Stéphane Huet
email: stephane.huet@univ-avignon.fr

Fabrice Lefèvre
email: fabrice.lefevre@univ-avignon.fr

Unsupervised Alignment for Segmental-based Language Understanding

Recent years' most efficient approaches for language understanding are statistical. These approaches benefit from a segmental semantic annotation of corpora. To reduce the production cost of such corpora, this paper proposes a method that is able to match first identified concepts with word sequences in an unsupervised way. This method based on automatic alignment is used by an understanding system based on conditional random fields and is evaluated on a spoken dialogue task using either manual or automatic transcripts.

Introduction

One of the very first step to build a spoken language understanding (SLU) module for dialogue systems is the extraction of literal concepts from word sequences hypothesised by a speech recogniser. To address this issue of concept tagging, several techniques are available. These techniques rely on models, now classic, that can be either discriminant or generative. Among these, we can cite: hidden Markov models, finite state transducers, maximal entropy Markov models, support vector machines, dynamic Bayesian networks (DBNs) or conditional Markov random fields (CRFs) [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting and labeling sequence data[END_REF]. In [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF], it is shown that CRFs obtain the best performance on a reference task (MEDIA) in French [START_REF] Bonneau-Maynard | Semantic annotation of the MEDIA corpus for spoken dialog[END_REF], but also on two other comparable corpora in Italian and Polish. Besides, the comparison of the understanding results of manually vs automatically transcribed utterances has shown the robustness of CRFs.

Among the approaches evaluated in [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF] was a method using log-linear models comparable to those used in stochastic machine translation, which turned out to have lower performance than CRF. In this paper, we further exploit the idea of applying automatic translation techniques to language understanding but limiting ourselves to the objective of obtaining a segmental annotation of training data.

In many former approaches literal interpretation was limited to list lexical-concept relations; for instance this is the case of the PHOENIX system [START_REF] Ward | Understanding spontaneous speech: the Phoenix system[END_REF] based on the detection of keywords. The segmental approach allows a finer-grained analysis considering sentences as segment sequences during interpretation. This characteristic enables the approach to correctly connect the various levels of sentence analysis (lexical, syntactic and semantic). However, in order to simplify its practical application, segments have been designed specifically for semantic annotation and do not integrate any constraint in their relation with the syntactic units (chunks, phrasal groups, etc.). Not only it simplifies the annotation process itself but as the overall objective is to use the interpretation module inside a spoken dialogue system, transcribed speech data are noisy and generally bound the performance of syntactic analysers (due to highly spontaneous and ungrammatical utterances from the users, combined with errors from the speech recognizer).

Among other interesting proprieties, segmental approaches offer a convenient way to dissociate the detection of a conceptual unit from the estimation of its associated value. The value corresponds to the normalisation of the surface form. For instance, if the segment "no later than eleven" is associated with the concept departure-time, its value is "morning"; the same value is associated with the segments "between 8 and noon" or "in the morning". The value estimation requires a link between concepts and sentence words. Then it becomes possible to treat the normalisation problem by means of regular expressions or concept-dependent language models (allowing an integrated approach such as described in [START_REF] Lefèvre | Dynamic bayesian networks and discriminative classifiers for multi-stage semantic interpretation[END_REF]). In the case of global approaches (not segmental), value detection must be directly incorporated in the conceptual units to identify, as in [START_REF] Mairesse | Spoken language understanding from unaligned data using discriminative classification models[END_REF]. The additional level is a real burden and is only affordable when the number of authorised values is low.

Obviously a major drawback of the approach is its cost: associating concept tags with a dialogue transcription is already a tedious task and its complexity is largely increased by the requirement for a precise delimitation of the support (lexical segment) corresponding to each tag. The SLU evaluation campaign MEDIA has been the first opportunity to collect and distribute a reasonably-sized corpus endowed with segmental annotations.

Anyhow the difficulty remains unchanged each time a corpus has to be collected for a new task. We propose in this study a new method that reduces the effort required to build training data for segmental annotation models. Making the assumption that the concepts evoked in a sentence are automatically detected beforehand or provided by an expert, we study how to associate them with their lexical supports without prior knowledge. A conceptual segmental annotation is obtained using alignment techniques designed to align multilingual parallel corpora in the machine translation domain. This annotation can be considered as unsupervised since it is done without a training corpus with links between word sequences and concepts.

We present in the paper the necessary adaptations for the application of the alignment techniques in this new context. They have been kept to their minimal so as to maintain the highest level of generality, which in return benefits from the availability of existing software tools. Using a reference annotation, we evaluate the alignment quality from the unsupervised approach in two interesting situations depending on whether the correct order of the concepts is known or not. Finally, the end-to-end evaluation of the approach is made by measuring the impact of the alignments on the CRF-based understanding system.

After a brief recall of the conceptual decoding principles in Section 2, the principles of automatic alignment of parallel corpora are described in Section 3 along with the specificities due to the alignment of semantic concepts. Section 4 presents the experiments and comments on the results, while Section 5 concludes the paper.

Segmental conceptual decoding

If literal interpretation can be seen as the translation of natural language to the set of semantic tag sequences, then the methods and models of machine translation can be used. Since the number of concepts is generally much lower than the vocabulary size, this particular type of translation can also be considered as a mere classification problem in which the conceptual constituents represent the class to identify. Interpretation can thus be performed by methods and models of classification.

Discriminant approaches model the conditional probability distribution of the semantic constituent sequence (or concepts) c 1 . . . c n considering a word sequence w 1 . . . w T : P (c n 1 |w T 1 ). In generative approaches, the joint probability P (c n 1 , w T 1 ) is modelized instead and can be used to compute inferences either for prediction/decoding or parameter training.

Generative models (such as hidden Markov models) have been first introduced to address the understanding problem with stochastic approaches [START_REF] Levin | Conceptbased spontaneous speech understanding system[END_REF].

Recent variants offer more degrees of freedom in modeling (see for instance [START_REF] He | Spoken language understanding using the hidden vector state model[END_REF] or [START_REF] Lefèvre | Dynamic bayesian networks and discriminative classifiers for multi-stage semantic interpretation[END_REF]). Since then log-linear models have clearly shown their superiority for tasks of sequence tagging [START_REF] Hahn | Comparing stochastic approaches to spoken language understanding in multiple languages[END_REF].

Several variants of log-linear models differ in their conditional variable independence assumptions and use different normalisation steps. CRFs [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting and labeling sequence data[END_REF] represent linear chains of random independent variables, all conditioned over the entire sequence and the normalisation is global over the sequence.

Some generative approaches such as DBNs make inferences in multi-level models [START_REF] Lefèvre | Dynamic bayesian networks and discriminative classifiers for multi-stage semantic interpretation[END_REF] Figure 1: Example of an alignment of words with their conceptual units. and intrinsically take into account segmentation. For models unable to handle multi-level representations (as CRF), it is convenient to represent segments directly at the tag level. For this purpose the BIO formalism can be used: B is added to tags starting a segment, I to tags inside a segment and O to out-of-domain tags (if these are not already handled through a specific NULL tag). In the case displayed in Figure 1, the concept sequence becomes:

B-cmd-task I-cmd-task I-cmd-task B-null I-null B-loc-town I-loc-town I-loc-town I-loc-town I-loc-town B-time-date I-time-date B-time-date I-time-date I-time-date.

Semantic concept alignment

Automatic alignment is a major issue in machine translation. For example, word-based alignments are used to generate phrase tables that are core components for many current statistical machine translation systems [START_REF] Koehn | Moses: Open source toolkit for statistical machine translation[END_REF]. The alignment task aims at finding the mapping between words of two sentences in relation of translation. It faces several difficulties:

• some source words are not associated with a translated word;

• others are translated by several words;

• matched words may occur at different positions in both sentences according to the syntactic rules of the considered languages.

Several statistical models have been proposed to align two sentences [START_REF] Peter | The mathematics of statistical machine translation: Parameter estimation[END_REF]. One of their main interests is their ability to be built in an unsupervised way from a parallel corpus aligned at the sentence level, but not at the word level. Formally, from a sentence S = s 1 . . . s m expressed in a source language and its translation T = t 1 . . . t n expressed in a target language, an IBM-style alignment A = a 1 . . . a m connects each source word to a target word (a j ∈ {1, ..., n}) or to the so-called NULL token which accounts for untranslated target words. IBM statistical models evaluate the translation of S into T from the computation of P (S, A|T ); the best alignment  can be deduced from this criterion using the Viterbi algorithm:

 = argmax A P (S, A|T ) .
(1)

IBM models differ according to their complexity level. IBM1 model makes the strong assumption that alignments are independent and can be evaluated only through the transfer probabilities P (s i |t j ). The HMM model, which is an improvement over IBM2, adds a new parameter P (a j |a j-1 , n) that assumes a first-order dependency between alignment variables. The next models (IBM3 to IBM5) are mainly based on two types of parameters:

• distortion, which measures how words of T are reordered with respect to the index of the words from S they are aligned with,

• fertility, which measures the usual number of words that are aligned with a target word t j .

In order to improve alignments, IBM models are usually applied in both translation directions. These two alignments are then symmetrized by combining them. This last step is done via heuristic methods; a common approach is to start with the intersection and then iteratively add links from the union [START_REF] Franz | Improved alignment models for statistical machine translation[END_REF]).

If we have at our disposal a method that can find concepts contained in an utterance, segmental annotation can be obtained by aligning words S = w T 1 with the found concepts T = c n 1 (Fig. 1). Concepts are ideally generated in the correct order with respect to the word segments of the analysed utterance. In a more pragmatic way, concepts are likely to be produced as bag-of-concepts rather than ordered sequences.

Statistical alignment methods used in machine translation are relevant in our context if we consider that the target language is the concept language. There are nevertheless differences with genuine language translation. First, each word is aligned to at most one concept, while a concept is aligned with one word or more. Consequently, it is expected that word fertilities are one for the alignment of words toward concepts and concept fertilities are one or more in the reverse direction. Another consequence is that NULL words are useless in our context. These specificities of the alignment process raise some difficulties with regard to IBM models. Indeed, according to the way probabilities are computed, the alignment of concepts toward words only allows one word to be chosen per concept, which prevents this direction from having a sufficient number of links between words and concepts.

Another significant difference with translation is related to the translated token order. While word order is not random in a natural language and follows syntactic rules, it is not the case anymore when a word sequence have to be aligned with a bag-ofconcepts. HMM and IBM2 to IBM5 models have parameters that assume that the index of a matched source word or the indices of the translations of the adjacent target words bear on the index of target words. Therefore, the randomness of the concept indices can disrupt performance obtained with these models, contrary to IBM1. As shown in the next section, it is appropriate to find ways to explicitly re-order concept sequences than to let the distortion parameters handle the problem alone.

Experiments and results

Experimental setup

The evaluation of the introduced methods was carried out on the MEDIA corpus [START_REF] Bonneau | MEDIA : évaluation de la compréhension dans les systèmes de dialogue[END_REF]. This corpus consists of human-machine dialogues collected with a wizard of Oz procedure in the domain of negotiation of tourist services. Produced for a realistic task, it is annotated with 145 semantic concepts and their values (more than 2k in total for the enumerable cases). The audio data are distributed with their manual transcripts and automatic speech recognition (ASR) hypotheses. The corpus is divided into three parts: a training set (approxi-matively 12k utterances), a development set (1.2k) and a test set (3k).

The experiments led on the alignment methods were evaluated on the development corpus using MGIZA++ [START_REF] Gao | Parallel implementations of word alignment tool[END_REF], a multi-thread version of GIZA++ [START_REF] Franz | A systematic comparison of various statistical alignment models[END_REF] which also allows previously trained IBM alignments models to be applied on the development and test corpora. 1The conceptual tagging process was evaluated on the test corpus, using WAPITI [START_REF] Lavergne | Practical very large scale CRFs[END_REF] to train the CRF models. Several setups have been tested:

• manual vs ASR transcriptions,

• inclusion (or not) of values during the error computation.

Several concept orderings (before automatic alignment) have also been considered:

• a first ideal one, which takes reference concept sequences as they are, aka sequential order;

• two more realistic variants that sort concepts either alphabetically or randomly, in order to simulate bag-of-concepts. Alphabetical order is introduced solely to show that a particular order (which is not related to the natural order) might misled the alignment process by introducing undue regularities.

To give a rough idea, these experiments required a few minutes of computing time to train alignment models of 12k utterances, a few hours to train CRF models (using 8 CPUs on our cluster of Xeon CPUs) and a few seconds to apply alignment and CRF models in order to decode the test corpus.

Experimental results for alignment

Alignment quality is estimated using the alignment error rate (AER), a metric often employed in machine translation [START_REF] Joseph | A comparison of alignment models for statistical machine translation[END_REF]. If H stands for hypothesis alignments and R for reference alignments, AER is computed by the following relation:2

AER = 1 - 2 × |H ∩ R| |H| + |R| . ( 2
)
In our context, this metrics is evaluated by representing a link between source and target identities by (w i , c j ), instead of the usual indices (i, j). Indeed, alignments are then used to tag words. Besides, concepts to align have positions that differ from the ones in the reference when they are reordered to simulate bags-of-concepts.

As mentioned in the introduction, we resort to widely used tools for alignment in order to be as general as possible in our approach. We do not modify the algorithms and rely on their generality to deal with specificities of the studied domain. To train iteratively the alignment models, we use the same pipeline as in MOSES, a widely used machine translation system [START_REF] Koehn | Moses: Open source toolkit for statistical machine translation[END_REF]:

1. 5 iterations of IBM1, 2. 5 iterations of HMM, 3. 3 iterations of IBM3 then 4. 3 iterations of IBM4.

To measure the quality of the built models, the model obtained at the last iteration of this chain is applied on the development corpus.

All the words of an utterance should normally be associated with one concept, which makes the IBM models' NULL word useless. However, in the MEDIA corpus, a null semantic concept is associated with words that do not correspond to a concept relevant for the tourist domain and may be omitted by counting on the probability with the NULL word included in the IBM models. Two versions were specifically created to test this hypothesis: one with all the reference concept sequences and another without the null tags. The results measured when taking into account these tags (AER of 14.2 %) are far better than the ones obtained when they are discarded (AER of 27.4 %), in the word → concept alignment direction. 3 We decided therefore to keep the null in all the experiments.

Table 1 presents the alignment results measured on the development corpus according to the way concepts are reordered with respect to the reference and according to the considered alignment direction.

3 For a fair comparison between both setups, the null concept was ignored in H and R for this series of experiments.

The three first lines exhibit the results obtained with the last IBM4 iteration. As expected, the AER measured with this model in the concept → word direction (second line), which can only associate at most one word per concept, is clearly higher than the one obtained in the opposite direction (first line). Quite surprisingly, an improvement in terms of AER (third line) over the best direction (first line) is observed using the default MOSES heuristics (called growdiag-final) that symmetrizes alignments obtained in both directions.

IBM1 models, contrary to other models, do not take into account word index inside source and target sentences, which makes them relevant to deal with bag-of-concepts. Therefore, we measured how AER varies when using models previously built in the training chain. The results obtained by applying IBM1 and by symmetrizing alignments (last line), show finally that these simple models lead to lower performance than the one measured with IBM4 or even HMM (last line), the concepts being ordered alphabetically or randomly (two last columns).

The previous experiments have shown that alignment is clearly of lower quality when algorithms are faced with bags-of-concepts instead of well-ordered sequences. In order to reduce this phenomenon, sequences are reordered after a first alignment A 1 generated by the symmetrized IBM4 model. Two strategies have been considered to fix the new position of each concept c i . The first one averages the indices of the words w i that are aligned with c i according to

A 1 : pos 1 (c j ) = is.t.(i,j)∈A 1 i Card({(i, j) ∈ A 1 }) . (3)
The second one weights each word index with their transfer probabilities determined by IBM4:

pos 2 (c j ) = is.t.(i,j)∈A 1 i × f (w i , c j ) is.t.(i,j)∈A 1 f (w i , c j ) (4)
where

f (w i , c j ) = λP (c j |w i ) + (1 -λ)P (w i |c j ) (5)
and λ is a coefficient fixed on the development corpus.

Training alignment models on the corpus reordered according to pos 1 (Tab. 2, second column) or pos 2 (third column) leads to a significant improvement of the AER. This reordering step can be repeated as long as performance goes on improving. By proceeding like this until step 3 for the alphabetic order and until step 7 for the random order, values of AER below 20 % (last column) are finally obtained.

It is noteworthy that random reordering has better results than alphabetic reordering. Indeed, HMM, IBM3 and IBM4 models have probabilities that are more biased in this latter case, where the same sequences occur more often although many are not in the reference.

Experimental results for spoken language understanding

In order to measure how spoken language understanding is disturbed by erroneous alignments, CRFs parameters are trained under two conditions: one where concept tagging is performed by an expert and one where corpora are obtained using automatic alignment. The performance criterion used to evaluate the understanding task is the concept error rate (CER). CER is computed in a similar way as word error rate (WER) used in speech recognition; it is obtained from the Levenshtein alignment between both hypothesized and reference sequences as the ratio of the sum of the concepts in the hypothesis substituted, inserted or omitted on the total number of concepts in the manual reference anno-tation. The null concept is not considered during the score computation. The CER can also take into account the normalized values in addition to the concept tags.

Starting from a state-of-the-art system (Manual column), degradations due to various alignment conditions are reported in Table 3. It can be noted that the absolute increase in CER is at most 8.0 % (from 17.6 to 25.6 with values) when models are trained on the corpus aligned with IBM models; the ordering information brings it back to 3.7 % (17.6 to 21.3), and finally with automatic transcription the impact of the automatic alignments is smaller (resp. 5.8 % and 2.0 %). As expected random order is preferable to alphabetic order (slight gain of 1 %).

In Table 4, the random order alignments are used but this time the n-best lists of alignments are considered and not only the 1-best hypotheses. Instead of training CRFs with only one version of the alignment for a concept-word sequence pair, we filter out from the n-best lists the alignments having a probability above a given threshold. It can be observed that varying this confidence threshold allows an improvement of the SLU performance (CER can be reduced by 0.8 % for manual transcription and 0.4 % for automatic transcription). However, this improvement is not propagated to scores with values (CER was reduced at best by 0.1 for manual transcription and was increased for automatic tran- scription). After closer inspection of the scoring alignments, an explanation for this setback is that the manually-designed rules used for value extraction are perturbed by loose segmentation. This is particularly the case for the concept used to annotate co-references, which has confusions between the values singular and plural (e.g. "this" is singular and "those" plural). This issue can be solved by an ad hoc adaptation of the rules. However, it would infringe our objective of relying upon unsupervised approaches and minimizing human expertise. Therefore, a better answer would be to resort to a probabilistic scheme also for value extraction (as proposed in [START_REF] Lefèvre | Dynamic bayesian networks and discriminative classifiers for multi-stage semantic interpretation[END_REF]).

The optimal configuration (confidence threshold of 0.3, 4th row of Table 4) is close to the baseline 1-best system in terms of the number of training utterances. We also tried a slightly different setup which adds the filtered alignments to the former corpus before CRF parameter training (i.e. the 1-best is not filtered in the n-best list). In that case performance remains pretty stable with respect to the filtering process (CER is around 21.4 % for concepts and 25.2 % for concept+value for thresholds between 0.1 and 0.7).

Conclusion

In this study an unsupervised approach is proposed to the problem of conceptual unit alignment for spoken language understanding. We show that unsupervised statistical word alignment from the machine translation domain can be used in this context to associate semantic concepts with word sequences. The quality of the derived alignment, already good in the general case (< 20 % of errors on the word-concept associations), is improved by knowledge of the correct unit order (< 15 %). The impact of automatic alignments on the understanding performance is an absolute increase of +8 % in terms of CER, but is re-duced to less than +4 % in the ordered case. When automatic transcripts are used, these gaps decrease to +6 % and below +3 % respectively. From these results we do believe that the cost vs performance ratio is in favour of the proposed method.

Table 1 :

 1 AER (%) measured on the MEDIA development corpus with respect to the alignment model used and its direction.

			Sequential order Alphabetic order Random order
	word → concept IBM4	14.4	29.2	28.6
	concept → word IBM4	40.9	51.6	49.0
	symmetrized IBM4	12.8	27.3	25.7
	symmetrized IBM1	33.2	33.2	33.1
	symmetrized HMM	14.8	29.9	28.7
		Initial 1 st reordering iteration Last reordering iteration
			pos 1	pos 2	pos 2
	Alphabetic order 27.3 22.2	21.0	19.4
	Random order	25.7 21.9	20.2	18.5

Table 2 :

 2 AER (%) measured on the MEDIA development corpus according to the strategy used to reorder concepts.

Table 3 :

 3 CER (%) measured for concept decoding on the MEDIA test corpus with several alignment methods of the training data. Inside parenthesis, CER for concepts and values.

With previousa, previoust, previousn, etc parameters.

2 This equation is a simplification of the usually provided one because all alignments are considered as sure in our case.

Acknowledgements

This work is partially supported by the ANR funded project PORT-MEDIA. 4


Research Ethics and Integrity Training for Doctoral Candidates: face-to-face is better! by Catherine Tessier

The University of Toulouse and Inria have set up face-to-face training in research ethics and integrity for doctoral candidates based on debates about their own theses.

In order to meet the requirements of 2016 French legislation about doctoral studies, face-to-face training in research ethics and integrity for doctoral students has been conducted at Toulouse University since 2016 and at Inria since 2017. The training programme is based on the work [START_REF]Commission de réflexion sur l'éthique de la recherche en sciences et technologies du numérique d'Allistene (CERNA)[END_REF] of CERNA, the French ethics committee for research in ICTs.

Based on the conviction that doctoral candidates should be able to express themselves and question and discuss issues of research ethics and integrity that directly concern themselves and their own theses, the training is organised as a single-day (i.e., six to seven hours) session with a maximum of twenty doctoral candidates per session. The originality of the approach lies in the fact that, on the one hand, classes can be multidisciplinary since they are open to doctoral students from any doctoral school; and on the other hand that the training is facilitated by two trainers preferably with different backgrounds.

The training material includes a set of slides, a collection of dilemma exercises and a trainer's guide [START_REF]une formation proposée par l'école des docteurs de Toulouse[END_REF]. The trainers commit to respecting the structure of the course i.e., to using the slides, supervising a dilemma exercise, asking each participant to write an ethical or integrity issue about their thesis and hosting class discussions about the issues that are put forward.

The slide-based lecture includes headline news examples of cases raising ethical or integrity issues, information about laws and codes of ethics, philosophical and historical background of ethics in science and useful vocabulary and concepts. The doctoral candidates' issues are collected and organised by the trainers in three topics: research integrity, publication and research ethics. Each set of issues is discussed by the whole class, and the trainers refer to the relevant available slides as the discussion goes along e.g., best research practices and misconducts, relationships with thesis supervisors (research integrity), authors, reviewers, citations (publications) and value conflicts during the thesis (research ethics). The slides are sent to the doctoral candidates after the training so that they can refer to them and visit the numerous links that are provided to relevant documents.

In order to increase the size of the pool of trainers, the training of trainers is done in situ, i.e., most often a pair of trainers includes an experienced trainer and a "new" trainer who may in turn train another colleague during a future session. The side effect of this structure is that the people involved as trainers are actually made aware of the issues of research ethics and integrity.

After more than two years' experience and over 1000 trained doctoral candidates, the following points are worth highlighting:  It is not necessary for a trainer to be an ethics "expert". Nevertheless they must become familiar with the concepts, ask themselves about their own research practice and pay attention to scientific news and associated issues. Indeed a trainer enriches the course with their own experience and thinking.

 A trainer should not be afraid of disturbing issues, difficult situations, conflicting points of view, and where appropriate should be able to cope with emotion. Indeed sensitive issues are likely to be raised by doctoral candidates.

 Classes must be composed exclusively of doctoral candidates so that the issues that they raise can be addressed without the pressure of "senior" colleagues.

 In order for doctoral candidates to talk freely, it should be announced at the beginning of the session that everyone (both the trainers and the doctoral candidates) should commit to confidentiality; furthermore a doctoral candidate may remain anonymous when raising an issue about their thesis (questions are put down in writing on stickers provided by the trainers).

 Trainers should announce that some issues -especially research ethics issues -do not have simple black and white answers and that the main point is to become familiar with ethical thinking and deliberation.

 Computers and cell-phones should be banned so that everyone can be involved in the debates.

Feedback from doctoral candidates about the training includes some very isolated but interesting comments:

 The dilemma exercise, which aims at creating ethical deliberation through the confrontation of moral values, can be regarded as aggressive and provoke rejection. On the contrary, it can be judged irrelevant because it is considered as being far from reality or pointless.

 Philosophical and historical references can provoke sharp or even hostile reactions, depending on each person's culture and beliefs.

Nevertheless, the overwhelming majority of feedback is positive, strengthening the pedagogical approach that has been adopted i.e., a face-to-face doctoral training in research ethics and integrity, in small classes, based on debates that are focused on the issues that are raised by the doctoral candidates themselves. Most doctoral candidates report that they have become aware of best practices concerning publications and experiments, that they have learnt to consider their work from an ethics point of view, and that they have appreciated being able to raise issues about their own theses. Clearly this costs money and the recruitment and remuneration policy of the trainers is an integral element of the (ethical) debate about this kind of training.


Quoc Bao Nguyen 1, a) and Cyril Touzé In cymbal making, taper refers to the gradual change in thickness from the centre to the rim, and is known to be a key feature that determines the tone of the instrument.

It is generally used in conjunction with shape variations in order to enable the cymbal to play a bell-like sound when hit near its centre, or a crash sound when struck close to the edge. The von Kármán equations for thin plates with thickness and shape variations are derived, and a numerical method combining a Rayleigh-Ritz approach together with a Störmer-Verlet scheme for advancing the problem in time, is detailed.

One main advantage of the method is its ability to implement easily any frequencydependent loss mechanism which is a key property for sound synthesis. Also, the accuracy of the computation of the nonlinear restoring force is especially preserved.

The method is employed to synthesize the sounds of cymbal-like instruments. The impact of taper is addressed and the relative effects of both thickness and shape variations, are contrasted. a) quoc-bao.nguyen@u-pec.fr; Also at: Laboratoire Modélisation et Simulation Multi Echelle (MSME), CNRS -UPEC -UPEM, Marne-la-Vallée, France; Also at: Institut de Chimie et des Matériaux Paris-Est, CNRS -UPEC, Thiais, France. b) cyril.touze@ensta-paristech.fr; Corresponding author.

I. INTRODUCTION

Cymbals belong to the category of percussion instruments having a long history, and used in various contexts, from orchestral symphonic music to pop-rock and jazz music [START_REF] Fletcher | The Physics of musical instruments[END_REF][START_REF] Rossing | Science of Percussion Instruments[END_REF] .

Having different shapes, names and tones and a wide range of playability, sounds and colors, they serve as a basis of all drum kits in modern amplified music, jazz bands and percussion ensembles. The usual terminologies to designate their roles and sounds are known as: ride, splash and crash, the last two names of which are evidently onomatopoeic. Ride cymbals are the largest ones, generally used to play a steady rhythmic pattern, whereas crash and splash are smaller and used to play dramatic accents.

In cymbal making, taper refers to the thickness variation encountered from the centre to the edge of a cymbal. Indeed, for most of the cymbals, taper is very important in order to confer the cymbals the ability to have two distinct tones. On the one hand, cymbals shall play a clear tone with a dominant pitch when hit near the centre, in the region known as the bell. This bell-like sound is essential to mark the beat, and is favoured by two different physical characteristics : the protruding dome manufactured at centre, and the thickness which is more important in this area. On the other hand, a non-tonal, bright glittering sound is awaited when strongly striking crash and splash cymbals at the edge.

A series of measurements have been realized in two different drum stores in Paris with an electronic vernier calliper in order to quantify more precisely the thickness at centre and edge. It has been found that for crash cymbals, the thickness at edge h e ranges from 0.9 mm to 0.5 mm, the minimal values being obtained for series known for their explosive sounds: The Paiste fast crash 14" (35 cm diameter) and the Sabian AA thin crash (30 cm diameter), and also for the Paiste "paperthin formula 602", generally used in hi-hat. The mean measured value for crash is 0.65 -0.7 mm. The thickness at centre h c is larger, with measured values ranging from 1.4 mm to 0.9 mm. For ride cymbals, h c can go up to 1.5 mm while h e ranges between 1.4 mm to 0.8 mm. Splash cymbals are the smallest ones, and h e has been found to be of the order of 0.55 mm to 0.65 mm, with a minimal value h e =0.4 mm for a Sabian AAX splash 8" (20 cm diameter).

The bright shimmering sound of gongs and cymbals is obtained thanks to strongly nonlinear vibrations [START_REF] Fletcher | The Physics of musical instruments[END_REF]3 occuring because the amplitude of the vibrations is larger than the thickness.

Recent studies revealed that the phenomenon at hand is wave turbulence, resulting from the geometric nonlinearity, and also at work in gongs and thunder plates [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF][START_REF] Touzé | Transition scenario to turbulence in thin vibrating plates[END_REF][START_REF] Düring | Weak turbulence for a vibrating plate: Can one hear a Kolmogorov spectrum?[END_REF][START_REF] Boudaoud | Observation of wave turbulence in vibrating plates[END_REF][START_REF] Cadot | Wave turbulence in vibrating plates[END_REF] . For the gongs, the energy cascade driven by the wave turbulence can be relatively slow and take up to half a second to fully develop and attain the highest frequencies. This results in the characteristic sound of gongs with a blow-up of higher frequencies occuring shortly after the strike. In the case of cymbals (and more specifically for crash and splash), taper strongly favours the nonlinearity by reducing the thickness close to the edge. Consequently, the cascade of energy and the excitation of high frequencies is very fast, so that the maximal frequency is obtained right after the strike, without an audible delay.

Numerical simulations and sound synthesis based on physical models face a number of difficulties while solving for the vibrations, the nonlinearity and the presence of the wave turbulence effect being the most important ones. However, successful results have been obtained in the last years, where a key component has been the derivation of energy-conserving schemes for von Kármán problems describing geometrically nonlinear vibrations of thin plates [START_REF] Bilbao | A family of conservative finite difference schemes for the dynamical von Kármán plate equations[END_REF][START_REF] Bilbao | Conservative numerical methods for the full von Kármán plate equations[END_REF] .

The temporal scheme has then been used together with finite difference methods [START_REF] Bilbao | Percussion synthesis based on models of nonlinear shell vibration[END_REF] , or with a modal approach 4 , the advantage of which is to offer an ability to easily implement losses with any frequency-dependent law, and to give a better control on the accuracy of the computation of the nonlinear terms. These two points are key to the realism of synthesized sounds. Other methods have been used where the kinematics of plates and shells is simplified by using ad-hoc bending models that are easier to implement for finite element approaches, hence opening the doors to create more complex geometries for a wide variety of shells [START_REF] Chadwick | Harmonic shells: a practical nonlinear sound model for near-rigid thin shells[END_REF] . This last work has been more deeply investigated recently [START_REF] Cirio | Multi-scale simulation of nonlinear thin-shell sound with wave turbulence[END_REF] , with an interesting improvement to efficiently simulate the energy cascade due to wave turbulence thanks to the phenomenological model proposed [START_REF] Humbert | Phenomenological model for predicting stationary and non-stationary spectra of wave turbulence in vibrating plates[END_REF] .

Even though the sound of gongs has been recovered with accuracy and realism by using the modal approach 4 , simulations of cymbal-like circular plates failed to reproduce the very fast cascade and the immediate emergence of the broadband Fourier spectrum, which gives to crash and splash cymbals their specific sparkling and explosive sound. A main reason is the absence of taper in earlier simulations, thus not covering the ability of cymbals to play two different sounds when hit near the centre or close to the edge. The goal of this article is thus to introduce a nonlinear model for plate vibrations including both taper (thickness variations) and curvature (also denoted as bow in cymbal description terminology, i.e. shape variations), and to develop a numerical method for solving the problem in time, based on earlier works [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF][START_REF] Camier | Non-linear vibrations of imperfect free-edge circular plates and shells[END_REF][START_REF] Touzé | Asymmetric non-linear forced vibrations of freeedge circular plates, part I: theory[END_REF][START_REF] Thomas | Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance[END_REF] . Once the model is established, numerical simulations are drawn in order to show the important effect of taper on the nonlinear vibrations, and to contrast both effects of shape and thickness variations, as well as to distinguish the bell sound from the crash sound obtained with strikes at the edge.

II. MODEL AND METHODS

In this section, the equations of motion for a thin plate with variable thickness are first recalled. A linear analysis is then performed for the two unknowns, namely the transverse displacement w and the Airy stress function F . The problem is subsequently discretized using a Rayleigh-Ritz approach and the time integration method is detailed. Finally a complete model including thickness together with shape variations, modeled as an imperfection of the plate at rest, is highlighted.

A. Von Kármán model for thin plates with variable thickness

In this paper a circular plate of radius R d , made of a homogeneous material of volume density ρ, Young's modulus E and Poisson's ratio ν, is considered. The surface of the plate is denoted as S = {(r, θ) ∈ [0, R d ] × [0, 2π]}. In order to simplify the presentation, the thickness of the plate h is assumed to depend only on the radius coordinate r. The von Kármán model for geometrically nonlinear vibrations of thin plates is used to compute the solutions for the transverse displacement w(r, θ, t) and the Airy stress function F (r, θ, t). It reads [START_REF] Nayfeh | Nonlinear oscillations[END_REF][START_REF] Thomas | Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties[END_REF][START_REF] Jones | Optimal control of plates using incompatible strains[END_REF][START_REF] Denis | Improvement of the acoustic black hole effect by using energy transfer due to geometric nonlinearity[END_REF]

ρh(r) ẅ + ∆(D(r)∆w) -(1 -ν) L(D(r), w) = p(r, θ, t) -R( ẇ) + L(w, F ), (1a)
∆(B(r)∆F ) -(1 + ν) L(B(r), F ) = - 1 2 L(w, w). (1b)
In these expressions, D(r) = Eh 3 (r)/12(1 -ν 2 ) is the flexural rigidity, B(r) = 1/Eh(r), p stands for the normal pressure loading and R( ẇ) is a generic expression accounting for the losses. ∆ is the laplacian operator, and L is the von Kármán bilinear operator, which reads, in polar coordinates for two arbitrary functions f (r, θ) and g(r, θ):

L(f, g) = f ,rr g ,r r + g ,θθ r 2 + g ,rr f ,r r + f ,θθ r 2 -2 f ,rθ r + f ,θ r 2 g ,rθ r + g ,θ r 2 • (2)
As compared to the equations of motion with constant thickness [START_REF] Touzé | Asymmetric non-linear forced vibrations of freeedge circular plates, part I: theory[END_REF] , the main incidence of the variable thickness onto the equations of motion is an added complexity to derive the linear terms. On the other hand, the nonlinear terms are not affected by the thickness variations, which is in line with the definition of the geometric nonlinearity. As applications to cymbals are targeted, a free-edge boundary condition is selected. It reads [START_REF] Touzé | Asymmetric non-linear forced vibrations of freeedge circular plates, part I: theory[END_REF] , ∀t, ∀θ ∈ [0, 2π], and for r = R d :

w ,rr + ν R d w ,r + ν R 2 d w ,θθ = 0 , (3a)
w ,rrr + 1 R d w ,rr - 1 R 2 d w ,r + 2 -ν R 2 d w ,rθθ - 3 -ν R 3 d w ,θθ = 0 , (3b)
F ,r + 1 R d F ,θθ = 0 , F ,rθ + 1 R d F ,θ = 0. (3c)

B. Linear analysis

The aim of this section is to analyse the linear part of Eqs. (1). As already underlined, the linear terms are deeply modified when taking the variable thickness into account, hence most of the added work focuses on calculating these new terms. In the course of this section, the results will be derived without referring to a particular case of boundary conditions, for the sake of generality. However when turning to numerical results, the free-edge boundary condition presented in the previous section, is used. The linear parts of Eqs. (1a) and (1b) correspond to two distinct linear problems in terms of the two unknowns w and F , respectively. Each of these two problems is tackled separately, by using the Rayleigh-Ritz method with the appropriate boundary conditions.

Transverse vibration

The linear solution for the transverse vibration is first derived. The problem at hand reads :

ρh(r) ẅ + ∆(D(r)∆w) -(1 -ν) L(D(r), w) = 0. ( 4
)
Let us assume that the eigenmodes of this problem with constant thickness (i.e. a perfect flat plate with h 0 a reference thickness, ) are known and denoted as Φ p (r, θ) for the eigenfunctions and ω p for the eigenfrequencies, meaning that we have :

∆∆Φ p (r, θ) = ρh 0 ω 2 p Φ p (r, θ). (5)
In the computations, h 0 is defined from the thickness variation profile as the value at the plate centre. The eigenfunctions Φ p (r, θ) verify the boundary conditions for the transverse displacement w at the edge, and they are assumed to be normalized such that S Φ p Φ q dS = δ pq , with δ pq the Kronecker symbol. Following the Rayleigh-Ritz approach, the problem ( 4)

is discretized in space with the expansion w(r, θ) = N Φ X p (t)Φ p (r, θ). Substituting this last expression in (4), multiplying by another basis function Φ j and integrating over the surface S, the problem can be rewritten as :

M Φ Ẍ + K Φ X = 0, (6)
with X = [X 1 , X 2 , ..., X N Φ ] T the vector of generalized coordinates and N Φ the number of transverse modes retained in the truncation. The complete expressions for the mass and stiffness matrices M Φ and K Φ are given in Appendix A. The advantage of using the Rayleigh-Ritz method in the case of a circular plate where the modes of the perfect plate Φ p (x) are analytic is that the resulting coefficients appearing in M Φ and K Φ , such as the finite-difference scheme where the convergence is more difficult to be achieved up to high frequencies [START_REF] Denis | Improvement of the acoustic black hole effect by using energy transfer due to geometric nonlinearity[END_REF] .

Airy stress function

The same methodology is applied to the second linear equation (1b), which reads:

∆(B(r)∆F ) -(1 + ν) L(B(r), F ) = 0. ( 7
)
This problem in case of constant thickness is associated with in-plane eigenmodes Ψ j which verify ∆∆Ψ j = ζ 4 j Ψ j , together with the associated boundary conditions. Expanding the unknown F (r, θ) = N Ψ Y p Ψ p (r, θ), the Rayleigh-Ritz approach applied to (7) leads to

K Ψ Y = 0, (8)
with

Y = [Y 1 , Y 2 , ..., Y N Ψ ]
T the vector of generalized coordinates and N Ψ the number of in-plane modes retained in the truncation. Note that no mass matrix is present here as the longitudinal inertia is neglected in the von Kármán model. The complete expressions of the entries of K Ψ are given in appendix B.

Comparison with a finite element computation

In order to validate eigenmode computation with the Rayleigh-Ritz method, a comparison is drawn with the results obtained from a commercial finite element (FE) software, namely ANSYS. The case study is that of a free edge circular plate, with radius R d = 0.2 m, and material parameters as E = 2.10 11 Pa, ν = 0.38 and ρ = 7860 kg/m 3 . Two thickness variations are selected and represented in Fig. 1(a-b). In the first case the thickness is constant and equal to 1 mm from the centre to r = 0.04 m, then it decreases linearly from that point to the edge, with a final value at edge h e which is a parameter ranging from h e = 1 mm (case of a plate with constant thickness) to h e = 0.4 mm. In the second case shown in 1(b), a parabolic dependence of the thickness on the radius is selected, with the same range of variation for the thickness at edge h e . In this numerical test, only the transverse eigenfrequencies corresponding to ( 6) can be easily computed as no direct output relative to the in-plane problem can be found. However since the coding of the two problems share numerous similarities, it has been found sufficient to test only the transverse problem. In the Rayleigh-Ritz method, the convergence is controlled by the number of modes N Φ used as a projection space to build the matrices M Φ and K Φ . It has been found that N Φ = 150 was sufficient to obtain converged values for the first 31 eigenfrequencies shown in Fig. 1(c-d). .

Regarding the computations realised with ANSYS, the FE solution has been obtained by using the element SHELL181, a four-node element with six degrees of freedom at each node implementing a Reissner-Mindlin kinematics. These computations have reached a fine convergence by using up to 33387 nodes (33091 elements).

Fig. 1(c-d) shows the results obtained by varying h e . When h e = 1 mm, the thickness is uniform and one retrieves the eigenfrequencies of a perfect circular plate. Decreasing h e leads to the reduction of the local stiffness of the plate, so that the eigenfrequencies are also lessened, explaining the general trend observed on the curves. Finally one observes a very good match between the eigenfrequencies calculated with the two methods. Whereas the results are perfectly similar in the case 2 (parabolic variation of thickness), case 1

(linear variation with two different slopes) shows some slight discrepancies between the two methods. They are attributed to the modeling difference and in particular to the fact that transverse shear is taken into account in the FE model.

C. Nonlinear analysis and time integration

We now address the time integration of the full problem given by Eqs. (1), including the nonlinear terms and the losses. Following the Rayleigh-Ritz procedure started in the previous section to solve the linear part, i.e. using the expansions w(r, θ) = N Φ X p (t)Φ p (r, θ) and

F (r, θ) = N Ψ Y j (t)Ψ j (r, θ)
, one can rewrite Eqs. (1) as :

M Φ Ẍ + K Φ X + C Φ Ẋ = N Φ (X, Y ) + P f , (9a)
K Ψ Y = - 1 2 N Ψ (X, X). (9b)
Note that, as underlined before, the thickness variation modifies only the linear parts of the von Kármán equations of motion. Consequently, in the semi-discrete equations ( 9), the new terms appearing are : the force vector P f , of which the entry k reads :

P f k = S p(r, θ, t)Φ k (r, θ)dS, ∀ k ∈ [0, N Φ ]
; the damping matrix C Φ which will be detailed later, and the two quadratic nonlinear terms N Φ (X, Y ) and N Ψ (X, X). These two nonlinear terms are computed following earlier studies on plates with uniform thickness, see e.g. [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF][START_REF] Touzé | Asymmetric non-linear forced vibrations of freeedge circular plates, part I: theory[END_REF][START_REF] Thomas | Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties[END_REF] .

They are two vectors with the entry k reading as

N Φ (X, Y ) k = N Φ i=1 N Ψ j=1 E k ij X i Y j , (10a) N Ψ (X, X) k = N Φ p=1 N Φ q=1 H k pq X p X j , (10b)
where the introduced coefficients E k ij and H k pq write

E k ij = S Φ k L(Φ i , Ψ j )dS, (11a)
H k pq = S Ψ k L(Φ p , Φ q )dS. (11b)
Note that Eqs. ( 9) can be condensed by replacing (9b) into (9a), highlighting the fact that a cubic nonlinearity is at hand for the transverse displacement variable X. Also, the von Kármán operator L has some symmetry properties [START_REF] Thomas | Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties[END_REF] , such that for some particular boundary conditions, the relationship E l m,n = H n m,l is fulfilled [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF] . The free-edge boundary conditions considered in this paper belong to the category where the relationship holds, so that subsequent gain in pre-computing time can be saved thanks to this symmetry property.

The damping matrix C Φ has to be populated with ad-hoc values. As underlined in [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF] , one advantage of the present approach is that one can select modal damping factors at ease, following any frequency dependence. However, as the problem is semi-discretized by using the eigenmodes of the plate with constant thickness that are not the eigenmodes of the problem considered, the relationship between the modal damping matrix C = diag(µ k ) (with µ k = 2ξ k ω k the modal loss coefficient associated to the modal loss factor ξ k ) and C Φ has to be used. Even though the equations of motion will be integrated in time using Eqs. ( 9), the idea is to use the modal loss coefficients µ k as input parameters of the simulations, since the physical meaning of these values is more tractable. Then, using P the matrix of normalized eigenvectors computed from ( 6), the matrix C Φ is computed using the relationship C = P T C Φ P .

To integrate the problem in time, a Störmer-Verlet scheme is used. This is a symmetric and symplectic method of order two [START_REF] Hairer | Geometric numerical integration: structurepreserving algorithms for Ordinary differential equations[END_REF] . It is simply deduced from Eqs. ( 9) by replacing the second-order time derivative by the centred finite difference operator

δ tt ≡ 1 k 2 (e t+ -2 + e t-),
where k is the time step, and e t+ and e t-are respectively the forward and backward shift operators. The first-order time derivative for the damping term is replaced by the centred finite difference operator δ t• ≡ 1 2k (e t+ -e t-), and all the other terms are computed at the current time step. The method has a stability condition reading f S > πf N Φ , with f S = 1/k is the sampling rate and f N Φ the eigenfrequency of the last transverse mode retained in the truncation.

D. A complete model with shape imperfection

A complete model for the simulation of cymbal vibrations can be derived from the previous one by adding a static, geometric imperfection to the plate model, representing the position of the structure at rest. Let us denote w 0 (r, θ) this geometric imperfection. In order to comply with von Kármán assumptions [START_REF] Thomas | Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance[END_REF][START_REF] Thomas | Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties[END_REF] , w 0 should not be too large so as to obtain a shallow shell. The full model equations can be derived easily following for example [START_REF] Camier | Non-linear vibrations of imperfect free-edge circular plates and shells[END_REF][START_REF] Ostiguy | Effects of initial imperfections on dynamic behaviour of rectangular plates[END_REF] , and read

ρh(r) ẅ + ∆(D(r)∆w) -(1 -ν) L(D(r), w) = L(w, F ) + L(w 0 , F ) + p -R( ẇ), (12a)
∆(B(r)∆F ) -(1 + ν) L(B(r), F ) = - 1 2 L(w, w) + 2 L(w, w 0 ) (12b)
Note that taking this shape imperfection into account modifies only the nonlinear terms appearing in the right-hand side. The complete model can be semi-discretized in the same manner, following the Rayleigh-Ritz method and using the eigenmodes of the perfect plate without imperfection nor thickness variation as the expansion basis. The shape imperfection w 0 has also to be expanded as [START_REF] Camier | Non-linear vibrations of imperfect free-edge circular plates and shells[END_REF] :

w 0 (r, θ) = N k k=1 a k Φ k (r, θ) + z g , (13)
where z g is the centre of mass offset due to imperfection, and the coefficients a k are the semi-discrete representation of the shape. They read

a k = S (w 0 -z g )Φ k dS, (14)
z g = S w 0 dS A p , (15)
where A p is the area of the perfect plate. The two unknowns w and F are expanded using the same procedure as in the previous sections. This leads to a semi-discrete problem reading

M Φ Ẍ + K Φ X + C Φ Ẋ = N Φ (X, Y ) + N Φ (a, Y ) + P f , (16a)
K Ψ Y = - 1 2 [N Ψ (X, X) + 2N Ψ (a, X)] . (16b)
In these equations, the nonlinear terms N Φ and N Ψ have the same expressions as in Eqs. (10), the only difference being that they are also applied to the imperfection represented by the vector a = [a 1 , a 2 , ..., a N Φ ] T . Consequently the Y vector, linked to the Airy stress function, is a function with linear and quadratic dependence on X. Substituting (16b) into (16a), one observes now that the problem from the transverse motion arising from the right-hand side of (16a) shows a linear, quadratic and cubic dependence on X. The linear dependence can be solved alone so as to obtain the eigenmodeshapes and eigenfrequencies of the complete problem. All these computations follow closely those presented [START_REF] Camier | Non-linear vibrations of imperfect free-edge circular plates and shells[END_REF] , where only the shape imperfection was taken into account. The method was compared to finite element simulations to assess the results, and this step has been repeated here, showing a perfect agreement. Finally, to integrate the semi-discrete problem in time, the Störmer-Verlet scheme is used, as in Section II C.

III. NUMERICAL SIMULATIONS A. Effect of variable thickness

Numerical simulations are conducted to highlight the effect of the thickness variation on the nonlinear vibrations of circular plates, and in particular on the explosiveness of the cymbal sound at the very beginning of a strong hit. Indeed, cymbals are known to produce on a very short time scale a very rich spectrum with a broadband frequency content, which is typical of their bright shimmering sound. In order to quantify the effect of thickness variations, a circular plate of radius R d = 0.2 m has been selected, with material parameters typical for a metallic alloy : Young's modulus E = 2.10 11 Pa, Poisson's ratio ν = 0.38 and mass density ρ=7860 kg.m -3 . The thickness variation has been selected with a constant thickness h 0 = 1 mm, for r ∈ [0, 0.05] m, and then a linear decrease to a value h e at r = R d .

Two cases, namely h e = 0.7 mm and h e = 0.4 mm, are investigated and compared to the plate with uniform thickness h 0 . These two cases have been selected from the thickness measurements realized on a number of cymbals: while h e = 0.7 mm corresponds to a mean value found for crash cymbals, h e = 0.4 mm is taken as the minimal thickness that can be reasonably attained and which has been measured on a splash cymbal. The modal damping coefficients have been selected following the rule : µ k = 0.007ω 0.6 k + 2. The constant term entails a small constant amount of damping even for the very low frequency modes, while the power-law form has already been used in [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF] ; it is based on measured experimental values identified in very large thin plates [START_REF] Humbert | Wave turbulence in vibrating plates: The effect of damping[END_REF] and has been found to give realistic results for the sound synthesis of gongs.

To assess the explosiveness of the sound, a strike is given as an input force to the plate.

This strike is located at a given point x 0 = (r 0 , θ 0 ) so that p(r, θ, t) = δ(x -x 0 )g(t), and the temporal content is a raised cosine :

g(t) =            pm 2 [1 + cos (π(t -t 0 )/T wid )] if |t -t 0 | ≤ T wid ; 0 if |t -t 0 | > T wid . (17)
The input parameters of the force are thus the time t 0 , the temporal width of the interaction T wid and the amplitude of the force p m in N. All the computations have been implemented

using the software VK-gong 25 , an open source code developed to handle nonlinear vibrations of plates and built from previous works by the second author. The first eigenfrequency of the plate with constant thickness h 0 = 1 mm is equal to 32 Hz, while the 800 th transverse mode frequency is 18861 Hz. Consequently, all the calculations have been performed with N Φ = 800 so as to ensure an almost complete covering of the frequency band of human hearing. For the number of in-plane modes, N ϕ = 60 has been selected following the convergence guidelines given [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF] . Note that this number is given as the cardinal of the subset of in-plane modes having non-vanishing coupling terms. This is another advantage of the method used to compute the coupling coefficients, especially for the case of free-edge circular plates where the modes are analytic and vanishing coefficients can be predicted from analytical formulas resulting in significant computational time savings. The sampling rate f S has to be larger than 59 kHz in order to comply with the stability condition for the Störmer-Verlet scheme, i.e. f S > πf N Φ with N Φ = 800. For all the computations, the sampling rate has been selected as f S = 100 kHz, well over the stability limit.

A first set of simulations is realized with an amplitude of the strike as p m = 90 N located at the edge of the plate (r 0 = R d ), and a short interaction time selected as T wid = 1 ms, in order to mimick a strong hit given by a drumstick. Fig. 2 shows the results obtained for the three cases investigated : the uniform plate, and the two plates with variable thickness, respectively with h e = 0.7 mm and h e = 0.4 mm. The first row, Fig. 2(ac) shows the displacement at an arbitrary point located at r = 0.1792 m (close to the edge), and θ = 0.52 rad, while the second row presents the spectrograms of the velocity of the same output point. The effect of the thickness variation is huge and can be clearly assessed both on the temporal displacement signal and the velocity spectrogram. For the displacement, one can observe that for the plate with uniform thickness, the maximum value of the displacement is reached at the beginning and is equal to 2 mm, while for h e = 0.7 mm the maximum is 3 mm and finally 4 mm for the thinnest plate with h e = 0.4 mm.

The fast spectral enrichment can be more evidently assessed with the velocity spectrograms shown in Fig. 2(d-f). For the plate with uniform thickness, most of the energy is concentrated below 5000 Hz, while the rapid generation of high frequencies is much more pronounced for the two cases with thickness variations. In particular when h e = 0.4 mm, frequencies up to 15 kHz are created in the very first milliseconds of the vibration. This results in a more explosive sound which is typical for a cymbal, while the sound produced by the plate with uniform thickness has not the same properties. The readers are invited to hear at the associated sound files, corresponding to the velocity resampled at 44.1 kHz, which are available as supplementary materials. Note that all the sounds corresponding to the simulations presented in the figures are available as WAV files.

Fig. 3 compares the displacement field of the plates with uniform thickness 1 mm (left column) to that obtained with the variable thickness and h e = 0.4 mm, at different instants.

For these figures, the amplitude of the excitation force p m has been raised to 150 N in order to enhance the vibration amplitudes. The temporal width of the strike is kept as T wid = 1 ms, which means that, according to the raised cosine formula (17), the input force lasts In order to have more quantitative results on the high-frequency content generated by the vibration, the following characteristic frequency f c is introduced as

(c) (d) (e) (f) (g) (h) displacement [m] displacement [m] displacement [m] displacement [m] displacement [m] displacement [m] displacement [m] displacement [m]
f c = f S /2 f =0 a(f ) 2 f df f S /2 f =0 a(f ) 2 df , (18)
where a(f ) is the Fourier amplitude (evaluated at frequency f ) of the velocity signal obtained from the output point at r = 0.1792 m, and f S = 100 kHz is the sampling rate. This characteristic frequency has already been used for quantifying the frequency content of a turbulent cascade of energy 26 , as well as for the energy transfer observed in the contact dynamics of strings against frets [START_REF] Issanchou | String/frets contacts in the electric bass sound: Simulations and experiments[END_REF] . Fig. 4 shows the behaviour of this characteristic frequency as function of the time t of the computed velocity output, and for 9 simulations, with 3 different plates (uniform thickness, h e = 0.7 mm and h e = 0.4 mm) and 3 different striking amplitudes (p m = 20, 90 and 150 N); thus giving a quantitative point of view to the frequency behaviours displayed by the spectrograms shown in Fig. 2(d-f). In particular, one can clearly see the considerable effect of decreasing the thickness at edge from 1 to 0.4 mm, as the characteristic frequency is almost equal in the cases h e = 1 mm, p m = 150 N and h e = 0.4 mm, p m = 20 N. Concentrating on the case p m = 150 N, one can observe that the characteristic frequency is multiplied by a factor of 2 when h e is decreased from 1 mm to 0.7 mm, and once again a factor of 2.3 by decreasing from 0.7 to 0.4 mm.

B. A complete model for cymbal vibrations

In this section, a complete model including both shape and thickness variations, is considered. The shape of the imperfection has been measured on a real Zildjian "custom rock" crash cymbal having a radius R d = 0.2 m, and shown in Fig. 5. The height at the centre H c has been found equal to 3.4 cm, however it will be used as a free parameter in order to compare cymbals with different shallowness. In particular, simulations with H c = 1.7 cm (half of the real measurement, shown as a dashed line in Fig. 5), will be addressed. The main aim of the section is to compare the relative effects of shape and thickness variations on the sound produced by cymbals, and in particular the ability of producing two distinctive sounds when hit on the bell or at the edge.

A first comparison is drawn out in Fig. 6, which shows the velocity spectrograms of two different cases simulated. The first case, Fig. 6(a), is that of a plate having uniform thickness h = 1 mm, and only shape variation following the profile shown in Fig. 5 with a maximum height at centre H c = 3.4 cm. In the second case, the same shape variation is considered, and a thickness dependence following the guidelines used in the previous section, is taken into account: a linear thickness variation, as in Fig. 1 spectrograms shown in Fig. 6 are thus complementary to those shown in Fig. 2 and must be compared altogether. In particular, contrasting Fig. 6(a) with 2(d), i.e. the two cases with the uniform thickness h = 1 mm and with/without shape variations, one can observe that the energy transfer is slightly more pronounced in the case with shape variation. Indeed, the maximum frequency attained in the spectrogram of Fig. 6(a) is around 750-800 Hz whereas it is only 500 Hz in 2(d) for the flat plate. This phenonemenon is easily explained by the presence of quadratic nonlinearity in the case of shape imperfections, which is key to generating more easily energy transfers [START_REF] Touzé | Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates[END_REF] . This effect is aurally particularly striking (see companion wav files) : the enrichment brought by the shape and the quadratic nonlinearities is really noticeable, especially because it happens in the most sensitive frequency band for the ear. However, shape variation alone is not sufficient in order to create a rich spectrum as that obtained in Fig. 2(f). Adding the thickness variation allows retrieving the bright spectrum with frequencies up to 14 kHz, as shown in Fig. 6(b). Hence, variation of thickness seems to be really needed in order to build the explosive sound and excite very high frequencies beyond 10 kHz; however it is undoubted that shape variation is already important and adds a brightness to the sound produced by adding frequencies in the sensitive band of the ear.

For a more quantitative comparison, Fig. 7 shows the characteristic frequencies for a number of test cases, mixing both shape and thickness variations. Interestingly, it is found that when the thickness is uniform, h = 1 mm, and with the maximum height of the shape imperfection being half the measured one, H c = 1.7 cm; a better energy build-up is obtained than with H c = 3.4 cm. This can be explained that this case offers a better compromise between the appearance of quadratic nonlinearity and the stiffening effect produced by the curvature. Increasing H c results in a more and more stiff structure with increasing eigenfrequencies, making the couplings between modes and energy transfer more difficult. Anyhow Fig. 7 highlights that adding a shape imperfection always favours the energy buildup, but the thickness variation is also needed in order to attain very high values for this characteristic frequency. Indeed, all the curves with h e = 0.4 mm are always the upper one for the characteristic frequency f c .

Another aspect not investigated yet with the thickness decrease is related to the pitch glide. Indeed, decreasing the thickness at the edge results in geometric nonlinearities more easily excited since they are proportional to the ratio between the transverse displacement and the thickness. The pitch glide is related to the backbone curve of the modes of the structure, and is more pronounced if the transverse displacement is large as compared to [START_REF] Bilbao | Percussion synthesis based on models of nonlinear shell vibration[END_REF][29][START_REF] Touzé | Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes[END_REF][START_REF] Jossic | Effects of internal resonances in the pitch glide of chinese gongs[END_REF] . problem one may encounter if the plate is too thin, is that the pitch glide will be too important, resulting in a sound which is not pleasant to the ear and will more resemble that of a membrane. This effect can be easily simulated with the methods used in this paper, showing the need to stiffen the structure and explaining why very thin flat plates are not used as cymbals. This is illustrated in Fig. 8 nonlinear effect resulting in rapid energy build-up, and minimizing the pitch glide which shall not be too pronounced and unpleasant to the ear.

h 3,
The last simulation investigates the behaviour of the cymbal when hit at centre. Indeed, as stated before, a bell sound has to be obtained when striking the cymbal near the centre. and h e = 0.4 mm. It shows that even with moderate strikes the nonlinear regime is too rapidly excited, and the broadband spectrum is at hand, even though the cymbal is hit at centre. This shows undoubtedly that this configuration is not and cannot be used for a cymbal since the two distinctive sounds (bell and crash) are out of reach. Fig. 9(c-d)

shows the results when only shape variation is considered, with H c = 3.4 cm, and a constant thickness h = 1 mm. In this case the spectrograms are almost unchanged and the sounds are completely equivalent to the ear. This is always the case if the strike is increased up to p m = 150 N, meaning that linear vibrations are at hand up to very large amplitudes of input forces. This case appears to be too stiff, preventing the drummer some colours to the sound found when striking the bell. Finally, Fig. 9(e-f) shows the results obtained with both shape and thickness variations, where H c = 3.4 cm and h e = 0.4 mm. This case seems to show a correct compromise, the sound produced when striking at centre being almost linear with a small excitation of nonlinearity for p m =90 N, giving a nice colour to the sound.

IV. CONCLUSION

This paper investigates the nonlinear vibrations of cymbals with a particular emphasis on the presence of the thickness variations in a physical model. While the model of flat plates with uniform thickness derived has been shown to be effective for simulating the sound of gongs [START_REF] Ducceschi | Modal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals[END_REF] , the explosive sound of cymbal was more difficult to obtain. Indeed, thickness and shape variations play important roles in cymbal making, particularly in order to obtain two different behaviours when the excitation is at the centre or at the edge. When a cymbal is hit at its centre, a bell-like sound is awaited so that nonlinearities should not be too much excited. On the contrary, a scintillating sound with a rapid build-up of energy and generation of a large broadband Fourier spectrum in the first miliseconds is expected from vigorous strikes at the edge. One of the main outcomes of the present study is to derive such a model, based on the von Kármán assumptions for thin plates, and accounting for both effects of thickness and shape variations. An ad-hoc numerical method is adopted, from earlier studies, to handle the complexity brought by these new terms for a time-domain simulation, using the Störmer-Verlet scheme for time integration and the Rayleigh-Ritz method, which allows one to give frequency-dependent modal damping factors as inputs. The substantial effect of thickness variations has been underlined and a set of simulations has been conducted in order to get a comprehensive view of the two different effects, resulting from shape and thickness variations. Simulations and synthesized sounds show that shape variations are very important the energy build-up, a result that was not expected but could easily be understood thanks to the quadratic nonlinearity. However, imposing only shape variations results in too stiff cymbals, so that only linear behaviour is observed when hit at centre, and the energy build-up is less important than that obtained with thickness variations. On the other hand, thickness variations alone make the plate too compliant in general, causing too high pitch glides that are not pleasant to the ear. Both of the physical features are thus needed, but the fact that shape variations already have an important effect, which is especially prominent in the frequency band where the ear is the most sensitive, shows that the thickness variations may be at a moderate level, a conclusion that is confirmed by the values of the edge thickness found in most of the cymbals.

1 JASA/

 1 1, b IMSIA, ENSTA ParisTech-CNRS-EDF-CEA, Université Paris Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France Plates with variable thickness, sound synthesis of cymbals Geometrically nonlinear vibrations of thin plates and shells with variable thickness are investigated numerically with the purpose of synthesizing the sound of cymbals.

FIG. 1 .

 1 FIG. 1. Comparison of eigenfrequency computation for two different cases of thickness variation. (a) thickness constant near centre then linearly decreasing. (b) parabolic thickness. (c-d) eigenfrequency variations for h e ∈ [0.4, 1] mm, corresponding respectively to cases (a) and (b). Red circles: Rayleigh-Ritz method, Blue triangles: Finite-element model.

FIG. 2 .

 2 FIG. 2. Ouput displacements (first row, a-c) and velocity spectrograms (second row, d-f) for the simulations of the nonlinear vibrations of a thin circular plate with a uniform thickness h=1 mm (a,d) ; with a linearly varying thickness down to h e = 0.7 mm (b,e), and to h e = 0.4 mm (c,f). Displacement of an arbitrary point located at r = 0.1792 in m, and spectrogram of the velocity derived from the same displacement signal, in dB with a 90 dB dynamic.

  from t = 0 to 3 ms. The first row shows the displacement field just after the beginning of the strike, at t =0.5 ms. One can see that the difference in the local stiffness between the two plates (local thickness 1 mm vs 0.4 mm), gives rise to very distinct magnitudes of displacements at the striking point (0.5 mm for the uniform thickness vs 2.2 mm). The second row shows the displacement fields at t = 3.3 ms, i.e. right after the end of the input t=0

FIG. 3 .FIG. 4 .

 34 FIG. 3. Snapshots of the displacement fields. Comparison between the plate with uniform thickness h 0 = 1 mm (left column, a,c,e,g) and the plate with thickness at edge h e = 0.4 mm (right column, b,d,f,h), and at four different instants: t = 0.5 ms (a,b), t = 3.3 ms (c,d), t = 13 ms (e,f) and t =

FIG. 5 .

 5 FIG.5. Profile of the shape imperfection used in the simulations. The shape has been measured from a real cymbal, with a height at centre H c =3.4 cm. An intermediate case with H c =1.7 cm (dashed line), is also be considered.

FIG. 6 .

 6 FIG. 6. Spectrogram of velocity with a 90 dB range. Strike at t = 0 s, with amplitude p m =90 N, and contact time T wid = 1 ms. (a) cymbal with uniform thickness h = 1 mm, shape variation only with height at centre H c = 3.4 cm. (b) cymbal with the same shape variation, H c = 3.4 cm, and linear decrease of the thickness, h e = 0.4 mm.

FIG. 7 .

 7 FIG.7. Characteristic frequency f c as a function of the time t for twelve different cases reported in inset, mixing shape and thickness variations, amplitudes of strike at the edge 90 and 150 N.

FIG. 8 .

 8 FIG. 8. Velocity spectrograms in the low frequency range for a strike at the edge of magnitude p m = 150 N, and for three different cymbal configurations. Restrained range of 50 dB in order to highlight the frequency peaks. (a) only linear thickness variation down to h e = 0.4 mm, flat plate without shape imperfection. (b) shape variation only with height at centre H c = 3.4 cm, and uniform thickness h=1 mm. (c) shape and thickness variation with H c =3.4 cm and h e = 0.4 mm.

FIG. 9 .

 9 FIG. 9. Velocity spectrograms, cymbal hit at r=4 cm from the centre. First row : soft strike, p m =20 N, second row, hard strike, p m =90 N. First column (a-b) : variable thickness only, h e = 0.4 mm. Second column, (c-d) : Shape variation only, uniform thickness h=1mm and height at centre H c =3.4 cm. Third column (e-f) : H c =3.4 cm and linear thickness variation down to h e = 0.4 mm.

Fig. 9

 9 Fig.9shows the velocity spectrograms for three different cymbal configurations and two

APPENDIX A: LINEAR ANALYSIS FOR THE TRANSVERSE PROBLEM

In this section the complete formula for computing the mass and stiffness matrices M Φ and K Φ , related to the linear problem for the transverse vibrations, are given. The mass matrix comes from the inertia term in (4). A simple calculation shows that the entries M Φij read:

In order to compute easily the current term in the stiffness matrix, the following identity is used:

Consequently the stiffness matrix can be decomposed as

, where the entries of each term read:

These equations can be used for an arbitrary boundary conditions once the eigenmodes Φ i and eigenfrequencies ω i are known. In the case of a free edge, the eigenmodes write Φ kn (r, θ) = R kn (r) cos kθ (first configuration) and Φ kn (r, θ) = R kn (r) sin kθ (second configuration), where the index k refers to the number of nodal diameters and n to the number of nodal circles. When k = 0 the modes are axisymmetric and no dependence on θ is found.

When k = 0, the eigenfrequency is degenerated with a multiplicity of 2, and the two configurations, one in sine and the other one in cosine, are present. R kn (r) is a combination of the previous equations, one can separate the radial part from the angular part, and all the expressions are analytic. The integrals are then computed numerically from the analytic expressions, leading to a perfect control of the accuracy of the results.

APPENDIX B: LINEAR ANALYSIS FOR THE IN-PLANE PROBLEM

Using the same identity (A2) to develop the terms arising from (7), one can write

, where the elements of each matrix read: