Claude Daubert
email: jcdaubert@orange.fr

MD Philippe Mabo

MD Christophe Leclercq

Do novel non-invasive ECG techniques improve patient selection for CRT?

Conflict of interest statement : Nothing to declare Guidelines recommend cardiac resynchronization therapy (CRT) to reduce morbidity and mortality for patients with symptomatic heart failure (HF), a reduced left ventricular ejection fraction and evidence of ventricular dyssynchrony. In the absence of validated markers of

mechanical dyssynchrony, the current recommendations are based on surface ECG with a prolonged QRS duration (QRSd) [START_REF] Tracy | ACCF/AHA/HRS Focused Update of the ACCF/AHA/HRS 2008 Guidelines for Device-Based Therapy of Cardiac Rhythm Abnormalities[END_REF][START_REF] Brignole | ESC Guidelines on cardiac pacing and cardiac resynchronization therapy: the Task Force on cardiac pacing and resynchronization therapy of the European Society of Cardiology (ESC)[END_REF] . Although none of the landmark studies used QRS morphology as inclusion criteria, guidelines indicate that morphology is an important determinant of response and therefore make a stronger recommendation in patients with typical left bundle branch block (LBBB). QRS duration and QRS morphology by standard 12-lead ECG measures are the current cornerstones of patient selection to CRT [START_REF] Daubert | Avoiding non-responders to cardiac resynchronization therapy: a practical guide[END_REF] .

Surface ECG is a simple, cheap and widely available tool to study electrical dyssynchrony but it has limitations. Because of the limited number of leads, it only allows gross approach of the activation process and may not capture the entire activation sequence. Although QRSd measurement had a good reproducibility in controlled studies with core center analysis (mean intra-and inter-observer variability of 1.6% and 1.4% on native QRS 4 ), non-controlled studies reported significant variations according to the lead(s) used, a poor reproducibility and a low concordance between manual and computerized measures 5,6 . There is therefore a need for novel non-invasive ECG techniques, more efficient and less investigator-dependent A first alternative could be the automated vectocardiogram (VCG) that contains 3D information of the cardiac electrical forces. It can easily be derived from the standard 12-lead ECG and analyzed automatically by customized MATLAB software. [START_REF] Engels | Why QRS duration should be replaced by better measures of elecrtical activation to improve patient selection for cardiac resynchronization therapy[END_REF] In the context of CRT, the value of VCG was extensively investigated by the Maastricht group [START_REF] Engels | Why QRS duration should be replaced by better measures of elecrtical activation to improve patient selection for cardiac resynchronization therapy[END_REF][START_REF] Van Deursen | Vectocardiography as a tool for easy optimization of cardiac resynchronization therapy in canine left bundle branch block hearts[END_REF][START_REF] Rad | Vectorcardiographic QRS area identifies delayed left ventricular lateral wall activation determined by electroanatomic mapping in candidates for cardiac resynchronization therapy[END_REF][START_REF] Van Deursen | Vectorcardiographic QRS area as a novel predictor of response to cardiac resynchronization therapy[END_REF][START_REF] Végh | T-wave area as biomarker of clinical response to cardiac resynchronization therapy[END_REF] . In a study in canine LBBB hearts, van Deursen 8 showed that maximum QRS vector amplitude (VA QRS ) was closely correlated with electrical and mechanical ventricular dyssynchrony. They also suggested that VCG may be a reliable and easy tool for individual optimization of CRT. In patients candidate for CRT, they showed that VA QRS and the QRS complex area (QRS AREA ) identified delayed LV lateral wall activation (assessed using coronary venous electroanatomic mapping) better than QRSd and/or QRS morphology on standard ECG . By imaging activation sequences and measuring spatial differences in activation times, BSEM also allows precise evaluation of electrical asynchronies. Several ventricular dyssynchrony indices can be derived from intrinsic maps, in particular electrical dyssynchrony index (ED) [START_REF] Shah | Body surface electrocardiographic mapping for non-invasive identification of arrhythmic sources[END_REF] , LV total activation time (LVAT) and ventricular electrical uncoupling (VEU), defined as the difference between LV and RV activation times 15

.

The system was used first for exploratory studies in CRT candidates to characterize the type and degree of ventricular dyssynchrony according to baseline characteristics : etiolology, QRSd and QRS morphology 14-16 . The value of BSEM to predict the 6-month clinical response to CRT using a clinical composite score was assessed by Ploux et al [START_REF] Ploux | Noninvasive electrocardiographic mapping to improve patient selection for cardiac resynchronization therapy: beyond QRS duration and left bundle branch block morphology[END_REF] . Electrical dyssynchrony was quantified using standard deviation of activation times (SDAT). The value of baseline SDAT and its change after CRT to predict the 6-month echocardiographic response was assessed in 66 CRT patients [START_REF] Gage | Changes in electrical dyssynchrony by body surface mapping predict left ventricular remodeling in cardiac resynchronization therapy patients[END_REF] . Results showed that SDAT and its changes predicted CRT response better than QRS duration. A native SDAT >35 msec could be the best predictor.

In summary, VCG and BSEM are both candidates to dethrone the 12-lead surface ECG for patient selection to CRT. In small size observational studies, dyssynchrony indices derived from the two techniques better predicted CRT response than QRSd and/or QRS morphology. These preliminary results must now be confirmed in larger multicenter prospective studies evaluating the clinical benefit and cost-effectiveness of these novel non-invasive ECG techniques compared with12-lead surface ECG .

Ideally, VCG and BSEM should be evaluated in the same trials to determine which technique is the best

9 . 11 . 12 .

 91112 In a single-center observational study, QRS AREA[START_REF] Van Deursen | Vectorcardiographic QRS area as a novel predictor of response to cardiac resynchronization therapy[END_REF] but also T-wave AREA[START_REF] Engels | Why QRS duration should be replaced by better measures of elecrtical activation to improve patient selection for cardiac resynchronization therapy[END_REF] were shown to predict the 6-month volumetric response to CRT (decrease>15% in LVESV) better than QRS measures on 12-lead surface ECG. Similar results were observed with SAI QRST, an averaged arithmetic sum of absolute areas under the QRST curve in a retrospective analysis of the SMART AV trial[START_REF] Tereshchenko | Novel measure of electrical dyssynchrony predicts response in cardiac resynchronization therapy: Results from the SMART-AV Trial[END_REF] . In the same way, a retrospective analysis of 335 CRT patients showed that T-wave AREA was the better predictor of a combined clinical endpoint of death, HF hospitalization or heart transplant/LVAD implantation over a 36-month follow-up period The value of VCG to predict the12-month echocardiographic response is currently assessed in a prospective multicenter observational study (ClinicalTrials.gov: NCT01519908)Another alternative is body surface electrocardiographic mapping (BSEM). The original system uses a multielectrode vest recording BS potentials from 250 sites around the entire torso and a thoracic computed tomography (CT) scan providing epicardial-surface and torso geometries BS potentials and CT images are merged and processed to reconstruct epicardial potentials, electrograms, and isochrones on the heart surface during a single beat. The system was originally developped to identify arrhythmic sources and showed remarkable performances 13


Houda Jmila
email: houda.jmila@telecom-sudparis.eu

Kaouther Drira
email: kaouther.drira@telecom-sudparis.eu

Djamal Zeghlache
email: djamal.zeghlache@telecom-sudparis.eu

A self-stabilizing framework for dynamic bandwidth allocation in virtual networks

Keywords: Virtual network embedding, dynamic bandwidth allocation, elasticity, self stabilization, local view, cloud, distributed and parallel algorithms;

This paper addresses dynamic bandwidth allocation for virtual network (VN) resources to respond to increasing or decreasing applications requirements in cloud environments. A distributed and local-view framework, composed of a controller and three algorithms running in substrate nodes, is proposed to deal with all types of bandwidth demand fluctuations in embedded virtual networks. The framework is based on the Self-Stabilization concept to drive the system back to a "stable state" when new bandwidth demands drift the system away into an "unstable state". Performance evaluation results demonstrate the effectiveness of our proposal in handling bandwidth demand fluctuations in convergence speed and cost.

I. INTRODUCTION

Network virtualization allows multiple virtual networks to coexist on a shared physical infrastructure. A critical issue in network virtualization is virtual network resource provisioning. This deals with the optimal allocation of the VN onto the substrate network (SN), and the management of the allocated resources throughout the life-cycle of the VN.

The current state of the art has extensively addressed the static provisioning where a fixed amount of resources is allocated to the VN for its entire lifetime. But cloud services and environment require the elastic allocation of resources according to increasing/decreasing applications demands. Moreover, traditional resource provisioning solutions employ centralized and total-view based techniques that face serious limitations when applied to large and highly dynamic cloud environments. In centralized solutions, maintaining dynamically up-to-date description of real-time substrate resources states induces high latency during analysis and enforcement of decisions because of the management traffic overhead with the central entity. This leads to low responsiveness to VNs evolution and affects in fine the cloud user satisfaction.

In this paper, we propose a distributed and parallel framework to manage bandwidth demand fluctuation without needing a global view of available resources to avoid costly updating and monitoring. The proposal is based on the selfstabilization [START_REF] Dijkstra | Self-stabilizing systems in spite of distributed control[END_REF] concept that guaranties the convergence of the system to a stable state in a finite time regardless of its initial state. In fact, we view bandwidth requirements fluctuations as events that drift the system (the substrate network) into an unstable state, and our self stabilizing algorithms, executed locally on each substrate node, as actions that take it back to a stable state where all new bandwidth demands are satisfied.

Section II of this paper presents the background and related work. Section III defines the dynamic bandwidth allocation problem in the context of network virtualization while section IV briefly introduces the concept of Self-Stabilization. Section V describes our model and proposal. The effectiveness of our algorithms is assessed in section VI where performance evaluation is reported.

II. RELATED WORK

To the best of our knowledge, only [START_REF] Mijumbi | Design and evaluation of learning algorithms for dynamic resource management in virtual networks[END_REF], [START_REF] Mijumbi | Neural network-based autonomous allocation of resources in virtual networks[END_REF], [START_REF] Mijumbi | A neuro-fuzzy approach to self-management of virtual network resources[END_REF] and [START_REF] Seddiki | Automated controllers for bandwidth allocation in network virtualization[END_REF] address the VN bandwidth demand fluctuation in a distributed manner. In [START_REF] Mijumbi | Design and evaluation of learning algorithms for dynamic resource management in virtual networks[END_REF], authors propose a decentralized multi-agent resource management system based on Reinforcement Learning. Each agent (substrate node/link) dynamically adjusts the allocated resources to avoid under utilization of the substrate network and to satisfy new demands. The agents choose an action among a pre-defined set of actions (Decrease/increase allocated resources by 50/37/25/12.5 percent or maintain the currently allocated resources). The choice is made according to the results of a decentralized Q-learning based algorithm that iteratively approximates the state action values. The main drawbacks of this proposal are i) a limited set of actions that affects the algorithm efficiency and action granularity, and ii) the need for a high number of learning episodes to determine an optimal decision policy.

Authors of [START_REF] Mijumbi | Neural network-based autonomous allocation of resources in virtual networks[END_REF] improve the efficiency of the previous system by conceiving an autonomous system based on Artificial Neural Networks (ANN). They represent each substrate resource as an ANN whose input is the network resource usage status and the output is an allocation action. An error function is used to evaluate the desirability of ANN outputs and hence perform online training. Note that even if neural networks are important for their learning and generalization capabilities, they do not have a clearly defined way on how the number of layers as well as the number of neurons in each layer are determined.

Work in [START_REF] Mijumbi | A neuro-fuzzy approach to self-management of virtual network resources[END_REF] is an extension of the previous model using an adaptive hybrid system composed of Neural Networks, Fuzzy systems and Reinforcement Learning. The system dynamically adjusts both the substrate network usage and structure by adding or removing substrate resources. The SN is modeled as a distributed system of autonomous agents. For each agent, an initial knowledge base is defined using supervised learning subsequently improved with Reinforcement Learning.

Unfortunately, authors still limit the action granularity of the algorithm.

Authors of [START_REF] Seddiki | Automated controllers for bandwidth allocation in network virtualization[END_REF] use a distributed controller system to adapt bandwidth allocations to the VN dynamic workloads. The system predicts the optimal request for each VN and adjusts the offered bandwidth accordingly.

All the previous work scales up and down the allocated resources but fixes the VN topology. This prior art does not address bandwidth demand fluctuation with topological changes: i.e. when adding/ deleting virtual links to/from the VN. The authors do not take advantage of virtual resources migration (re-allocation) either. We devise a self-stabilizing framework to deal with all types of bandwidth fluctuations without the need for a learning phase, unlike [START_REF] Mijumbi | Design and evaluation of learning algorithms for dynamic resource management in virtual networks[END_REF]- [START_REF] Mijumbi | A neuro-fuzzy approach to self-management of virtual network resources[END_REF]. To the best of our knowledge, this is the first distributed and localview based framework that manages all types of bandwidth demand fluctuations; i.e. increasing/diminishing bandwidth needs of embedded virtual links, new bandwidth requests to connect nodes or the need to free link bandwidth no longer needed between nodes.

III. PROBLEM FORMULATION

Virtual network embedding (VNE) can be decomposed into two stages: the initial VNE and a dynamic resource management phase. The first stage provides an efficient mapping of virtual resources (nodes and links) onto the physical hosts (nodes and substrate paths), while the second manages the resource demand fluctuation of already embedded VNs.

A. Initial VNE

Let us represent the substrate network by a weighted undirected graph Ĝ = ( N , L). N (resp. L) is the set of substrate nodes (resp. links). Each substrate node n ∈ N is characterized by an amount of available resources nav (typically CPU and memory) and a unit cost cost(n). Each substrate link l ∈ L is associated with an available bandwidth lav and a unit cost cost( l).

Similarly, Let G = (N, L) be a weighted undirected graph that represents the VN request topology. N (resp. L) is the set of required virtual nodes (resp. links). Let req n (resp.req l ) denote the minimum required capacity (resp. bandwidth) of the virtual node n ∈ N (resp. link l ∈ L).

Figure 1 shows an example of VN and SN. The numbers in rectangles next to the substrate (resp. virtual) nodes represent the amount of available (resp. requested) node resources and the numbers next to the substrate (resp. virtual) edges represent the available (resp. required) bandwidth.

The initial VNE stage is out of the scope of this paper, diverse approaches can be found in the literature (centralized [START_REF] Zhu | Algorithms for assigning substrate network resources to virtual network components[END_REF], [START_REF] Fajjari | Vne-ac: Virtual network embedding algorithm based on ant colony metaheuristic[END_REF] and distributed [START_REF] Houidi | A distributed virtual network mapping algorithm[END_REF], [START_REF] Beck | A distributed, parallel, and generic virtual network embedding framework[END_REF] ). We focus exclusively on the second stage and more specifically on managing bandwidth demand fluctuations by adapting links and network topology.

B. Management of resource demand fluctuation

The first stage, successful initial VNE, results in multiple VNs running simultaneously over the SN. This is the starting Over time, the VN end user requirements can change, for instance when staring/ completing a new task/ application, or when their required resources change, consequently, the corresponding VN characteristics (topology and resources requirements) will change.

We enumerate four bandwidth demand fluctuation scenarios: i) partially releasing no more required bandwidth ii) completely removing virtual links iii) adding new virtual links and iv) allocating more bandwidth to embedded links. To cope with scenarios i) and ii) the substrate network provider should release some bandwidth. For cases iii) and iv), the provider should allocate more bandwidth on the hosting paths when feasible, otherwise find new paths to support the requested bandwidth.

To model a virtual link l bandwidth demand fluctuation we define allo l , the amount of bandwidth allocated to l. Note that req l = allo l when the bandwidth demand is met. Else, we formulate the four bandwidth demand fluctuation scenarios as follows:

• i) Decrease in Bandwidth Requirement (DBR): allo l > req l > 0 (the required bandwidth is lower then the allocated).

• ii) Link Removal (LR): 0 < allo l and req l = 0 (link l does not require the allocated bandwidth any more and the link should be removed).

• iii) Link Addition (LA): allo l = 0 and 0 < req l (link l requires an amount of bandwidth not allocated yet, it should be added to the VN topology).

• iv) Increase in Bandwidth Requirement (IBR: 0 < allo l < req l (the required bandwidth is higher then the allocated).

We opt for a self-stabilizing approach to manage bandwidth demand fluctuation. We will first introduce the Selfstabilization concept and motivate this choice, then, we will describe our solution.

A. Introduction to Self-Stabilization

Self-Stabilization is related to Autonomic Computing [START_REF] Parashar | Autonomic computing: An overview[END_REF], which entails several "Self-*" attributes: self-management, self-configuration, self-healing, etc. It is the property of an autonomous process to reach a "stable state" no matter what initial state is given. Hence a self-stabilizing system will eventually correct itself automatically without the need for an outside intervention.

In a self stabilizing system, each of the individual entities (nodes) composing it maintains local variables determining its state that can be either stable or active. A node can execute an action to make a move and change its state. The system is said to be stable when all the nodes are stable. Hence all the active nodes should make moves to reach the system stability. The number of moves (or rounds if nodes make moves simultaneously) required to reach the stable state is often used to measure the efficiency of the algorithm.

Central to the theory of self-stabilization is the notion of daemon [START_REF] Dubois | A taxonomy of daemons in self-stabilization[END_REF], it is the entity responsible for setting the actions/moves execution order. In fact, in each round, it selects some of/ or all active nodes to make a move, and the process continues until there are no more active nodes, i.e. until reaching stable state. In the self stabilization literature, a daemon is often viewed as an adversary to the system that tries to prevent stabilization by scheduling the worst possible nodes for execution.

However, in our work, this definition seems unnecessarily restrictive. In fact, most resource management systems ( [START_REF] Seddiki | Automated controllers for bandwidth allocation in network virtualization[END_REF], [START_REF] Carter | Resource allocation in a distributed computing environment[END_REF]) employ a manager (or controller) to supervise and harmonize the distributed entities behavior. In the same perspective, we replace the daemon by a scheduler that helps the system to converge by setting the best nodes execution scheme. In the following, this scheduler is called controller.

Unlike resource allocation traditional central entities, our controller does not require the dynamic description of physical resources to make scheduling decisions. Only the substrate network topology (generally static) and the list of the physical nodes scheduled to execute are needed. In fact, the nodes are the only responsible of choosing the actions they need to perform, the controller simply sets the actions execution order using an elementary scheduling algorithm that sorts the nodes according to their action priority.

B. Motivation for Self-stabilization

A self stabilizing system i) is local-view, hence maintaining a global up-to-date description of the physical resources is not required, ii) allows simultaneous nodes execution, hence favors managing multiple and different demand fluctuations simultaneously, iii) is suitable to the topology changing networks as modification of the code when the system topology evolves during operation is not required and vi) does not require initialization as the system system converges irrespective of its original state. All this reasons motivated us to explore the self-stabilization concept.

V. A SELF-STABILIZING MODEL FOR BANDWIDTH ALLOCATION

A. System model

This section extends the network model proposed in III with self stabilizing elements. Namely, we expand the virtual link and substrate node models.

1) Virtual link description: Let src l and dst l denote the source and destination virtual nodes of virtual link l, and let P l denote the ordered list of substrate nodes composing l's hosting path. In Fig. 1, for l = (a, c), we have src l = a, dst l = c and P l = â, b, ĉ, d, ê .

2) Substrate node description: We suppose that each substrate node knows the local variables of its neighbors (through periodic message passing or a shared memory ( [START_REF] Gambette | Introduction l'algorithmique rpartie et l'autostabilisation[END_REF])), yet it can only modify its local variables. We model each substrate node local-view of its environment as follows:

Local view of mapping: for each virtual link l passing through or incident to substrate node n, let nx denote the feature x describing l and saved in (seen by) n (x ∈ l, allo l , src l , dst l , P l ). For example, nallo l describes the amount of bandwidth allocated to l as seen by n. In other terms, each substrate node saves "a copy" of the parameters describing the virtual links l ∈ L n. When the system is in the stable state, all the substrate nodes in P l share the same l's description, else end to end update is required.

Local view of available resources: let N eigh(n) denote the set of n neighbors (adjacent substrate nodes). Let m ∈ N eigh(n) be such a neighbor. To describe the substrate link ( m, n) connecting n and m in a distributed manner, node n (resp. m) holds a variable (n, m) av (resp ( m, n) av ) defining the available bandwidth on ( m, n). When the SN is stable, (n, m) av = ( m, n) av i.e. the substrate nodes n and m share the same view on substrate link (n, m) available bandwidth, otherwise an update is required.

B. The Self stabilizing framework

We propose a framework to deal with bandwidth demand fluctuation of embedded virtual networks. The framework contains algorithms, composed of a set of actions executed locally by the substrate nodes, and a controller defining the actions execution scheme. We propose three different algorithms to deal with the four bandwidth fluctuation types described previously. We will first describe the controller role and the general execution plan, then we will describe the algorithms.

1) Controller description: The controller is responsible of setting the actions execution plan. It holds a local database containing a subset of substrate nodes, called active nodes, scheduled to execute one or many actions. An active node requires the controller permission to perform. Each action A is associated with a "priority" p A defining its precedence.

The framework execution plan is organized into rounds. In each round, the controller examines the list of active nodes in its database and allows the nodes scheduled with the most urgent action/task to run simultaneously. At the end of the round, the controller updates its database (it removes the executed actions and keeps the others for the next round). This process continues until reaching system stability. Note that the local views of substrate nodes are combined with the controller to reduce load on this central point of failure that usually has a back up for fault tolerance. One of the substrate nodes can be elected as a central node as needed.

Two reasons can activate a substrate node: either a new bandwidth request is submitted, or the node is solicited by a neighboring node. When a virtual link l requires an amount of bandwidth req l ( = allo l ), the substrate node hosting src l (host(src l )) is activated: an entry of the form (host(src l ), A, l, req l ) is added to the controller database to define i) the activated node, ii) the task to run, iii) the concerned virtual link and vi) the new request. Substrate nodes cooperate to reach stability, to do so a node n can activate its neighbor m ∈ N eigh(n) to perform an action A (concerning a virtual link l with req l = allo l ). Node n asks the controller to add ( m, A, l, req l ) in its database. The different possible actions are described in the following.

2) Algorithms description: Three algorithms manage the four bandwidth fluctuation types. Each algorithm is composed of actions. The first algorithm concerns scenarios i) Decrease in Bandwidth Requirement and ii) Link Removal. The second manages the Link addition (scenario iii). The third handles the Increase in Bandwidth Requirement (scenario iv). Due to lack of space, only the two first algorithms will be detailed.

Algorithm1, Decrease in Bandwidth Requirement or Link Removal: This algorithm deals with the two following cases: i) a virtual link l requires less bandwidth or ii) should be completely removed from the VN topology. Hence, no more required bandwidth should be freed and the virtual link mapping should be updated along the hosting path.

To do so, the origin substrate node (hosting src l ) is activated upon receiving the new request. If selected by the controller, the node executes the following action: Trigger Allocation, to i) update its state (mapping and available resources) and ii) activate the next node in P l . If selected by the controller in the next round, the latter runs an other action: De-allocate; it updates its local description to synchronize with the soliciting node, then activates the next node in P l . This process continues ∈ N n (if n is not the last node in P l ) then 4:

(n, ô)av += (ô allo l -req l ) where ô = P l next (release no more required bandwidth in (n, ô), ô is the next node in P l ) 5:

Add (ô, Deallocate, l, req l ) to the controller database to activate ô 6: end if 7: if req l = 0 (i.e. in case of virtual link removal) then 8:

L n.remove(l) (i.e. remove l from n's mapping list) 9: end if until reaching the end (extremity) of the substrate path. The action Trigger Allocation (resp. De-allocate) is associated with a priority p T A (resp. p D ), such that p T A > p D . The idea behind this choice is to favor managing multiple requests simultaneously. Details can be found in Algorithm 1.

Figure 2 depicts a detailed example of the algorithm execution. We use the VN and SN of figure 1. For each round, we show two tables. The first describes the most relevant substrate node parameters and the second illustrates the controller database (at the start/ end of the round). The second table is a 4 column table defining i) the active nodes ii) the scheduled actions, iii) the concerned virtual links and vi) the new bandwidth requests. In each round, the executing substrate nodes and the main changes are colored in red.

Suppose that the virtual link l = (a, b), initially requesting 20 bandwidth units and hosted by the substrate path P l = (â, ê, d), is now requiring a new amount of bandwidth = 10 units, and let us run through the algorithm.

Initially, the substrate network is stable: the required bandwidth is met for all the embedded virtual links, in particular, âallo l = req l = 20, where â = host(src l ). Moreover, (â, ê) av = (ê, â) av = 50 and the controller database is empty.

After receiving l's new request req l = 10, the substrate node â is activated to run the Trigger de-allocation action. As there is only one node in the controller database, â is selected to run in the first round. It i) updates l's mapping to have âallo l = 10, ii) frees the no longer required bandwidth in the substrate link (â, ê) ((â, ê) av = 60), and finally iii) activated the next node in P l : ê is added to the controller database.

In the second round, node ê is selected by the controller to execute the action De -allocate. It i) updates l mapping and the available bandwidth in (ê, â) to synchronize with â, ii) releases bandwidth from (ê, d) and finally iii) activates d. In the last round, node d runs the same steps as ê except activating the the next node in P l as the path's end is reached. Note that the system has reached stability in 3 rounds, which corresponds to the number of substrate nodes composing P l . Moreover, all the substrate nodes in P l have the same new l description at the end of execution, and the amount of available bandwidth is updated through the entire substrate path. Algorithm 2, Link Addition:This is the case where a new virtual link l should be added to connect two embedded virtual nodes.

To do so, a substrate path connecting the substrate nodes hosting src l and dst l , and having enough available bandwidth (> req l ) should be found. We define the cost of embedding a virtual link l as the sum of costs of the substrate links hosting l, as in [START_REF] Jmila | An adaptive load balancing scheme for evolving virtual networks[END_REF]. The aim is to find the most cost effective substrate path, in a distributed, and self stabilizing manner.

To achieve this, the algorithm runs in two steps: first it i) searches all available paths (having enough bandwidth) to connect the two end nodes and ii) saves the required bandwidth in each of them. Second, it i) selects the most cost effective substrate path among the K first arriving path proposals (where K is a tuning variable), ii) maps the virtual link l in the best path, and iii) releases previously reserved bandwidth in non selected paths. Four actions compose this algorithm:

Action 1, Trigger searching and reserving bandwidth: This action concerns the source substrate node (hosting src l ). It triggers the first algorithm step. A node n executing this action checks if all its attached substrate paths are saturated, if it is the case, the new bandwidth request is immediately rejected as no path connecting host(src l ) and host(dst l ) can be found. Else, n starts building a path proposal by adding n to P l , initially empty. Then, for all available attached substrate links, n reserves bandwidth and activates the corresponding nodes to do so.

Note that the existence of an available substrate path to connect host(src l ) and host(dst l ) is not guarantied, hence, we risk to search infinitely for a nonexistent substrate path and never reach stability. To avoid this, we define a timer T imer l , (n, m)av -= req l , P l .add(n) (i.e. synchronize with the soliciting node and continue building the path) 7:

for for all available links (n, ô) with no risk to produce a loop do 8:

(n, ô)av -= req l (i.e. reserve bw in the link) 9:

Activate the neighbor ô to Reserve BW 10:

end for 11:

end if 12: else 13:

(i.e. the path end is reached and this is the Rth path proposal) 14:

(n, m)av -= req l , P l .add(n) (i.e. synchronize with the soliciting node and continue building the path) 15:

if R < K then 16:

Save Add ( m, Release BW, l, req l ) to the controller database (i.e. activate m to Release BW) 34: end if to limit the path search phase duration. It can be set in terms of rounds, and depends on the SN dimension. T imer l is launched when n starts searching for a path. If T imer l expires and no path is found, the request is rejected. Detailed algorithm can be found in Algorithm 2, action 1. Action 2: Search and reserve bandwidth:

This action concerns the other substrate nodes searching for an available path for l. A node n executing this action runs different operations depending on the situation: IF n is not the end node, there are two cases: in the first case, all adjacent substrate links are unavailable or we risk to produce a loop if we add the neighboring node to l's path (the node is already in P l ), we conclude that n is the end of a no-through path! and previously reserved bandwidth should be released from the saved path. To do so n reactivates the soliciting node to perform a Release bandwidth action, described below, that will be spread back through the reserved path. In the second case:, node n synchronizes with its soliciting node and goes on reserving bandwidth on available substrate links, then solicits corresponding neighbors to do so. ELSE the path end is reached.

Recall that n must wait K path proposals to select the most cost effective path, hence n synchronizes with soliciting node then runs the following actions depending on the rank R of the arriving proposal:

• if R < K: node n simply saves the path proposal.

• If R = K : node n i) selects the most cost effective path, ii) assigns it to l and re-activates the previous node in the selected path to spread back an Allocate bandwidth action, described below, finally iii) for all non selected paths, the reserved bandwidth is released, and the previous node in each path is activated to spread back a Release bandwidth action. Node n informs the controller that a path proposal was found for l to stop reserving other paths. • If R > K: the proposal is arriving late, thus it is rejected: reserved bandwidth is released and a Release bandwidth action is triggered among the reserved path. Note that if a path proposal is found or the link timer has expired, a node activated to execute this action will not run the previously described steps, but simply spreads back a Release BW action to free the previously reserved path.

Action 3: Allocate bandwidth: Node n updates mapping and if it is not the origin node, it activates the previous node in P l to spread back the Allocate bandwidth request.

Action 4: Release bandwidth: Node n releases previously reserved bandwidth and if it is not the origin node, it releases reserved bandwidth in (n, m) (with m the previous node in P l ) and activates m to spread back the Release bandwidth request.

Actions priorities: We choose the following priority order for previous actions: p SR < p Rel < p All < p T SR where p T SR is for action Trigger searching and reserving BW, p SR for Search and reserve bandwidth, p All for Allocate bandwidth and p Rel for Release bandwidth action.

The motivation is threefold: first we give the highest priority to Trigger searching and reserving bandwidth action to favor dealing with new arriving requests and thus handle many demands simultaneously. Allocate bandwidth action comes next in order to allocate the virtual links as soon as an available path proposal is found. Finally, by giving Release bandwidth a higher priority then Search and reserve bandwidth, we release bandwidth before searching for available paths to increase the number of path solutions. Note that the controller selects a set of non adjacent active nodes to execute Trigger searching and reserving BW or Search and reserve bandwidth actions to avoid conflicts.

Figure 3 depicts a detailed example of the algorithm 2 execution. We use the same VN, SN and notations of the first example. Imagine that a new virtual link l = (a, c) has to Note that the algorithm converges in 6 rounds, but a path solution was found since the 4th round.

Algorithm3, Increase in bandwidth requirement: This is the case where an already embedded virtual link l requires more bandwidth. To handle such request, we propose a two step algorithm: first, the algorithm checks if there is enough bandwidth on the path hosting l to meet the new demand. If it is the case, the required bandwidth is allocated and the request is satisfied. Else we move to the second step that consists of i) finding a new path for l, and ii) de-allocating l from its old hosting path, in other terms, we perform a virtual link migration. To do so, a slightly modified version of already defined actions in previous Algorithms is used.

Due to lack of space, these actions will not be detailed.

VI. SIMULATION RESULTS AND EVALUATION

In this section, we will evaluate the effectiveness of our algorithms by conducting extensive simulations. We will describe the simulation environment and then present the results.

A. Simulation environment

We adjusted the C++ simulator used in previous work [START_REF] Jmila | An adaptive load balancing scheme for evolving virtual networks[END_REF] to fit our scenario: substrate nodes with local view, and a round per round algorithm execution. The GT-ITM tool ( [START_REF] Zegura | How to model an internetwork[END_REF]) is used to generate random topologies of the substrate and VN networks.

The SN size is set to 50 nodes and each pair of substrate nodes is randomly connected with probability 0.5. The node resource capacity and edge resource capacity are randomly drawn between 0 and 50 for nodes and between 0 and 100 for links. The per unit node and edge resources costs are selected randomly between 0 and 50.

The VNs requests have between 2 and 10 virtual nodes in their topologies with an average connectivity also set to 0.5.The node resource capacity is randomly selected between 0 and 20 and the edge resource capacity between 0 and 50.

In order to initialize the scenario and start the system from a typical situation we map the virtual nodes greedily and follow with the shortest path algorithm to map edges. This step leads to suboptimal embedding that can reflect the state of a SN subject to multiple virtual link evolutions. The central performance metric will be the the number of rounds required to reach the stable state (i.e. to converge), called Convergence T ime. Other metrics will be presented later.

B. Simulation results

1) Algorithm1: Decrease in Bandwidth requirement (DBR) or Link Removal (LD): We simulate two scenarios to assess the effectiveness of this algorithm:

Scenario 1: Only one bandwidth request is managed. In accordance with the example of Fig. 2, Fig. 4.a shows that Convergence T ime is equal to the number of substrate nodes supporting the substrate path hosting the evolving virtual link.

Scenario 2: Multiple bandwidth fluctuations are handled simultaneously. To create a highly dynamic environment and unpredictable states, we select randomly N virtual links among the 96 hosted by the SN as virtual links with fluctuating bandwidth demands. To each link l, we associate a Decrease in Bandwidth Requirement or Link Removal request randomly. For the DBR, we set the new bandwidth demand to req l = allo l /2.

If multiple bandwidth requests are managed simultaneously, Convergence T ime is at least equal to the convergence time when handling the the longest path among those hosting the evolving links. With this idea in mind, we evaluate the performance of our algorithm in handling N bandwidth fluctuations at the same time.

We measure both i) the Convergence T ime when handling all the requests, called CT Multiple Requests and ii) the Convergence T ime when managing only the virtual link with the longest supporting path, called CT One Request and compare.

Figure 4.b depicts the results of 200 averaged runs and shows that both convergence times increase with the number of bandwidth requests. Moreover, the gap between the two curves is small compared to the number of requests handled simultaneously ( CT M ultiple Requests/CT One Request < 1.4 for N=30). In other terms, managing multiple bandwidth fluctuations is at most 1.5 more time expensive then managing only one request. The two convergence times are even equal for low N values.

2) Algorithm 2: Link addition (LA): Recall that this is the case where a new virtual link is added to the VN topology. We will examine three metrics to evaluate the algorithm: the virtual link embedding cost, the convergence time and the controller load. To simulate dynamic and unpredictable LA requests, we select randomly a pair of virtual nodes (among the 133 hosted in the substrate network) as source and destination of the new virtual link. Note that the pair of nodes belong to the same VN and are non adjacent (not connected by a virtual link, in order to avoid multi-graphs).

Embedding cost Recall that the end node of a newly added virtual link should wait for K path proposals to select the most cost effective one. We measure the virtual link embedding cost for K = 1 and K = 2, for different bandwidth demands req l , and compare to the shortest path cost, found with a global view of the system.

Figure 4.c shows the ratio of the embedding cost found with our algorithm, and that of the shortest path algorithm, called Costs Ratiofor different values of req l /Average available BW for 100 accepted requests, where Average available BW is the average of available bandwidth in the substrate network.

Note that the ratio decreases with K : the most path proposals we wait, the most chance we have to find the best path. Moreover, for small values of req l , our algorithm fails in finding the shortest path as there are many path solutions that can meet the demand, however, when req l increases, the number of available paths decreases and the gap between the two algorithms costs decreases.

Convergence time Recall that this algorithm is composed of two steps: the first searches for available paths, and the second assigns the most cost effective path to the virtual link and releases bandwidth from other paths. We measure two time durations: the first phase duration, called T ime to f ind a path and the total convergence time, i.e. the two steps duration, called Convergence T ime. We set K to one, and T imer to 100 rounds, and make evaluation for three SN configurations: ALS = 78%, 64% and 50%, where ALS is the average substrate links saturation, as defined in [START_REF] Jmila | An adaptive load balancing scheme for evolving virtual networks[END_REF]. We measure the time in seconds for more precision. Figures 4.d,[START_REF] Mijumbi | A neuro-fuzzy approach to self-management of virtual network resources[END_REF].e, 4.f depict the results of 100 averaged runs and show that both times decrease when req l /Average available BW increases. In fact, when the required bandwidth is high, there are few available links, hence few substrate nodes will be activated to search for a path, and the system will reach stability more rapidly. In contrary, when req l is small, the majority of the substrate nodes in SN will be activated to search for a path, and the system needs more time to stabilize.

Moreover, looking at Figures 4.4, 4.5, 4.6 jointly, notice that both times decrease with the average link saturation (ALS), for the same reasons explained above. Finally, note that the gap between the two curves increases with ALS, in fact, when there are more available substrate resources (ALS low), more substrate paths will be reserved during the first phase of the algorithm, hence, after finding a path solution, releasing bandwidth from reserved paths will take more time.

Controller load analysis: in each round, the controller selects and notifies nodes to execute part of the bandwidth adaptation. A loose upper bound on controller load (measured by the number of exchanged messages) is obtained when all nodes are involved in each round. The bound is given by the product: substrate network size × the total number of rounds. Figures 4.d,4.e,[START_REF] Mijumbi | A neuro-fuzzy approach to self-management of virtual network resources[END_REF]f show that the controller load curve has nearly the same shape as that of the convergence time since it is proportional to the number of activated nodes. It decreases with the new bandwidth demand for the reasons explained above.

VII. CONCLUSION AND FUTURE WORK

In this paper, a self stabilizing framework was proposed to deal with bandwidth demand fluctuation in embedded virtual networks. The solution is composed of a central controller, and three parallel, distributed and local view algorithms running in each substrate node to handle all types of bandwidth demand fluctuations. Simulation results show that many requests can be managed simultaneously at low cost and in a time effective way. Future work concerns demand fluctuation in embedded virtual nodes. We aim at extending the actual framework to handle all types of resource fluctuations in virtual networks.

Fig. 1 .

 1 Fig. 1. Example of initial VN embedding

Fig. 2 .

 2 Fig. 2. A step by step example of Algorithm 1 execution

Algorithm 1 : 2 :

 12 Decrease in Bandwidth Requirement/Link Removal Inputs:1: The executing node : n The concerned virtual link : l 3: The new required bandwidth : req l Action 1: Trigger de-allocation (Priority=p T A > p D ) Steps: 1: (n, ô)av += (n allo l -req l ) where ô = P l next (i.e. release no more required bandwidth in the substrate link (n, ô) where ô is the next node in l's hosting path) 2: nallo l = req l (update the amount of resources allocated to l) 3: Add (ô, De -allocate, l, req l ) to the controller database (i.e. activate the next node in P l ) 4: if req l = 0 (i.e. In case of virtual link removal) then 5: L n.remove(l) (i.e. remove l from n mapping list) 6: end if Action 2: De-allocate (Priority=p D ) Steps: 1: (n, m)av += (n allo l -req l ) where m = P l previous (i.e. update available bandwidth in (n, m) to synchronize with the the soliciting node m, i.e. the previous node in hosting path) 2: nallo l = req l (update the amount of resources allocated to l) 3: if dst l /

Fig. 3 .

 3 Fig. 3. A step by step example of algorithm 2 execution

Fig. 4 .

 4 Fig. 4. Simulation results

  Search and reserve BW (Priority=p SR )Steps: 1: if T imer l has not expired and no path is found yet for l then 2: if dst l / ∈ N n (i.e. if n is not the end node) then

	Algorithm 2: Link Addition
	Inputs:1: The executing node :	n
	2: The concerned virtual link : l
	3: The new required bandwidth : req l
	Action 1: Trigger searching and reserving BW (Priority=p T SR )
	Steps: 1: Launch T imer l
	2: if all connected substrate links are saturated then
	3:	Reject the New bandwidth request
	4: else
	5:	P l .add(n) (i.e. start building a path proposal)
	6:	for all available attached links (n, ô) do
	7:	(n, ô)av -= req l (i.e. reserve bandwidth in (n, ô))
	8:	Activate	ô
	9:	end for
	10: end if
	Action 2: 3:	if all attached substrate links are saturated, or we risk to
		produce a loop then
	4:	Reactivate the soliciting node to Release BW
	5:	else
	6:


Antoine Muller
email: antoine.muller@irisa.fr

Marvin Chauwin

Charles Pontonnier

Georges Dumont

INFLUENCE OF PHYSIOLOGICAL CONSTRAINTS ON A SUBJECT-SPECIFIC BSIP CALIBRATION

INTRODUCTION

Calibration of body segment inertial parameters (BSIP) is crucial in biomechanical studies. To avoid strenuous protocols, identification methods based on rigid body dynamics laws have been proposed [2,3]. Thanks to a motion capture system and force platforms, these methods optimize BSIP by minimizing errors in the equations of motion. These errors arise from estimated BSIP but also from kinematics and force plate measurements that may introduce overfitting in the calibration. To prevent this, [3] added physiological constraints in the calibration. On one case study, the goal of this current work is to analyze the influence of added physiological constraints in a BSIP calibration on results. The different calibration results are compared to the widely used anthropometric data proposed by [1].

METHODS

For this study, one male participant (183 cm, 80kg) performed a range of motion-type motion which activates sequentially each degree of freedom. 47 motion capture markers were placed on standardized anatomical landmarks and captured thanks to a Vicon ® motion capture system (125 Hz). Two force platforms (1000 Hz) were used to access external forces applied on each foot. The motion is reconstructed thanks to a whole body model. Then, the BSIP calibration method consists in minimizing the dynamic residuals as proposed by [2]. To improve the results consistency, each limb is assimilated to a stadium solid [4]. So, the calibration finds the better stadium solid characteristics of each limb to minimize the dynamic residuals.

Three different BSIP are compared:

 anthropometric data proposed by [1] (C0);  results of the BSIP calibration computed without any constraint on solids geometry (C1);  results of the BSIP calibration computed with physiological constraints close to those used by [3] (C2). They consist in limiting the asymmetry and in limiting the BSIP variation from C0.

RESULTS AND DISCUSSION

Figure 1 shows the results of the different calibrations compared to the anthropometric data C0. First, it is easy to see that results of C1 are completely inconsistent (the mass of the torso is very low compared to the other limbs, the symmetry is not respected). Results of C2 seem more consistent with anthropometric data and correspond to the trends obtained by [3] (i.e. masses of torso and arms slightly lower than C0 and masses of pelvis, head and legs slightly higher than C0).

Thus, even if the optimization is better in C1 (in terms of cost function value), it appears to lead to overfitting. Additional physiological constraints are therefore essential in this BSIP calibration. Complementary studies have to be achieved to support these results. First, the BSIP calibration has to be tested with different morphologies especially with those distant of the 50 th percentile. It allows to refine the choice of the physiological constraints to be added to the optimization problem. Secondly, obtained calibrated BSIP has to be compared with a ground truth estimation, for example with 3D scanning technologies.

CONCLUSIONS

The purpose of this study was to analyze the influence of additional physiological constraints on a BSIP calibration, based on an identification method. Results obtained without any additional constraint turned out to be inconsistent, revealing overfitting. Addition of physiological constraints, close to those used by [3], yields to more consistent results that have be thereafter validated with more accurate approach like 3D scanning.

Figure 1 :

 1 Figure 1: Results of BSIP comparison. (a) shows the mass of the different body parts; (b) represents the limbs BSIP, each assimilated to the close cone frustum in terms of BSIP. 1 st result (black) is C0, 2 nd result (grey) is C1 and 3 rd result (yellow) is C2.


Simon Bardy
email: simon.bardy@cea.fr

Bertrand Aubert

Laurent Berthe

Patrick Combis

David Hébert

Emilien Lescoute

Jean-Luc Rullier

Laurent Videau

Numerical study of laser ablation on aluminum for shock-wave applications: development of a suitable model by comparison with recent experiments

Keywords: laser, shock, aluminum, hydrodynamics, elasto-plasticity, fracturation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

For the last decades, industrial applications of shocks generated by laser have arisen. They are based on lasers delivering from 1 to 100 J with pulse durations of several tens of ns. Focusing this type of laser beam on a target with a focal spot of some mm² enables heating the matter from solid to plasma state by inverse Bremsstrahlung absorption [START_REF] Mora | Theoretical model of absorption of laser light by a plasma[END_REF]. Once created at the surface, the plasma tends to expand in all directions and an ablation pressure (P abl ) of some GPa is created at the ablation front, in reaction to the ablated matter which is ejected in the normal direction to the front face. This ablation pressure is applied to the front face in some nanoseconds and generates therefore a shock wave in the material (Figure 1). Using a confining dielectric layer maximizes the shock pressure and duration in comparison to non-confined experiments which are traditionally led in vacuum atmosphere [START_REF] Anderholm | Laser-generated stress waves[END_REF]; hence confined configurations are preferred for industrial applications. Monitoring the rear surface of thin targets with velocity measurement system is usually employed to characterize the shock amplitude on the rear surface. Mastering shockwaves within the material pave the way to industrial outcomes. In all cases of application, the use of laser sources enables repeatability and eases the development of automatic processes. Nowadays, the LASer Adhesion Test (LASAT) and the Laser Shock Peening (LSP), identified as the two main laser shock processes, are well known in the industrial community.

The LASAT process aims to assess for the quality of interfaces within multi-layered structures by steering shock waves generated by laser [START_REF] Berthe | State-of-the-art laser adhesion test (LASAT)[END_REF]. The LSP is seen as an alternative to classical shot peening which uses high speed micro particles [START_REF] Fabbro | Physics and applications of laser-shock processing[END_REF]. If the LASAT is expected to start its development phase in various industrial sectors in the upcoming years, the LSP has already proven its efficiency at industrial scale since the beginning of laser-driven shock understanding.

In addition, shocks induced by laser are seen as an alternative tool for characterizing solid materials behavior under high speed loading up to 10 7 s -1 without having to take the impactor behavior into account [START_REF] Seisson | Dynamic fragmentation of graphite under laser-driven shocks: Identification of four damage regimes[END_REF] [START_REF] Laporte | Analysis of the response of bonded assemblies under shock wave loading -Experiments and modelling[END_REF]. However, many challenges appear in the laser-matter interaction at this moderate intensity range (10-500 GW/cm 2 ) and even more at low intensities for confined configurations (0.1 -6 GW/cm²). Some phenomena related to laser-matter interaction or mechanical behavior of materials, such as elasto-plasticity, may be negligible at higher regimes and have a determining influence in the laser shocks applications.

The development of a predictive numerical code for the applications mentioned above is the best way to design process parameters for related applications. Building this predictive code implies validation by experiments to identify key parameters for the development of new or existing models to be implemented within the considered code. This paper presents the first step of such approach which is the validation of Esther in a non-confined configuration. This is achieved coupling Esther code simulations and rear free velocity measurements of a reference material (Pure Aluminum) under laser shocks. Sensitivity of the code to material properties is also investigated with a different grade of aluminum (6061 Aluminum).

Numerical tool

Esther is a Lagrangian mono-dimensional code which manages both laser-matter interaction and shock propagation into the target [START_REF] Colombier | Hydrodynamic simulations of metal ablation by femtosecond laser irradiation[END_REF] [8] [9] [START_REF] Seisson | Dynamic cratering of graphite: experimental results and simulations[END_REF]. It solves the Helmholtz equation for the deposition of laser energy in the depth of skin and describes the evolution of material state from solid to plasma. It achieves hydrodynamic calculations on finite volumes using tabulated equations of state. Calculations include also radiative transfer and heat conduction models. A mechanical model (S-C-G [START_REF] Steinberg | A constitutive model for metals applicable at high strain rate[END_REF]) has been added to reproduce elasto-plasticity of materials. A model for damaging and fracturation of materials (Johnson [12]) is also integrated to simulate spall experiments. This model describes the evolution of porosity in the material from an initial existing porosity to a maximal porosity that triggers spallation. This process is driven by stresses and enables the description of shock wave mechanics within the damaged material. Esther gives the opportunity to generate the temporal ablation pressure profile corresponding to a given temporal laser impulsion profile. This pressure profile can then be used in 2D-mechanical codes for studying shock propagation with bi-dimensional considerations.

Experimental set-up

We have performed experiments on the Transportable Laser Shock Generator (GCLT) of the CEA (Alternative Energies and Atomic Energy Commission) which delivers 1064-nm pulses up to 40 J with 3 to 100 ns durations. This flexibility in terms of pulse duration allows various studies of laser shock, either dedicated to laser-matter interaction or experimental evaluation of mechanical properties. The laser pulse temporal profile (I(t)) can be shaped in order to create an on-demand pulse shape such as Gaussian-like, Triangle or Top-Hat pulses. A vacuum facility enables direct ablation experiments with a control of the on-target spatial distribution of intensity by using phase plates. This configuration is available for spots diameter (d foc ) from 0.25 mm to 4.4 mm. A 532-nm VISAR system [START_REF] Barker | Laser interferometer for measuring high velocities of any reflecting surface[END_REF] allows rear surface velocity measurement on the target rear surface, as shown on Figure 2. Typically, rear free surface velocity is characterized by successive oscillations corresponding to shock waves going back and forth within the target [START_REF] Tollier | Study of the laser-driven spallation process by the velocity interferometer system for any reflector interferometry technique. I. Laser-shock characterization[END_REF]. The period of these oscillations is given by equation [START_REF] Mora | Theoretical model of absorption of laser light by a plasma[END_REF]. on both materials. A factor 3 exists between both materials since Pure Aluminum and 6061

Aluminum exhibit a hardness of 40 and 117 respectively (Table 1). This implies a difference of the same order of magnitude in terms of strength and a similar difference of yield strength. This static mechanical information is correlated to the elastic precursors visible on the rear surface velocity profiles, which show also a difference factor of the same order of magnitude (30 m/s for Pure Aluminum vs. 100m/s for 6061 Aluminum). Simulations realized with Esther show very good accordance with the experimental curves.

Elastic precursors, related to the elasto-plasticity behavior of materials, are correctly defined.

The maximal velocity peak, impacted by both laser-matter interaction and mechanics is also well reproduced considering the uncertainty on incident intensity (+/-10%) and its influence on the rear surface velocity. Taking into account a good definition of elasto-plasticity in simulations and the accordance on the maximal rear surface velocity, the laser-matter interaction is then well defined by Esther. The cut-off point on pure Aluminum and the unloading phase of the Aluminum 6061 are also correctly reproduced in simulations, even if perfectible. These observations count for a good modelling of materials damaging occurring for both materials.

Esther reveals here its ability to reproduce laser shock experiments on materials with very different mechanical behaviors.

Same laser parameters have been used for generating a shock in a thicker sample. Both experimental and simulation results are described on Figure 5. intensity on the results of Figure 6. In this configuration, the maximal rear surface velocity is slightly higher than the elastic precursor (200 m/s and 100 m/s for the beginning of elastic precursor). Simulations curves representing +/-10% of uncertainty on intensity are very close to each other (20 m/s of difference on the peak value). They reveal the small dependence of rear surface velocity to intensity in this regime. The resulting behavior must then be largely influenced by the mechanical model of the material and poorly affected by laser-matter interaction. This must imply that a large part of induced energy is converted into elastic strain energy and hardening of the material. A very small part is dedicated to hydrodynamic plastic Elastic precursor shock. The elastic precursor is accurately simulated by Esther, confirming the reliability of the mechanical model. The first peak is reasonably well reproduced but the unloading phase could be better fitted with a slightly lower maximal rear surface velocity. After 100 ns, bi-dimensional effects can be considered to significantly impact the rear surface velocity. After this time, the large discrepancy between simulation and experimental results can be attributed, for a nonnegligible part, to 2D effects [START_REF] Boustie | Study of damage phenomena induced by edge effects into materials under laser driven shocks[END_REF].

Impact of modulations in I(t)

The experimental I(t) profiles shows modulations intrinsic to the GCLT laser source (Figure 7).

These modulations are about 10-15% around the intensity of a perfect Top-Hat pulse which has the same experimental fluence (F eq ). As shown on Figure 8, these modulations are reported in the ablation pressure profile (P abl (t)), numerically determined with Esther. While the shock wave propagates within the target thickness, these modulations are attenuated. Beyond 200 μm of propagation, the form of the pressure profile (P(t)) generated with the experimental I(t) corresponds to the theoretical Top-hat one. The first peak of the experimental profile is higher than the maximal intensity of the theoretical Top-Hat which explains a higher maximum in P abl (t). However, the here above-mentioned modulations are responsible of multiple release waves within the pressure profile. They increase the attenuation of the pressure peak along the propagation within the thickness. At 250 μm of propagation, both experimental and theoretical Top-Hat profiles have the same maximal pressure value. Beyond this value, the experimental profile has a lower maximal, generating a maximal rear surface velocity (v(t)) slightly smaller than the one obtained with a theoretical Top-Hat (Figure 9). This numerical study highlights the diminution of modulations in the temporal loading profile with the propagation in the material. It shows that these perturbations have no significant impact on the shape of the rear face velocity peak and a very limited impact on the maximal rear face velocity. Using a perfect Top-Hat impulsion for numerical predictions in the frame of future experimental campaigns should give a good approach of what is expected with real profiles.

Validating laser-matter interaction model with Esther

As mentioned before, Esther simulates the laser-matter interaction to generate an ablation pressure used as mechanical solicitation for modelling the behavior of material under shock loading. In order to validate the ability of Esther to generate adequate ablation pressure, P-I points obtained with Esther must be compared to a well-known P=f(I) law such as the one developed by Phipps from the works of Caruso [START_REF] Caruso | Ionization and heating of solid material by means of a laser pulse[END_REF], [START_REF] Caruso | Some properties of the plasmas produced by irradiating light solids by laser pulses[END_REF] , Nemchinov [START_REF] Nemchinov | Steady-state motion of radiation-heated vapors of a material in the presence of lateral spreading flow[END_REF] and Kidder [START_REF] Kidder | Application of lasers to the production of high-temperature and high-pressure plasma[END_REF]:

8 / 1 4 / 1 4 / 3         I A Pabl (2)
It has been identified by Phipps as the best law for representing experiments [START_REF] Phipps | Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers[END_REF]. In this study A was chosen as equal to 1. P abl (t) are obtained with Esther, using the model used for fitting experiments results with simulations (IV). This numerical study is achieved with a range of intensities from 50 to 500 GW/cm² for 5-ns, 10-ns and 50-ns for Top-Hat and Gaussian pulses. The maximum value of P abl (t) is extracted and plotted versus the maximum value of I(t). Results are described on Figure 10. For Gaussian pulses obtained with Esther, an increase in P with a decrease of pulse duration is to be noticed. This is corroborated by Phipps' law. The overall trend of all points obtained with Esther for Gaussian profiles show the same progression of P than Phipps law. Despite some fluctuations, this agreement is good, especially for rather high intensities (>250 GW/cm²). P-I points with Top-Hat impulsions obtained with Esther are very close to the one obtained with Phipps law for 5-ns impulsions. For these Top-Hat pulse configurations, maximal values of P abl (t) seem to not be influenced by the pulse duration (τ) since simulations realized with Esther generates exactly the same maximum in ablation pressure, whatever τ. In the considered regimes, τ seems to have an influence on the loading duration but the maximum value is generated by the very first ns of interaction. Hence, at equal intensity, a Top-Hat laser impulsion generates equal maximum of P abl .

Conclusion

The use of Esther for reproducing experiments has been demonstrated. The results presented here show a coherent laser-matter interaction and accurate simulation of mechanics, shock behavior, damaging and fracturation of materials. In the considered configurations, this code demonstrates its ability and can therefore be used for predicting and designing future laser experiments in nonconfined configurations. Future work has to be done in order to develop Esther for materials such as polymers, composites and alloys under laser shock in confined configuration. Other biographies are not available.

Biographies

Table Caption List Table 1 Results of Vickers micro-hardness measurements led on samples

Fig. 1

 1 Fig. 1 Schematic view of the generation of a shock wave induced by laser

Fig. 2

 2 Fig. 2 Schematic view of the experimental set-up used on GCLT

Fig. 3

 3 Fig. 3 Space-time diagram without fracturation model (a), with fracturation model (b) and rear surface velocity profile (c) obtained by simulation (+/-10% on E) with fracturation (red), without fracturation (blue), and compared to the experimental signal -Pure Aluminum -E=18.3 J -t=250 μm -τ=10 ns -dfoc=1 mm (I=170 GW/cm²)

  the cases without fracturation model described here (Figure3 a), Figure4 a) and blue curves on Figure3 c) and Figure4 c)), the period of oscillations linked to shockwaves going back and forth within the target thickness is about 90 ns. It is typical of shockwaves travelling through 250 µm-thick aluminum targets. Maxima 1 and 2 correspond to shocks coming out onto the target free surface and minima are related to the end of unloading of the free surface by release waves.If damage occurs within the target (Figure4 b) and red curves on Figure4 c)), the maximum after the first peak (2' -180 m/s) is attenuated in comparison to the case without fracturation (2 -310 m/s). This may be due to energy dissipation involved in the damaging process. The increase of porosity simulated by the fracturation model can be seen as spending energy into the creation of many new surfaces. If fracturation occurs (Figure3 b) and red curves on Figure3 c)), there is then apparition of two new free surfaces by spallation. In this case, the resulting rear surface velocity maxima (2', 3', 4', 5', 6') refers to back and forth of shock and release waves within the spall, hence a higher frequency of oscillations (Period ≈ 20 ns). Simulation results prove here the existence of damaging of Aluminum 6061 and 1-D spallation of the pure Aluminum sample.

Fig. 4

 4 Fig. 4 Space-time diagram without fracturation model (a), with fracturation model (b) and rear surface velocity profile (c) obtained by simulation (+/-10% on E) with fracturation (red), without fracturation (blue), and compared to the experimental signal -Aluminum 6061 -E=20.3 J -t=250 μm -τ=10 ns -dfoc=1 mm (I=180 GW/cm²)

Fig. 5

 5 Fig. 5 Rear surface velocity profile obtained by simulation (+/-10% on E) with fracturation (red) and without fracturation (blue) and compared to the experimental signal -Aluminum 6061 -E=18.2 J -t=335 μm -τ=10 ns -dfoc=1 mm (I=170 GW/cm²) Comparing Figure 4 and 5 allows having an insight of the effects of thickness on the behavior of

Fig. 6

 6 Fig. 6 Rear Surface velocity profile obtained by simulation (+/-10% on E) with fracturation (red) and without fracturation (blue) and compared to the experimental signal -Aluminum 6061 -E=6.1 J -t=250 μm -τ=10 ns -dfoc=1 mm (I=50 GW/cm²)Figure 4 and 6 refer to the same experimental conditions, except about 3 times less induced

Fig. 7 I

 7 Fig. 7 I(t) for a 10-ns GCLT Top-Hat (full line), and 10-ns numerical Top-Hats defined with a fluence equal to the measured experimental fluence (dotted line)

Fig. 8

 8 Fig. 8 Pabl(t) (yellow) and P(t) at 50 μm (red), 100 μm (magenta), 150 μm (pale blue), 200 μm (dark blue) and 250 μm (green) of propagation for a 10-ns GCLT Top-Hat (full lines) and 10-ns numerical F eq Top-Hats (dotted line)

Fig. 9 v

 9 Fig. 9 v(t) for a 10-ns GCLT Top-Hat (full line) and a 10-ns numerical Top-Hat defined with a F eq (dotted line)

Fig. 10

 10 Fig. 10 P=f(I) with P(t) defined with Esther and from the equation (2)

  Simon BARDY received a Master degree from the European School of Material Sciences and Engineering in 2014. After having worked 1 year in a mechanical testing department of a major producer of civil aircrafts, he is currently PhD student in daily collaboration with Dr. Laurent VIDEAU at the Alternative Energies and Atomic Energy Commission (CEA, DAM, DIF), under the supervision of Dr. Laurent BERTHE (PIMM-ENSAM ParisTech). His works aim at developing the LASer Adherence Test (LASAT).

Fig. 1 Fig. 2 Fig. 3

 123 Fig. 1 Schematic view of the generation of a shock wave induced by laser

Fig. 4

 4 Fig. 4 Space-time diagram without fracturation model (a), with fracturation model (b) and rear

Fig. 5 Fig. 6 Fig. 7 I

 567 Fig. 5 Rear surface velocity profile obtained by simulation (+/-10% on E) with fracturation

Fig.

  Fig. P abl (t) (yellow) and P(t) at 50 μm (red), 100 μm (magenta), 150 μm (pale blue), 200 μm

Table 1

 1 Results of Vickers micro-hardness measurements led on samples

			HV 50g
		Pure Aluminum	6061 Aluminum
	Average Value	40	117
	Standard Deviation	2.0	7.9
	Coefficient of Variation	4.9%	6.8%