Damien Simonnet

Nathalie Girard

Eric Anquetil

Mickaël Renault

Sébastien Thomas

Evaluation of Children Cursive Handwritten Words for e-Education

As part of an innovative e-education project, a digital workbook is being developed to help teach handwriting at school for children aged three to seven. The main objective of this project is to offer an advanced digital writing experience at school by using pen-based tablets. In this paper, an automatic qualitative analysis process of cursive handwriting words is presented. This approach is original because the goal is not to recognise the word that was handwritten by children (it is an explicit instruction) but to design a precise evaluation of the quality of his handwriting production to give them a real-time feedback. The presented method is based on a specific explicit elastic letter spotting segmentation able to deal with the imprecision of the handwriting of young children. This approach is suited to automatically and precisely highlight the difficulties encountered by children (adding or missing letters, incorrect shapes...). The validation of the proposed approach has been done on a dataset collected in French preschools and primary schools from 231 children. Beyond quantitative results, this paper reports the very positive impact of using this digital workbook that allows children to work independently with online and real-time feedbacks.

Introduction

Several studies show that digital devices can help students and teachers in a learning context. For example, [START_REF] Chickering | Seven principles for good practice in undergraduate education[END_REF] have extracted seven principles to improve teaching, like Good practice uses active learning techniques, Good practice gives quick feedback. Then, in [START_REF] Chickering | Implementing the seven principles: Technology as Lever[END_REF], some ways to use new technologies are described to apply these principles in a learning context. This article focuses on handwriting learning by young children (three to seven years old) in primary schools. In an overview of the overall criteria used in scales for handwriting evaluation, [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF] highlight that as a consensus about which criteria constitute the critical components of handwriting readability, most researchers accept the criteria of size (height, width); slant; spacing (spaces between letters/words); the degree of line-straightness; shape (letter form and shape); and the general merit of the writing. They also conclude that computer based analysis are more accurate, sensitive, and reliable than the subjective analysis, but they also observe that the current practical applications are still limited. In fact, these applications were not yet capable of making a global decision as to the legibility of the writing product. Since this study, digital devices have been improved. Especially, efficient pen-based tablets have been designed. This kind of tablet allows writing on screen with a sensation very similar to writing on paper, in particular because the user can hold the pen in the same way, and can also lay the palm of the hand on the screen as on paper. Moreover, [START_REF] Jolly | Evaluation of a tactile training for handwriting acquisition in french kindergarten children: A pilot study[END_REF] have presented a comparative study about the acquisition of handwriting between training with digital devices and paper. Results show a significant improvement of children trained on the digital device compared to children trained on paper, notably in terms of fluency (decreasing of the 'in-air' time, and stopping time). In this study, children writings are mainly analysed according to recorded time, velocity, etc. but letter height, width, form and shape are not considered. With pen-based tablet, various criteria such as shape, direction, order, pressure, fluidity can also be analysed. In this objective, a digital notebook was designed to help writing learning, see Fig. 1(a).

Moreover, the digital notebook gives immediate and personalised feedback to children. This prompt feedback helps them to be more autonomous. Another advantage of the notebook is the ability to adapt the educational progression to each child.

Since handwriting learning involves cognitive, kinesthetic and perceptual-motor components [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF][START_REF] Djeziri | Learning handwriting with pen-based systems: computational issues[END_REF][START_REF] Plamondon | Recent developments in the study of rapid human movements with the kinematic theory: Applications to handwriting and signature synthesis[END_REF], the notebook is based on several modules dedicated to different required skills, and is defined alongside pedagogical experts. The notebook is currently composed of six modules linked to different learning steps in primary schools: Block Letter Writing, Digit Writing, Word Identification, Graphical Identification, Preparation to the Cursive Writing and Cursive Writing. Moreover, in each module, special attention is paid to the feedback returned to children. Actually, prompt (i.e. real-time) and adaptive feedback are very important in learning processes [START_REF] Kluger | The effects of feedback interventions on performance: A historical review, a meta-analysis, and a preliminary feedback intervention theory[END_REF][START_REF] Shute | Focus on formative feedback[END_REF].

This paper focuses on the cursive writing module in which . In this module (and for all modules), the objective is to evaluate the children's gesture (i.e. to identify if they use the right order and direction to write the letter, and if the shape of each letter is legible) in order to explain them how to correct and improve their writings.

To do so, especially when more than one letter is written, the evaluation implies to deal with the segmentation of a word in letters, and to identify matching, missing and added letters compared to the reference model. In the literature, detection of a letter or a word as part of a sentence or a text has been extensively studied for handwriting recognition [START_REF] Tappert | The state of the art in online handwriting recognition[END_REF][START_REF] Plamondon | Online and off-line handwriting recognition: a comprehensive survey[END_REF][START_REF] Cheriet | Character Recognition Systems: A Guide for Students and Practitioners[END_REF]. But this challenge is all the more complex in the context of the initial learning of the writing skills by children who have a very approximate handwriting. Indeed, they frequently add, distort, and even forget letters in the writing of a word. Moreover, children frequently combine several kinds of mistakes on the same word which makes the analysis task especially complex. As examples, Fig. 2(a) shows a distorted letter problem in the writing of the word an by a child, for the same word, Fig. 2(b) shows an addition of stroke that can induce confusion between n and m. Other common errors lie in missing letters as illustrated in Fig. 2(c) where the expected word is ours (i.e. bear). Fig. 2(d) shows a letter transformation, where the last e of the pseudo-word alette has been transformed in an r. Finally, Fig. 2(e), where the expected pseudo-word is alette, shows mistakes combination with missing letters and letter distortion.

Therefore, to tackle these issues, we propose to strengthen the common segmentations with verification steps based on specific analysis methods. The proposed workflow allows an evaluation of handwriting quality knowing the expected word, i.e. the teacher's instruction. In Section 2, the specific evaluation of children handwriting is presented. This approach has been tested on a real dataset and experimental results are pre-

Evaluation of Children Handwritten Word Quality

In this section, a new approach is presented to evaluate the children's handwriting with regards to a targeted sequence, i.e. a grapheme or a word. The handwritten words are on-line signals captured on digital touch-screen devices with styluses. Since the main objective is to return corrective feedback to children, the system must detect correct letters, as well as incorrect writings, additional strokes and missing letters. To do so, as summarised in Fig. 3, the evaluation of the handwritten word quality consists of three main steps: at first, extraction of primary segmentation hypotheses, then extraction of letter hypotheses and finally word hypothesis extraction and evaluation.

As shown in Fig. 3, the evaluation is based on a standard workflow from an initial segmentation to word extraction with a recognition process. But as detailed in subsection 2.2, the standard recognition processes are inefficient on distorted writings like those of children. Thus, the letter hypotheses extraction step is reinforced to reduce error propagation by adding a verification step through a supervision (see the gray sub-block 2.2 in Fig. 3). The ranking of the word hypotheses is also redefined with a new scoring (gray sub-block 3.2 in Fig. 3). And, finally, we propose a new evaluation of the quality of the written word combining elastic matching and writing analysis score (gray sub-block 3.3 in Fig. 3).

As mentioned before, in our context the expected word is known, and refers to the sequence that should be written by children. Hereafter, the expected word is denoted X = (x i ) i∈ 1,m where x i represents a character, and the analysed word is denoted Y = (y i ) i∈ 1,n where y i represents a letter.

The rest of this section details each block in the workflow. Each subsection focuses on the proposed strengthening.

Extraction and Organisation of Segmentation Hypotheses

The first part of the evaluation of the children's handwriting consists in extracting segmentation hypotheses corresponding to sub-strokes of the original drawing: a letter, a sub-part of a letter, or more generally a sub-stroke that could be an expected letter. Segmentation hypotheses are build from the primary segmentation and organised in a graph.

Primary element extraction

The primary segmentation corresponds to a partition of the on-line handwritten signal based on stable parts of the handwriting: down-strokes (Anquetil and Lorette, 1997a). As an example, for the input word lune (i.e. moon) in Fig. 4(a), the primary segmentation is illustrated in Fig. 4(d). More precisely, this segmentation is spatially ordered along the x-axis, and has been built by considering significant down-strokes as indivisible (see green parts in Fig. 4(b) matching with green nodes in Fig. 4(d)). Cutting points are based on y-extrema (see each cross in Fig. 4(c)) and points at one third of the curvilinear distance for up-strokes non associated to loops (in red in Fig. 4(c)). This segmentation represents all possible cutting in the on-line signal around significant down-strokes.

Graph construction

Similarly to (Anquetil and Lorette, 1997b), a segmentation graph (Fig. 4(e)) is used to represent all partitions of the input handwritten signal. This graph contains nodes corresponding to segmentation hypotheses that are valid for the cursive Latin alphabet when there are between one and three significant downstrokes. The first level of the segmentation graph results from the primary segmentation in which each down-stroke has been merged with adjacent up-strokes resulting in the level where only one down-stroke is used. The next levels of the graph are built by creating segmentation hypotheses corresponding to the merger of all consecutive hypotheses of the previous level. This process through levels is depicted on Fig. 4(e) by node colourisation, where the lowest gray node is the merger of the two gray nodes of the previous level, which are themselves the merger of the gray nodes of the previous level and so on.

After the graph construction, the identification process of letter hypotheses is conducted in each graph node. The next section presents this step.

Identification of Word Letter Hypotheses

The identification process consists in finding the most relevant letter hypotheses associated to a segmentation hypothesis knowing the expected word. This process combines two complementary steps: prediction and verification. The former, associated to a recognition process, predicts the most relevant symbols of the alphabet for a given segmentation hypothesis. The latter checks the correctness of predicted hypotheses through two analysis processes: local and global.

More precisely, for each segmentation hypothesis the recognition process gives a recognition confidence for each letter of the alphabet. On the contrary, the analysis process takes as input a segmentation hypothesis with a letter hypothesis, and returns an analysis confidence.

Local Prediction of Letter Hypotheses

The local prediction process consists in finding the most relevant letter hypotheses associated to a segmentation hypothesis (i.e. a graph node) based on the recognition confidences. In a standard approach, the best letter based on recognition confidence would be used as correct hypotheses. However, as illustrated in Fig. 5(a) where the expected letter is an 'e', the recognition process (here an EVOLVE classifier (Almaksour andAnquetil, 2011, 2013), see Section 3.2 for details) recognises as best hypothesis an 'l' due to the degraded writing of child. If this letter is considered, the feedback to child will be inadequate.

In our context, the expected word is known and can be used to select the most relevant hypotheses in a verification step. Moreover, this verification step should guide the letter hypotheses extraction to allow useful feedback. This original verification step is explained in the following section.

Local and Global Verification of Letter Hypotheses

The verification step is fundamental to analyse finely handwriting that allows to understand degraded children's handwrit- ing.

The local verification process consists first in selecting the most relevant hypotheses with regards to the expected sequence X and then in checking their correctness with an analysis process. As an example, in Fig. 5(b), the expected letter is an 'e', the analysis score is computed based on this knowledge and letter hypotheses are re-ranked according to this score (here the analysis score is computed based a confidence-based classifier [START_REF] Simonnet | Multi-criteria handwriting quality analysis with online fuzzy models[END_REF], see Section 3.2 for details). This re-ranking takes into account the letter confusion children can have in a learning context. It also allows to pull correct letter up in order to give useful feedback. Therefore, hypotheses are selected from two sources: the expected letters (n a h best) corresponding to the context and the recognition process ( n r h best) to handle confusion. Then, an analysis process is used to evaluate the quality and the relevance of the selected hypotheses. If the analysis score is smaller than the correct analysis rate θ a , the hypothesis is discarded as the writing is too degraded to be considered as a valid hypothesis.

The global verification process considers all letter hypotheses resulting from the local verification process and select the most adapted to the expected sequence X knowing that some parts of the children writing may not be legible. The first part of the selection keeps the n best letter hypotheses of each character in the sequence X. As an example, in Fig. 5(c) and Fig. 6(c), 'e' is expected in the word and its analysis score is high compared to other 'e' letter hypotheses in the graph, therefore this letter hypothesis is kept in this node. However, due to non legible parts of children writing, the previous process may remove all letter hypotheses in a graph node. A second process is therefore used to add letter hypotheses with the best recognition score, which implies that all nodes of the graph have at least one letter hypothesis. Thus, this last step allows to identify unexpected letters written by children, i.e. letters which are not in the teacher's instructions.

Selection of the Best Word Segmentation Path

The last part of the handwriting evaluation consists in extracting the most probable partition of the handwriting input signal with regards to the expected word called best word segmentation path. The previous letter hypotheses combined with the segmentation path of the graph allow to build a set of word segmentation path (WSP). To find the best WSP, an analysis score is computed for each hypothesis, and finally this score is used with an elastic matching method to extract the best analysis path.

In the rest of the section, first the notion of segmentation path is presented, followed by the process of selection of WSP hypotheses. Finally, the best WSP hypothesis is selected with an elastic matching method.

Segmentation Path

A segmentation path is a path in the graph corresponding to a partition of the handwriting's on-line signal. More formally, let P a n = (S i l i ,k i ) i∈ 1,n be a partition of the handwriting's on-line signal where S i l i ,k i corresponds to a substroke of the input signal associated to the level l i and position k i of the segmentation graph, and n is the size of the segmentation path.

Selection of Word Segmentation Path Hypotheses

The first step in the extraction of the best word segmentation path builds a set of hypotheses reflecting children writing based on the segmentation path of the graph and letter hypotheses selected. The selection process based on an analysis path score s ap detects hypotheses corresponding to various scenarios of the children writing (noisy strokes, confused or forgotten letters).

The analysis path score s ap combines an analysis score s a representing the correctness of each segmented element with consecutive coherence scores (n-gram and inter-letter matching) as follows:

s ap (P a n ) = 1 2 s a (P a n ) + 1 4 s ng t (P a n ) + 1 4 s im t (P a n ) (1)
The first coherence score s ng t corresponds to a coherence in relation with the expected word by penalising gradually bi-gram different from the expected word as expressed in equation ( 2).

s ng t t(i, i -1) =          1.0 if (i, i -1) is a bi-gram of X 0.7 if i or i -1 is a letter of X 0.4 otherwise
(2) The second coherence score s im t is related to the spatial coherence of letters as in (Anquetil and Lorette, 1997b).

Coherence scores are computed as the conjunction of transition scores s t(i,i-1) as presented in equation (3).

s t (P a n ) = n-1 Π n i=2 s t(i,i-1) (3)
Finally, the analysis score s a represents the average correctness of each element of the partition. It is defined mathematically by equation (4) where s a (i) is the analysis score of the ith path element.

s a (P a n ) = n Π n i=1 s a (S i l i ,k i ) (4)
The N WS PH best paths according to the analysis path score s ap are kept as candidates to the selection of the best hypothesis.

As shown in Fig. 7(a) and Fig. 7(b), the analysis scores for the first letters of the word would suggest an 'a' followed by a 'w' but the information about the expected word 'ours' (bear) will increase the score of the bi-gram 'o-u' over the score of the bi-gram 'a-w' (Fig. 7(c)).

Selection of the Best Hypothesis

A customised elastic matching method is used to match the expected word X asked by teacher with the analysed word Y that is associated to an analysis path P a n . This step makes it possible to identify correct letters, substitutions, missing and additional letters. More precisely, a Damerau Levenshtein edit distance [START_REF] Damerau | A technique for computer detection and correction of spelling errors[END_REF] is computed between X and Y, with optimised substitution costs learnt from the analyser. Insertion and deletion costs are proportional to the number of downstrokes. The substitution cost is composed of two parts: the first is learnt from the analyser and the second is a penalisation when the number of significant down-strokes between the two letters to substitute is different. The learnt substitution cost corresponds to an analysis confusion score defined by equation ( 5) where s l a (k) represents the analysis score of a symbol k as a symbol l, and K, L are respectively the training set for the symbol k and l.

c l substition (k, l) = 1 -k∈K s l a (k) l∈L s l a (l) (5)
The final substitution cost is given by equation ( 6) where α k,l is the penalisation factor defined in equation ( 7) and depends on the absolute difference of significant down-strokes d a k,l .

c substition (k, l) = α k,l × c l substition (k, l) (6)
α k,l = 1.0 if d a k,l = 0 0.5 + d k,l else (7)
Finally, the best analysis path is chosen as the analysis path hypothesis with the minimum edit distance.

Experiments and Results

This section presents experimental results of the analysis segmentation algorithm. First, the testing dataset is introduced followed by the recognition and analysis processes used for these experiments. Then, the evaluation protocol and results are explained.

Datasets

The testing dataset was collected during in-class experiments in a context of a user-centered design approach that uses feedback of children and teachers to improve the application. More specifically, during digital workshops of twenty minutes with six tablets children were drawing sequences and letters. This dataset, composed of 1012 pseudo-words from two to eight letters, was collected in French preschools from 231 children from four to five years old. The list of all the sequences used in this dataset is ab, alette, am, an, ap, apin, di, es, qu, je, jeu, jeudi (= Thursday), ji, lu, lune (= moon), lundi (= Monday), mardi (= Tuesday), oel, onne, ours (= bear), oyeux, ph, rs, se, ux, ve, ven, vendredi (= Friday), samedi (= Saturday). In Fig. 8, there are some samples of this dataset containing writing such as additional strokes, missing letters, substitutions and deformed words. A segmentation ground truth is created semiautomatically for each sequence that can contain missing or additional strokes. First of all, segmented areas corresponding to letters are manually annotated, then an automatic process generates the segmented partition corresponding of the handwritten signal input including notably parts that do not correspond to segmented letters.

Recognition and Analysis Processes

For these experiments, the recognition process is based on the discriminative properties of a classifier that rank hypotheses of the alphabet. In this work, an EVOLVE classifier (Almaksour andAnquetil, 2011, 2013) is trained with HBF49 [START_REF] Delaye | HBF49 feature set: A first unified baseline for online symbol recognition[END_REF] features on the letters of the Latin alphabet. More precisely, HBF49 is a generic set of features designed for handwriting symbols recognition. It is composed of dynamic features that depend on the writing process (e.g. starting and ending positions, proportion of down-strokes trajectory, angle of the initial vector, inflexions), and visual features that focus on the appearance of the written results (e.g. 2D histogram of point, k-perpendicularity, k-angle).

The analysis process uses cluster models built with HBF49 features for each letter of the alphabet represented as a mean vector and a covariance matrix (Almaksour andAnquetil, 2011, 2013). More specifically, a confidence-based classifier [START_REF] Simonnet | Multi-criteria handwriting quality analysis with online fuzzy models[END_REF] based on discriminative and generative (cluster) models with HBF49 features is used to compute a normalised analysis score. This score is based on the calibration of the Mahalanobis distance to have a confidence score.

Evaluation Protocol

The analysis segmentation algorithm presented in this paper makes it possible to identify correct and incorrect parts of the drawing for a given input handwritten signal by returning an analysis path P a n = (S i l i ,k i ) i∈ 1,n . The evaluation process quantifies the coherence of the segmentation with regards to the expected sequence asked.

It is noted that a written sequence can have several correct segmented partitions (e.g. an expected sequence 'an' written as a 'am' has three valid partitions as depicted in Fig. 9). Let G m = (S i g ) i∈ 1,m be a ground truth partition of size m, and G the set of ground truth partitions associated to an input handwriting signal.

First, the evaluation process selects the closest ground truth to the analysis path by maximising the matching score defined in equation ( 8) where d (i, j) tp , d (i, j) f p , d (i, j) f n correspond respectively to the curvilinear distance of true positive, false positive and false negative strokes between the substrokes S i g and S j . The association between substrokes of a ground truth partition and an analysis path is done by first selecting the pair with the best matching score. m(S i g , S j ) =

d (i, j) tp d (i, j) tp + d (i, j) f p + d (i, j) f n (8)
From this process, two evaluation metrics are defined: the letter average matching score and the correct word segmentation ratio, that correspond respectively to the average of all the matching scores, and to the ratio of sequences correctly segmented, i.e. sequences having a matching score for all letters greater the τ m threshold.

Results

This section presents quantitative and qualitative results of the analysis segmentation algorithm. In each node, the number of analysis and recognition hypothesis are respectively set up to n a h = 2 and n r h = 2. These two values have been chosen empirically to reduce the number of word path hypotheses. The correct analysis rate θ a to reject incorrect analysis hypothesis is equal to 0.25 because in [START_REF] Simonnet | Multi-criteria handwriting quality analysis with online fuzzy models[END_REF] a score lower corresponds to an incorrect writing. The number of hypotheses kept corresponding to each letter of the sequence to analyse is empirically fixed to n best = 3 to reduce the number of word path hypotheses. The maximum number of word segmentation path hypotheses kept for the elastic matching is empirically fixed to N WS PH = 100. The matching threshold for the evaluation is τ m = 0.7.

The quantitative results on the testing dataset correspond to a correct segmentation ratio of 0.90 in which a letter average matching score is of 0.95. In Fig. 10 and Fig. 11 qualitative results are presented respectively for sequences and words where the segmentation of the handwriting is represented with different greyscales. Green, blue and red bounding boxes correspond respectively to a letter with a correct matching, a substitution and a deletion. In Fig. 10, results show that the presented method is able to segment children handwriting, and to identify additional strokes (#15 and #17) and some substitution (#20). However, although the segmentation is correct, some detected substitutions are incorrect (#14, #16) due to deformed letters. Results on words presented in Fig. 11 show the ability of the presented method to deal with longer sequences. Fig. 11(b) presents the segmentation graph of the word 'samedi' with the analysis path detected. Errors in the segmentation presented in Fig. 12 are mainly related to low handwriting quality of children (#0, #1, #2). In #3, the segmentation of the letter u is false because the additional part after the u is not detected as a segmentation hypothesis since it is composed of up-strokes only.

Conclusion

This paper presents an original analysis method for handwriting quality evaluation, that is able to deal with children cursive handwriting. The proposed approach was defined to be well adapted for degraded handwriting analysis. Indeed, a common recognition method is reinforced with specific verification steps, and useful information extracted from a customised elastic letter spotting is used to give relevant feedback to children. This proposal allows a detailed evaluation of children's handwriting (added strokes, missing strokes and incorrect writing). The proposed analysis enables the definition of real-time feedback needed in learning systems, especially to make children more autonomous in their learning. This method gives very relevant results and is currently tested on a large scale (in 40 primary schools with 1000 children).

Fig. 1 .

 1 Fig. 1. First in-class experiment of the e-education innovative project (a) with tablet touch-sensitive devices, and the cursive writing module scenario (b) (c) (d).

Fig. 2 .

 2 Fig. 2. Children's approximate writings

Fig. 3 .

 3 Fig. 3. Automatic Cursive Handwriting Evaluation overflow in which the expected word is 'lune' (moon).

  Fig. 4. Segmentation Graph (e) and Primary Segmentation (d) of an input gesture lune (a) based on loops and y-extrema (c), and down-strokes (b). Loops and down-strokes are represented by red and green drawing, crosses represent y-extrema.

Fig. 5 .

 5 Fig. 5. Letter hypotheses computation from the last node of the second level of the graph in Fig. 4(e) with recognition scores and ranking in (a), analysis scores and ranking in (b), and global letter hypothesis selection in (c)

Fig. 6 .

 6 Fig. 6. Letter hypotheses computation from letter 'e' in word 'je' ('i') with recognition scores and ranking in (a), analysis scores and ranking in (b), and global letter hypothesis selection in (c)

Fig. 7 .

 7 Fig. 7. Segmentation path hypotheses computation from word 'ours' (bear) with first letter analysis scores and ranking in (a), second letter analysis scores and ranking in (b), and some bigram coherence scores s ng t in (c)

Fig. 8 .

 8 Fig. 8. Sample words in the testing dataset written by children with additional strokes (#0, #5), missing letters (#7), substitution (#4, #6) and deformed words (#1, #2, #3). The word that children should write is displayed on the top right corner.

Fig. 9 .

 9 Fig. 9. Multiple word segmentations of an input handwriting signal (a) corresponding to an expected sequence an. (b) represents the correct letters with regards to the expected sequence. The three correct segmentations (c) of (a) correspond to sequences with addition and substitution of letters.

Fig. 10 .

 10 Fig. 10. Sequences Qualitative Results. In (a), the segmentation of the handwriting is represented with different greyscales. Green, blue and red bounding boxes correspond respectively to a letter with a correct matching, a substitution and a deletion. (b) and (c) are the segmentation graph of the sample #4 and #19. The score corresponds to the recognition path score defined in equation (1).

Fig. 12 .

 12 Fig. 11. Words Qualitative Results

Acknowledgments

This work takes place in the context of a three year research project founded by the French government as part of innovative projects (BPI): IntuiScript (http://intuiscript.com/). This work has also been supported by the LabCom 'ScriptAndLab' founded by the ANR (ANR-16-LVC2-0008-01). The authors are grateful to their industrial partner (Script&Go), educational experts of Rennes academy, ESPE, Brittany region and LOUS-TIC laboratory for their collaboration.


P Blanc-Vannet
email: pierre.blanc-vannet@airliquide.com

S Jallais

B Fuster

F Fouillen

D Halm

T Van Eekelen

S Welch

P Breuer

S Hawksworth

FIRE TESTS CARRIED OUT IN FCH JU FIRECOMP PROJECT, RECOMMENDATIONS AND APPLICATION TO SAFETY OF GAS STORAGE SYSTEMS

In the event of a fire, composite pressure vessels behave very differently from metallic ones: the material is degraded, potentially leading to a burst without significant pressure increase. Hence, such objects are, when necessary, protected from fire by using thermally-activated devices (TPRD), and standards require testing cylinder and TPRD together. The pre-normative research project FireComp aimed at understanding better the conditions which may lead to burst, through testing and simulation, and proposed an alternative way of assessing the fire performance of composite cylinders. This approach is currently used by Air Liquide for the safety of composite bundles carrying large amounts of hydrogen gas.

NOMENCLATURE

Type IV pressure vessels, typically made of a carbon fibre-epoxy composite shell wrapped around a polymeric liner, are considered a mature way of storing hydrogen at high pressure. Their use is increasing for applications requiring lightness and/or high pressure, such as on-board storage for cars or large vessels mounted on trailers. Though they can already be operated safely, a number of material issues and R&D challenges still limit their optimisation potential, including fire behaviour [START_REF] Barthelemy | Hydrogen storage: Recent improvements and industrial perspectives[END_REF].

When exposed to fire, composite pressure vessels exhibit a completely different behaviour than metallic ones. For the latter, the heat is rapidly transferred to the gas, leading to a pressure increase, and ultimately cylinder burst when the pressure gets too high. On the other hand, the pressure in composite vessels tends not increase much more than about 10%, whereas vessels are designed with safety factors of 2.25 to 3. The degradation of the composite material can lead directly to a burst, or the melting of the plastic liner can lead to a leakage through the vessel's body. Whichever happens first depends on fire conditions, materials properties and tank geometry [START_REF] Barthelemy | Hydrogen storage: Recent improvements and industrial perspectives[END_REF][START_REF] Bustamante-Valencia | Thermal History Resulting in the Failure of Lightweight Fully-Wrapped Composite Pressure Vessel for Hydrogen in a Fire Experimental Facility[END_REF]. Due to the high energy content, the burst must be avoided, and the cylinders are protected with thermally activated pressure relief devices (TPRD), which must be placed in a way that ensures they detect the fire and vent the gas as required [START_REF] Saw | The EU FireComp Project and Risk Assessment of Hydrogen Composite Storage Applications using Bow-tie Analysis[END_REF].

In current standards addressing the qualification of composite cylinders, either for on-board storage [START_REF]Regulation (EC) No 79/2009 of the European Parliament and of the Councilon type-approval of hydrogen-powered motor vehicles[END_REF] or for transportable applications [START_REF]NF EN 12245+A1 Transportable gas cylinders -Fully wrapped composite cylinders 6[END_REF], it is required to test the vessel and its TPRD together in a bonfire test. The test is deemed a success if the gas vents through the TPRD and no burst occurs. For transportable applications in Europe, the use of TPRD is not mandatory: in that case, the cylinder must not burst before 2 minutes. A disadvantage of this method is that it does not provide much information on what the vessel alone can endure.

The European project FireComp [6, 7], a three-year FCH JU funded pre-normative research project which ended in June 2016, aimed at providing a better understanding of the failure of unprotected composite pressure vessels in fire. It brought together partners with different expertise: a hydrogen supplier as coordinator (Air Liquide), a composite vessels manufacturer (Hexagon), experts of industrial risks and safety (INERIS and HSL), experts in thermal radiation and mechanical behaviour of composites (CNRS, Samtech), in thermal degradation and reaction-to-fire (University of Edinburgh) and in collaborative research projects management (Ayming). Within the project an extensive bonfire test campaign was performed on 19L and 36L cylinders. Models were developed for heat transfer through the composite thickness [START_REF] Hidalgo | A framework for evaluating the thermal behaviour of carbon fibre composite materials[END_REF] and loss of material strength with temperature [START_REF] Mary | Residual strength of wound composite pressure vessels subjected to fire exposure[END_REF][START_REF] Benelfellah | Residual mechanical properties of a carbon/epoxy wound composite exposed to fire, 22ème Congrès Français de Mécanique[END_REF]. Used on real cylinder geometry, they proved able to accurately reproduce the experimental results and capture the transition between leak and burst failure modes [START_REF] Benelfellah | Simulation of burst of hyperbaric hydrogen tanks in fire conditions[END_REF].

This paper presents the fire test setup that was used, how the thermal aggression was calibrated using a metallic tube, the experimental results obtained, and how this knowledge is used for the fire protection of composite trailers.

BONFIRE TESTS SETUP AND CALIBRATION

The risk analysis underlined three possible types of fire: "punctual" jet fire (e.g. from an ignited hydrogen jet from piping), localised fire exposing a part of the cylinder (e.g. 30 to 50% of the surface), and fully engulfing fires [START_REF] Saw | The EU FireComp Project and Risk Assessment of Hydrogen Composite Storage Applications using Bow-tie Analysis[END_REF]. Jet fires are known to lead to very quick burst of composite vessels [START_REF] Chaineaux | Sûreté des dispositifs de stockage de l'hydrogène sous haute pression équipant des véhicules routiers[END_REF] and are hardly detected by TPRDs; so the project focused on the two other types. Fully engulfing bonfire tests were retained, to create a conservatively severe thermal aggression.

Such fire tests were carried out by INERIS. The experimental programme comprised two steps. The first aimed at determining the fire conditions to test the thermo-mechanical behaviour of composite vessels in real conditions: metallic cylinders were instrumented in order to determine the incident equivalent heat flux, to calibrate the setup, and to record experimental data to be compared to the modelling results.

The second part of the study was performed on real composite vessels, using the thermal aggression determined in the first step. The results presented were obtained on type IV composite cylinders of 19 L and 36 L. They were manufactured by Hexagon, by filament winding [START_REF] Peters | Composite filament winding[END_REF] around a polymeric liner and comprise one metallic boss at each end. The boss allows connecting the tank to the piping; the liner ensures the tightness of the vessel; and the role of the composite shell is to withstand the inner pressure. The service pressure is 700 bar. During manufacturing, thermocouples have been inserted in the composite wall, at different depths and locations. It was shown that they decrease a little the resistance of the composite (time to burst of one cylinder equipped with thermocouples was found to be 22% shorter than for one cylinder without), which is conservative for the approach considered below. These thermocouples were used to check the reproducibility of the fire and to provide data to compare with numerical simulations.

Hydrogen gas burner and instrumentation

Bonfire tests are typically performed using a hydrocarbon pool, where the quantity of liquid is regulated to ensure a constant flame regime [START_REF] Bustamante-Valencia | Thermal History Resulting in the Failure of Lightweight Fully-Wrapped Composite Pressure Vessel for Hydrogen in a Fire Experimental Facility[END_REF][START_REF] Ruban | Fire risk on high-pressure full composite cylinders for automotive applications[END_REF]. The drawbacks of this type of test are that a large quantity of heptane mist is generated, which may lead to secondary explosion after cylinder burst and the necessity to extract fumes to allow visibility during the test. By contrast, hydrogen gas fires have significant assets. Firstly, hydrogen gas fire is a realistic scenario when assessing safety of these pressure vessels, as it is a combustible which may always be found near hydrogen storage. Secondly, it is important to note that gas fires are not excessively complex to calibrate, whereas standard pool bonfires involve repeatability issues due to ventilation effects, soot deposition and the high impact of the vessel size on the flame.

In order to increase the amount of energy that can be transmitted to the cylinder during gas fires, and improve the homogeneity of the thermal aggression around it, a confinement is placed above the steel cylinder. The material used to make this confinement is standard aerated concrete. Since the tests can be done with cylinders of various dimensions (19 L & 36 L were used), the geometrical definition of the confinement is located relative to the external walls of the cylinder. This approach allows complete definition of the confinement, regardless of the size of the tested cylinder. A picture of the experimental setup is displayed in Fig. 1.

The cylinder is exposed to fire by the means of four injectors. They are symmetrically located under the cylinder as presented in Fig. 1. This position guarantees a satisfactory engulfment of the cylinder.

On each injector, a calibrated flow rate of hydrogen and oxygen mixture can be controlled. The different elements implemented are presented in Fig. 2. The red parts are the elements where the injection of oxygen is performed on each injector. The hydrogen gas fire tests are carried out with 1.5 g/s hydrogen and 0.5 g/s oxygen flow rates per injector.

Calibration of thermal aggression

Preliminary tests were performed using a steel cylinder in order to calibrate the heat load measurement method and study the reproducibility of the fire. Thanks to the use of metallic vessels in this preliminary study, there are no issues related to thermal decomposition of the specimen and the thermal properties are well known. The external dimensions of the steel cylinder are the same as those of the 36 L composite cylinders provided by Hexagon. The sealing of the cylinders is ensured by copper seals, to enable pressure measurements. The weight of the metallic cylinder is about 196 kg. In addition to the pressure measurement, the evolution of temperature is recorded by 24 thermocouples marked by red points in Fig. 3: -8 on the internal face of the cylinder, -8 on the external face of the cylinder, -8 in the flame region. The evolution of the temperature inside the vessel allows determining the net heat flux absorbed by the mock-up, which behaves like a calorimeter. By differentiating the thermal energy stored by the gas , which can be calculated thanks to the measured temperature of the gas , the mass of gas , and its specific heat capacity , it is possible to calculate the heat flux transmitted by the internal wall surface :

(1) with [START_REF] Bustamante-Valencia | Thermal History Resulting in the Failure of Lightweight Fully-Wrapped Composite Pressure Vessel for Hydrogen in a Fire Experimental Facility[END_REF] It is then possible to estimate the average temperature of the internal wall of the cylinder, provided the areal transfer coefficient does not vary appreciably during the test. This coefficient can be estimated at about 4.4 kW/m²/K.

(3)

The thermal gradient in the thickness of the vessel wall is calculated thanks to the heat flux absorbed by the cylinder, its thermal conductivity (32 W/m/K at 1 000°C) and its thickness :

(4)
Even if the cylinder is assumed to absorb the totality of the heat flux of the gas fire, the thermal gradient in the thickness of the wall remains at a low value of 70°C compared to the evolution of the internal wall temperature, which is about 650°C. It is then acceptable to consider that the temperature is constant in the thickness of the cylinder wall, and is equal to the internal wall surface temperature. Thanks to this assumption, the net heat flux absorbed by the vessel can then be calculated thanks to the evolution of the internal wall surface temperature , the cylinder mass , its external surface and its specific heat capacity .

(

) 5
The evolution of the net heat flux received by the cylinder is plotted in Fig.

Influence of filling media

It is expected that replacing hydrogen by helium or nitrogen does not strongly change the behaviour of the vessel but relaxes many safety concerns and simplifies the interpretation of the internal pressure evolution (since helium or nitrogen have a behavior closer to a perfect gas). To check this assumption, the influence of the filling media was studied in a first series of tests.

Six tests were performed with 19 L vessels at two different inner pressures: 700 bar and 467 bar. For each pressure, three gases were used: hydrogen, helium, nitrogen. As hydrogen was used, bags of sand, concrete walls and a metallic protection were put around the cylinder (Fig. 6) to avoid dangerous projectiles due to the fireball and overpressure that will be generated after burst. The most important results in this phase are the evolution of the inner pressure and the time to burst.

The results obtained are presented in Fig. 7 (a) for a 700 bar initial pressure, and Fig. 7 (b) for a 467 bar initial pressure. It appears that there is not an obvious law that governs the behaviour of the cylinders when the initial pressure varies. At 700 bar, the earliest burst occurs with hydrogen, and the latest with nitrogen, with a 15 bar (2%) and 20 s (12%) relative difference. At 467 bar, nitrogen and hydrogen results are similar, while the test with helium proceeds to burst 35 s later (20%), and 15 bar higher (3%). These facts suggest that the filling media does not have a significant influence on the behaviour of the cylinder, in comparison with the impact of the variability of cylinder strength due to normal uncertainties linked with their fabrication process.

As a result, the subsequent tests were performed with nitrogen, which presents the advantage of being non flammable, thus decreasing the costs and increasing the safety of the experimental campaigns.

Tests with different inner pressures

The tests presented in the following part were performed on 36 L vessels. According to the previous conclusions, they were carried out using nitrogen gas. The test with initial pressure 700 bar was performed twice on different vessels to assess the reproducibility of the fire aggression. The results are presented in Fig. 8, and it can be seen that both the temperature increase through the composite thickness and the inner pressure increase are very close from one test to the other. The times to burst are 238 s and 241 s, with pressure increases of around 15 bar. During the first 100 s, it appears that internal pressure slightly decreases (by 2.5 bar), and then increases until burst. This phenomenon can be explained by the fact that vessels were filled just before the test, and stabilization of pressure and temperature was not fully achieved. These curves confirm a well known result regarding the failure mode of composite cylinders: burst does not occur because of the pressure increase, but because the cylinder loses its mechanical properties.

Three other tests were carried out on 36 L vessels with different initial internal pressure: 100 bar, 250 bar and 525 bar. Results are presented on Fig. 9 for convenience, only one of the 700 bar tests is representedand the failure times and modes are reported in Table 1. No burst was observed during tests at lower pressures, but break of slopes after 400 and 490 s respectively indicate the moment when the body of the cylinder starts to leak sufficiently to observe a pressure decrease. This behavior of type IV cylinders has also been reported on completely different cylinders [START_REF] Bustamante-Valencia | Thermal History Resulting in the Failure of Lightweight Fully-Wrapped Composite Pressure Vessel for Hydrogen in a Fire Experimental Facility[END_REF] and is attributed to the melting of the polymeric liner. Figure 9. Evolution of inner gas pressure during the four bonfire tests with different initial pressures

Recommendations to Regulations, Codes and Standards

The results presented above are very valuable information for determining a fire safety strategy. For a given application using composite cylinders, if burst is not an acceptable optionwhich is most often the casethe performance of the bare cylinder can be used in the design of thermal shields and calculation of a depressurisation strategy, to ensure the cylinder is kept in a pressure vs. time zone in which no burst occurs. As an example, the safe pressure zone provided by the tests results of Fig. 9 is displayed in Fig. 10 an arbitrary safety margin is taken on the time to burst for the example, but the correct value for this margin should be further discussed.

Figure 10. Safe pressure zone determined from the four previous fire tests It was then recommended by the FireComp project consortium [START_REF] Blanc-Vannet | Recommendations to Regulations, Codes and Standards, FireComp project webinar[END_REF][START_REF] Heggem | Behaviour of composite cylinders in fire -Recommendations to Regulation, Codes and Standards bodies provided by the EU project FireComp, Pressurized Hydrogen and Storage Equipment conference[END_REF][START_REF] Heggem | Industrial perspective on fire safety of composite cylinders in hydrogen applications[END_REF] to separate:

A) on one side, the fire performance of the cylinder itself, exposed to a significant heat flux such as performed with the setup presented above B) on the other side, the assessment of the safety of the complete structure (car, trailer, stationary storage,...)

The goal of test A) is to provide the cylinder user with the most reliable information of what the cylinder can endure. As illustrated above, it can be achieved with fire tests at different initial pressures; but this presents the drawback of a very high cost due to multiple expensive tests. An alternative can be considered using only one cylinder and a controlled emptying. Within FireComp, partners Samtech and CNRS developed models, with thermal properties from University of Edinburgh tests, which proved able to accurately reproduce the time to failure and to accurately capture the transition between burst and leak modes, if the materials, geometry and stacking sequence of the cylinder are known.

Such a modelling approach can be used by the cylinder manufacturer to anticipate a pressure relief curve, and choose a combination of (time to activation; depressurisation rate) to be tested with a cylinder. Unlike when using a TPRD, the pressure relief is manually triggered, to provide a safe pressure zone such as displayed in Fig. 11. This method presents the advantage of requiring only one fire test.

Figure 11. Method for obtaining a safe pressure zone with only one test. Experimental pressure curve in red, safe pressure zone in greenan arbitrary safety margin on time is displayed as an example

Then, the purpose of part B) is for the integrator or end-user of the cylinder to choose and test their fire safety elements according to the risk analysis of the application considered. The advantages of this method are that:

-only one full scale fire test on a composite vessel is required; it provides more information than actually required by the norms; and need not be repeated in case of changes to the pressure relief system, -the fire safety elements (shields, pressure relief devices...) can be adapted according to the real risks, -the functions of the pressure relief system (detection of the fire and proper opening) can be tested separately.

APPLICATION TO THE FIRE SAFETY OF A MULTI-ELEMENT GAS CONTAINER

This section gives an example of how the previous results can be used to define a fire safety strategy for a specific application: a bundle using composite cylinders for delivery of high pressure hydrogen, displayed in Fig. 12. Four composite cylinders of volumes approx. 150 L and service pressure 700 bar are connected together, transported on the road and delivered as stationary storage for a power generation unit. In case of fire, the effects of a burst of one of the cylinders have been assessed. Considering the speed of the burst event, it can be assumed that the four cylinders will burst separately and that the safety distances to be considered are the same as for a single cylinder. The safety distance associated to a 50 mbar overpressure (French threshold for irreversible effects) was calculated using dedicated engineering tools [START_REF] Houssin-Agbomson | Développement d'outils d'ingénieurs pour l'évaluation du risque hydrogène[END_REF] to be 34 m, which was not acceptable for this application. A pressure relief strategy using a TPRD was decided upon.

The performance of the cylinder alone was assessed using the methodology presented above. The results obtained are gathered in Fig. 13, from reference [START_REF] Bustamante-Valencia | Thermal History Resulting in the Failure of Lightweight Fully-Wrapped Composite Pressure Vessel for Hydrogen in a Fire Experimental Facility[END_REF]. It was decided to install one TPRD for the complete bundle, which should vent all four cylinders simultaneously in case of fire. A metallic frame was used around the bundle for two fire-related reasons: 1) to protect the cylinders from localised jet fires and ensure heat would be transferred to the TPRD, and 2) to delay the fire aggression on the cylinders and increase the time to burst, allowing more time for the pressure relief process, hence limiting the flame length to acceptable levels in case of ignited hydrogen release. It also protects the cylinders from mechanical impacts. A fire test on a cylinder integrated in a bundle was performed and the time to burst at 700 bar was delayed to 13 minutes. The required pressure relief rate was then used to calculate the diameter of the calibrated orifice at the exhaust of the TPRD. The capacity for the TPRD to open when exposed to heat was tested separately, to ensure satisfactory reaction times. The resulting numerical simulation of the pressure relief of the complete bundle is displayed in Fig. 14, which summarises the final safety strategy.

Figure 14. Summary of the fire safety strategy for the bundle. The performance of the unprotected cylinder is known, a metallic frame facilitates delaying the failure, and the flow rate is calibrated so that pressure always remains within the safe envelope.

CONCLUSIONS

A large programme of full-scale fire tests on type IV composite cylinders was performed within the project FireComp. An experimental setup using hydrogen gas burners as a fire source was specified, and proved able to reach an equivalent level of heat stress as with conventional bonfire test systems. A method for calibrating the fire was proposed, which led to fire tests of very good reproducibility. Moreover, the setup can easily be adapted to different sizes or shapes of cylinders.

Results of fire tests carried out on pressurised cylinders without any protection allowed exploration of their behaviour. It was confirmed that below a certain pressure level, in an engulfing fire, the failure mode switches from burst to leak through the body. This is due to a competition between on the one side, the degradation of the composite material, which leads to a burst when the pressure can no longer be contained; and on the other side, the heat transfer to the liner, which leads it to melt and creates a pathway for the gas to exit through the cylinder. A potential path of research could be to design a pressure vessel for which the only failure mode in fire is a leak, by increasing the thermal conductivity while limiting the loss of strength of the composite layup. However, so far the leak through the cylinder body is not fully mastered and cannot be guaranteed in all fire conditions, so the use of a pressure relief system is mandatory when burst must be avoided.

In order to design a pressure relief system well adapted to how the cylinder is used, it was proposed in FireComp to separate the testing of the cylinder performance for the testing of the system performance. In the first step, independently of the application, an unprotected cylinder is exposed to fire until failure, or until a controlled emptying, which provides a safe pressure vs. time chart. In the second step, the capacity of the complete system (e.g. stationary bundle or vehicle storage system) to actually detect fire and safely release the pressure is assessed. An advantage of this method is to provide more information for the choice of a safety strategy adapted to the specificities of the application; meanwhile reducing the number of fire tests required on composite cylinders, as no new qualification of the cylinder is needed if a safety element is changed. A limitation is that the second step, i.e. assessment of the safety at system level, is harder to standardise; but this may be inevitable considering the wide range of possible applications for the same cylinder.

TPRD

  Thermally activated Pressure Relief Device Thermal energy stored by the gas Temperature of the gas Mass of gas Mass of the cylinder Specific heat capacity Specific heat capacity of the steel / Internal / External wall surface Heat flux transmitted by the internal wall surface Net heat flux absorbed by the vessel Average temperature of the internal wall of the cylinder Areal transfer coefficient between gas and inner wall Heat flux absorbed by the cylinder Thermal

Figure 1 .Figure 2 .

 12 Figure 2. Definition of gas injection circuit

Figure 3 .

 3 Figure 3. Thermocouples placed inside and around the steel cylinder used for fire calibration

4 .

 4 The value of this flux fluctuates over test time around a mean level. An average value of 90 kW/m² can be retained, considering that after 400 s the temperature of the cylinder exceeds 400 °C and increases the radiative heat flux emitted by the external surface.

Figure 4 .

 4 Figure 4. Net heat flux absorbed by the cylinder Fig. 5 displays the evolution of the external temperatures recorded by the external thermocouples at 5 cm of the vessel. It can be observed that the temperature fluctuates between 800°C and 1000°C, with a dispersion of very good quality compared to what was measured during bonfire tests with a heptane pool [2].

Figure 5 .

 5 Figure 5. Temperatures recorded at 5 cm outside of the cylinder 3.0 RESULTS AND RECOMMENDATIONS

Figure 6 .

 6 Figure 6. Outdoor experimental setup for hydrogen use (confinement not yet in place in left picture)

Figure 7 .

 7 Evolution of internal pressure for the tests with He, N 2 , H 2 at (a) 700 bar (b) 467 bar

Figure 8 .

 8 The evolution of (a) temperature through the composite thickness and (b) inner gas pressure during both of the 700 bar fire tests show a very good reproducibility

  Figure 12. (a) Picture of the hydrogen bundle, (b) without the external metallic frame, (c) in use

Figure 13 .

 13 Figure 13. Results from fire tests

Table 1 .

 1 Summary of fire tests results for different initial pressures.

	Initial nitrogen pressure (bar)	Failure mode	Time to failure (s)	Pressure increase
	700	Burst	238	15 bar (2%)
	525	Burst	311	40 bar (8%)
	250	Leak	400	55 bar (22%)
	100	Leak	490	35 bar (35%)

ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Union's Seventh Framework Program (FP7/2007-2013) for the Fuel Cells and Hydrogen Joint Technology Initiative under Grant Agreement No.325329.

Parts of the work presented received financial support from OSEO H2E project (2009-2016).


Patrice Hauret

Patrick Le Tallec

Two-scale Dirichlet-Neumann preconditioners for boundary refinements

.

Introduction

Let Ω ⊂ R d be the reference configuration of a body, partitioned into a coarse region Ω 0 where the properties of the material are rather smooth and where a coarse approximation should be sufficient, and into small disjoint boundary regions denoted by (Ω k ) 1≤k≤K where a fine discretization is required (e.g. geometrical refinements, fine behavior of the material). Such a situation typically occurs for tires, the internal structure and the surface sculptures playing the role of the coarse and fine zones, respectively. Let us denote by Γ D a part of the boundary of Ω where displacements are prescribed and by Γ N = ∂Ω \ Γ D its complementary part. Denoting by H 1 * (Ω) := {v ∈ H 1 (Ω) d , v| ΓD ∩∂Ω = 0} the space of admissible displacements, our model elastostatic problem consists in finding u ∈ H 1 * (Ω) such that:

a(u, v) := Ω E ijkl ε(u) kl ε(v) ij = Ω f • v + ΓN g • v =: l(v), ∀v ∈ H 1 * (Ω).
Here E denotes the fourth order elasticity tensor, f ∈ L 2 (Ω) d and g ∈ L 2 (Γ N ) d the loading forces, and (v) = 1 2 (∇v + (∇v) t ) is the linearized strain tensor. Considering that the solution must be computed with a multi-scale approach in order to respect the characteristics of the problem, the strategy proposed in this paper consists in using:

1. mortar formulations [START_REF] Bernardi | Domain decomposition by the mortar element method[END_REF][START_REF] Wohlmuth | Discretization methods and iterative solvers based on domain decomposition[END_REF] on the interfaces Γ 0k = ∂Ω 0 ∩ ∂Ω k enabling to use independent approximations in the coarse and fine regions respectively, 2. efficient Dirichlet-Neumann preconditioners [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF], which we adapt so that the computational cost of the full algebraic problem remains independent (or at least weakly dependent) of the number and the size of the fine subdomains (Ω k ) 1≤k≤K .

The sequel is organized as follows. After the introduction of a mortar formulation (section 2), we propose two possible Dirichlet-Neumann preconditioners and state their two-scale properties (section 3). In particular, the second enhanced preconditioner makes use of a coarse space counterbalancing the effect of essential boundary conditions imposed on the boundary sculptures. A simple numerical test shows its increased efficiency for a simple academic problem.

A broader perspective on the subject as well as complete proofs are given in [START_REF] Hauret | Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles (Numerical methods for the dynamic analysis of two-scale incompressible nonlinear structures[END_REF][START_REF] Hauret | Dirichlet-neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary[END_REF].

Non-conforming formulation

For every 0 ≤ k ≤ K, let (T k;h k ) h k >0 be a sequence of meshes of the substructure Ω k , h k denoting the maximal diameter of its elements. The corresponding finite-element spaces of order q are denoted by (

V k;h k ) h k >0 ⊂ H 1 * (Ω k ).
As in [START_REF] Brezzi | Error estimates for the three-field formulation with bubble stabilization[END_REF][START_REF] Hauret | A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems, part ii: discontinuous Lagrange multipliers[END_REF], for stability purpose when using a discontinuous mortar formulation, interface bubbles can be added on the fine subdomains. As a consequence, we introduce the potentially enriched spaces of displacements X k;h k = V k;h k ⊕B k;h k for every 1 ≤ k ≤ K and X 0;h0 = V 0;h0 . For each interface Γ 0k , W k;h k will stand for the trace of the local space X k;h k on this interface. In order to impose a weak displacement continuity between Ω 0 and Ω k , a space of Lagrange multipliers M k;h k is introduced on the mesh T k;h k over Γ 0k . Actually, various choices of continuous or of discontinuous polynomial functions of degree r can be used [START_REF] Bernardi | Domain decomposition by the mortar element method[END_REF][START_REF] Seshaiyer | Non-conforming hp finite element methods[END_REF][START_REF] Wohlmuth | A mortar finite element method using dual spaces for the Lagrange multiplier[END_REF][START_REF] Kim | Multiplier spaces for the mortar finite element method in three dimensions[END_REF][START_REF] Hauret | A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems, part i: abstract framework[END_REF][START_REF] Hauret | A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems, part ii: discontinuous Lagrange multipliers[END_REF] but in any case, they must satisfy the following fundamental assumptions:

Assumption 1 [Coercivity]. Let u 0 ∈ H 1 (Ω 0 ) d and u k ∈ H 1 (Ω k ) d be rigid motions, i.e. ε(u 0 ) = 0 in L 2 (Ω 0 ) d×d and ε(u k ) = 0 in L 2 (Ω k ) d×d , satisfying the weak continuity requirement Γ 0k (u 0 -u k ) • µ = 0 for every µ ∈ M k;h k . Then u 0 = u k almost everywhere on Γ 0k . Assumption 2 [Inf-sup condition]. There exists a mapping π k : L 2 (Γ 0k ) → W k;h k such that for all v ∈ L 2 (Γ 0k ), Γ 0k (π k v) • µ = Γ 0k v • µ, ∀µ ∈ M k;h k , satisfying π k v k, 1 2 ≤ C v k, 1 2 .
The mesh dependent norm • k, 1 2 introduced above is defined as in [START_REF] Le Tallec | Une méthode d'éléments finis hybrides en décomposition de domaines[END_REF][START_REF] Wohlmuth | Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers[END_REF] by

v 2 k, 1 2 = K∈T k;h k diam(K ∩ Γ 0k ) -1 K∩Γ 0k v 2 .
Assumption 3 [Accuracy]. The total degree r of Lagrange multipliers is bounded from below by r ≥ q -1, q being the total degree of the displacement shape functions.

Then, the mortar formulation of the problem of interest can be written as

finding u = (u 0 , u 1 , ..., u K ) ∈ K k=0 X k;h k and λ = (λ 1 , ..., λ K ) ∈ K k=1 M k;h k satisfying for every v ∈ K k=0 X k;h k and µ ∈ K k=1 M k;h k , a 0 (u 0 , v 0 ) + K k=1 b 0k (v 0 , λ k ) = l 0 (v 0 ) a k (u k , v k ) - b k (v k , λ k ) = l k (v k ), 1 ≤ k ≤ K b 0k (u 0 , µ k ) - b k (u k , µ k ) = 0, 1 ≤ k ≤ K. (1)
The above problem uses the obvious notation

a k (u k , v k ) = Ω k E ijmn ε(u k ) mn ε(v k ) ij , l k (v k ) = Ω k f • v k + ΓN ∩∂Ω k g • v k , b 0k (v 0 , µ k ) = Γ 0k v 0 • µ k and b k (v k , µ k ) = Γ 0k v k • µ k .
3 Two-scale preconditioners

In matrix notation, after elimination of the Lagrange multipliers λ k in the first equation of (1), the system becomes

     S 0 U 0 = L 0 , K k U k Λ k = L k -B 0k U 0 , 1 ≤ k ≤ K, (2)
where

S 0 = A 0 - K k=1 B t 0k R k K -1 k R t k B 0k is the Schur complement matrix,
and

L 0 = L 0 - K k=1 B t 0k R k K -1 k L k 0
the corresponding right hand side. In these definitions, the local stiffness matrix K k and restriction operator R k are given by

K k = A k -B t k -B k 0 , R k U k Λ k = Λ k .
An iterative solver can be efficiently used to solve [START_REF] Bernardi | Domain decomposition by the mortar element method[END_REF] if one is able to define a preconditioner S0 of the exact Schur complement S 0 which is spectrally equivalent to S 0 , with constants independent of the number and the size of the small subdomains. When L 0 , .., L K are given, the application of such a preconditioner consists in the following operations:

1. Compute L 0 by solving Dirichlet problems on the small subdomains prescribing zero displacements on the interfaces (Γ 0k ) 1≤k≤K , 2. Solve the extended Neumann problem S0 Ũ0 = L 0 , 3. Compute ( Ũk , Λk ) over each Ω k by solving the Dirichlet problem:

K k Ũk Λk = L k -B 0k Ũ0
.

The most natural -and rather efficient-preconditioner consists in simply using S0 = A 0 . This is a standard Dirichlet-Neumann preconditioner for which we prove [START_REF] Hauret | Dirichlet-neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary[END_REF]:

Proposition 1.
Assuming that A 0 is invertible, i.e. Γ D ∩ ∂Ω 0 has a positive measure, the following spectral equivalence holds for all U 0 :

W 1,h S 0 U 0 , U 0 ≤ A 0 U 0 , U 0 ≤ S 0 U 0 , U 0 , with: 1 W 1,h = 1 + C max k∈I1 C k c 0 + max k∈I2 C k L 0 α 0 L k ,
where I 1 (resp. I 2 ) is the set of indices k ≥ 1 such that Ω k is not fixed on its boundary (resp. is fixed on a part of its boundary), the positive constants

c k and C k are such that c k |ξ| 2 ≤ E ijmn ξ mn ξ ij ≤ C k |ξ| 2
over Ω k for every symmetric matrix ξ ∈ R d×d , α 0 is the coercivity constant of the bilinear form a 0 and L k = diam(Ω k ). The constant C > 0 is independent of the number K and the size of the subdomains.

This simple choice will lack of efficiency in two simple situations:

1. a fine subdomain Ω k (k ≥ 1) has a small size L k << L 0 and is fixed on a part of its boundary (k ∈ I 2 ); in this situation, because of its size, the substructure will have a rather large stiffness to interface rigid body displacements, 2. a fine subdomain Ω k (k ≥ 1) has several stiff modes involving interface motions (rigid links, incompressibility).

Assuming that these directions of localized interface stiffness be in very small number N k (this is indeed the case for interface rigid body motions), we then propose a modification of the previous preconditioner enabling to correct such a lack of efficiency.

For all k ≥ 1 such that Ω k is fixed on a part of its boundary, we denote by (e i k ) 1≤i≤N k (with N k = 6 in general) the interface rigid motions of Γ 0k or rigid links and introduce Wk = span{e i k , i = 1, .., N k }. To each interface rigid body motion e i k , we associate its local a k -harmonic extension (u

i k , λ i k ) ∈ X k;h k × M k;δ k solution of        a k (v, u i k ) - Γ 0k v • λ i k = 0, ∀v ∈ X k;h k , - Γ 0k u i k • µ = - Γ 0k e i k • µ, ∀µ ∈ M k;δ k . (3)
These solutions span two small local spaces

Xk = span{u i k , i = 1, .., N k } ⊂ X k;h k , Mk = span{λ i k , i = 1, .., N k } ⊂ M k;δ k . If k ≥ 1 is such that Ω k is not fixed on its boundary, we adopt Wk = Mk = {0}.
Then, instead of finding U 0 such that S 0 U 0 = L 0 , we propose to compute

u 0 ∈ X 0;h0 , (u k ) ∈ ( Xk ) 1≤k≤K , (λ k ) ∈ ( Mk ) 1≤k≤K solution of the coupled problem                  a 0 (u 0 , v 0 ) + K k=1 Γ 0k v 0 • λ k = l 0 (v 0 ), ∀v 0 ∈ X 0;h0 , a k (u k , v k ) - Γ 0k v k • λ k = 0, ∀v k ∈ Xk , 1 ≤ k ≤ K, - Γ 0k u k • µ k = - Γ 0k u 0 • µ k , ∀µ k ∈ Mk , 1 ≤ k ≤ K. (4)
This amounts to reduce the local substructure response to the harmonic extension of its stiff interface modes, which belongs to Xk . We introduce the matrix

I 0k = Λ t k B 0k where Λ t k = Λ 1 k , .., Λ N k k t
is the matrix built with the multipliers computed in (3), and the restriction Åk of the displacement stiffness matrix A k to the local space Xk

Åk ij = (U i k ) t A k U j k = a k (u j k , u i k ) = Γ 0k u j k • λ i k , (5)
where (3) has been used. Exploiting [START_REF] Hauret | Dirichlet-neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary[END_REF] to reformulate (4)-2,(4)-3, the system (4) can be rewritten after some algebraic elimination as

S0 U 0 = L 0 , (6)
with a new approximate Schur complement given by

S0 = A 0 + K k=1 I t 0k Å-t k I 0k (7) = A 0 + K k=1 B t 0k Λ k Å-t k Λ t k B 0k .
The complexity of its inversion is much smaller than solving S 0 U 0 = L 0 because each local problem (3) used in the construction of S0 only involves a subspace of displacements of dimension N k . Moreover, we prove in [START_REF] Hauret | Dirichlet-neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary[END_REF] that:

Proposition 2. For all U 0 , the following spectral equivalence holds

W 1,h S 0 U 0 , U 0 ≤ S0 U 0 , U 0 ≤ S 0 U 0 , U 0 , with 1 W 1,h = C 1 + max 1≤k≤K C k c 0 .
The constant C > 0 is independent of the number K and the size of the subdomains.

Numerical illustration

Let us consider here a two-scale beam (as represented on figure 1) whose both tips are clamped. The material is elastic, isotropic, homogeneous in each substructure, and the displacements under loading are computed by a preconditioned conjugate gradient method. Figure 2 illustrates the advantage of the enhanced Dirichlet-Neumann preconditioner when two small substructures are clamped. In conformity with the announced results, the gain in efficiency is independent of the ratio of Young moduli between the fine and coarse zones. Moreover, a factor 3 improvement is achieved in the number of iterations, and roughly speaking in the time of computation. Finally, it is shown in [START_REF] Hauret | Dirichlet-neumann preconditioners for elliptic problems with small disjoint geometric refinements on the boundary[END_REF] that such a preconditioner can be used as an efficient quasi-tangent operator in the nonlinear framework as soon as boundary geometrical details are sufficiently soft.

Conclusion

The domain-decomposition based preconditioners proposed here achieve scaleindependent performances. They should be extended to cases where the details overlap the coarse region in the line of the fictitious domains approach, and also to cases where the details are not disjoint but constitute a continuous belt along the boundary.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Maximal stress distribution on a deformed configuration of our two-scale model problem where two of the details are clamped on their lower face.

Acknowledgement. The authors gratefully acknowledge the financial support of Michelin Tire Company during the completion of this work at the Center of Applied Mathematics (CMAP), Ecole Polytechnique, and Professor François Jouve for providing his finite element code in which the presented ideas have been implemented.