D Toe

A Mentani

L Govoni

F Bourrier

G Gottardi

S Lambert

Introducing meta-models for a more efficient hazard mitigation strategy with rockfall protection barriers

Keywords: Rockfall mitigation, Barrier, Meta-model, Deterministic model

The paper presents a new approach to assess the effecctiveness of rockfall protection barriers, accounting for the wide variety of impact conditions observed on natural sites. This approach makes use of meta-models, considering a widely used rockfall barrier type and was developed from on FE simulation results. Six input parameters relevant to the block impact conditions have been considered. Two meta-models were developed concerning the barrier capability either of stopping the block or in reducing its kinetic energy. The outcome of the parameters range on the meta-model accuracy have been also investigated. The results of the study reveal that the meta-models effective in reproducing with accuracy the response of the barrier to any impact conditions, providing a formidable tool to support the design of these structures. Furthermore, allowing to accomodate the effects of the impact conditions on the prediction of the block-barrier interaction, the approach can be successfully used in combination with rockfall trajectory simulation tools to improve rockfall quantitative hazard assessment and optimise rockfall mitigation strategies.

Introduction

The mitigation of rockfall hazard often relies on passive protective structures such as embankments, galleries or protection barriers (Lambert and Bourrier, 2013;[START_REF] Calvetti | Rockfall impacts on sheltering tunnels: real-scale experiments[END_REF][START_REF] Gentilini | Design of falling rock protection barriers using numerical models[END_REF]. Rockfall barriers are the most widely used structural countermeasures for intercepting rock blocks on their route down to the elements at risk. Barriers are typically made of different metallic components including posts, net, cables and other connecting components which makes their mechanical response very complex to predict.

As for other passive rockfall protection structures, the design of barriers, as well as the definition of the optimum protection strategy for a given site, relies on trajectory simulation results. In particular, stochastic trajectory simulation models provide statistics associated to the rock blocks paths along the slope as well as their reach probability. The design aims at reducing the latter down to a targeted value while considering the former. More precisely, relevant statistical descriptors associated to the blocks passing heights and kinetic energies are considered for determining the required protective structure characteristics in terms of interception height and kinetic energy absorption capacity respectively (Lambert et al., 2013). In practice, the barrier design for a given site is mainly based on the comparison of the statistical descriptor of the block kinetic energy barrier nominal capacity. This capacity is often obtained following the European guideline ETAG 027 [START_REF] Eota | ETAG 027 : Guideline for European Technical Approval of Falling Rock Protection Kits[END_REF], which provides detailed indications on how to test and assess the performance of a barrier and to obtain the CE marking. Nevertheless, the impact conditions in such test, which essentially consist in an impact in the center of a 3-spans barrier by a block without any rotation, can be considered not representative of the wide variety of impact loading cases as resulting from the interception of blocks on-site. This issue has been long debated over the last ten years and several research works have suggested that a test in such conditions may not be the most critical, as it neglects the effects of parameters such as the impact point location or the incident angle of the block trajectory [START_REF] Cazzani | Dynamic finite element analysis of interceptive devices for falling rocks[END_REF][START_REF] Cantarelli | Modelling rockfall protection fences[END_REF][START_REF] Lambert | Experimental study of the impact response of geocells as components of rockfall protection embankments[END_REF][START_REF] Chanut | Dynamic Behavior of Rock Fall Protection Net Fences: A Parametric Study[END_REF]. This suggests that current design approaches might be inadequate in accounting for the global ability of barriers in arresting the blocks, considering all the possible trajectories.

Improving the design of barriers as well as assessing their efficiency in reducing the hazard at the elements at risk requires better accounting for their actual mechanical response to blocks impacts. This may be undertaken making use of suitable numerical tool, among the various ones that have been developed over the last 20 years, with increasing complexity with either finite or discrete element models (FEM or DEM resp.) [START_REF] Nicot | From a constitutive modelling of metallic rings to the design of rockfall restraining nets[END_REF][START_REF] Volkwein | Numerical Simulation of Flexible Rockfall Protection Systems[END_REF][START_REF] Gottardi | Full-scale Modelling of Falling Rock Protection Barriers[END_REF][START_REF] De Miranda | A simple model to simulate the fullscale behaviour of falling rock protection barriers[END_REF][START_REF] Bertrand | Full-Scale Dynamic Analysis of an Innovative Rockfall Fence Under Impact Using the Discrete Element Method: from the Local Scale to the Structure Scale[END_REF][START_REF] Gentilini | Three-dimensional numerical modelling of falling rock protection barriers[END_REF][START_REF] Gentilini | Design of falling rock protection barriers using numerical models[END_REF][START_REF] Escallòn | Parameter identification of rockfall protection barrier components through an inverse formulation[END_REF][START_REF] De Miranda | Virtual testing of existing semi-rigid rockfall protection barriers[END_REF][START_REF] Bourrier | Accounting for the variability of rock detachment conditions in designing rockfall protection structures[END_REF][START_REF] Mentani | Numerical Modelling of a Low-Energy Rockfall Barrier: New Insight into the Bullet Effect[END_REF] curate. Nevertheless, a relatively high computational cost limits their use in view of investigating the response of barriers varying many parameters, either concerning the structure or the impact conditions.

A promising alternative of accounting for the mechanical response of barriers, for both design and hazard reduction assessment purposes, consists in using meta-models. Such approache proposes surrogate models, so called meta-models, of more complex mechanical models, embedding their complexity but are more efficient in terms of computational time [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansions[END_REF][START_REF] Blatman | Efficient computation of global sensitivity indices using sparse polynomial chaos expansions[END_REF][START_REF] Mollon | Probabilistic Analysis of Pressurized Tunnels against Face Stability Using Collocation-Based Stochastic Response Surface Method[END_REF]. In the context of rockfall protection structures, the surrogate models are computationally cost-effective tools dedicated to statistical analysis of the structure response to varying impact conditions. Meta-models are widely used in civil engineering [START_REF] Jin | Comparative studies of metamodelling techniques under multiple modelling criteria[END_REF][START_REF] Farhang-Mehr | Bayesian meta-modeling of engineering design simulations: a sequential approach with adaptation to irregularities[END_REF][START_REF] Gonzalez-Perez | Metamodelling for Software Engineering[END_REF][START_REF] Toe | Analysis of the effect of trees on block propagation using a DEM model: implications for rockfall modelling[END_REF]. Application in the field of rockfall protection structures was first considered by [START_REF] Bourrier | Accounting for the variability of rock detachment conditions in designing rockfall protection structures[END_REF] with the aim of investigating the failure occurrence of a barrier via a performance function and then by [START_REF] Mentani | A New Approach to Evaluate the Effectiveness of Rockfall Barriers[END_REF].

This article proposes the use of meta-modelling approaches for improving the design of passive rockfall protection structures with a specific focus on a barrier intended for low kinetic energies frequently encountered in the Alpine arc. This barrier type features an interception structure made of an hexagonal wire mesh supported by longitudinal cables passing through steel posts. A finite element (FE), three-dimensional, non-linear model of the barrier has been developed and subjected to impact simulations by varying simultaneously 6 impact parameters over a wide and a comparatively narrower range. The results of the analyses provides a thorough insight of the protection barrier behaviour in terms of failure mechanisms and enables to explore comprehensively its effect on the energy possessed by the impacting block. Meta-models of the barrier response in terms of block arrest and block kinetic energy reduction have been developed, considering two sets of impact condition parameters. The first set has been defined based on the widest possible range for each parameter. The second set has been adapted to the barrier capacity evaluated in standard conditions.

The article is organised as follows. In the first section, details of the FE models and FE analyses are given and results are discussed in terms of block-structure interaction, revealing the complexity of the barrier response to impact. In the second section, the development of meta-models of the barrier response is detailed. Then, the meta-model results are presented and discussed in terms of accuracy. The discussion addresses the influence of the parameter ranges on the meta-models accuracy, the benefits in using these meta-models compared to current design approaches, and their application to real cases.

Finite element modelling of a cable-net barrier

This section provides the details of the finite element (FE) modelling of a cable-net barrier, developed using the commercial code Abaqus (Abaqus, 2013). For this barrier type, the interception structure is made of longitudinal cables, connected to steel posts fully restrained at the base.

In general, this structure type is also provided with a secondary hexagonal meshwork fastened to the longitudinal cables. Although widely used, information on the response to impact of this barrier type are scarce. The results of the FE analyses then offer a new insight on the barrier mechanical behaviour, while providing the necessary base to the development of a meta-model of the block-barrier interaction.

FE model

As described in Fig. 1a, the study considers a three spans, 5 m spaced, cable-net barrier of 3.2 m nominal height. Longitudinal cables, 12 mm in diameter, pass through the internal posts (IPE 200) and are knotted to the external posts (IPE 300), connected to the ground by side cables of 18 mm diameter. A secondary meshwork, made of a double twisted hexagonal mesh is connected to the top and bottom longitudinal cables with steel wires. The FE model of the barrier is three-dimensional and made of one-dimensional elements, whose behaviour is governed by elasto-plastic constitutive laws. The mechanical response of the barrier elements was described based on available results of laboratory tests in de [START_REF] De Miranda | Virtual testing of existing semi-rigid rockfall protection barriers[END_REF]. Particular attention was devoted to model the behaviour of the wires within the hexagonal mesh, following data of experiments carried out on mesh portions [START_REF] Mentani | Numerical Modelling of a Low-Energy Rockfall Barrier: New Insight into the Bullet Effect[END_REF][START_REF] Thoeni | Discrete modelling of hexagonal wire meshes with a stochastically distorted contact model[END_REF]. A representative scheme of the stress-strain curve for each structural element is given in Fig. 1b, where the relevant model parameters are also inserted. As depicted, the posts behave following an elastic-perfectly-plastic law up to a failure limit, cables harden in the plastic phase and may undergo indefinite deformations once a second yielding threshold is attained and mesh wires soften prior to fail.

FE simulations and results

The barrier model was subjected to non-linear dynamic simulations. According to the reference system and notation introduced in Fig. 1, simulation were carried out, by impacting the barrier model with a prismatic test block of known volume V; at a position of coordinates X and Y; with an incident angle α; a translational velocity v; and a rotational velocity ω.

The first simulations were performed in accordance to the procedure described in the Annex A of ETAG 027 [START_REF] Eota | ETAG 027 : Guideline for European Technical Approval of Falling Rock Protection Kits[END_REF], to provide the cable-net barrier model with a reference capacity evaluated in standard conditions. In these simulations a translational velocity of 25 m/s and no rotational velocity were considered. The maximum block mass for which all the Guideline requirement were fullfilled, was found equal to 640 kg, yielding a reference capacity of 200 kJ for the cable net barrier. Further analyses were then run to provide the data necessary to the development of meta-models of the barrier response. To this purpose, parameters related to the block were varied. Tab. 1 collects these parameters along with their variation ranges, according The wide range (WR) set was considered in agreement with possible output of rockfall trajectory simulations and results of field tests (Bourrier et al., 2009a;[START_REF] Toe | Analysis of the effect of trees on block propagation using a DEM model: implications for rockfall modelling[END_REF]. As it is observed in Tab. 1, a freeboard on the barrier top was inserted, to avoid direct impacts of blocks on the top cable. A total of 280 simulations were carried out using combinations of the input parameters. The programme of tests resulted from a Latin-Hypercube sampling, assuming a uniform distribution of values within the parameter ranges [START_REF] Sacks | Design and Analysis of Computer Experiments[END_REF][START_REF] Fang | Design and Modeling for Computer Experiments[END_REF].

The results of this set of simulations provided new evidence of the barrier response considering a wide variety of realistic loading conditions. In particular, four types of block-barrier interactions were observed, which can be described as follows: i) the block is arrested by the barrier; ii) the block passes the barrier by rolling over it; iii) the block passes the barrier as a result of the perforation of the secondary hexagonal meshwork; iv) the block passed the barrier as a result of the failure of the whole structure. response within these threshold values, and thus supplying more data for the development of a meta-model of the barrier ability to arrest a block. In these new FE tests, the translational velocity and volume were varied within this comparatively narrower range, according to the limit value indicated in Table 1. As for the wide range, the virtual test programme was designed based on the Latin-Hypercube sampling procedure.

Over the 280 simulations, the barrier succeeded to stop the block in 61 cases. In 219 cases the barrier failed to stop the block : in 78 cases due to the block rolling over mechanism, in 66 cases due to mesh perforation and in 75 cases due to global failure.

Fig. 3b gather the 560 simulation results from the analyses conducted considering the narrow and wide range sets. The locus of kinetic energy equal to the determined reference capacity (isoenergy line at 200 kJ), is also inserted in Fig. 3b. The vast majority of the points corresponding to the arrested blocks falls below this line. However, cases of barrier failures in arresting the block are also found below this line. On total, the ratio of observed failure cases to the total number of cases below this line is as high as 52%. This comment holds for moderate block volumes and velocities. Indeed, restricting the analysis to cases where the block size is less than 1/3 the barrier height (thus to block volumes less than 0.7 m 3 and as suggested by ETAG 027 (EOTA, 2013)), this ratio equals 33.7%. This suggests that the use of a unique reference capacity value might be unconservative for this barrier.

Effects of the impact parameters on the barrier response

Fig. 4 illustrates the influence of the other impact parameters on the barrier response : the incident angle (Fig. 4a), the rotational velocity (Fig. 4b), the impact position (Fig. 4c -4d). For negative values of incident angle (upward trajectory), the rolling over mechanism is the prevailing block-barrier interaction mode (Fig. 4a). On the velocity plane, although symbols related to arrested blocks tend to concentrate in the area close to the origin, several points also prove the ability of the barrier to stop impacting blocks with high rotational velocities, up to 35 rad/sec (Fig. 4b). As shown in Fig. 4c, the points relevant to the general failure mechanism tend to concentrate close to the posts, whose location is highlighted with two hatched areas along the y-axis (position X in Fig. 1). This is particularly clear for translational velocities less than 20 m/s and general failures are observed at velocities down to 5 m/s. The influence of the impact position of the block along the y-axis is depicted in Fig. 4d. As it is observed, high values of block velocity and position tend to result in a rolling over mechanism, whereas no significant effects of the position parameter is observed on the barrier ablity of arresting the block, which remains prevalently driven by the block speed. This is also due to the considered freeboard introducing a threshold in the impact height with respect to the barrier height.

In Fig. 5, the results of the simulations are plotted on the volume-kinetic energy plane up to 600 kJ and compared with the iso-energy line. In Fig. 5a the kinetic energy is computed by considering the sole contribution of the translational component, while in Fig. 5b the total kinetic energy is used, accounting for both the translational and the rotational kinetic energies.

Comparison shows that, although the upward shifting of the data points from Fig. 5a to Fig. 5b produces a migration of some arrested block points above the nominal energy line, there are still several cases below the line, in which the barrier failed in arresting the block, due to mesh perforation, global failure and block rolling over mechanisms.

Overall, the results show a strong dependency of the barrier response to the impact conditions. Analysis of the data also showed that, although some trends could be observed, a clear correlation between input parameters and relevant block-barrier interaction mechanisms could not be established with certainty. These results bring to light the shortcomings of deterministic approaches based on a single impact assessment test, towards the evaluation of a structure effectiveness, as not adequately accounting for the impact conditions. This crucial aspect can be more successfully accommodated by reliability probabilistic approaches, through the use of meta-models which allow for the prediction of the barrier response as a function of the impact conditions, as described in the following sections.

The meta-modeling approach

This section provides the essential details of the meta-modelling strategy used in this study.

Meta-models can be defined in this context as mathematical operators describing the response envelop of a given structure, by considering a large number of variables. With reference to a rockfall protection structure, a meta-model can be developed from an optimized number of numerical simulations, performed using a detailed numerical model of the protection work, allowing for the prediction of its response to any set of variables, without running extra simulations. Due to its mathematical structure and computational cost-effectiveness, a meta-model can be integrated into probabilistic rockfall trajectory models rather easily.

Within the context of rockfall hazard assessment, it is important for a meta-model to capture with accuracy the effects of the structure on the falling block in terms of trajectory and postimpact kinetic energy. In the light of these general observations, this study focuses on two essential and correlated aspects of the block-barrier interaction, which are the barrier ability to arrest a block and, in case of failure, the block post-impact kinetic energy. To the scope, two different meta-models were developed for these two aspects using classical meta-modeling techniques.

The first meta-model is concerned with two classes: success and failure in arresting the block.

Under the first class, referred to as B S ucc , the arrested block cases are grouped, while the passing block cases are found to the second class, B Fail . As dealing with two classes, a Support Vector Machine (SVM), was used for creating the meta-model [START_REF] Brereton | Support Vector Machines for classification and regression[END_REF][START_REF] Kausar | A Review of Classification Approaches Using Support Vector Machine in Intrusion Detection[END_REF]. The second meta-model was developed with reference to the B Fail cases with the aim to predict the block kinetic energy reduction due to the block-barrier interaction. The value of energy reduction is given as the difference between the total (translational plus rotational) kinetic energy posses by the block just prior to the impact and the total kinetic energy possessed by the block just after the impact (E incident -E out = E RED ). The total kinetic energy was considered in order to account for possible coupling between translational and rotational velocities, after and before impact. To this aim, the Kriging method has been used [START_REF] Kleijnen | Kriging metamodeling in simulation: A review[END_REF][START_REF] Martin | Computational Improvements to Estimating Kriging Metamodel Parameters[END_REF].

Main features of these two meta-modeling approaches are given in the following sections.

The two meta-models were developed using the results of the FE simulations carried out on the cable-net barrier, by considering the input parameters within wide (WR) and narrow (NR) ranges separately. For each range set, a meta-model was created based on the SVM approach to predict the success or failure of the barrier in stopping the block. Another meta-model was built based on the Kriging approach to estimate the post-impact total kinetic energy of blocks passing the barrier. The Kriging approach based meta-model was developed based on the data of the FE simulations in which the barrier failed in arresting the block.

Support Vector Machine

The Support Vector Machine (SVM) approach is based on statistical learning theory [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], and can be used to build a meta-model which can predict the class of an output data.

This method has been used in many different fields of study as for examples remote sensing [START_REF] Mountrakis | Support vector machines in remote sensing: A review[END_REF], shape recognition [START_REF] Ma | Face Detection Based on Cost-Sensitive Support Vector Machines[END_REF], genomics recognition [START_REF] Sonnenburg | Large Scale Genomic Sequence SVM Classifiers[END_REF] and spam detection [START_REF] Wang | SVM-Based Spam Filter with Active and Online Learning[END_REF]. This method is adapted for binary or multi-class recognition and is here used for the former (success/failure of the barrier).

The basic SVM approach (M S V M ) consists of defining, in a space of input parameters, the optimal hyper-plane separating the regions associated with the different classes (success/failure of the barrier in this context). For that purpose, among all points of the space only those that are closest to the hyperplane, called support vector, are considered. The optimal hyperplane is defined as the hyperplane whose margin, i.e. distance from these closest points is maximal. It is thus calculated by maximizing the distance from the hyperplane to the closest points on each side.

The optimal definition of the hyperplane can require non linear transformation of the data to another space of potentially higher dimension using kernel functions [START_REF] Baudat | Kernel-based methods and function approximation[END_REF].

In this study, the space of the input parameters corresponds to the different parameters associated with the impact conditions. Linear and radial kernels have been used to build accurate meta-models (function svm in R (V 3.2.3) package e1071).

Kriging

The Kriging (Gaussian process modeling) approach is a procedure of interpolation which is used in various engineering and applied mathematical problems [START_REF] Simpson | Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization[END_REF][START_REF] Sudret | Meta-models for structural reliability and uncertainty quantification[END_REF][START_REF] Zhang | A Kriging Model for Dynamics of Mechanical Systems With Revolute Joint Clearances[END_REF]. This method is well adapted for approximating results of deterministic models, such as the post-impact kinetic energy of the block in this study [START_REF] Martin | Computational Improvements to Estimating Kriging Metamodel Parameters[END_REF]. Kriging

Error quantification

The accuracy of the developed meta-models was estimated by comparison with the data obtained from the FE simulations described and illustrated in Section 2. The prediction error for both meta-models was estimated using the leave-one-out cross validation method [START_REF] Allen | The Prediction Sum of Squares as a Criterion for Selecting Predictor Variables[END_REF].

For the SVM approach based meta-model, n results M(x i ) from the FEM simulations are considered. For each parameters combination x i , a meta-model is created using all FEM simulation results except M(x i ). The meta-model prediction for x i (M i S V M (x i )) is compared to the remaining result M(x i ) observed from the FEM simulations. This comparison is repeated for all x i ranging between x 1 and x n . The global accuracy of the meta-model (Q(M S V M )) is evaluated as follows :

Q(M S V M ) = 1 -( 1 n n i=1 M(x i ) -M i S V M (x i )) (1)
The results of the SVM based meta-model were further discussed with reference to the misclassification rate defined as follows. With reference to the FE observations, the SVM based meta-model can provide bad (false, F) or good prediction (true, T ). As described in Tab. For the Kriging approach based meta-model, a similar leave-one-out cross validation method as for the SVM approach based meta-model was used. The values of the residual block kinetic energy predicted by the meta-model M i E,K (x i ) were compared to those obtained from the FE simulations (M E (x i )). The accuracy of the meta-model is evaluated using the mean (Mean Err ) and standard deviation (S d Err ) of the residual error. Mean Err was calculated as:

Mean Err = 1 n n i=1 (M E (x i ) -M i E,K (x i )) (2)
S d Err is calculated as:

S d Err = n i=1 (M E (x i ) -M i E,K (x i )) 2 n -Mean 2

Err

(3)

Meta-models of the cable-net barrier

In this section the results of the meta-models of the cable-net barrier described in Section 2 are illustrated and discussed. The validation of the meta-models was pursued by comparison with the data obtained by the FE simulations. Focus is placed on the influence of the parameters range on the performance of the considered meta-models.

Wide range based meta-models

This section presents and discusses the results from the two meta-models developed based on the wide range set FE simulations (WR, Tab. 1).

Results of the meta-model addressing the ability of the barrier in stopping the block are given in Fig. 6 where symbols in grey stand for good prediction by the meta-model compared to FE simulations results. Bad predictions are grouped as positive (red) or negative (yellow) according to the definition given in Tab. 2. In particular, as described in Tab. 3, the meta-model failed to predict 8 out of 26 barrier success (FN r = 32%) and 5 out of 254 barrier failure (FP r =2%).

Over the 5 misclassified failures of type FP, 1 was related to mesh perforation, 3 were related to global failure and 1 was related to block rolling over the barrier. It can be concluded that the barrier efficiency in arresting the block is slightly overestimated as only 2% of the failure cases are not predicted by the meta-model. The global accuracy of the meta-model, Q(M S V M ), was found equal to 95% according to eq. 1. The meta-model dealing with the block kinetic energy reduction was created excluding the 26 simulations in which the block were stopped by the barrier. According to eq. 2 and 3, the mean error, Mean Err , and standard deviation, S d Err of the M K , are 0.52 kJ and 200 kJ respectively. The former value indicates that the meta-model prediction is unbiased, with the practical implication that there are as many unconservative predictions than conservative ones. The later value appears rather high compared to the barrier nominal capacity. The black curve on Fig. 7 gives the cumulative distribution of the difference between the energy reduction observed in the FE simulations (M(x i )) and the corresponding value as predicted by the meta-model (M k (x i )) normalised by the incident block kinetic energy (E in ). Cases where the block kinetic reduction predicted by the meta-model is higher than values observed in the FE simulations correspond to negative ratio values, and reveal an error on the unconservative side. On the contrary, error on the conservative side consists in cases where the predicted energy reduction is less than the observed value, corresponding to positive ratio values. In 5% of the cases, the overstimation of the barrier capacity in reducing the block kinetic energy prediction by the meta-model exceeds 50 % of the incident block kinetic energy.

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Narrow range based meta-models

The meta-models for the narrow range analysis were created using the second plan of experiments consisting of 280 combinations of the 6 input parameters (NR, Tab. 1).

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Cumulative distribution of the ratio between the prediction error (M(x i ) -M K (x i )) of the block energy reduction after the impact on the barrier and the total incident block kinetic energy Ein for the wide (black) and narrow (green) ranges sets.

The accuracy of the model dealing with the barrier block arresting ability was evaluated according to eq. 1 and resulted in M S V M equal to 92%. The meta-model failed to predict 16 barrier success over 61 (FN r = 27%) and failed to predict 6 barrier failures over 219 (FP r =3%) (Table . 4). Over these 6 misclassified cases, 5 are related to mesh perforation and 1 is related to global failure (Fig. 8).

Here again, the meta-model overestimates the barrier capacities and 3% of the failure cases are not predicted by the meta-model. As for the meta-model concerning the block kinetic energy reduction, the mean error, Mean Err , and standard deviation, S d Err , of the M K are -5.21 kJ and 111 kJ respectively. The same comments as for the wide range results hold. Neverthless, the lower standard deviation indicates that this meta-model is more efficient in predicting the block kinetic energy reduction.

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Discussion

The aim of this article being to propose using meta-models for assessing rockfall protection barriers efficiency, this section compares the two approaches (WR and NR) and discusses the advantages over classical design approaches in using meta-models.

Influence of the range considered

The meta-models have been developed considering two ranges of values for the input parameters. Intuitively, a meta-model developed for a range of input parameters fitted to the barrier nominal capacity is expected to provide much better results.

Similar trends in terms of model quality are observed for both meta-models and the difference between meta-models developed for narrow and large ranges is not that pronounced. As for the M S V M almost the same global accuracy is obtained for wide and narrow ranges (92/95.3%). A significantly smaller FP r rate is observed for the narrow range (27/32 %). In addition, the model developed for the narrow range accounts for a larger number of arrested blocks over the 300 simulations. Its ability in detecting success cases is thought to be higher.

As for the the prediction of the block kinetic energy reduction, both models have a Mean Err around 0 kJ. However, the predictions M K are significantly more accurate for the narrow range compared to that for the wide range. Indeed, the error standard deviation of the meta-model based on the narrow range is almost half that of the meta-model based on the wide range (111 kJ compared to 200 kJ).

In the end, the difference between the two meta-models appear rather small, the narrow range resulting in a slightly more accurate meta-model. Results obtained using the narrow range are considered in the following section.

Benefits of the meta-models

The current design practices are mainly based on the barrier nominal capacity. In this study, the nominal capacity was considered as the reference value obtained from impacts following the recommendations of the European guideline ETAG 027 [START_REF] Eota | ETAG 027 : Guideline for European Technical Approval of Falling Rock Protection Kits[END_REF]. A straightforward design for this specific barrier would consider that all the block having a kinetic energy less than 200 kJ are stopped. Similarly, the block kinetic energy reduction by this barrier would be computed as the block incident kinetic energy minus 200 kJ.

As for the efficiency of the barrier in arresting the block, results presented in section 2 have shown the limits of an assessment based on the barrier nominal capacity, while section 4 has suggested the interest of meta-models. A detailed analysis of the results presented in Figure 8 shows that the prediction by the meta-models below the the iso-kinetic energy line results in 4.8% of False Positive cases while considering the barrier nominal capacity as a criterion led to a value of 52% (see Figure 3). Restricting the comparison to a block size of 0.7m, leads to values of 4.54% for the meta-model compared to 33.7 % for the barrier nominal capacity based approach. This means that the later is far too optimistic with respect to the ability of the barrier in stopping the blocks and that the later is more realistic, demonstrating the benefit in using this meta-model for design or hazard assessment purpose.

As for the kinetic energy reduction, Figure 9 compares estimates based on the barrier nominal capacity and that from the meta-model. The former is obtained substracting 200 kJ to the block total incident kinetic energy. The full curve is not shown as for large kinetic energies no difference is observed from one curve to the other. Negative block output kinetic energies results from the fact that in some cases the block incident kinetic energy is less than the barrier nominal capacity. These cases are accounted for in the figure but should be neutralised setting the kinetic energy to zero. The same comment holds for results from the meta-model. This figure shows that the meta-model M K fits rather well with the simulation results (in green). On the contrary, results based on the barrier nominal capacity shows important difference with the simulation results, between 0 and 150 kJ. In fact, the kinetic energy reduction is overestimated when using the barrier nominal capacity. The overestimation by the meta-model is much less and is limited to the 0-50kJ range. Beyond 200 kJ, the two approaches are similar.

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Output block kinetic energy (kJ) In the previous section, the benefit in using meta-models has been demonstrated, in particular by comparison with approaches based on the direct use of the barrier nominal capacity. The study has shown, that, at least, for the barrier considered, meta-models are much more efficient in assessing the barrier response.

It is worth highlighting that the considered impact conditions did not consider biased trajectories nor rotational velocities around all block axes. This simplification is thought not to call into question the conclusions drawn and is assumed to be of negligible influence on the developed meta-models accuracy.

The meta-models were developed considering all the possible impact conditions, and thus are able to mimic the barrier response whatever the impact case. However, the different impact scenario within the given parameters ranges were considered equiprobable, which is not realistic.

The impact conditions depend on the block trajectories as observed on real sites. This means that the statistics in relation to the barrier response that are presented in the previous section, including the comparison with the barrier nominal capacity, should not be considered for a real site. In lieu the statistics associated to the ability of the barrier in arresting the blocks for a given site should be based on the distribution of trajectories of that site. As a consequence, the next steps will consist in assessing the efficiency of this barrier for a real site. In this aim, the next development will consist in implementing the meta-models in rockfall trajectory simulation code for a fast and accurate assessment of the barrier effect on the blocks trajectory.

The second meta-model was developed with the aim of evaluating the reduction in kinetic energy of the block, in case of barrier rupture. Such an event would also result in a change of the block propagation direction. Implementation of the kinetic energy reduction meta-model in a rockfall trajectory simulation tool would require making assumptions on the block trajectory changes after impact. Significant improvement would consist in a meta-model of the block velocity changes, but tackling the problem of building such a meta-model is a very difficult task because it entails input and output random variables that are statistically dependent. It is a very difficult problem that was already tackled in 2D for rebound [START_REF] Bourrier | Bayesian stochastic modeling of a spherical rock bouncing on a coarse soil[END_REF].

Conclusion and perspectives

In this article the application of meta-modelling techniques to rockfall protection structures, focusing on a specific barrier intended for low block kinetic energies has been proposed.

The results of 560 FE simulations showed that the barrier efficiency in arresting the block depends not only on the block volume and its translational velocity but it is also controlled by other parameters related to the block trajectory. As a consequence, quantifying the barrier efficiency without accounting for their influence may lead to unconservative estimates. For instance, 33.7 % of the impact cases below the nominal barrier capacity, as deduced from a normal-to-the-fence and centered impact, in fact leads to barrier failure in arresting the block.

Two meta-models have been developed, based on the results of the 560 FE simulations: one concerning the ability of the barrier to arrest the block, the other concerning its ability in reducing the block kinetic energy in case of barrier failure in arresting the block. Two parameters ranges were considered for creating these two meta-models. The one closer to the barrier nominal capacity appears to be slightly more accurate. Nevertheless, the difference being small, no optimisation is required with respect to the definition of the ranges for creating a reliable meta-model. Overall, the meta-models have been shown to provide an accurate prediction of the barrier response. In particular, the meta-model unconservative error associated to the ability of the barrier in arresting the block is less than 5%, compared to 33.7 % following a straightforward design approach. This clearly demonstrates that meta-models represent a promising approach for improving the design of protective structures, and consequently the rockfall risk mitigation.

One possible limitation in the followed methodology is the number of barrier response simulations required for creating a meta-model. In this case, 280 FE simulations were used. One perspective to this work would be to reduce the necesssary numerical simulations without altering the meta-model accuracy.

The meta-models have been developed with the final aim of quantifying the real efficiency of the barrier in reducing the hazard or the block kinetic energy downhill. The next step will consist in introducing the meta-models in rockfall trajectory simulations. This will allow accounting for the real distributions of the various parameters describing the possible block trajectories and will represent a significant improvment in quantitative rockfall hazard assessment in presence of a protective barrier [START_REF] Corominas | Quantitative assessment of the residual risk in a rockfall protected area[END_REF].

Meta-models may also be used for helping in the optimisation of the design of rockfall barri-ers, allowing fro the identification of detrimental mechanisms leading to structure failure. In this case parameters related to the design of the structures may be considered, such as the position and initial tension of the cables, post spacing, position of energy dissipating device, if present. This does represent an inspiring perspective for manufacturers, designers and researchers.

Fig. 1 .

 1 Fig. 1. Geometry and impact conditions for the cable-net protection barrier: a) back view, b) stress-strain behaviour of cables, posts and net and c) side view.

Fig. 2 .

 2 Fig. 2. Block-barrier interaction mechanisms: a) block rolling over; b) mesh perforation and c) global failure.

Fig. 2 Fig. 3 .

 23 Fig. 2 illustrates the front and side views of the deformed barrier through the last instants of three selected impact simulations, where the elements which have entered the plastic domain are coloured in red. In all these FE tests, the barrier resulted unable to stop the impacting block. In particular, Fig. 2a provides an example of the rolling over mechanism, with the block overcoming the barrier, Fig. 2b depicts the barrier failing due to the perforation of the hexagonal net at a side span. Finally, global failure of the barrier is shown in Fig. 2c, with clearly visible formation of plastic hinges at one external post, detachment of longitudinal cables and ruptures within the secondary meshwork.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Results of the analyses as a function of: a) incident impact angle, b) rotational velocity, c) and d) impact position.

  models are generally described as the combination of a deterministic component, defined by a regression model, with a stationnary Gaussian process associated with a constant variance and a correlation function. Contrary to polynomial regression models, Kriging models do not only assume an underlying global functional form. They can approximate arbitrary functions with high global and local accuracies.In this study, the meta-model based on a Kriging approach (M K ) is developed using the matlab tool box UQLab[START_REF] Marelli | UQLab: A Framework for Uncertainty Quantification in Matlab[END_REF] which enables the creation of an efficient Kriging predictor based on small number of data. A 3 rd order polynomial regression model was used for the deterministic component of the Kriging model. Following recommendations for a default use in UQLab[START_REF] Marelli | UQLab: A Framework for Uncertainty Quantification in Matlab[END_REF], an ellipsoidal Matern function was set as correlation function.

  2, a false prediction can be positive (FP) if a barrier success (B S ucc ) is estimated for a case in which failure (B Fail ) was observed; a false prediction can be negative (FN) if a barrier failure (B Fail ) is estimated for a case in which a success (B S ucc ) was observed. In a similar way, good prediction can be positive when the barrier success (B S ucc ) is both estimated and observed and negative when the barrier failure (B Fail ) is both estimated and observed. Based on these definitions, two indicators were used to discuss the performance of the SVM based meta-model: the false negative rate (FN r = FN FN+T P ) and the false positive rate (FP r = FP FP+T N ). In the context of this study, the false positive rate (FP r = FP FP+T N ) is the most relevant to deal with as it focuses on the most critical situation. In fact, a high FP r value is associated to an overestimation of the barrier capacity by the meta-model, with the meta-model erring on the unconservative side.

Fig. 6 .

 6 Fig. 6. Prediction of the of B S ucc and B Fail as a function of the block volume and translational velocity for the wide range scenario.

Fig. 8 .

 8 Fig. 8. Prediction of the of B S ucc and B Fail for the narrow range scenario. Good and bad predictions are indicated by grey and red symbols receptively. The shape of the symbols indicate the mode of failure.

Fig. 9 .

 9 Fig. 9. Comparison between the cumulative distribution of the output kinetic energy predicted by the FE model (E in -M(x i ) in green), the narrow range meta-model (E in -M K (x i ) in blue) and the nominal capacity of the barrier (E in -E Nom in red).

Table 1 :

 1 Input parameters for loading conditions.

	Input parameter	unit Wide Range Narrow Range
			(WR) min-max (NR) min-max
	Translational velocity, v m/s	5 -40	5 -22.5
	Rotational velocity, ω rad/s	0 -35	0 -35
	Volume of the block, V m 3	0.03 -4	0.03 -2.5
	Incident angle, α	deg	-60 -60	-60 -60
	Impact position, X	m	0 -7.5	0 -7.5
	Impact position, Y	m	1 -2.5	1 -2.5
	to the notation introduced in Fig. 1. As indicated in this table, two sets of parameters with
	different ranges were considered in this study for generating virtual test programmes: a wide
	range set (WR) and a narrow range set (NR) .

Table 2 :

 2 Definition of cases for assessing the meta-models performance

		SVM prediction
	FE observation B Fail	B S ucc
	B Fail	T N	FP
	B S ucc	FN	T P

Table 3 :

 3 SVM based meta-model: wide range results

		Prediction
	Observation B Fail B S ucc
	B Fail	249	5	FP r = 2%
	B S ucc	9	17	FN r =35%

Table 4 :

 4 Quality evaluation for the meta-model created for narrow range values.

		Prediction
	Observation B Fail B S ucc
	B Fail	213	6	FP r = 3%
	B S ucc	16	45	FN r =27%