 4, where the poloidal magnetic flux map is shown on the left together with profiles of plasma pressure (P), flux surface averaged toroidal current density (Jphi) and safety factor (q). The time traces for the on axis safety factor is also shown on the right panel.

FIG. 4 -Plasma equilibrium reconstruction at t=47.45s using magnetic only for #89140 (left) and time traces of the computed safety factor on axis (right).

Good agreement is found with EFIT in the plasma boundary (<1cm for EQUAL, <2cm for EQUINOX at outer midplane) and divertor strike points (<2.5cm) is found and fitting errors in the magnetics are below the experimental 5mT error (see Fig. 5). Even with magnetics only, the regularisations chosen in EQUAL allowed for some relaxation in the edge profiles and, in view of the ELMy character of the discharge, indicate some traces of a pedestal at the edge. This is however insufficient to resolve the pressure profile at the edge and kinetic pressure assisted reconstructions are mandatory which shall be included in future versions of the workflow.

FIG. 5 -Fitting errors on magnetic sensors for equilibrium reconstruction at t=47.45s for #89140. Red signal indicates the absolute errors on the measurements.

When MSE data is included as a constrain, a clear improvement on the overall agreement in the q-profile (<5% difference in q(0) among the 3 codes) is obtained as shown in Fig. 6. The q-profile is also in agreement with preliminary estimates of the q=1 surface from the inversion radius in electron temperature profile as observed before and after the sawtooth crash occurring at t~47.55s. Considering the relatively high experimental error on the magnetics, although mostly an "internal constrain" (edge channels not used as constrains), MSE assisted reconstructions may lead to some changes on the geometry of the plasma separatrix of the best fitted solution, with differences in X-point location and strike points observed to vary by less than 1.5cm when comparing magnetics only for instance with EQUAL code. The constant time traces and jump at t=49.5s indicated by EQUINOX are simply due to too coarse grid resolution and thus a very rough estimate.

FIG. 6 -Details on divertor region for equilibrium reconstruction at t=47.45s for #89140 with MSE constrain (left) and time evolution of the lower X-point Z-coordinate with magnetics only and MSE assisted using EQUAL and EQUINOX (right). The jump at t~49.5 observed in EQUINOX with MSE data is simply a grid resolution artifact.

The JET discharge (#84600) was considered next in order to compare EFIT++, EQUAL and NICE codes, all running within IMAS software infrastructure, using only magnetics data. Although the experimental dataset used by the 3 codes is the same, different functional types (B-splines for EQUAL/NICE and first order polynomial for EFIT++) are used for dP/dψ and FdF/dψ (ψ is the poloidal magnetic flux) and slightly different regularisations (but all penalizing sharp gradients at the edge) are used. As shown in Fig. 7, discharge #84600 is characterized by plasma current of ~2MA, toroidal magnetic field of ~1.9T and NBI assisted heating starting at 9.5MW and dropping to ~6.5MW. Similarly to #89140, the discharge showed frequent (~200ms period during 6.5MW NBI stage) sawtooth, eventually disappearing by t~52.5s.

FIG. 7 -Details on JET plasma discharge #84600 (left), toroidal mode number perturbed magnetic field amplitudes (middle). and spectrogram of Mirnov magnetic coil (right). Units and scale of plasma current (Ip) are in MA, toroidal field (B vac ) in Tesla and NBI power (P NBI ) in 10MW.

Similarly to #89140, the chosen time instant (t=50.0s) sits in between 2 consecutive sawtooth crashes, indicating that the q=1 surface should be present in the plasma. In addition, as the discharge evolves and the sawteeth become smaller in amplitude and eventually vanish, one should expect an increase of the on axis safety factor q(0). In Fig. 8 one shows the equilibrium reconstruction results at t=50s, showing the flux map and relevant profiles as well as the time traces of the on axis safety factor. As it is easily observed, all three codes correctly reproduce the increase in core safety factor that is consistent with mitigation of the sawteeth. The sawtooth inversion radius as inferred from High Resolution Thomson scattering indicates that the q=1 surface is located at normalised rho poloidal radius around 0.4, in closer agreement with the estimates from EQUAL. Similarly to the case of #89140, results on the plasma geometry including the separatrix are in close agreement among the three codes as inferred from Fig. 9. However, an offset of 2.5cm on radial position of Low Field Side strike point is found when comparing the code results with estimates from Infra Red camera peak heat flux data taking into account cross field transport.

FIG. 9 -Details on divertor region for equilibrium reconstruction at t=50.0s for #84600 with magnetic only (left) and time evolution of the lower X-point Z-coordinate (right).

CONCLUSIONS

In this paper first results on the plasma equilibrium reconstructions of JET discharges using the IMAS modelling infrastructure installed at JET as well as the EU-IM platforms were shown. A Kepler workflow was developed which performs routine equilibrium reconstructions over the whole pulse, using at present only magnetic diagnostics and MSE measurements as constrains. The results on JET discharges #89140 and #84600 primarily demonstrate the modular workflow approach, with a reasonable agreement between the several reconstruction codes involved even though slightly different regularisations in the reconstructions are used. Plasma boundary characteristics e.g. X-point, strike points, outer and inner gaps show differences between the codes within a ~2cm range and q-profile estimation in the core is shown to be sensitive to the regularization used but with q=1 surfaces clearly present during sawteeth activity. Motional Stark Effect is also clearly beneficial in providing more accurate estimations of the magnetic flux and field distributions in the core of the plasma. It is worth noting that the results are preliminary, far from optimal and do not include the error bars on the time traces or profiles. As such they should be interpreted primarily as a practical demonstration of the workflow. Future developments to the workflow include the addition of a dedicated actor that connects to experimental database, imports the diagnostic data and fills in the appropriate IDSs, include Stokes polarimetry as a constrain (recent development on NICE code [START_REF] Faugeras | Equilibrium reconstruction at JET using Stokes model for polarimetry[END_REF]) and also thermal plasma pressure. The work presented here also demonstrated the portability of IMAS, as the same workflow design and implementation can work seamlessly on different computer clusters (EUROfusion Gateway within MARCONI-HPC and FREIA cluster at JET) running a different software environment with IMAS infrastructure installed and IDS and Kepler compliant versions of the physics codes.

FIG. 8 -

 8 FIG. 8 -Plasma equilibrium reconstruction at t=50.0s using magnetic only for #84600 (left) and time traces of the computed safety factor on axis (right).

ACKNOWLEDGEMENTS

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. IST activities also received financial support from "Fundação para a Ciência e Tecnologia" through project UID/FIS/50010/2013. The views and opinions expressed herein do not necessarily reflect those of the European Commission.