Nazih Mechbal

Eurípedes G O Nóbrega

Spatial 𝑯 ∞ Approach to Damage Tolerant Active Control

Keywords: Damage Tolerant Active Control, Spatial 𝐻 ∞ Control, Fault Tolerant Control, Structural Health Monitoring, Active Vibration Control, Piezoelectric, Smart Structures, Finite elements

Damage tolerant active control is a new research area targeting to adapt fault tolerant control methods to mechanical structures submitted to damage. Active vibration control is a mature engineering area, using modern control methods to address structural issues that may result from excessive vibration. However, the subject of structural vibration control under damage represents a novel subject in the literature. There are some difficulties to adapt regular controller designs to active control, which may not result in good performance even for healthy structures. Fault detection and diagnosing research has conducted to development of the fault tolerant control area, whose methods are equally hard to translate to damaged structure control. Spatial active vibration control encompasses some techniques that present important features to structure control, however this is not necessarily true in the general control design area, where spatial constraints are normally not involved. We propose in this paper an investigation of these spatial techniques, applied to structural damage control. Several new strategies for vibration control are presented and analyzed, aiming to attain specific objectives in damage control of mechanical structures. Finite element models are developed for a case study structure, considering healthy and three different damage conditions, which are used to design controllers, adopting an approach based on a 𝐻 ∞ spatial norm, and according to some of the proposed strategies. Discussion of the achieved results contributes to clarify the main concepts related to this new research area, and controller performance analysis permit to foresee successful real case application of the techniques here described.

INTRODUCTION

Development of new technologies and materials, aiming to extend the operative life of mechanical structures and increase the level of safety standards, constitute a modern challenge that calls for the use of advanced tools, methods and procedures. New generations of mechanical structures based on smart materials possess the ability to change their physical properties in response to specific stimulus. Embedding smartness to structures improves its intrinsic or extrinsic characteristics and entails innovations beyond the ingenious design of final products. It also brings significant impact on processes and systems adopting the new structures. For example, smart composites can present the ability to monitor its structural health and therefore to guarantee better control of its passive, active or adaptive functions. Smart structure development in general opens new fascinating frontiers for applied research and product engineering.

1* Corresponding author, PIMM (UMR -CNRS 8006), Arts et Métiers ParisTech (ENSAM), Paris, France. E-mail: nazih.mechbal@ensam.eu 2 E-mail: egon@fem.unicamp.br objectives may be adopted: vibration suppression in the whole structure or only in some specific parts. In this paper, an approach to DTAC system design, based on spatial H ∞ norm, is presented, aiming to maintain acceptable performance for a possibly damaged controlled structure. Active vibration control of structures is basically concerned with spatially distributed systems. These system models are governed by partial differential equations that can be discretized using a modal analysis procedure, which leads to mode shapes that describe the spatial characteristics of the structure. In the case of irregular geometries and non-trivial boundary conditions, analytical solutions are not available to generate this type of model. Finite Element (FE) and system identification methods are often used to represent spatially distributed models. Numerical simulations provide the ability to understand the relative effects of property changes in great detail, such as brought by damage, and are thus quite fundamental for design. In this work, FE models, for healthy and damaged smart structures, are used to design spatial controllers intended to manage vibration suppression in the presence of structural damages.

The common approach to active vibration control design is to assume an input disturbance in one or more specific positions, with authority to locally excite dominant modes. In this case, and especially if a damage is present, one needs to adjust the controller for each considered vibration mode, in order to not increase unexpected vibration modes at some structure regions (damaged or not). Furthermore, it is common to have some specific position to measure the vibration to be controlled, often in a collocated configuration of inputs and outputs. Applying a 𝐻 ∞ based approach, which is currently the preferred design method for active vibration control, early work of the authors [START_REF] Mechbal | Application of a Combined Active Control and Fault Detection Scheme to an Active Composite Flexible Structure[END_REF] has investigated a combined scheme allowing vibration rejection and sensor faults detection on smart structures. As in these approaches, one use only specific points for input and output, it does not adequately take into account the spatial restrictions of the structure. This opens the way to spatial norm definitions and respective controller design methods [START_REF] O R Moheimani | Spatial balanced model reduction for flexible structures[END_REF]. To control (healthy) structures, some early works introduced the application of spatial norms [26] [27]. The authors proposed to use spatially distributed models instead of these mentioned pointby-point framework or, as generally called, pointwise models. These norms have been employed to elaborate collocated optimal 𝐻 2 and 𝐻 ∞ spatial controllers that ensure vibration reduction over entire structures in a spatially average sense. This is done by employing the spatial information embedded within the models of structures that result from modal analysis. Robustness of collocated controllers have been investigated and implemented effectively on real structures using piezoelectric transducers (PZT) by [START_REF] Halim | Spatial resonant control of flexible structures-application to a piezoelectric laminate beam[END_REF] [START_REF] Halim | Spatial H2 control of a piezoelectric laminate beam: experimental implementation[END_REF] and [START_REF] Halim | Experimental Implementation of Spatial H∞ Control on a Piezoelectric-Laminate Beam[END_REF]. The improved performance obtained using spatial approach to control healthy structure vibration is well established now [START_REF] Barrault | High frequency spatial vibration control for complex structures[END_REF] [START_REF] Mazoni | A decentralized and spatial approach to the robust vibration control of structures[END_REF]. The objective of this paper is to further explore issues in vibration analysis for smart structure testing and control, considering the occurrence of damage. It presents a spatial approach to perform DTAC on damaged smart structures. We propose to investigate 𝐻 ∞ spatial control applied to damaged structures with non-collocated PZT sensors and actuators, under some strategic treatment of DTAC goals. Comparing collocated and non-collocated configuration, the last one present restricted stability performances and do not have necessarily the same number of inputs and outputs [START_REF] Preumont | Vibration Control of Active Structures, An Introduction[END_REF] [START_REF] Rastgaar | Orthogonal Eigenstructure Control with Non-collocated Actuators and Sensors[END_REF]. Structures always present flexibility between the actuator position and the region where the vibration must be attenuated, which entails the collocated concept only for theoretical applications. Composite smart structures, constructed with the active elements embedded in non-isotropic composite layers, are clearly the non-collocated case. Discussions of FE simulated results are here presented, adopting the spatial approach and new control strategies, which are applied to a case study smart structure.

The outline of the rest of the paper is the following: in Section 2 DTAC different strategies and respective controller architectures, which are going to be explored in the examples, are presented; in Section 3, the Finite Elements models used in the simulations are given, including details about transducers and damages representations; in Section 4, the mathematical model approach for spatial norm based controller design is presented; in Section 5, simulated results for a cantilevered structure is developed, including a spatial robust controller and damage effects; in Section 6, application of some DTAC configurations and strategies are analyzed; a discussion of the results is presented in Section 7 and final remarks and future perspective are drawn in the last section.

STRUCTURES AND DAMAGE CONTROL

DTAC systems mean essentially active control of mechanical vibrations of structures aiming to maintain an adequate performance under the possibility of structural damage. Applications demand a network of transducers to produce measurement data to feedback the control loop, but also to feed specialized modules implementing structural integrity analysis techniques, in order to guarantee that the DTAC controller attains its goals. Ergo, the structure has to be smart to embed a DTAC system. A brief insight of smart structure constitution is given in the sequence.

Smart functions in structures

Advanced structures with improved self-capabilities have been intensively studied over the last four decades [START_REF] Srinivasan | Smart Structures: Analysis and Design[END_REF]. A smart structure needs the ability to respond to internal changes and also to environmental conditions. It means that built-in sensors and actuators are necessary, to monitor and diagnose these changes. Smart structures emulate some characteristics of a biological system as sensing, actuation, adaptability and self-repair. Development of smart structures is fuelled by the on-going technological progress and evolution of performance demands.

Different materials can be used on smart structures to act and sense. Among possible current transducers, piezoelectric materials are those widely used, due to their adaptable properties. Because piezoelectric elements can be used for mechanical to electrical transduction and viceversa, they can be adopted as sensors and/or actuators, which makes it easier to integrate the smart functions to the structure. Commonly used piezoelectric materials are semicrystalline polymer film PVDF (polyvinylidene fluoride), and PZT ceramic (lead zirconate titanate). Nowadays, and particularly in aeronautical industry, composite materials are increasingly used due to their strength properties. Because of its multilayer structure, they are suitable to host smart materials. Indeed, embedded sensors and actuators could be easily incorporated into the composite as a smart layer, during or even after the manufacturing phase of the composite panels.

Therefore, based on the adopted objectives to perform structural active control, different strategies may be considered, depending if the structure is in a healthy condition or if damage was detected. The respective framework is presented in Fig. 1, where the strategies depend if the structure has an identified damage or not. If the structure is a healthy one, there are two strategies:

 Strictly tolerant active control;  Preventive active control; If damage is identified there are two other strategies to be applied:

 Evolving damage active control;  Adaptive tolerant active control. It is also possible that a DTAC system may combine these strategies, but this will not be treated here. A description of the four above possibilities is presented in the following. A robust control is the first step to face damage that may occur in the future to the structure. The basic objective here is to design an active controller that rejects specifically some considered modes. Indeed, targeting the vibration level of the most damaging mode of some structures, subjected to known severe stress conditions, can enhance its longevity. As an example, one can refer to [START_REF] Chomette | Damage reduction of on-board structures using piezoelectric components and active modal control -Application to a printed circuit board[END_REF], where the authors proposed an active logic for the damage reduction of printed circuit boards by means of piezoelectric actuators and sensors. A strictly tolerant active controller (STAC) is the strategy to be used to design a controller robust enough to guarantee a minimal acceptable performance even if damage occurs. The resultant controller is fixed, meaning that it will not necessarily respond to parameter changing in the structure, beyond its already included robustness capacity. The applicability of this strategy is limited, and basically depends on the predictability of future damages and its respective severity. Nevertheless, the compromise between robustness and performance may conduct to a poor controller behavior for a not damaged structure. A preventive active control (PAC) is the strategy to be used when the controller is designed to avoid or retard the occurrence of damage. An example of this strategy is presented by [START_REF] Ambrosio | An optimal vibration control logic for minimising fatigue damage in flexible structures[END_REF], where a cost function includes an expression that take into account material fatigue, which is then minimized in the controller design process. The main goal here is to guarantee a longer structure life based on the use of a PAC system. Again the controller does not change if damage is detected.

If damage is detected and identified, an evolving damage active controller (EDAC) is the strategy to design a controller to protect the structure. The main goal is to minimize the vibration energy flow to the damaged region in order to avoid the damage evolution. It may be typically used in a smart structural repair, which now can monitor and control the extent of damage [START_REF] Chapuis | Structural health monitoring of bonded composite patches using Lamb waves[END_REF]. The last strategy general approach is to accommodate a detected damage and at the same time maintain an adequate level of performance. Consider a smart structure with a network of sensors and actuators including an SHM module and an active vibration controller. When damage is detected, the controller shall be adapted to face the new structural condition. This is the adaptive tolerant active control (ATAC) novel strategy, mainly addressed in this paper. Some different system configurations are examined briefly in the next paragraphs. The first and simplest ATAC configuration is to depend on a human interaction to design a new controller. An existing SHM module must provide enough information so that an engineer may be able to design a new controller to guarantee the required performance. Evidently, this may not be applicable for emergency situations, where a rapid response is necessary. In this case, an automatic reconfiguration of the controller is indicated. The automatic reconfiguration mechanism and the FDD module interaction is the subject of a future publication, demanding its proper space to detail the reconfiguration method. To be able to perform online the reconfiguration, and ensure a fast response of the controller to a sudden change of structural behavior, the FTC main configurations [START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF] [36] may be adopted. The main difference between FTC and DTAC are the specific methods to be adopted for controller design, which must take into account spatial restrictions necessary to provide effective control of structural vibrations. The FTC typical framework is depicted for DTAC in Fig. 2. Comparing the block diagram seen in Fig. 2 with a regular feedback controller diagram, it may be noticed the introduction of two new blocks, respectively SHM and ATAC, which includes the reconfiguration algorithm and the active controller. The SHM module must provide information to the reconfiguration algorithm to be able to change the active controller online. Signals that come from the SHM block may be a residue vector, generated by recent input and output signal measurements, weighed against a validated signature of the healthy structure. However, for DTAC systems, more information may be necessary, such as severity and localization of damage, or other data required by the reconfiguration algorithm, for example pole deviations due to damage, or semantic information for an artificial intelligence based reconfiguration.

Considering that even if this controller reconfiguration mechanism should respond as fast as possible, eventually the implicit time delay may bring some difficulty to its application when a really fast response is necessary, for example, in the case of serious and sudden aircraft damage. New elaborations over this scheme may avoid catastrophic consequences of time delay, but they will be treated in future publications.

ACTIVE SMART STRUCTURE AND FE MODEL

The FE models of the cantilevered structure represented in Fig. 3, equipped with non-collocated and surface-mounted piezoelectric patches, are developed in the next sections. Models of the healthy structure, including representation of the piezoelectric patches, are used to assess the undamaged performance. Developed models of the structural damages are also based on patches, and used to simulate the closed-loop behaviour under the action of different DTAC controllers.

Adopted composite smart structure

We adopt in this study an active cantilevered laminate plate in a horizontal position, according to dimensions (920 × 192 × 1.2 𝑚𝑚) and elements in Fig. 3, which is modeled with the complete six degree-of-freedom (DOF) at each node in a dense FE grid, in order to have the complete dynamics represented as realistic as possible. The structure consists in four epoxy/carbon layers with orientation angle [0°/-45°/+45°/0°] . The actuators are circular PZT patches from NOLIAC ® and sensors are flexible macro fiber ceramic (MFC), from Smart Materials ® . Material properties for the layers, piezoelectric elements and respective glue are given in the Appendix.

Piezoelectric transducers

A key challenge is to achieve a good validated model for the piezoelectric patches, and their interactions with the composite structure. Thereby, we tested and identified the mechanical properties of composite specimens, for the transducers' parameters in the FE models [START_REF] Massot | Structural Health Monitoring Process for Aircraft Nacelle[END_REF]. Accordingly, piezoelectric patches have been glued on these specimens and electromechanical measurements have been performed for correlation studies [START_REF] Massot | Structural Health Monitoring Process for Aircraft Nacelle[END_REF]. The smart structure FE models developed in this work use the Structural Dynamics Toolbox for MATLAB ® (SDT) [START_REF] Balmes | Structural Dynamics Toolbox & FEM Link, User's Guide[END_REF], based on piezoelectric Mindlin shells, taking into account the viscoelasticity of the composite core, the glue, and the piezoelectric coupling equations. Electrical degrees of freedom (DOF) are included in addition to the nodal displacement [START_REF] Balmes | Modeling structures with piezoelectric materials, Theory and SDT tutorial[END_REF]. For piezoelectric shell elements, electrical DOFs correspond to the difference of potential on the electrodes while the corresponding load is the electrical charge. For volume elements, one defines multiple point constraints to enforce equal potential on nodes linked by a single electrode and sets one of the electrodes to zero potential. To represent sensors, the configuration corresponds to cases where the voltage remains very small and an electrical charge is generated basically due to mechanical deformations. For the actuator configuration the electrical charge corresponds to the resulting force associated with the voltage, applied through a voltage driven amplifier [START_REF] Florens | Test/Analysis comparison of piezoelectric patch local behavior for vibroacoustic active control application[END_REF].

Considering dynamic measurements, one expects that the first in plane mode of the patch, due to Poisson's effect, will generate compression through the thickness and thus a charge distribution on the surface [START_REF] Benjeddou | Piezoelectric smart structures and systems[END_REF]. To bypass the need to compute all modes in the frequency range of interest, the proposition is to compute the direct frequency response on a given range and to use the resulting shapes to build a Ritz basis. Further details on this method may be found in [START_REF] Balmes | Effects of temperature on the impedance of piezoelectric actuators used for SHM[END_REF], but the approach guarantees good results at the retained frequencies and is deemed most efficient for the considered applications.

FE patches modeling

Referencing to Fig. 4a, to account for material properties, one considers a volume mesh distinguishing an orthotropic base plate (shown in blue), a glue layer (EPO-TEK) assumed to be an isotropic elastic material, (in orange), and a transversely isotropic piezoelectric material (shown in green). The electrodes are placed on the circular surfaces and polarization is in the direction normal to the surface.

Based on a Stiffness Reduction Method (SRM) [START_REF] Gopalakrishnan | Computational techniques for structural health monitoring[END_REF], a parametrical damage is introduced through the parameter Υ. Indeed, in SHM context, a common and simple way of modeling damage is to incorporate a stiffness loss in the region of the damage by modifying the respective materials properties. We represent this change using the vector 𝑃 𝑚𝑎𝑡 (Young's modulus, shear modulus, density etc.) to the product Υ𝑃 𝑚𝑎𝑡 with Υ ≤ 1 [START_REF] Gopalakrishnan | Computational techniques for structural health monitoring[END_REF]. Therefore, to introduce this damage, an automatic approach has been developed using SDT software where a damage-patch is generated with a specific mesh. The dimension and the material properties of this damage-patch could then be changed and adjusted. The introduction of this patch before changing its properties does not alter the modal properties of the structure. Moreover, nodes inside this damage-patch could be removed to create crack damages. It is possible to see in Fig. 4b two examples of this automatic generated approach to represent damage in FE models. Notice also that there is a mesh for the structure, a different one for the damage boundaries and a third for the damage itself.

SPATIAL NORM-BASED CONTROL DESIGN

An active vibration control methodology based on a spatial 𝐻 ∞ norm definition [START_REF] Halim | Experimental Implementation of Spatial H∞ Control on a Piezoelectric-Laminate Beam[END_REF] has been studied in the last two decades, starting from the initial work of [START_REF] Moheimani | Spatial control for active vibration control of piezoelectric laminates[END_REF]. This approach may be very useful to DTAC systems, and is explored further in this paper. It must be emphasized that application of these concepts to damage control is probably unique to DTAC, representing a control methodology that may not be used by a regular FTC system. The idea is to control the vibration in predefined regions of the structure, instead of having pointwise controllers, i.e. a norm minimization at specific points where the structure has performance indicators or sensors.

In pointwise active control, the controller is designed based only on the information of some positions along the structure [START_REF]Simulations and Experiments on Active Vibration Control of a Composite Beam with Integrated Piezoceramics[END_REF]. Notwithstanding, vibration characteristics at other locations along the structure are not indeed accounted for. This may cause problems if vibration reduction over the entire structure is needed. To deal with such issues, [START_REF] Halim | Spatial H2 control of a piezoelectric laminate beam: experimental implementation[END_REF] have extended the 𝐻 2 and 𝐻 ∞ norms to spatially distributed systems. Before introducing the spatial controller, we address first and briefly the general robust design framework.

Modal model

Second-order differential modal equations are commonly used to represent structure behavior, because it is basic for a FE model, and can be easily converted to state-space formulation used in control area. Beginning with the nodal second-order approach, the concentrated parameters matrix equation of a generic mechanical structure is modelled as 𝑀 𝑝 𝑝̈+ 𝐷 𝑝 𝑝̇+ 𝐾 𝑝 𝑝 = 𝐵 0 𝐹 𝑎 (1) where 𝑝 is the displacement vector for the adopted nodes, 𝑀 𝑝 , 𝐷 𝑝 and 𝐾 𝑝 are respectively the inertial, damping and stiffness matrices, 𝐹 𝑎 is the vector of external forces acting on the structure and 𝐵 0 reflects its position distribution. Mechanical structures have an infinite number of vibration modes, and modelling them inevitably limits it to an adequate number, in order to represent the problem at hand. The modal matrix is defined as

Φ = [𝜙 1 𝜙 2 … 𝜙 𝑚 ]
where 𝛷 𝑖 , 𝑖 = 1, … 𝑚 are the first m natural modes. Considering the transformation for modal coordinates obtained using 𝑝 = Φ𝑞 and pre-multiplying (1) by Φ 𝑇 , it results

a T p T p T p T F B q K q D q M 0             
(2) that may be written as 𝑀 𝑚 𝑞̈+ 𝐷 𝑚 𝑞̇+ 𝐾 𝑚 𝑞 = 𝐵 𝑚 𝐹 𝑎 (3) where 𝑀 𝑚 = Φ 𝑇 𝑀 𝑝 Φ, 𝐷 𝑚 = Φ 𝑇 𝐷 𝑝 Φ, 𝐾 𝑚 = Φ 𝑇 𝐾 𝑝 Φ, 𝐵 𝑚 = Φ 𝑇 𝐵 0 , commonly known as modal matrices, with 𝑀 𝑚 and 𝐾 𝑚 diagonal matrices. In order to get also a diagonal modal damping matrix, which is very convenient for the modal analysis, it is common to adopt the proportional form, 𝐷 𝑝 = 𝛼𝑀 𝑝 + 𝛽𝐾 𝑝 , where 𝛼 and 𝛽 ≥ 0. This assumption gives rather approximate values considering that flexible structures have very small damping factors in general.

State-space model

Starting from (2), it is usual to choose the modal matrix Φ such as the modal mass matrix results an identity matrix, leading to (4), for each mode, 𝑞̈𝑖 + 𝑑 𝑚,𝑖 𝑞̇𝑖 + 𝑘 𝑚,𝑖 𝑞 𝑖 = 𝑏 𝑚,𝑖 𝐹 𝑎,𝑖 , (4) where 𝑖 = 1,2, . . . 𝑚, is the mode index, 𝑑 𝑚 and 𝑘 𝑚 are the damping and stiffness coefficients for the respective mode. Even if a finite number of modes are considered, a model reduction approach is yet necessary to get a manageable model. Consider for the moment this reduced number 𝑛 < 𝑚 as the number of modes to control explicitly.

Adopting the state vector definition 𝑥(𝑡) = [𝑞(𝑡) 𝑞̇(𝑡)] 𝑇 corresponding to modal displacements and velocities, the respective state-space model derived from (4) may be written as

𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵 𝑤 𝑤(𝑡) + 𝐵 𝑢 𝑢(𝑡) 𝑦(𝑡) = 𝐶𝑥(𝑡) (5)
where the four piezoelectric transducers seen in Fig. 3 are being considered as one control input 𝑢(𝑡) (Piezo 1), two measured outputs vector 𝑦(𝑡) (Piezo 2 and Piezo 3) and disturbance input (Piezo 4). The system matrix 𝐴 has dimension 2𝑛 × 2𝑛, and is given by

𝐴 = [ 0 𝑚×𝑚 𝐼 𝑚×𝑚 𝐾 𝑚 𝐷 𝑚 ]
where 𝐷 𝑚 and 𝐾 𝑚 are the diagonal matrices obtained from (4) but limited to only 𝑛 modes. Notice that the 𝐵 and 𝐶 matrices must reflect the position of the respective transducers. We are here neglecting a possible feedthrough term between inputs and outputs.

Spatial H ∞ controller design

It is convenient to represent the adopted structure, in order to design its active controller, by the general MIMO system framework seen in Fig. 5. There, it may be seen two blocks and four signals, respectively the structural plant and controller blocks and disturbance and control signal vectors input signals 𝑤(𝑡) and 𝑢(𝑡), and the time and spatial dependent performance index vector 𝑧(𝑟, 𝑡) and measured output vector 𝑦(𝑡). In state space form, the plant to be controlled may be represented by [START_REF] Carden | Vibration based Based Condition Monitoring: A Review[END_REF], where 𝑥 𝑝 (𝑡) is the state vector. All vectors and matrices must have adequate dimensions, regarding the number of inputs and outputs and the order of the plant. Considering the closed-loop system, its input is solely the disturbance vector and the performance vector is the output. In general, performance index depends only on time. But to include the spatial restrictions we need to formulate the 𝐻 ∞ spatial control problem where the performance output vector depends also on the spatial position. Considering initially only the time dependence, the state-space formulation is represented in [START_REF] Carden | Vibration based Based Condition Monitoring: A Review[END_REF]:

𝑥̇𝑝(𝑡) = 𝐴 𝑝 𝑥 𝑝 (𝑡) + 𝐵 𝑤 𝑤(𝑡) + 𝐵 𝑢 𝑢(𝑡) 𝑧(𝑡) = 𝐶 𝑧 𝑥 𝑝 (𝑡) + 𝐷 𝑧𝑤 𝑤(𝑡) + 𝐷 𝑧𝑢 𝑢(𝑡) 𝑦(𝑡) = 𝐶 𝑦 𝑥 𝑝 (𝑡) + 𝐷 𝑦𝑤 𝑤(𝑡)

Given a state-space controller as 𝑥̇𝑘(𝑡) = 𝐴 𝑘 𝑥 𝑘 (𝑡) + 𝐵 𝑘 𝑦(𝑡) 𝑢(𝑡) = 𝐶 𝑘 𝑥 𝑘 (𝑡) + 𝐷 𝑘 𝑦(𝑡) ,

the 𝐻 ∞ problem is to find the controller matrices which satisfy the infinity norm, stated as an optimization objective function for the transfer matrix 𝑇 𝑧𝑤 between the disturbance 𝑤(𝑡) a nd performance 𝑧(𝑡) according to [START_REF] Carden | Vibration based Based Condition Monitoring: A Review[END_REF], which is a well-known problem.

𝐽 ∞ = ∫ 𝑧(𝑡) 𝑇 𝑧(𝑡)𝑑𝑡 ∞ 0 ∫ 𝑤(𝑡) 𝑇 𝑤(𝑡)𝑑𝑡 ∞ 0 (8)
To take into account a spatial region Ω where we want to minimize the 𝐻 ∞ spatial norm, a space dependent weighing matrix 𝑄(𝑟) , where 𝑟 ∈ Ω is the spatial vector, is introduced according to (9):

𝐽 ∞ = ∫ ∫ 𝑧(𝑡) 𝑇 𝑄(𝑟)𝑧(𝑡)𝑑𝑟 Ω 𝑑𝑡 ∞ 0 ∫ 𝑤(𝑡) 𝑇 𝑤(𝑡)𝑑𝑡 ∞ 0 (9)
Considering the objective function in [START_REF] Massot | Structural Health Monitoring Process for Aircraft Nacelle[END_REF], the new performance vector may now be written as 𝑧 𝑟 (𝑡) = Π𝑥 𝑝 (𝑡) + Θ 𝑤 𝑤(𝑡) + Θ 𝑢 𝑢(𝑡). It is then possible to write the spatial objective function as

𝑧
𝐽 ∞ = ∫ [𝑥 𝑝 (𝑡) 𝑤(𝑡) 𝑢(𝑡)]Γ 𝑇 Γ[𝑥 𝑝 (𝑡) 𝑤(𝑡) 𝑢(𝑡)] 𝑇 𝑑𝑡 ∞ 0 ∫ 𝑤(𝑡) 𝑇 𝑤(𝑡)𝑑𝑡 ∞ 0 , (13)
conducting to ( 14), now similar to [START_REF] Hajrya | Principal Component Analysis and Perturbation Theory Based Robust Damage Detection of Multifunctional Aircraft Structure[END_REF],

𝐽 ∞ = ∫ 𝑧 𝑟 (𝑡) 𝑇 𝑧 𝑟 (𝑡)𝑑𝑡 ∞ 0 ∫ 𝑤(𝑡) 𝑇 𝑤(𝑡)𝑑𝑡 ∞ 0 . ( 14
)
which is similar to a regular 𝐻 ∞ objective function. Then, the spatial 𝐻 ∞ problem may be solved using known methods for the 𝐻 ∞ problem.

Model reduction

As mentioned before, real structures have an infinite number of vibration modes or degrees-offreedom (DOF). To elaborate a useful mathematical model, we first choose an adequate number of nodes, which implies a consequent number of DOFs, reflecting a bandwidth of interest. To account for local geometrical details, especially to model damages and their spatial variations, FE method is the most used to model structures. FE method leads in general to significantly large models, impacting on the controller order, and eventually resulting in numerical difficulties. To solve this conflict, a model reduction technique is necessary. The simplest approach for model reduction is by direct truncation, projecting the nodal displacements on a truncated modal matrix model. This pure truncation of high frequency modes is often a poor approximation. The neglected modes are ignored by assuming that their collective impact on the in-bandwidth modeled dynamics is minor. However, for control design purposes a controller based on such a model can alter zeros of the system, and, as closed-loop performances are largely dictated by open-loop zeros, one can expect differences between simulated and experimental results. It is, therefore, important to improve the reduced model of the system so that high performance controllers can be designed. The approach used here is based upon including additional modes with resonant frequency far from the desired bandwidth, such that the flexibility of this single mode is equivalent to that of all out-of-bandwidth modes, omitted from the expansion. During the last decades, several publications have proposed to obtain a feedthrough term to capture the effect of truncated modes in an optimal way [START_REF] O R Moheimani | Spatial balanced model reduction for flexible structures[END_REF] [START_REF] Moheimani | Minimizing the truncation error in assumed modes models of structures[END_REF]. In [START_REF] Halim | Spatial H2 control of a piezoelectric laminate beam: experimental implementation[END_REF] and [START_REF] Halim | Experimental Implementation of Spatial H∞ Control on a Piezoelectric-Laminate Beam[END_REF], the authors have extended this idea to the spatial model. In this case, the feedthrough term is found by minimizing the weighted 𝐻 ∞ and 𝐻 2 spatial norms of the error between the infinite dimensional and the truncated model [START_REF] Moheimani | A convex optimization approach to the mode acceleration problem[END_REF]. However, from a FE simulation point of view, the modal basis with augmented states is generally preferred to using a feedthrough term. It induces only marginal increase of numerical cost.

In this work, given the position of actuators and sensors, the finite-dimensional model is obtained after a choice of a restricted subspace (or modes) that sufficiently describe the global dynamic behavior in the bandwidth of interest. Following the preceding discussion, the resulting reduced model is augmented by two second order poles, placed at least one decade beyond the bandwidth of interest. This model results in a system transfer function that maintains the appropriate pole-zero relationship.

SPATIAL 𝑯 ∞ CONTROL AND STRUCTURE DAMAGE

Next subsection presents details of the FE modeling of the healthy structure, for the complete and nominal models. In the sequence, the spatial 𝐻 ∞ controller is designed and its performances are shown through comparison between structure response to disturbance and a controlled response. Finally, the last subsection shows the effect of a small damage on the structure response.

FE simulation of the healthy structure

Following the model reduction approach previously described, two second order out-ofbandwidth terms is included in the model, to ensure that the high frequency response content are close to that of the actual structure response. Damping value of 𝜉 𝑖 = 0.7 and frequencies 𝜔 1 = 5000 𝑟𝑎𝑑/𝑠 and 𝜔 2 = 5400 𝑟𝑎𝑑/𝑠 were adopted for these two terms. Two models are adopted for the numerical simulation: for the first one, which is used to design the controllers, only the 12 first modes are adopted, and it is here called the nominal model; and the second one, which is based on the first 40 modes and is used to simulate the performance of the designed controllers, we called the complete model. Fig. 6 shows the magnitude Bode diagram of the complete and the nominal transfer functions from the control input 𝑢 to outputs 𝑦 1 and 𝑦 2 . Some natural frequencies are given in Table 1. Fig. 6 presents FE simulations including the piezoelectric patches and responses to modes 1, 4 and 10.

It may be seen Fig. 6a that the natural frequencies of the reduced nominal model, with the additional two frequencies, coincide with the complete model in-bandwidth low frequencies, and also the effect for additional modes in the high frequency response. It is to be also noticed that, depending on the considered input output path, some modes are not excited and so do not appear in the respective frequency response.

The spatial 𝐻 ∞ control of the healthy structure

The controller is designed to reduce the effect of the disturbance in a spatial 𝐻 ∞ sense. The state space model for the nominal plant, including the spatial performance output is written as where the spatial weighting matrix 𝑄 𝑤 (𝑟) is chosen according to the region of the structure where we want to control the vibration. Fig. 7 indicates the positions of the measured output vector 𝑦 𝑛 (𝑡) and one arbitrarily chosen displacement output 𝑧(𝑡), which is only used to give a comparison reference of the controller performances. This performance output could be experimentally measured using a laser vibrometer. The respective controller, designed based on the healthy structure nominal model, is called here the first controller. The impact area, also indicated in Fig. 7, is the region where the subsequent damage will be included. The first simulation refers to the healthy structure with unitary spatial weighting function, which means that the vibration reduction effort is equally performed along the plate. An adequate value of 𝑅 is chosen (𝑅 = 0.3) as a compromise between performance and stability. Fig. 9 shows the magnitude Bode diagram of the open loop and the closed loop system for the performance and the measured outputs. To highlight the controller effect on the vibration attenuation of the structure nominal modes we used in this case the nominal model to close the loop and obtain the frequency responses shown in Fig. 8.

For the next simulations, the controller is designed using the nominal plant model but we adopt the complete model to close the loop, in order to evaluate its robustness and spillover avoiding capabilities. The modes in Fig. 8a are numbered respecting the frequencies given inTable 1.Considering the 𝑧 output, it is possible to notice that the peak reduction for mode 1 is more than 20 𝑑𝐵, what is true also for mode 2, with a smaller attenuation for modes 6, 8 and 10, and less attenuation at mode 4, near 5 dB. Modes 3, 5, 7 and 11 are not present in any response, open or closed loop. The 𝑦 1 and 𝑦 2 responses are similar to the 𝑧 response.

Considering Fig. 9, where the complete model is used to close the loop, the in-bandwidth frequencies present the same behavior, and the highest frequencies did not show significant variation. We can notice that spillover is not present and this first controller shows good results. The corresponding spatial frequency response of the system for the output 𝑦 1 is presented in Fig. 10, where third axis represents the position on the beam. These 3D diagrams permit to assess the controller spatial effect over the healthy structure. They show the simulated spatial frequency responses of the uncontrolled and the controlled structure. Here, we have taken the 𝑦 position at its middle, i.e. 𝑦 = 96 𝑚𝑚, and measured the 𝑥 along the structure while the frequency response (vertical axis) is in terms of the structure's transverse displacement. One should notice the mitigation of vibration of the entire structure, which has been also illustrated by the Bode regular diagram in Fig. 9. Fig. 11 presents the 𝐻 ∞ norm evaluated for the uncontrolled and controlled systems, in order to complement the analysis of the spatial effect of the controller over the healthy structure. It is clearly visible the effect of vibration reduction on the norm caused by the adopted spatial controller. Time domain responses for both outputs for a Schroeder disturbance input signal can be seen in Fig. 12, for open and closed-loop, which complements the analysis of the system behaviour. The controller effectiveness can be easily assessed in these figures, considering the linear amplitude and also the linear sweep of the excitation signal from DC to 1570 rad/s in the shown 200 seconds period. It is visible also the attenuation in the in-bandwidth frequencies for the nominal model and that, for the extra frequencies of the complete model, the control signal increases or decrease the peak amplitudes by a small factor, but maintains the stability of the system.

Including damage

In this section we consider the effect of a barely visible impact damage (BVID), admitting 𝚼 = 2 × 10 -3 , on the plant dynamics. Damage is represented, as described before, by a circular patch with diameter ∅ = 20 𝑚𝑚, centered at position (570,140) 𝑚𝑚 (see Fig. 13b). Fig. 13a illustrates the amplitude response of the complete model of the structure after including this damage. It is possible to see that the damage consequences on the plant model are small. As mentioned before, there are several strategies to be adopted in a DTAC framework. For the present study, we are beginning in the next subsection with the basic approach where the controller is sufficiently robust to guarantee performance to some damage. In the sequence a controller reconfiguration scheme to accommodate the fault is presented and in the last subsection a controller to prevent evolution of damage is the subject, always based on 𝐻 ∞ spatial control.

Robust active controller

The spatial 𝐻 ∞ scheme has intrinsic robustness properties that can be straightforward used here to face small structural damages. Using the first 𝐻 ∞ spatial controller, simulations of the healthy and the damaged systems are presented for the measured output vector, 𝑦 1 (𝑡), and the spatial performance output 𝑧(𝑡). Fig. 14 shows the open-loop time response of the plane model to a Schroeder disturbance for the previous BVID damage, the closed-loop for the healthy plate response and the closed-loop where the first controller is being used after the occurrence of the damage.

Considering the damaged closed-loop response (Fig. 14) and comparing with the healthy closedloop case, it is possible to notice a small increase in modes 4, 6 and 10, but a bigger increase in mode 8, both in the performance output and the measured output y 1, . But, considering the openloop response, there is yet attenuation due to the controller anyway. For the out-of-bandwidth frequencies, it is possible to notice amplification of the response, but also by a small factor.

Based on results in Fig. 14, showing that the differences between the healthy and the damaged closed-loop structure response are not significant, it is possible to conclude that a robust controller may deal with small damages. Obviously, this is a specific case, but it may be expected that similar results be achieved for several small level damages. Furthermore, if a damage study is performed first, the controller may then be designed to specifically reject the main damaging modes. That being said, the effect of a severe damage may not be so easily treated. In the following subsection, we study the effect of a severe damage on the closed loop responses.

Reconfigurable Active Controller

The impact on closed-loop performance due to a severe damage may be much more significant than the previous case, conducting even to instability. Also, it may be expected that, for some well-localized damage, even if it is small, the consequence may be very significant. For these cases, an adaptive tolerance is required, demanding a DTAC strategy. A severe damage is considered here, adopting 𝚼 = 2 × 10 -5 . Fig. 15 shows the effect of this damage on response to the same Schroeder disturbance input response, using yet the first controller.

It may be seen in Fig. 15 that now the first controller is not so effective. The performance output presents a significant increase in the level of several modes, and mainly in the higher frequencies. In the measured output, this is expected with the reduction of stiffness due to damage. A reconfiguration to attenuate this damage effect is indicated, and in this case a new controller design needs to be conducted, which would be based on an experimental identification of the new structure. For an online redesign, this module is necessarily included in the framework, as described in Fig. 2. This reconfiguration module design demands some involved modeling and also a SHM method and we will be following here the simple approach. In the present case, a new controller was designed, based on the damaged FE model. Results for this new controller are shown in Fig. 16, where it is clearly visible how the new controller improves vibration mitigation, recovering a performance near to the one of the first controller. Fig. 17 presents 𝐻 ∞ spatial norm distribution for the same simulation, and it is very useful to understand the reconfiguration controller effect.

In Fig. 17 it may be seen the previous three closed-loop systems. In the upper panel the healthy plate model with the first controller. In the middle panel, it may be seen the first controller facing the severe damage and in the lower panel, the second controller after reconfiguration. Damage effect in the plate under the strictly robust controller is clearly visible in Fig. 17b, showing high amplitude peaks located in the damage region. This effect is also seen in Fig. 17c, but the respective peaks are here considerably attenuated, showing only a fraction of the energy from the previous case. It is implied that reconfiguration of the spatial controller achieved its goal, and we may expect that the operative life of the structure will be extended.

Evolving active controller

The case where damage has been detected and localized, and we want to control its evolution, is addressed in this subsection, applying EDAC strategy. The idea is to try protecting the damaged area, in order to avoid the growth of the damage, attenuating vibration in the region where it is localized. This can be done by emphasizing the controller performance around the damaged area, albeit at the expense of vibration rejection performance over other regions. The spatial approach studied here is especially interesting to this strategy, providing this capacity through the spatial weighing function 𝑄(𝑟). To show the effectiveness of the EDAC strategy, three weighing functions and a point-wise controller are simulated and their performances compared on assessing crack propagation. A likely situation has been simulated by considering that the former severe damage has provoked crack propagation along one of the composite plies. Indeed, we have introduced a crack that begins from the border of the first impact damage to the border of the structure along the 45° direction of the ply. Thereby, a specific damage-patch has been added to the FE model with the impact damage, as shown in Fig. 18. The crack measures 68 𝑚𝑚 and its effect on frequency response outputs are presented in Fig. 19. The changing due to this crack damage is easily seen as a significant one for the 𝑦 2 response. We can also see in Fig. 20 how this crack affects the spatial 𝐻 ∞ norm of the nominal system showing a big vibration increase, which could lead to instability. The open loop response shows a minimum at the damage region, and then increases to the tip of the plate. The pointwise controller obtains the smaller norm, but only at the specific considered point, and after this point increases the norm in a rate bigger than the open loop system. At the region near the clamp of the plate the norm is similar to the others, but in the center it is higher. Considering the spatial controller with a constant weighing function, it presents a regular performance no the complete plate, and only in a region near the performance point the pointwise norm is smaller. Comparing this with the other two spatial function controllers, result of the constant weigh function controller is a little better in general, except in the region of the damage. Considering that the objective here is to attenuate vibration on this region, the better one is the Gaussian weighing function controller, but both functions protect the damaged area, and the Gate function has a performance close to the Gaussian. It is possible to notice that the efficiency of the Gaussian weighing function is maximal around the center of the Gaussian. The first controller corresponds to the simple strategy to design a robust spatial 𝐻 ∞ controller that may respond well to a structural damage. Results have shown that it is possible to expect that for small damages this may work with some loss of performance. Also, for a bigger damage, the loss of performance may be beyond an expected level, and it is possible to expect also that instability may come as a consequence of the damage localization, even though this did not occur in the simulations. The second controller was designed according to the strategy that is supposed to become the most common, when a controller is reconfigured to attend a detected and localized damage. This control scheme may present a SHM module to provide these functions, and a reconfiguration mechanism to use data so generated to change the controller parameters in order to face the damage. These two modules are necessary to guarantee an automatic online performance, and it is not treated here because goes beyond the scope of this paper. Also, it is possible to engage an engineer to provide the controller reconfiguration, what was adopted in this case. Results shown that it is possible to recover a good performance, similar to the original healthy controller, when the spatial 𝐻 ∞ controller is used to face the damaged structure, under a severe damage.

Finally, the third analyzed strategy relates to the case when damage was detected and localized in some structure, and a controller is designed to maintain the respective damage region under low vibration, in order to guarantee an operative life extension to the structure. In this case, the spatial norm controller has shown specific features that confirm its vocation for DTAC application. Using four different optimization objectives, varying the spatial norm regions constraint, it was clearly shown the protection capacity of the controller over the known region of the damage, in this case corresponding to a Gaussian weighing function presenting the better performance. It is possible to conclude also that the third strategy could be mixed with the first one, which becomes the second strategy presented in Fig. 1, the preventive active controller. Considering that we may know in advance the more probable region to show damage, the structure could be protected using an adequate weighing function, in order to prevent a future damage to occur, or at least retarding it.

CONCLUSIONS

This work aims to elaborate over some fundamental concepts and main strategies of DTAC application, presenting several 𝐻 ∞ spatial norm controller design approach. Based on detailed FE models, developed to represent healthy and damaged structures, these controllers and strategies are assessed through various simulated results and diagrams.

Although FTC methods can be adapted to DTAC systems, peculiar characteristics of mechanical structures demands specific design solutions, which detach these two areas. In particular, spatial constraints, normally present on requirements of active control of structure vibration, indicate 𝐻 ∞ spatial norm as a base for control design techniques of great interest for DTAC application. Those characteristics are illustrated on the analysis of the results here presented.

For the sake of completeness, mathematical modelling is briefly presented, from the structural control standing point. Simulated results of the addressed DTAC strategies exemplify the interesting properties of the spatial norm approach to face structural damage consequences. As future development perspective, new methods to implement modules to provide online efficient reconfiguration and switching of controllers, and respective stability analysis, are under investigation. The mitigation of fatigue damage and how to take it into account in the design of the control algorithm are DTAC's topics that are also currently explored.

APPENDIX

We present here the material properties of the active composite structure presented in Section 3.1. Hence, the materials properties of each composite layer of the structure are given in Table A1, whereas those of the two piezoelectric types in Table A2 and Table A3 respectively. The four piezoelectric patches were bonded using Epoxy glue (302-3M, produced by EPO-TEK) with properties given in Table A4.

Fig. 1 :

 1 Fig. 1: Structural active controllers under possible damage

Fig. 2 :

 2 Fig. 2: DTAC scheme in a reconfigurable design

Fig. 3 :

 3 Fig. 3: Adopted composite structure with active elements

Fig. 4 :

 4 Fig. 4: (𝑎) Geometric representation of the glued piezo-patch : (𝑎1) Top view; (𝑎2) zoom with a side view. (𝑏) Example of damage-patches.

Fig. 5 :

 5 Fig. 5: Active control problem framework

  (𝑟, 𝑡) = 𝐶 𝑧 (𝑟)𝑥 𝑝 (𝑡) + 𝐷 𝑧𝑤 (𝑟)𝑤(𝑡) + 𝐷 𝑧𝑢 (𝑟)𝑢(𝑡) (10) or, 𝑧(𝑟, 𝑡) = [𝐶 𝑧 (𝑟) 𝐷 𝑧𝑤 (𝑟) 𝐷 𝑧𝑢 (𝑟)][𝑥 𝑝 (𝑡) 𝑤(𝑡) 𝑢(𝑡)] 𝑇 (11) Defining an auxiliary matrix 𝛤 such that Γ 𝑇 Γ = ∫ [𝐶 𝑧 (𝑟) 𝐷 𝑧𝑤 (𝑟) 𝐷 𝑧𝑢 (𝑟)] 𝑇 𝑄(𝑟)[𝐶 𝑧 (𝑟) 𝐷 𝑧𝑤 (𝑟) 𝐷 𝑧𝑢 (𝑟)]𝑑𝑟 Ω (12) and the new performance index as 𝑧 𝑟 (𝑡) = Γ[ 𝑥 𝑝 (𝑡) 𝑤(𝑡) 𝑢(𝑡)] 𝑇 =[Π Θ 𝑤 Θ 𝑢 ] [𝑥 𝑝 (𝑡) 𝑤(𝑡) 𝑢(𝑡)] 𝑇 , it results

Fig. 6 :

 6 Fig. 6: (𝑎) Bode diagram of the complete and the nominal model; (𝑏) Mode shapes 1, 4 and 10.

Fig. 7 :

 7 Fig. 7: Selected outputs, inputs and the impact zone to be monitored

Fig. 8 :Fig. 9 :

 89 Fig. 8: (𝑎) Bode diagram of the uncontrolled and controlled nominal structure. (𝑏) Zoom on mode 4

Fig. 10 :

 10 Fig. 10: Spatial transfer function response of the nominal system

Fig. 11 :

 11 Fig. 11: Spatial 𝑯 ∞ norm of the controlled and uncontrolled system (𝑸(𝒓) = 𝟏)

Fig. 12 :

 12 Fig. 12: Schroeder disturbance signal output responses of the healthy structure Different response effects may result from different damages, which depend on several factors. It is possible for instance that the damage can lead to a decrease in the structure vibration on some regions. However, for the problem at hand, and considering this increase in the vibration, we aim to address the several possible DTAC strategies, to analyze how to change the controller in order to get the better possible performance of the respective controller.

Fig. 13 :

 13 Fig. 13: (𝑎) Comparing the effect on the complete model including a small damage. (𝑏) BVID Damage centered at position (𝑥 = 570, 𝑦 = 140) 𝑚𝑚

Fig. 14 :

 14 Fig. 14: Effect of a small damage on the first controller performance for a Schroeder disturbance response

Fig. 15 :

 15 Fig. 15: Effect of a severe damage on the first controller performance

Fig. 16 :Fig. 17 :

 1617 Fig. 16: Controller reconfiguration for a Schroeder disturbance

Fig. 18 :

 18 Fig. 18: Severe damage crack along the 45° direction

Fig. 19 :Fig. 20 :

 1920 Fig. 19: Effect of crack damage on the frequency responses

Fig. 21 :

 21 Fig. 21: Gauss (𝑎) and Gate (𝑏) spatial weighing functions 𝑄(𝑟) Fig. 22 presents results obtained for these two weighing functions, and also for a constant spatial weight for the complete beam, the pointwise 𝐻 ∞ controller and the open-loop system, in terms of the 𝐻 ∞ norm. It is possible to see that the open-loop 𝐻 ∞ norm is growing along the 𝑥-axis, with an inflection due to the damage, and getting a high level after that. All controlled signals clearly show the attenuation in the vibration. To analyze these results, it is important to notice the region where damage is located and also the performance point used for the pointwise controller.The open loop response shows a minimum at the damage region, and then increases to the tip of the plate. The pointwise controller obtains the smaller norm, but only at the specific considered point, and after this point increases the norm in a rate bigger than the open loop system. At the region near the clamp of the plate the norm is similar to the others, but in the center it is higher. Considering the spatial controller with a constant weighing function, it presents a regular performance no the complete plate, and only in a region near the performance point the pointwise norm is smaller. Comparing this with the other two spatial function controllers, result of the constant weigh function controller is a little better in general, except in the region of the damage.

Fig. 22 :

 22 Fig. 22: Simulated closed loop spatial 𝐻 ∞ norm for the nominal system as function of 𝑥 at the line 𝑦 = 150 𝑚𝑚

Table 1 :

 1 Some natural frequencies of the healthy structure

	Modes	Frequency (rad/s)	Modes	Frequency (rad/s)	Modes	Frequency (rad/s)	Modes	Frequency (rad/s)	Modes	Frequency (rad/s)
	𝟏	2.7	𝟔	93.3	𝟏𝟏	278.2	𝟏𝟔	448.0	𝟐𝟐	742.2
	𝟐	16.9	𝟕	140.2	𝟏𝟐	325.5	𝟏𝟕	475.9	𝟐𝟓	742.2
	𝟑	26.6	𝟖	152.7	𝟏𝟑	366.5	𝟏𝟖	493.1	𝟑𝟎	1011.5
	𝟒	47.2	𝟗	208.9	𝟏𝟒	405.4	𝟏𝟗	556.8	35	1184.1
	5	79.5	𝟏𝟎	228.3	𝟏𝟓	430.7	𝟐𝟎	564.3	𝟒𝟎	1297.8

Table A1 :

 A1 Mechanical properties of the composite material (23℃) Mechanical and electrical properties of the PZTs (NOLIAC ® )

	Property	𝜌 (𝑔/𝑚 3 )	𝐸 11 = 𝐸 22 (𝐺𝑃𝑎)	𝐸 33 (𝐺𝑃𝑎)	𝜐 12	𝐺 12 = 𝐺 13 = 𝐺 23 (𝑝𝐶/𝑁)
		Value	1154	69		8.1	0,03	4.8
	TableA2: Property Dim (𝑚𝑚)	𝜌 (𝐾𝑔/𝑚^3)	𝐸 11 (𝐺𝑃𝑎)	𝐸 33 (𝐺𝑃𝑎)	𝜐 12	𝑑 31 (𝑝𝐶/𝑁)	𝑑 33 (𝑝𝐶/𝑁)
	Value	28 × 14 × 0.1	7600	62,50	52,63	0,38	-195	460

Table A3 :

 A3 Mechanical and electrical properties of the MFCs (Smart Materials ® )

	Property	Dim (𝑚𝑚)	𝜌 (𝐾𝑔/𝑚^3)	𝐸 11 (𝐺𝑃𝑎)	𝐺 33 (𝐺𝑃𝑎)	𝜐 21	𝜐 12	𝑑 31 (𝑝𝐶/𝑁)	𝑑 33 (𝑝𝐶/𝑁)
	Value	∅30 × 0.1		5440	30.4	5.52	0,16	0,38	-170	400
				TableA4: 302-3M EPO-TEK glue properties
	Property	Operating temperature		Coefficient of thermal expansion	Dielectric constant	Dissipation factor	Storage modulus
	Value	-55℃ 𝑡𝑜 100 ℃	193 10 -6 ℃ -1	3,39 𝑎𝑡 1 𝑘𝐻𝑧	0,0061 𝑎𝑡 1 𝑘𝐻𝑧 1,73 𝑀𝑃𝑎 𝑎𝑡 23°𝐶


Roberto A Boto

J Contreras-García

Dr M Calatayud
email: calatayu@lct.jussieu.fr

The role of dispersion forces in metal-supported self-assembled monolayers

Keywords: SAM, DFT, NCI, platinum, amine

The self-assembly of organic molecules (octylamine) is studied by means of periodic DFT calculations including Grimme D2 dispersion scheme. It has been found that in the gas-phase dispersive forces are crucial when packing is considered, modifying considerably the potential energy surface compared to the case of a molecular dimer. In interaction with a metallic surface (platinum), the electronic interaction between adsorbate and substrate is predominant, dispersion forces mainly increasing the adsorption strength. A detailed study on the structural parameters of the moleculemetal system is reported to gain understanding on the factors affecting their stability. The intermolecular bonding in these systems is analyzed by means of the non-covalent interactions NCI index. The decomposition of the 2D packing along each direction (x,y) highlights the importance of the diagonal contributions in the stabilization of the square pattern. Moreover, the stabilizing nature of H-H interactions between neighboring molecules is supported in this model.

Introduction

The binding of long alkyl chain molecules to metal surfaces is being deeply investigated for application in many technological fields such as films, electronic junctions, tribology or coating [START_REF] Love | Self-Assembled Monolayers of Thiolates on Metals as a Form of Nanotechnology[END_REF].

The stability of such systems is mainly due to a competition between metal-molecule and moleculemolecule interactions, whose strength critically determines their physicochemical properties. For instance, in gold/thiol self-assembled monolayers (SAMs), the surface-molecule interaction must be strong to immobilize the alkyl chains on the metal surface [START_REF] Love | Self-Assembled Monolayers of Thiolates on Metals as a Form of Nanotechnology[END_REF][START_REF] Folkers | Self-assembled monolayers of alkanethiols on gold: comparisons of monolayers containing mixtures of short-and long-chain constituents with methyl and hydroxymethyl terminal groups[END_REF]. Another example is the synthesis of metal nanoparticles in mild conditions, where the alkylamine chains are used as surfactant to stabilize the metallic surface [START_REF] Salzemann | Influence of Hydrogen on the Morphology of Platinum and Palladium Nanocrystals[END_REF][START_REF] Chen | Small Adsorbate-Assisted Shape Control of Pd and Pt Nanocrystals[END_REF]. Here, the metal-surface strength must be intermediate: if it is too strong, the molecule blocks the surface site and the particle cannot grow further leading to small nanocrystals, but if it is too weak the particle surface is not stabilized and it grows isotropically leading to large spherical nanoparticles without control on the size or shape. Understanding the nature and the strength of the interactions between surfaces and long chain molecules is thus of great importance and may help in the design of systems with a specific molecule-surface and molecule-molecule strengths.

The intermolecular interactions in SAMs come from the vicinity of the neighboring molecules that form an ordered two-dimensional layer and are mainly caused by dispersion forces. The role of the dispersion forces between the alkyl chains is poorly understood although it may become crucial in the stabilization of the system. Dispersion forces are commonly estimated from different theoretical approaches. Semi-hybrid functionals have been used to study thiol/gold SAMs [START_REF] Ferrighi | Study of Alkylthiolate Self-assembled Monolayers on Au(111) Using a Semilocal meta-GGA Density Functional[END_REF]. Recently, the implementation of the Grimme approach [START_REF] Grimme | Semiempirical GGA-type density functional constructed with a long-range dispersion correction[END_REF][START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF] in solid state codes has allowed exploring their role in adsorption systems. It has been found that the inclusion of Grimme DN corrections can significantly affect the calculated geometry of adsorption as well as its strength [START_REF] Mercurio | Structure and Energetics of Azobenzene on Ag(111): Benchmarking Semiempirical Dispersion Correction Approaches[END_REF]. Also, the self-assembling of polymers in vacuo has been successfully described by using such approach [START_REF] Melis | Self-Assembling of Poly(3-hexylthiophene)[END_REF]. In the last years, dispersion in short alkyl chains assembly on metals has been studied with ab initio methods [START_REF] Abufager | Theoretical study of the structure of self-assembled monolayers of short alkylthiolates on Au(111) and Ag(111): the role of induced substrate reconstruction and chain-chain interactions[END_REF][START_REF] Canchaya | Study of the interaction between short alkanethiols from ab initio calculations[END_REF].

As regards bonding analysis on SAMs, the reactivity of the tail groups has been studied by conceptual DFT methods [START_REF] Tielens | Exploring the reactivity of mixed -functionalized undecanethiol self-assembled monolayers A DFT study[END_REF] and the chemical bond between surfactant and surface studied with the electron localization function ELF [START_REF] Andreoni | Density functional theory approach to thiols and disulfides on gold: Au(111) surface and clusters[END_REF]. However, little is known about the weak interactions involved in the packing of molecules, as in self-assembled monolayers on a solid surface. The aim of the present paper is to highlight the crucial role of dispersion forces in the organization of long alkyl molecules in two dimensions. It will be shown that in the gas-phase the energetic profile of a packing of molecules is dramatically affected by such interactions. Standard Density Functional Theory (DFT) calculations are used combined with Grimme D2 approach. A bonding analysis based on Non Covalent Interactions (NCI) index is applied to illustrate the nature of the intermolecular interactions.

Models and methods

The platinum (100)-octylamine (NH 2 C 8 H 17 ) system has been chosen because it has been recently reported to lead to stable cubic platinum nanoparticles [START_REF] Salzemann | Influence of Hydrogen on the Morphology of Platinum and Palladium Nanocrystals[END_REF][START_REF] Aguilera-Porta | Understanding How in Situ Generated Hydrogen Controls the Morphology of Platinum Nanoparticles[END_REF]. Such particles are synthesized in mild conditions at the interface between an organic and an aqueous phase. The octylamine is used as surfactant to stabilize the metal particle in the organic (toluene) phase. The particles obtained are cubic and the metal will be modeled by a (100)-terminated five layers-thick slab. The octylamine is investigated in the gas-phase as well as supported on the platinum slab in order to decompose the parameters affecting stability: molecule-molecule distance, molecule packing, surface-molecule distance and coverage.

The calculations are carried out with the revised Perdew Burke Ernzerhof [START_REF] Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Zhang | Comment on Generalized Gradient Approximation Made Simple[END_REF] functional as implemented in the VASP code [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Ab initio molecular-dynamics simulation of the liquid-metal-amorphoussemiconductor transition in germanium[END_REF]. Test calculations on the Pt fcc bulk lattice parameter and cohesive energy are shown in Figure S1, they support the choice of rPBE+D2 level of theory. Projector Augmented Wave (PAW) pseudopotentials [START_REF] Blochl | Projector augmented-wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] combined to plane waves (cutoff=400 eV) represent the electronic distribution. An appropriate set of k-points is used to sample the Brillouin zone. Periodic boundary conditions apply and a vacuum of at least 15 Å is included to avoid interaction between successive layers. Starting from the gas-phase optimized geometry at the rPBE level, calculations are done without geometrical optimization for the gas-phase studies and with the conjugate-gradient algorithm for the surface-molecule interaction studies, with the geometrical constrains specified in each case (see Table S1). The interaction energies are calculated by the formula:

A+ B = [AB] where E int = E [AB]-E A -E B
thus negative for exothermic reactions.

Dispersion interactions are included by using the Grimme D2 approach as implemented in the VASP code (parameters for Pt: radius=1.676 Å and C 6 =19.46 Jnm 6 mol -1 obtained from [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF]). As noticed in the literature [START_REF] Mercurio | Structure and Energetics of Azobenzene on Ag(111): Benchmarking Semiempirical Dispersion Correction Approaches[END_REF], the problem of metal screening of dispersive forces can be roughly corrected by including dispersion only on the uppermost slab layer. We have considered this possibility together with the pure rPBE and full D2, denoted as 1L, rPBE and D2 respectively. Tests on the propilamine dimer potential energy curve have been carried out at the MP2/6-311++g(3df,2pd), BSSE corrected, together with the rPBE and rPBE+D2 levels, and show very good agreement with the rPBE+D2 setting used in the present work, as can be seen in Figure S2.

It has been recently shown that although the energetic balance at weak interactions is cumbersome to detect from energetics, the density (r) reconstruction by those methods is accurate enough to reveal the interactions present in the system. This is the principle lying behind the recently reported index for detecting Non-Covalent-Interactions, NCI [START_REF] Johnson | Revealing Noncovalent Interactions[END_REF]. This index enables the identification and characterization of weak interactions of various strengths as chemically intuitive reduced density gradient isosurfaces that reveal both stabilizing (hydrogen bonds, van der Waals) and destabilizing (steric clashes) interactions. It can also be applied to ionic interactions [START_REF] Contreras-García | Ionic interactions: Comparative topological approach[END_REF]. In order to achieve a better comprehension of the nature of bonding and packing, representative systems will be studied by means of the NCI method. The NCI analysis has been carried out with the critic2 code [START_REF] Otero-De-La-Roza | Critic: a new program for the topological analysis of solid-state electron densities[END_REF] and visualized in VMD [START_REF] Humphrey | VMD -Visual Molecular Dynamics[END_REF].

Results and discussion

We have chosen to investigate independently the key factors in the stabilization of the system surface-molecule so we can derive the nature and the strength of the forces responsible for each interaction. Thus, we have computed the gas-phase molecule-molecule interaction as regards their interatomic distance, the relative orientation and the packing effect. Then, we have studied the surface-molecule interaction concerning the adsorption site, the distance surface-molecule, the tilting angle and the coverage.

The molecule-molecule interaction in the gas-phase

The potential energy surface of an octylamine dimer in the gas-phase is presented in Figure 1. It can be seen that the curve decays asymptotically and reaches a minimum for a distance of 5.5 Å (rPBE) and 5.0 Å (D2). The energy gain with respect to the isolated molecule in the gas-phase is almost athermic for rPBE and only slightly exothermic, -0.12 eV, for D2. Compared to the propilamine results (Figure S2), the energy gain is of 0.05 eV. Such gain is due to the longest alkyl chain that increases the dispersive interactions.

The packing effect of the molecule on the surface has been calculated by considering the molecule in a tetragonal unit cell repeated in three dimensions. The a parameter is modified so as to consider the distance between consecutive neighboring molecules. The curves obtained, presented in Figure 1, show a behavior similar to that of the dimer for the pure rPBE functional. Instead, a qualitative change is observed for the D2 calculations: a minimum in the curve appears at a distance of 4.5 Å, with an energy gain of -0.28 eV. Therefore, dispersion forces seem to be crucial for representing packing in two-dimensions in the gas-phase.

Molecule-surface interaction

The adsorption site preferred for the NH 2 C 8 H 17 molecule is a quasi-top site [START_REF] Aguilera-Porta | Understanding How in Situ Generated Hydrogen Controls the Morphology of Platinum Nanoparticles[END_REF], as found also for NH 3 [START_REF] Novell-Leruth | DFT Characterization of Adsorbed NHx Species on Pt(100) and Pt(111) Surfaces[END_REF]. Concerning the molecule orientation on the surface, previous calculations on Au/thiol SAMs show that the alkyl chains are not perpendicular to the surface but tilted [START_REF] Tielens | Characterization of omega-functionalized undecanethiol mixed self-assembled monolayers on Au(111): A combined polarization modulation infrared reflection-absorption spectroscopy/X-ray photoelectron spectroscopy/periodic density functional theory study[END_REF][START_REF] Tielens | The stability of binary SAMs formed by omega-acid and alcohol functionalized thiol mixtures[END_REF][START_REF] Tielens | Exploring the reactivity of mixed omega-functionalized undecanethiol self-assembled monolayers -A DFT study[END_REF], and is also the case in Ag/thiol systems [START_REF] Torres | Evidence for the Formation of Different Energetically Similar Atomic Structures in Ag(111)-(V7xV7)-R19.1 CH 3 S[END_REF]. Our calculated value for the tilting angle a (see Figure 2) is ~23 deg (rPBE), ~27 deg (D2) with a gain in energy of -0.33 eV (rPBE) -0.55 eV (D2) with respect to the molecule perpendicular to the slab. Also, the relative orientation of the molecule with respect to the surface sites has been explored through the radial angle beta, 45 deg being the optimal angle for all the methods (see Figure 2). The system is much less sensitive to this radial angle than to the tilting angle, the gain in energy is small (0.09 eV for rPBE, 0.05 eV for rPBE+D2). The coverage effect has been investigated for coverage values q from 0.11 to 0.5. The coverage is defined as the number of molecules per surface Pt atom as in Table 1. For all the coverages considered the adsorption occurs on a quasi-top position, the platinum atom to which the amine is bonded slightly moving upwards. The most favorable adsorption energy is found for q=0.125, and becomes less favorable as the coverage increases. For a coverage of q= 0.5 the amine is almost vertical to the surface, but as the coverage decreases the tilting angle increases reaching values around 30 deg. This is an indication of the higher intermolecular interaction occurring when the molecules are close to each other (high coverage). Low coverage involves almost exclusively molecule-surface interaction due to long distances between neighboring molecules. The Pt-N distance is higher than 2 Å in all cases. In order to investigate the interaction between the molecule and the surface we have selected the coverage q=0.25. At this coverage the adsorption energy is quite exothermic and the neighboring molecules interact at a distance of 5.65 Å which means that the adsorbate-substrate interaction is predominant. We have studied the potential energy curves for the molecule as a function of the distance with the surface, d Pt-N , keeping the geometry fixed. Figure 3 right displays the curve obtained. The rPBE calculations lead to a shallow minima (d Pt-N =2.2-2.4 Å, E int = -0.39 eV). The D2 calculations result in a deeper well (d Pt-N =2.2 Å, E int =-1.26 eV). The 1L calculations lie in between the curves (d Pt-N =2.2 Å , E int =-1.14 eV), closer to the full D2 values. It can be concluded that the equilibrium distance between the octylamine and the Pt slab is around 2.2 Å, which indicates a physisorption. The adsorption energy is moderate, around -1.2 eV when considering dispersion forces and -0.39 eV at the rPBE level. Dispersion is therefore more important in the energetics of adsorption, which results in a significant increase from -0.39 eV to -1.2 eV, than in the geometry, that results in a slightly shorter Pt-N bond upon inclusion of the Grimme correction.

Orientation

As shown in Section 3.1, Figure 1, the gas-phase packing of octylamine molecules leads to a significant stabilization of the square 2D arrangement. In the ordered superstructure the interaction between neighboring molecules depend on the relative orientation. A closer look at the geometries enables understanding that this difference comes from a different geometrical arrangement (Figure 4). In these figures, the y alignment is clearly different than the x one as regards H-H interactions. In order to gain more information on the role of the relative orientation of the two chains, we have considered polymers of octylamine in the x and y directions separately. Tests on the propilamine dimer at MP2 level are displayed in Figure S3. They indicate that the x dimer is slightly more stable than the y one (-0.054 eV for x, -0.04 eV for y), and the equilibrium intermolecular distance is longer for the former (4.7 Å vs 4.4 Å). The same system calculated at the rPBE+D2 level gives x and y dimers almost isoenergetic (-0.05 eV) with d NN distaces at equilibrium of 4.8 Å (x) 4.4 Å (y). This indicates a slight overestimation of the y interactions compared to the x ones in the D2 approach used.

Figure S4 shows the energetic profile of 1D polymers along x and y directions as a function of the distance between neighboring molecules d NN . It can be observed that the stabilization in the alignment along y direction is higher than in the x direction which is in part due to the method as we have seen above. Comparing now the 2D pattern to the addition of x and y polymers (Figure S3 right) the result is a net stabilization of the 2D square pattern with respect to the addition of x plus y, almost inexistent for rPBE but significant when using the D2 approach. Although the method used overstabilizes the interactions in the y direction, the overall stabilization of the 2D pattern compared to the sum of the polymers is significant. This result highlights the key role of second neighbors in the stabilization of the squared superstructure. The relative orientation of the neighbors plays thus a key role in stabilizing extended systems. In order to have a better insight on the energetics we have calculated the energy profile of a gas-phase octylamine dimer as a function of the relative orientation of the two molecules. It can be seen in Figure S4 that the molecules rotated c=80 deg in the z-axis is the most favourable situation. This orientation is similar to the one obtained for the y-polymer that would correspond to c=90 deg (Figure 4, right). On the other side, having the molecules face to face, c=0 deg, is less favourable and corresponds to the x-polymer in Figure 4 left. Note that both the rPBE and D2-corrected series give very similar energetic profiles.

Bonding

In order to understand the effects of these geometrical arrangements at the bonding level, we have carried out NCI calculations at the rPBE+D2 level. Figure 5 shows the isocontour plots of the NCI index of the octylamine molecule polymer model along the x and y directions. Note that NCI is very little affected by the level of calculation at a given geometry: the same features are found with the MP2 curves of the propylamine dimer, see Figure 5 right. The reduced density s vs r diagrams for the three systems can be found in Figure S5.

Figure 5: NCI isocountours (in green) for the interaction of octylamines in periodic systems as found for the the x and y polymers and the square 2D packed system.

As expected, NCI isosurfaces are anisotropic, differentiating between delocalized and localized interactions along x and y directions respectively. The symmetric arrangement along the y direction gives rise to localized H-H interactions between them, whereas delocalized H-CH 2 interactions appear along the x direction. The directionality of H-H interactions is characterized within the NCI approach by the small, round shape of the regions and the strength by the relatively large accumulation of electron density. Weak interactions such as the ones found in the x polymer give rise to interaction regions, rather than localized NCI domains, and present a relatively low electron density. In both cases, the colour of the NCI isosurfaces reveals the weak non-bonding nature of all the interactions, which agrees with the need to resort to D2 corrections to account for them.

Moreover, the stabilizing effect of H-H interactions along the y direction raises a fundamental problem in crystal packing, which is still an open question. These interactions are usually assumed to give rise to steric clashes, and contribute to the destabilization of the crystal. However, topological approaches such as QTAIM point towards their stabilizing nature.

The H H BCPs were first interpreted by Cioslowski and Mixon in kekulene [START_REF] Cioslowski | Weak bonds in the topological theory of atoms in molecules[END_REF] and ortho-substituted biphenyls [START_REF] Cioslowski | Topological properties of electron density in search of steric interactions in molecules: electronic structure calculations on ortho-substituted biphenyls[END_REF] and assigned to non-bonded repulsive contacts , a view supported by later studies with different energy partitions [START_REF] Dunitz | Molecular Recognition in Organic Crystals: Directed Intermolecular Bonds or Nonlocalized Bonding?[END_REF][START_REF] Poater | Hydrogen Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-In-Molecules Theory, Does Not Exist[END_REF]. QTAIM atomic energies, on the other hand, predict a stabilization caused by H H contacts of up to 10 kcal/mol in the general case of polybenzenoid molecules [START_REF] Matta | Hydrogen Hydrogen Bonding: A Stabilizing Interaction in Molecules and Crystals[END_REF] . It should be noted that the weakness of dihydrogen interactions and the arbitrariness in the choice of energy partition precludes determining how much these interactions contribute to the total binding energy of a crystal, so it would be extremely difficult to determine in an unbiased manner the strength and character of these interactions, except in those cases where the total stabilization is to a good approximation, only given by those interactions. This is the case of the y polymer. Being hydrogen interactions the main source of interaction in the y direction (very little dispersion is observed), the attractive interaction observed in this polymer can only be attributed to H-H interactions.

In the case of octylamine 2D square packing, the stabilizing nature of H-H interactions in the y direction is supported by i) the presence of clear NCI maps regions instead of atom-atom contacts and ii) the energetic curves obtained after inclusion of the D2 Grimme correction.

From the visual analysis of NCI isosurfaces of the linear polymers and the 2D arrangement (Figure 5), the x and y components are perfectly recovered, along with an extra region between interactions along x and y direction. This interaction between diagonal elements of the arrangement, may be identified as the responsible of its relative stability. In other words, the additive effect of the diagonal interactions is related to packing. It is interesting to note that once again, these interactions are dispersive-like, which explains why they are so difficult to track in solids. The use of NCI index allows therefore revealing the packing interactions in a visual way and provides a new tool to analyse dispersion interactions in extended systems.

Conclusion

A combined periodic DFT and NCI approach has been used to describe the geometry, energetics and bonding in a model of self-assembled monolayer octylamine-platinum. The dispersion effects are found to play a major role in the molecular packing energetics and a moderate effect in the adsorption geometry. We have been able to identify the stabilizing nature of dispersive H-H interactions in packing thanks to the NCI index. The NCI analysis combined with dispersive corrections in standard DFT calculations are thus powerful tools to understand molecular packing in extended systems.

Figure 1 .

 1 Figure 1. Energy profile of gas-phase octylamine systems. Energies are given relative to an octylamine

Figure 2 .

 2 Figure 2. Energy profile for the geometry of octylamine on a 2x2 Pt(100) slab as a function of the

Figure 3 .

 3 Figure 3. Energy profile for the molecule-surface interactions as a function of the coverage (left) and

Figure 4

 4 displays the relative orientation of two neighboring octylamines in the x and y directions. It can be observed that they are not equivalent since they involve different H-H interactions. Along the y direction, octylamine molecules are oriented in such a way that the hydrogen atoms in the chains face those of the neighboring molecule. Instead, there is no such orientation when molecules arrange along the x direction and hydrogens are not face to face.

Figure 4 .

 4 Figure 4. Octylamine molecules along x (left) and y(right) direction in a square 2D model

Table 1 .

 1 Geometric and energetic parameters considered for the unit cells given. Coverage q, cell parameter a in Å, adsorption energy E ads in eV, distance Pt-N in and tilting angle c in degrees.

				rPBE			D2		1L
	Unit cell	q	a	E ads	d Pt-N (a)	a	E ads	d Pt-N (a)	E ads	d Pt-N (a)
	1x1 (10 at)	0.5 3.975	1.24 2.024(21.0)	3.985	-0.59	2.190(5.5)	-0.47 2.187(5.5)
	V2V2 (20 at)	0.25 5.653 -0.39	2.145(26.0)	5.635	-1.26 2.142(26.0)	-1.14 2.134(26.0)
	2x2 (40 at)	0.125	8.00 -0.78	2.151(29.7)	7.969	-1.42 2.111(28.0)	-1.27 2.130(29.8)
	V3V3 (45 at) 0.111	8.48 -0.54	2.150(32.3)	8.453	-1.22 2.118(30.7)	-1.17 2.127(31.9)

Acknowledgements

This work was performed using HPC resources from GENCI-CINES/IDRIS (Grant 2013-x2013082131, Grant 2014-x2014082131) and the CCRE-DSI of Université P. M. Curie. Dr. B. Diawara is warmly acknowledged for the Modelview visualization program. This work undertaken partially in the framework of CALSIMLAB is supported by the public grant AND-11-LABX-0037-07 overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir" program (reference: ANR-11-IDEX-0004-02).