FR3117637A1 - Method for reducing the volume of training data intended for training a stress field classifier, and associated electronic devices - Google Patents

Abstract

Method for reducing the volume of training data intended for training a stress field classifier, and associated electronic devices Such a method: in which stress fields, for example temperature, of high resolution, exerted on a turbine blade element, give values of said stress at different nodes of a high resolution mesh of said blade element, a restricted number of nodes of said mesh are selected to form a group of relevant nodes, basic stress fields by restricting the high resolution stress fields to said group of relevant nodes, the relevant nodes are each selected on the basis of mutual redundancy information (Ij,j ') between the node (j) considered and any of the other nodes (j') of said mesh, said mutual redundancy information being determined from a spatial distance (dj,j') between the two x nodes considered. Figure to be published with abstract: Figure 6

Description

Method for reducing the volume of training data intended for training a stress field classifier, and associated electronic devices

TECHNICAL FIELD OF THE INVENTION

In general, the invention relates to the field of digital simulation of the mechanical behavior of mechanical parts, for example of their deformation and ageing. It concerns in particular the application of artificial intelligence techniques during a preliminary phase of simplification of the digital simulation to be carried out. The invention relates in particular to the determination of learning data intended for the training of a stress field classifier, for example a temperature field, this classifier making it possible to identify, in a pre-established list, a model of mechanical behavior simplified well adapted to the stress field considered.

TECHNOLOGICAL BACKGROUND OF THE INVENTION

Numerical simulation by finite elements, of the deformations undergone by a mechanical part subjected to various constraints, has made significant progress in recent years. Such simulations make it possible in particular to predict the way in which a particular mechanical part will age, for a long period of use (for example several years), which could not have been the subject of tests or direct measurements. More generally, these simulations make the design and prototyping phase of a mechanical system faster and more efficient.

But some situations remain difficult to simulate, because they require a very large calculation time.

This is particularly the case for the characterization of the aging of a turbine engine turbine blade, for example a high-pressure turbine located just downstream of the combustion chamber. In this case, a high temperature, and generally inhomogeneous, is exerted on the blades of the turbine, in addition to the pressure field and the centrifugal force undergone by these blades.

For a given temperature field, exerted on this blade, simulating the deformation and aging of such or such element of the blade, for several cycles corresponding to several flights of an aircraft, requires very significant computing resources.

Moreover, the temperature field exerted on such a turbine blade is in fact not perfectly known. To be able to precisely determine a duration during which a blade will remain usable, and an uncertainty associated with this duration, it is therefore necessary to repeat the numerical simulation (already very demanding in itself) several times, for a whole (statistical) distribution of light fields. temperature.

With a conventional finite element numerical simulation, the calculation time or the calculation resources for such a characterization (number of processors or calculation clusters to be made to work in parallel) then becomes prohibitive.

To reduce the calculation time necessary to carry out such simulations, an approach based on a clustering of different possible inputs (solicitations), and a classification to recommend a reduced model (for example a reduced model of mechanical behavior) has been proposed in the article “Localized Discrete Empirical Interpolation Method” by B. Peherstorfer et al., SIAM J. Sci. Computer. (2014). More recently, an approach based on such an order reduction of the mechanical problem and on a temperature field classifier has been proposed in the following paper: “Model order reduction assisted by deep neural networks (ROM-net)” by T Daniel et al., Adv.Model. and Sim. in Eng. Science. (2020) 7:16.

According to this approach, instead of determining the mechanical deformation at each node of a mesh of the blade element to be characterized, with a number of degrees of freedom (scalars) which would therefore be equal to at least three times the number of mesh nodes, the mechanical response of the blade element is studied on the basis of a simplified mechanical behavior model, M _k , parameterized by a smaller number of degrees of freedom.

Such a mechanical behavior model M _k describes for example the deformation of the turbine blade element considered in the form of a weighted sum of deformation modes, typical of this element (for example a pure bending mode, a mode corresponding to uniform stretching, another mode corresponding to localized stretching at one or other of the ends of the turbine blade element in question, etc.). The degrees of freedom, or in other words the parameters of this model of mechanical behavior then correspond, in this weighted sum, to the coefficients which weight the contributions of these different modes of deformation.

The mechanical response of the turbine blade element, subjected to a given temperature field T, is then determined over time for several flight cycles, by digital simulation, on the basis of this simplified model M _k . Thanks to the simplified nature of this model, it is possible to carry out a simulation corresponding to a long period of use, while keeping a limited calculation time.

Such a model of mechanical behavior M _k is sometimes called a “scale model” in this technical field. It is representative of the mechanical response of the turbine blade element (in particular of the deformation of this element) when this element is subjected to a given type of stress (for example at a given temperature field T). It is therefore in a way a typical response of the turbine blade element. Moreover, in what follows, such a model of mechanical behavior M _k is designated either by the expression “scale model” or by the expression “standard response”.

According to the approach proposed in the article by T. Daniel et al. mentioned above, before carrying out the numerical simulation in question, on the basis of the reduced model Mk, this reduced model Mk is chosen, as a function of the temperature field T exerted on the turbine blade element 1'. The reduced model Mk is selected, by a classifier of temperature fields 2', from among different reduced models M1, …Mk, …, MP, that is to say from among different models of mechanical behavior of the blade element turbine ( ). The reduced model Mk which is selected by the classifier 2' is, among the reduced models M1,...Mk,...,MP, the one which is the most representative of a complete mechanical response expected for the turbine blade element 1 ', when subjected to the temperature field T in question.

The classifier 2′, which for example comprises a neural network, is trained by artificial intelligence techniques during a training step 20 ( ). This training is carried out on the basis of learning data (TH,i, Mk), determined beforehand during a step 10 of determining learning data.

This step 10 comprises a first step 11 of digital simulation on a stress cycle of the turbine blade element ( ), corresponding for example to the stress to which this element is subjected during the entire flight of an aircraft powered by the turbine engine to which this blade element belongs. During step 11, the mechanical response of the turbine blade element is determined, over time, for this stress cycle, the blade element being subjected to a given temperature field TH, i.

Such a numerical simulation is not necessarily sufficient to determine the long-term evolution of the mechanical characteristics of the blade element (which would require simulation over numerous successive cycles), nor its lifetime. But it already makes it possible to identify the type of mechanical behavior of this element, when it is subjected to the field of temperature T _H,i in question.

This complete, but short-time, numerical simulation is carried out for different so-called “high resolution” temperature fields, T _H,i , i=1,…,N. These high resolution temperature fields each give temperature values at the different nodes j, j=1,…,Q _H of a mesh (so-called “high resolution” mesh) of the blade element.

This numerical simulation makes it possible to determine “complete” mechanical responses of the blade element, for these different temperature fields T _H,i .

Then, during a step 12, for each pair (T _{H,i 1} , T _{H,i 2} ) combining two of these high resolution temperature fields, a dissimilarity of mechanical behavior δ _i1,i2 is determined, or, in other words, an effective “distance” between the mechanical responses obtained for these two temperature fields T _{H,i 1} , T _{H,i 2} . This distance δ _{i 1,i 2} is for example representative of a difference between the two deformation fields obtained respectively for these temperature fields. Different metrics can be considered to determine this deviation: it can be determined as being equal to the square root of the sum of the squares of the deviations at each node of the mesh, or as the sum of the absolute values of the deviations at each node of the mesh, or even as a distance in the sense of Grassmann, as done in the aforementioned article by T. Daniel et al.

Then, during step 13, the various high-resolution mechanical responses determined in step 11 are grouped together in the form of several groups of mechanical responses C ₁ , …, C _k , …, C _P , which each bring together the mechanical responses which are similar to each other, that is to say which present, two by two, a weak dissimilarity δ _{i 1,i 2} (“distance” between two elements of the same group). This step is generally called “clustering” in English (that is to say “data partitioning”), and the groups of mechanical responses C ₁ , C ₂ , …, C _P are generally called “clusters”.

Then, in step 14, for each cluster C _k , k=1…P, a reduced model M _k representative of the complete mechanical responses of the cluster considered is determined (responses which are relatively similar to each other, by construction), using an order reduction technique.

• l’un des champs de température de haute résolution T_H,i, i=1,…,N,
• ainsi qu’une étiquette (c’est-à-dire un label) qui repère, parmi des réponses-types de l’élément d’aube M_k, k=1…P, la réponse-type M_kcorrespondant au champ de température T_H,iconsidéré (cette réponse-type étant la réponse-type associée au cluster C_kdans lequel se trouve la réponse mécanique complète, de haute résolution, obtenue en réponse au champ de température T_H,ien question).

The training data finally obtained comprise various “labeled” data, each labeled data (T _H,i , M _k ) comprising:

• one of the high resolution temperature fields T _H,i , i=1,…,N,
• as well as a label (that is to say a label) which locates, among the standard responses of the blade element M _k , k=1…P, the standard response M _k corresponding to the field of temperature T _H,i considered (this standard response being the standard response associated with the cluster C _k in which is found the complete mechanical response, of high resolution, obtained in response to the temperature field T _H,i in question).

The label in question corresponds for example to the index k which identifies the standard response M _k , in the set of standard responses of the blade element (this label is for example constituted by the index in question ).

This set of labeled data (T _H,i , M _k ), i=1,…,N, is then used for training, that is to say for learning the classifier 2'.

It is important to note that the temperature fields T _H,i are grouped into different clusters C _k , each associated with a given standard response, not on the basis of a similarity between temperature fields, but on the basis of a similarity between the mechanical responses corresponding to these different temperature fields.

This distinction is particularly important, because two apparently close temperature fields, similar to each other, can in fact lead to very different mechanical responses from each other, the interactions between thermal and mechanical loading being complex. and depending on the geometry of the turbine blade element.

Thus, two identical temperature fields except on a very small part of the blade element can in fact lead to very different mechanical responses from each other, if the zone in question is critical from a point of mechanical view (for example because it is thin, or very stressed mechanically, or subject to concentrations of mechanical stresses – “stress concentration” in English). Conversely, two temperature fields which differ from each other over a large but not very critical zone, and which on the other hand take the same values at the level of the critical zone in question, will probably lead to similar mechanical responses whereas these two temperature fields are apparently dissimilar to each other.

The training data (T _H,i , M _k ), constructed on the basis of a similarity between responses , therefore allow training of the classifier which is much more precise and reliable than with training data which would be based on a similarity between requests .

But obtaining the learning data (T _H,i , M _k ) requires on the other hand a significant calculation time, since a numerical simulation (certainly in short time, but on a complete mesh) is carried out for each field of high resolution temperature T _H,i . In practice, the number N of labeled data finally obtained is therefore limited.

For a mesh comprising approximately 40,000 nodes, obtaining only 500 labeled data already requires a very significant computation time. And if one wants to obtain approximately 10 ⁴ labeled data, even with a reduced mesh comprising only a few thousand nodes, more than one day of calculation is required by parallelizing the calculation on about fifty processors.

In this context, it would therefore be desirable to be able to obtain more reliable training data, without increasing the computation time too much.

Different techniques for augmenting learning data exist, in particular in the field of automatic image analysis. A known technique is based on the application of various transformations and deformations to a given image. For example, an image representing a car is stretched, flipped, blurred, cropped, etc. to obtain additional images, which always represent a car.

But this type of technique is not applicable to the case considered here. Indeed, deforming one of the temperature fields T _H,i would probably lead to a new temperature field giving a completely different mechanical response from that obtained with the initial temperature field T _H,i (in other words, l label would not be preserved, during such a transformation).

On the other hand, a difficulty is linked to the size of each labeled datum, in itself. In fact, to build the clusters C _k and the associated standard responses M _k in a suitable manner, while the mechanical behavior of the blade element is initially unknown, the simulations mentioned above must be carried out (“in time” simulations). short”) with a fine mesh. Indeed, we do not have any information in advance concerning the more or less critical character of such and such a zone, from a thermomechanical point of view. The temperature field T _H,i is therefore a high resolution temperature field, and the labeled data (T _H,i , M _k ) obtained in this way is therefore voluminous and cumbersome to process afterwards.

• dans laquelle des données d’apprentissage de haute résolution rassemblent des champs de contrainte de haute résolution exercés sur un élément d’aube de turbine et, pour chacun de ces champs, une étiquette associée au champ de contrainte considéré, chaque champ de contrainte de haute résolution donnant des valeurs de ladite contrainte en différents nœuds d’un maillage de haute résolution dudit élément d’aube,
• la méthode comprenant une étape de sélection d’un nombre restreint de nœuds du maillage de haute résolution, les nœuds sélectionnés formant un groupe de nœuds pertinents,
• les données d’apprentissage de base rassemblant des champs de contrainte de base qui sont obtenus en restreignant les champs de contrainte de haute résolution audit groupe de nœuds pertinents,
• et dans laquelle une partie au moins desdits nœuds pertinents sont sélectionnés chacun sur la base d’une valeur approchée d’une information mutuelle de redondance entre le nœud considéré et l’un quelconque des autres nœuds dudit maillage, ladite valeur approchée étant déterminée sur base d’une distance spatiale entre les deux nœuds considérés.

In this context, we propose a method to determine basic training data for training a stress field classifier,

• wherein high resolution training data gathers high resolution stress fields exerted on a turbine blade element and, for each of these fields, a label associated with the considered stress field, each high resolution stress field resolution giving values of said stress at different nodes of a high resolution mesh of said blade element,
• the method comprising a step of selecting a restricted number of nodes of the high-resolution mesh, the selected nodes forming a group of relevant nodes,
• the basic training data gathering basic stress fields which are obtained by restricting the high resolution stress fields to said group of relevant nodes,
• and in which at least part of said relevant nodes are each selected on the basis of an approximate value of mutual redundancy information between the node in question and any one of the other nodes of said mesh, said approximate value being determined on the basis of a spatial distance between the two nodes considered.

The stress fields in question can for example be temperature fields, or pressure fields exerted on the turbine blade element. For the simplicity of the explanations, the technical effects obtained thanks to this method are presented below in the case of temperature fields. But, as will appear to those skilled in the art, this method can also be applied to other types of stresses exerted on the turbine blade element, such as a pressure exerted on this element, for example.

Considering such mutual redundancy information when selecting the relevant nodes allows to remove a significant part of the nodes while losing only a minimal part of the information initially contained in the high resolution temperature fields.

But the evaluation of such mutual redundancy information is very heavy in terms of computation time, because of the number of values to be determined, equal to the number of pairs of nodes, for the high resolution mesh, number of pairs which varies as the square of the number of mesh nodes. Calculating each of these values of mutual redundancy information, by a direct statistical calculation, would require a very large calculation time.

Determining the different values of mutual redundancy information simply from the distance between the two nodes considered, in the form of a function of the distance between points, is much faster. In addition, this leads to an estimate that is admittedly approximate, but nevertheless relatively precise of the mutual redundancy information. Indeed, in practice, the mutual redundancy information turns out to be well correlated with the distance between points (as illustrated by the , For example).

The fact that the mutual redundancy information is well correlated with the distance between points can be explained as follows. The temperature fields, or more generally the stress fields, used for numerical simulations of the mechanical behavior of turbine blades are generally at least partially random fields, with Gaussian statistics. The temperatures at two nodes j and j', as given by this set of fields, can then be seen as Gaussian random vectors. And we can mathematically show that the mutual redundancy information between two such vectors X ¹ and X ² is expressed as: where ρ is the correlation between X ¹ and X ² .

Moreover, these random temperature fields usually have isotropic correlation functions. The correlation ρ then depends only on the distance between the considered nodes, which ultimately explains why the mutual redundancy information can be expressed, at least in an approximate way, as a function of the distance alone.

The link between mutual redundancy information, and geometric distance between points, is therefore a consequence of the type of temperature fields used (Gaussian, and with an isotropic correlation function). Note, however, that this causal relationship is in fact quite indirect. Indeed, the temperature fields are generally obtained by imposing surface temperature fields of known statistics on the surface of the blade element, then taking into account the propagation of heat in the turbine blade element to get the temperature at any point. The fact that the volumetric temperature fields thus obtained finally have a statistic leading to a good correlation between mutual redundancy information, and geometric distance between points, is therefore far from direct at first sight.

The method that has just been presented is implemented by a computer or any computer system capable of executing logical operations.

• ladite contrainte exercée sur l’élément d’aube de turbine est une température exercée sur cet élément, les champs de contrainte de haute résolution et les champs de contrainte de base étant respectivement des champs de température de haute résolution et des champs de température de base ;
• l’information mutuelle de redondance entre les deux nœuds considérés est représentative d’un niveau de redondance d’information entre :
  • les valeurs de ladite contrainte au premier des deux nœuds, telles que données par l’ensemble des champs de contrainte de haute résolution, et
  • les valeurs de ladite contrainte au deuxième des deux nœuds, telles que données par l’ensemble des champs de contrainte de haute résolution ;
• la méthode comprend une étape de calibration qui comporte les étapes suivantes :
  • pour chaque couple de nœuds d’un sous-ensemble regroupant une partie des nœuds du maillage de haute résolution :
    • détermination d’une valeur de référence de l’information mutuelle de redondance entre les deux nœuds considérés, par calcul statistique sur la base des valeurs de ladite contrainte, au niveau de ces deux nœuds, telles que données par les différents champs de contrainte de haute résolution, et
    • calcul de la distance spatiale séparant les deux nœuds considérés,
  • ajustement d’une courbe, de manière à ce qu’elle soit représentative d’un lien entre lesdites valeurs de référence de l’information mutuelle de redondance, et les distances spatiales correspondantes ;
• la valeur approchée de l’information mutuelle de redondance entre deux nœuds du réseau est la valeur donnée par ladite courbe, pour la distance spatiale séparant les deux nœuds considérés ;
• pour chaque champ de contrainte de haute résolution, l’étiquette associée au champ de contrainte de haute résolution considéré repère une réponse-type, représentative d’une réponse mécanique attendue pour ledit élément d’aube en réponse audit champ de contrainte de haute résolution, parmi différentes réponses-types dudit élément d’aube de turbine ;
• chaque nœud pertinent est sélectionné en outre sur la base d’un degré de pertinence de la valeur de ladite contrainte au nœud considéré, vis-à-vis du comportement mécanique dudit élément d’aube, ledit degré de pertinence étant représentatif d’un niveau de dépendance entre, d’une part, l’étiquette associée au champ de contrainte de haute résolution considéré, et, d’autre part, la valeur de ladite contrainte au nœud considéré pour ce champ de contrainte ;
• le groupe de nœuds pertinents contient initialement un premier nœud, qui est le nœud du maillage de haute résolution ayant le plus grand degré de pertinence, et l’on exécute ensuite une étape d’ajout, au cours de laquelle on ajoute un nœud supplémentaire au groupe de nœuds pertinents, le nœud supplémentaire étant sélectionné, parmi les nœuds dudit maillage qui ne font pas encore partie dudit groupe, de manière à maximiser une quantité égale : au degré de pertinence du nœud considéré, moins un niveau de redondance moyen entre ce nœud et les nœuds déjà présents dans le groupe de nœuds pertinents, l’étape d’ajout étant ensuite exécutée à nouveau plusieurs fois ;
• le niveau de redondance moyen, entre le nœud considéré et les nœuds déjà présents dans groupe de nœuds pertinents, est égal à la moyenne arithmétique des valeurs approchées de l’information mutuelle de redondance entre ledit nœud et les nœuds déjà présents dans groupe de nœuds pertinents, le nœud supplémentaire, ajouté au groupe de nœuds pertinents, étant le nœud qui maximise la quantité suivante : où i est l’indice repérant le nœud considéré, dans le maillage de haute résolution, et où est le nombre de nœuds contenus dans le groupe de nœuds pertinents lors de l’itération courante de l’étape d’ajout ;
• la méthode comprend en outre une étape d’augmentation de données au cours de laquelle des données d’apprentissage supplémentaires sont déterminées à partir desdites données d’apprentissage de base, chaque champs de contrainte de base étant assorti d’une étiquette du champs de contrainte de haute résolution correspondant, l’étape d’augmentation de données comprenant les étapes suivantes :
  • former au moins un sous-ensemble pur, qui regroupe des champs de contrainte de base associés à une même réponse-type de l’élément d’aube de turbine, et qui a une enveloppe convexe ne contenant aucun champ de contrainte de base qui soit associé à une autre réponse-type dudit élément,
  • pour l’un au moins desdits sous-ensembles purs, détermination d’une ou plusieurs données d’apprentissage supplémentaires, chaque donnée d’apprentissage supplémentaire comprenant un champ de contrainte supplémentaire, obtenu par combinaison linéaire de champs de contraintes de base du sous-ensemble pur considéré, affectés de coefficients de pondérations positifs et normalisés à 1, le champ de contrainte supplémentaire étant assorti d’une étiquette qui est la même que pour les champs de contraintes de base de ladite combinaison ;
• l’étape de formation d’un ou plusieurs sous-ensembles purs comprend, pour chacun de ces sous-ensembles, une étape de test de pureté du sous-ensemble considéré, au cours de laquelle on vérifie qu’aucun champ de contrainte de base associé à une réponse-type différente de la réponse-type commune aux champs de contrainte de base du sous-ensemble considéré, n’est situé à l’intérieur de l’enveloppe convexe dudit sous-ensemble ;
• chaque champ de contrainte base regroupe un ensemble de valeurs de ladite contrainte en différents nœuds d’un maillage donné comprenant un nombre total de nœuds Q, et pour chaque champ de contrainte de base, un sous-champ de contrainte associé est obtenu par projection du champ de contrainte de base sur un sous-espace comportant un nombre de degrés le liberté inférieur à Q, et au cours de l’étape de test de pureté :
  • on exécute un test par projection, au cours duquel on détermine si, pour lessous-champs de contrainte associés aux champs de contrainte de base du sous-ensemble à tester, une enveloppe convexe de l’ensemble regroupant cessous-champs de contrainte ne contient aucunsous-champ de contrainte associé à l’un des champs de contrainte de base pour lesquels la réponse-type est différente de réponse-type commune aux champs de contrainte de base du sous-ensemble à tester,
  • et, lorsque le résultat de ce test par projection est positif, on détermine que le sous-ensemble de champs de contrainte considéré est pur.

In addition to the characteristics mentioned above, the method which has just been presented may have one or more of the following additional characteristics, considered individually or in all technically possible combinations:

• said stress exerted on the turbine blade element is a temperature exerted on this element, the high resolution stress fields and the basic stress fields being respectively high resolution temperature fields and basic temperature fields ;
• the mutual redundancy information between the two nodes considered is representative of an information redundancy level between:
  • the values of said stress at the first of the two nodes, as given by the set of high resolution stress fields, and
  • the values of said stress at the second of the two nodes, as given by the set of high resolution stress fields;
• the method includes a calibration step which includes the following steps:
  • for each pair of nodes of a subset grouping part of the nodes of the high resolution mesh:
    • determination of a reference value of the mutual redundancy information between the two nodes under consideration, by statistical calculation on the basis of the values of said constraint, at the level of these two nodes, as given by the various high constraint fields resolution, and
    • calculation of the spatial distance separating the two nodes considered,
  • adjustment of a curve, so that it is representative of a link between said reference values of the mutual redundancy information, and the corresponding spatial distances;
• the approximate value of the mutual redundancy information between two nodes of the network is the value given by said curve, for the spatial distance separating the two nodes considered;
• for each high-resolution stress field, the label associated with the considered high-resolution stress field identifies a standard response, representative of an expected mechanical response for said blade element in response to said high-resolution stress field, among different typical responses of said turbine blade element;
• each relevant node is further selected on the basis of a degree of relevance of the value of said stress at the node considered, with respect to the mechanical behavior of said blade element, said degree of relevance being representative of a level of dependence between, on the one hand, the label associated with the high-resolution stress field considered, and, on the other hand, the value of said stress at the node considered for this stress field;
• the group of relevant nodes initially contains a first node, which is the node of the high resolution mesh having the greatest degree of relevance, and then an addition step is executed, during which an additional node is added to the group of relevant nodes, the additional node being selected, from among the nodes of said mesh which are not yet part of said group, so as to maximize a quantity equal to: the degree of relevance of the node considered, minus an average level of redundancy between this node and the nodes already present in the group of relevant nodes, the adding step then being executed again several times;
• the average level of redundancy, between the node considered and the nodes already present in the group of relevant nodes, is equal to the arithmetic mean of the approximate values of the mutual redundancy information between said node and the nodes already present in the group of relevant nodes , the additional node, added to the group of relevant nodes, being the node which maximizes the following quantity: where i is the index identifying the node considered, in the high resolution mesh, and where is the number of nodes contained in the group of relevant nodes during the current iteration of the addition step;
• the method further comprises a data augmentation step in which additional training data is determined from said basic training data, each basic stress field being provided with a label of the stress field corresponding high resolution, the step of increasing data comprising the following steps:
  • form at least one pure subassembly, which groups together basic stress fields associated with the same typical response of the turbine blade element, and which has a convex hull containing no basic stress field which is associated with another standard response of said element,
  • for at least one of said pure subsets, determination of one or more additional training data, each additional training data comprising an additional stress field, obtained by linear combination of basic stress fields of the sub- pure set considered, assigned positive weighting coefficients and normalized to 1, the additional stress field being matched with a label which is the same as for the basic stress fields of said combination;
• the step of forming one or more pure subsets comprises, for each of these subsets, a step of testing the purity of the subset considered, during which it is checked that no basic stress field associated with a response-type different from the response-type common to the basic stress fields of the considered subset, is located inside the convex hull of said subset;
• each base stress field groups together a set of values of said stress at different nodes of a given mesh comprising a total number of nodes Q, and for each base stress field, an associated stress sub-field is obtained by projection of the basic stress field on a subspace with a number of degrees of freedom less than Q, and during the purity test step:
  • a projection test is carried out, during which it is determined whether, for the stress sub -fields associated with the basic stress fields of the subset to be tested, a convex hull of the set grouping these stress sub -fields does not contains no stress sub- field associated with one of the basic stress fields for which the standard response is different from the standard response common to the basic stress fields of the subset to be tested,
  • and, when the result of this test by projection is positive, it is determined that the subset of stress fields considered is pure.

• une phase préalable d’entrainement comportant :
  • une détermination de données d’apprentissage de base, ou une détermination de données d’apprentissage de base et de données d’apprentissage supplémentaires conformément à la revendication, expliqué ci-dessus,
  • l’entrainement d’un classificateur de champs de contrainte sur la base desdites données d’apprentissage de base, ou sur la base desdites données d’apprentissage de base et desdites données d’apprentissage supplémentaires, et
• une phase d’utilisation comprenant les étapes suivantes :
  • identification de l’une des réponses-types dudit élément d’aube par le classificateur de champs de contrainte recevant en entrée ledit champ de contrainte,
  • et simulation numérique de la réponse mécanique dudit élément d’aube, soumis audit champ de contrainte, conformément au modèle de comportement mécanique simplifié défini par la réponse-type précédemment identifiée par le classificateur.

The invention also relates to a method for numerical simulation of the mechanical response of a turbine blade element subjected to a given stress field T _R comprising:

• a preliminary training phase comprising:
  • a determination of basic training data, or a determination of basic training data and additional training data according to the claim, explained above,
  • training a stress field classifier based on said basic training data, or based on said basic training data and said additional training data, and
• a use phase comprising the following steps:
  • identification of one of the standard responses of said blade element by the stress field classifier receiving said stress field as input,
  • and digital simulation of the mechanical response of said blade element, subjected to said stress field, in accordance with the simplified mechanical behavior model defined by the standard response previously identified by the classifier.

The invention also relates to an electronic device comprising at least one processor and one memory, programmed to determine basic learning data, or to determine basic learning data and additional learning data in accordance with the method described upper.

The invention also relates to an electronic device comprising at least one processor and one memory, programmed to execute the digital simulation method which has just been presented.

The invention and its various applications will be better understood on reading the following description and examining the accompanying figures.

BRIEF DESCRIPTION OF FIGURES

The figures are presented for information only and in no way limit the invention.

There schematically represents an operation of identification, by a temperature field classifier, of a standard response adapted to a given temperature field, exerted on a turbine blade element.

There schematically represents a step for determining training data and a step for training a classifier such as that of the .

There schematically represents steps of the training data determination step of the .

There schematically represents a method for determining learning and training data of a classifier such as that of the .

There schematically represents, in perspective, an example mechanical part for which the process of the is implemented.

There schematically represents values of mutual redundancy information between nodes of the mesh of the .

There schematically represents the steps making it possible to select relevant nodes, among the nodes of a complete mesh such as that of the .

There schematically represents the part of the , in perspective, with the set of relevant nodes selected thanks to the steps of the .

There schematically represents steps in a data augmentation process.

There is a schematic representation, in a reduced two-dimensional space, of two clusters of temperature fields, and of a pure subset identified in one of these clusters.

There is a schematic representation, in a reduced two-dimensional space, of different temperature fields each reduced to a point, these temperature fields belonging to different clusters.

There schematically represents the steps implemented to form a pure subset, such as that of the .

There schematically represents the steps implemented to test the purity of such a subset.

DETAILED DESCRIPTION

The embodiment described below corresponds to a case in which the stress exerted on the turbine blade element considered is a temperature exerted on this element (at different points of this element). In other embodiments, the constraint in question could be of another type. It could, for example, be a pressure exerted on this blade element by a fluid flowing in the turbine.

• une étape S0 de sélection de variables, et
• une étape A0 s’augmentation de données.

There represents a method for determining learning data, and for training a temperature field classifier, comparable to that described in the part relating to the technological background (with reference to FIGS. 1 to 3) but integrating in addition :

• a variable selection step S0, and
• a step A0 increases data.

Step 10 of the is the same as that described previously, with reference to FIGS. 2 and 3. It makes it possible to determine a first set of learning data (TH,i,Mk) bringing together the high resolution temperature fields TH,i mentioned above, and, associated with each of these temperature fields, the label which identifies the corresponding response-type Mk (that is to say the response-type Mk corresponding to the cluster Ck to which this temperature field belongs).

The variable selection step S0 is a step during which, for each high resolution temperature field T _H,i , a corresponding base temperature field T _i is determined, the base temperature field T _i giving temperature values at different nodes of a simplified mesh which includes fewer nodes than the high resolution mesh. This simplified mesh is constituted by a group of relevant nodes S , selected among the various nodes of the high resolution mesh. This simplified mesh can for example comprise a number Q of nodes 100 times, or even 400 times smaller than the number of nodes Q _H of the high resolution mesh.

The simplified mesh is determined here so that the basic temperature fields T _i , clearly simplified compared to the high resolution temperature fields T _H,i , are nevertheless the most relevant possible from the point of view of the thermomechanical response of element 1'; 1 of turbine blade whose behavior we want to simulate.

Step S0 thus makes it possible to obtain a set of basic learning data (T _i , M _k ), which gather together the basic temperature fields T _i in question each matched with their label, this label being the same as for the high resolution temperature field T _H,I which has been “compressed” in the form of the basic temperature field T _i considered.

Then, during step A0 of increasing data, additional training data (T′ _i , M _k ) are determined from the basic training data (T _i , M _k ). The additional training data (T' _i , M _k ) gather additional temperature fields T' _i , each with a label that identifies one of the standard responses M ₁ ,…M _k ,…,M _P of the turbine blade element.

Then, in step 20, the temperature field classifier is trained with the basic training data (T _i , M _k ) and the additional training data (T′ _i , M _k ) obtained previously.

In the following, variable selection step S0 is described in more detail, with reference to Figures 5 to 8. Data increase step A0 is then presented, with reference to Figures 9 to 13.

Step S0: selection of variables

There represents, in perspective, the mechanical part, 1, which serves here as an example to illustrate the implementation of this method for determining learning data. This figure shows the high resolution mesh, at QH nodes, mentioned above, as well as the positions ξ1, ξ2, and ξj of three nodes of this mesh. In this example, the number QH of nodes of the high resolution mesh is equal to 42445.

Inset (c) of the shows, in the form of thick dots, the positions of the different nodes of the group of relevant nodes S obtained for this same example, thanks to the variable selection technique in question. In this example, the number Q of nodes group of relevant nodes S is equal to 87.

• sur la base d’une information mutuelle deredondanceI_{j,j ’} entre nœuds, et
• sur la base d’un degré depertinence de chaque nœud, vis-à-vis de l’attribution de telle ou telle réponse-type Mk à un champ de température donné.

Here the relevant nodes are selected:

• on the basis of mutual information fromredundancyI_{not a word '} between nodes, and
• based on a degree ofrelevance of each node, with respect to the attribution of such and such a response-type Mk to a given temperature field.

The mutual redundancy information I _{j,j '} is representative of an information redundancy level between the temperature value at a first node j of the high-resolution mesh, and the temperature value at a second node j' of this mesh, for the distribution of temperature fields corresponding to the set of high resolution temperature fields.

The relevant nodes are selected so as to have a high degree of relevance, while being little redundant with the nodes already selected, already belonging to the group of relevant nodes S .

• parmi les nœuds du maillage de haute résolution qui ne font pas encore partie dudit groupe, et
• de manière à maximiser une quantité égale au degré de pertinence du nœud i considéré, moins un niveau de redondance moyen entre ce nœud et les nœuds déjà présents dans groupe de nœuds pertinentsS.

For that, one forms the group of relevant nodes S as follows. We first integrate into the group of relevant nodes S a first node, which is the node j of the high resolution mesh having the greatest degree of relevance _. Then, a step of adding a node is then executed several times in succession. During this step, we add to the group of relevant nodes S an additional node which is selected:

• among the nodes of the high-resolution mesh that are not yet part of said group, and
• so as to maximize a quantity equal to the degree of relevance of the node i considered, minus an average level of redundancy between this node and the nodes already present in the group of relevant nodes S .

More precisely, the additional node, added to the group of relevant nodes S during the addition step, is the node i which maximizes the following quantity: where i is the index identifying the node considered, in the high resolution mesh, and where is the number of nodes already contained in the group of relevant nodes S during the current iteration of the addition step.

Here, the iterations of the node addition step cease when the above quantity no longer evolves noticeably, during a few successive iterations. For example, if this quantity has not varied by more than 5% (in relative value) during the last 5 iterations, we stop repeating this addition step.

As a variant, it would also be possible to stop repeating this addition step when the quantity in question no longer notably decreases from one iteration of the step to the next. Provision could also be made to stop the iterations when the quantity in question becomes less than a given threshold, or when a previously fixed number of iterations is reached. We could still plan to stop the iterations when the relevance information of the node to be added becomes less than a given threshold. This criterion is interesting because the quantity to be maximized at each iteration (quantity above) can take on a high value for a node with low relevance information (and therefore not very relevant) but also having a low redundancy with the nodes already selected.

As already indicated, the mutual redundancy information I_{not a word '}is representative of the level of information redundancy between the temperature value T^Iat a first node j of the high resolution mesh, and the temperature value T_iinto a second node j' of this mesh. More precisely, this mutual redundancy information, which translates a greater or lesser interdependence between the temperatures T^Iand T^{I '}, is defined by the following relationship: Or :
- the domain of integration isR ²,
- is the probability density, for the distribution of temperature fields given by the set of high resolution temperature fields T_{H, 1}, … , T_H,N, to have a temperature t^Iat node j,
- is the probability density, for this same distribution, of having a temperature t^{I '}at node i, and where
- is the (joint) probability density, for this same distribution, of having a temperature t^Iat node j and a temperature t^{I '}at node i.

• d’une part, le numéro de l’étiquette (i.e . : l’indice k repérant la réponse-type appropriée), associée à un champ de température de haute résolution donné, et,
• d’autre part, la valeur de température au nœud j, donné par ce champ de température.

As for the degree of relevance of the node j, vis-à-vis the attribution of such and such a response-type M_kat a given temperature field, it is representative of a level of dependence between:

• on the one hand, the number of the label (ie .: the index k identifying the appropriate standard response), associated with a given high resolution temperature field, and,
• on the other hand, the temperature value at node j, given by this temperature field.

The degree of relevance is a particular form of mutual redundancy information, in the case where a first variable is the temperature T^Iat node j, while the second variable, Y, has a discrete value and represents the label number associated with the temperature field: with Or :
- the domain of integration isR,
- is the probability density, for the distribution of temperature fields given by the set of high resolution temperature fields T_{H, 1}, … , T_H,N, to have a temperature t^Iat node j,
- is the probability for the standard response to be M_k
- is the probability density for the temperature field distribution given by the set of high resolution temperature fields T_{H, 1}, … , T_H,N, to have a temperature t^Iat node j knowing that the typical response is M_k, and or
- is the probability density for the temperature field distribution given by the set of high resolution temperature fields T_{H, 1}, … , T_H,N, to have a temperature t^Iat node j and for the response-type to be Mk, simultaneously.

The degree of relevance , and the mutual redundancy information I _{j,j '} can be determined, on the basis of the above expressions, after a prior estimation of the various probability functions involved in these expressions.

These probability functions can for example be estimated, for the (discrete) distribution of temperature fields and labels constituted by the set of high resolution temperature fields T_{H, 1}, … , T_H,N, accompanied by their labels, by “binning” techniques (i.e. by sampling techniques).

By way of example, the following article describes such a “binning” technique, which can be used to calculate such mutual redundancy information from discrete data: “Estimating mutual information”, A. Kraskov, H. Stögbauer, and P. Grassberger, Phys. Rev. E 69, 066138 – June 23, 2004; Erratum: Phys. Rev. E 83, 019903 (2011).

Be that as it may, a determination of the mutual redundancy information I _{j,j '} between two nodes j and j', by such a statistical calculation based on the temperature values T ^j _H,1 , …, T ^j _{H ,N} , and T ^j' _H,1 , …, T ^j' _H,N at the level of these two nodes, is consuming in terms of computation time.

And the number of terms Ij,j' to be calculated is very large since it is the number of pairs grouping together any two nodes j,j' of the high-resolution mesh. In the numerical example corresponding to the , the number of pairs is of the order of 109, for example.

Also, to reduce the calculation time necessary for the evaluation of the mutual redundancy information I _{j,j '} between nodes, rather than determining it by direct statistical calculation (which would be very long), a value is determined here approached of this quantity from the spatial distance, d _{j,j '} , between the two nodes j and j' considered.

In other words, it is considered here that the mutual redundancy information I _{j,j '} can be expressed, to within a good approximation, in the form of a function of the distance d _{j,j '} in question: . This distance d _{j,j '} is equal to the geometric distance separating the two nodes j, j' in question. This distance is quick to calculate, from a numerical point of view. Evaluating the mutual redundancy information I _{j,j '} in this way makes it possible to form the group of relevant nodes S much faster than by evaluating all the terms I _{j,j '} by direct statistical calculation.

The one-variable function , which is used to calculate the approximate values from the distances dj,j', is determined beforehand, during a calibration step S11 (FIG. 7), by determining the mutual redundancy information Ij,j' in an "exact" manner, i.e. say by direct statistical calculation, for some nodes of the high resolution mesh, in order to calibrate the curve, or, in other words, the function .

• S12 : pour chaque couple de nœuds (j, j’) d’un sous-ensemble regroupant une partie des nœuds du maillage de haute résolution :
  • détermination d’une valeur de référence (valeur « exacte ») de l’information mutuelle de redondance I_{j,j ’}entre les deux nœuds j et j’ considérés, par calcul statistique sur la base des valeurs de température T^j _{H, 1}, …, T^j _{H, N}, T^j’ _{H, 1}, …, T^{j ’} _{H, N}, au niveau de ces deux nœuds, telles que données par les différents champs de température de haute résolution T_H,1, … , T_H,N, et
  • calcul de la distance d_{j,j ’}spatiale séparant les deux nœuds j et j’ considérés, puis
• S13 : ajustement de la courbe de manière à ce qu’elle soit représentative d’un lien entre les valeurs de référence ainsi obtenues, et les distances d_{j,j ’}spatiales correspondantes.

The calibration step S11 comprises the following steps:

• S12: for each pair of nodes (j, j') of a subset grouping part of the nodes of the high resolution mesh:
  • determination of a reference value (“exact” value) of the mutual redundancy information I_{not a word '}between the two nodes j and j' considered, by statistical calculation on the basis of the temperature values T^I _{H, 1}, …, T^I _{H, NOT}, Tⁱ _{H, 1}, …, T^{I '} _{H, NOT}, at these two nodes, as given by the different high resolution temperature fields T_H,1, … , T_H,N, And
  • calculation of the spatial distance d _{j,j '} separating the two nodes j and j' considered, then
• S13: curve fitting so that it is representative of a link between the reference values thus obtained, and the corresponding spatial distances d _{j,j '} .

The subset of nodes, for which the mutual redundancy information is calculated “exactly”, can comprise a number of nodes comprised for example between one thousandth and one twentieth of the total number Q _H of nodes of the high-resolution mesh .

The curve is a curve parametrized by one or more parameters which are adjusted (for example by a method of the least squares type) so that this curve is as representative as possible of the reference data ( , d _{j,j '} ).

Figure 6 shows reference values mutual redundancy information Ij,j' between nodes, obtained for part 1 of figure 5. In this figure, each point represents a pair (dj,j', ). This figure also shows the curve obtained by adjusting these reference data.

In this example, about 750 reference values have been determined, to calibrate the curve . In this case, the curve represents the following one-variable function: where H is the Heaviside step function, and where are the parameters adjusted in step S13.

As can be seen in this example, the mutual redundancy information I _{j,j '} is well correlated with the distance d _{j,j '} , and can effectively be expressed, to a good approximation, in the form of a function from this distance.

This property, exploited here in a particularly interesting way, can be explained as follows. The temperature fields used for numerical simulations of the mechanical behavior of turbine blades are generally at least partially random fields, with Gaussian statistics (this is notably the case for the high-resolution temperature fields used here). The temperatures at two nodes j and j', as given by this set of fields, can then be seen as Gaussian random vectors. And we can mathematically show that the mutual redundancy information between two such vectors X ¹ and X ² is expressed as: where ρ is the correlation between X ¹ and X ² .

Moreover, these random temperature fields usually have isotropic correlation functions. The correlation ρ then depends only on the distance d _{jj '} between the considered nodes, which finally explains why the mutual redundancy information I _{j,j '} can be expressed as a function of the distance d _{jj '} only.

Be that as it may, at the end of the calibration step S11, the curve is completely determined, to be used then for the selection of the relevant nodes.

The degree of relevance is calculated for its part during a step S14, for all the nodes of the high-resolution mesh. It is determined by direct statistical calculation. The computation time necessary to carry out this step nevertheless remains reasonable, because, contrary to the case of mutual information, it is a question of calculating a value per node (and therefore Q_Hdifferent values), and not one value per pair of nodes (i.e. Q_H.(Q_H - 1)/2 different values).

Then, during a step S15, the group of relevant nodes S is determined, as explained above, by selecting the relevant nodes on the basis of the degree of relevance and the mutual redundancy information. During the progressive growth of the group of relevant nodes S , the additional node added to the nodes already present in this group is the node i which maximizes the following quantity:

As already indicated, insert (c) of the shows the different nodes of the relevant node group S obtained for the high resolution mesh example of the . In this example, the node addition iterations were stopped once the above quantity stopped varying noticeably from one iteration to the next. 87 relevant nodes were thus selected (which shows in passing that a significant compression of the initial high-resolution mesh was possible). 303 seconds were needed to identify this group of 87 relevant nodes S (ie: to perform 86 successive executions of the addition step). By way of comparison, when the mutual redundancy information Ij,j' is evaluated by direct statistical calculation, it takes, on the same computer, 6469 seconds to execute only the first 6 iterations of this selection algorithm (and thus select 7 relevant nodes). This illustrates the very significant acceleration allowed by the approximate mutual information calculation technique presented here.

The insert (a) of the shows a cluster S' of 87 nodes, selected from the high-resolution mesh using another selection technique. In this case, the nodes of this group were selected on the basis of a purely geometric criterion, so as to obtain, for the selected nodes, a substantially homogeneous density of nodes over the whole of the part (for example, so that the volume of an area specific to each selected node, centered on this node and containing no other relevant node, varies little from one selected node to another). This selection is made without taking into account the mechanical responses of the part.

Inset (b) of the shows a group S'' of 87 nodes, selected from the high-resolution mesh using yet another selection technique, based on a univariate filter. As can be seen in this figure, the nodes selected in this way are concentrated mainly at the level of a central constriction that part 1 presents, and on the surface of this part (zone a priori quite critical from a thermomechanical point of view ). On the other hand, we note that almost no node is selected elsewhere than in this zone, and that many nodes are located (probably unnecessarily) in the same place.

The selection technique presented above, based on a degree of relevance corrected for redundancy, leads to a distribution of nodes (inset (c)), which, compared to the homogeneous distribution of inset (a), is denser at the central constriction, while covering the whole of this part (unlike the selection by univariate filter).

• (a) : répartition purement géométrique ; D=0,009 et R=0,1072 ;
• (b) : sélection par filtre univarié ; D=0,0671 et R=0,8129 ;
• (c) : maximisation de la pertinence tout en minimisant la redondance ; D=0,046 et R=0,1111.

The global relevance degree D of the group of relevant nodes S , S ' and S '', and its global redundancy level R are given below, for these three selection techniques:

• (a): purely geometric distribution; D=0.009 and R=0.1072;
• (b): selection by univariate filter; D=0.0671 and R=0.8129;
• (c): maximizing relevance while minimizing redundancy; D=0.046 and R=0.1111.

A purely geometric distribution therefore leads to low redundancy R, but also to low relevance D.

Univariate filter selection leads to high relevance D, but high redundancy R.

As for selection by maximizing relevance and minimizing redundancy, it leads to both high relevance D and low redundancy R.

The global degree of relevance D of a group of relevant nodes given above, is defined by the following formula:

The overall redundancy level R of a group of relevant nodes is defined by the following formula:

For the training data determination method described here, the selection technique used during the variable selection step S0 is technique (c), of maximizing relevance while minimizing redundancy.

As a variant, another selection technique, for example by univariate filter or by purely geometric distribution, could however be used during step S0.

Be that as it may, at the end of the variable selection step, we have a set of basic temperature fields T₁ ,…,T_I,…,T_NOT, together with their labels, less voluminous than the high resolution temperature fields T_H,1,…,T_H,i,…,T_H,Ninitials.

During the next step A0, additional learning data is determined, by data augmentation, from these basic learning data.

And ape A0 : data increase

• Etape A1 : former différents sous-ensembles pur S_l, et
• Etape A2 : au sein de chaque sous-ensemble pur, déterminer des données d’apprentissages supplémentaires par combinaisons linéaires convexes.

The data augmentation step A0 includes the following steps ( ):

• Step A1: form different pure subsets S _l , and
• Step A2: within each pure subset, determine additional learning data by convex linear combinations.

• à regrouper des champs de température de base T_iqui sont associés à une même réponse-type M_kde l’élément d’aube de turbine (i.e. : qui appartiennent à un même cluster C_k),
• et de manière à avoir une enveloppe convexe E_lqui ne contient aucun champ de température de base T_iqui soit associé à une autre réponse-type M_{k ’}, k’≠k, de l’élément d’aube de turbine (c’est-à-dire aucun champ de température de base T_iétrangers, appartenant à un autre cluster que le cluster C_k).

At step A1, each pure subset S _l is formed in such a way:

• to group basic temperature fields T _i which are associated with the same standard response M _k of the turbine blade element (ie: which belong to the same cluster C _k ),
• and so as to have a convex hull E _l which does not contain any base temperature field T _i which is associated with another typical response M _{k '} , k'≠k, of the turbine blade element (c ie no extraneous basic temperature field T _i , belonging to a cluster other than the cluster C _k ).

• l’un des champs de température supplémentaires T’_ien question, qui est obtenu par combinaison linéaire des champs de températures de base T_idu sous-ensemble pur S_lconsidéré, affectés de coefficients de pondérations positifs et normalisés à 1, et
• une étiquette (constituée par exemple par l’indice k repérant la réponse-type M_k), qui est la même que pour les champs de températures de base T_iqui ont servi à obtenir ce champs de température supplémentaire T’_i.

Then, during step A2, for each subset S ₁ , various additional training data are determined, each additional training data comprising:

• one of the additional temperature fields T' _i in question, which is obtained by linear combination of the basic temperature fields T _i of the pure subset S _l considered, assigned positive weighting coefficients and normalized to 1, and
• a label (consisting for example of the index k identifying the standard response M _k ), which is the same as for the basic temperature fields T _i which were used to obtain this additional temperature field T′ _i .

• Etape A10 : parmi les champs de température de base T_iappartenant à un même cluster C_k, sélection de « graines » SD_l, l=1…M, pour la croissance ultérieure des sous-ensembles purs S_l,
• Etape A11 : pour chaque graines SD_l, formation d’un sous-ensemble S_l, contenant initialement ladite graine, et que l’on fait croitre jusqu’à ce que sa pureté soit perdue.

Step A1, of formation of the pure subsets S _l , l=1..M, is now described in more detail. This step includes the following steps:

• Step A10: among the basic temperature fields T _i belonging to the same cluster C _k , selection of “seeds” SD _l , l=1…M, for the subsequent growth of the pure subsets S _l ,
• Step A11: for each seed SD ₁ , formation of a subset S ₁ , initially containing said seed, and which is grown until its purity is lost.

The seed selection step A10 is presented first, and the growth step A11, which includes a specific purity test, is described next.

Step A10: selection of seeds

Each seed SD _l is constituted by one of the basic temperature fields Ti belonging to the cluster C _k considered, judiciously selected.

Here, the seeds SD _l are chosen, within the considered cluster C _k , so as to be somewhat distant from the boundaries of the cluster. More precisely, they are chosen so as to be at least s far from the basic temperature fields belonging to the other clusters C _{k '} , k' ≠k. This distance is a distance in the sense of the distance δ _ij described above (in the part relating to the general technological background), that is to say a distance which translates a dissimilarity of the mechanical responses, obtained respectively for the two basic temperature fields T _i , T _j considered (dissimilarity of mechanical behavior).

Thus, in each cluster C _k , the seeds SD _l are selected so as to present, with respect to the different basic temperature fields T _{i '} of the other clusters C _{k '} , k'≠k, a dissimilarity δ _{i ,i '} of mechanical response greater than a given threshold δ ₁ .

In each cluster, it is interesting to thus choose seeds far from the basic temperature fields of the other clusters C _{k '} , because they are favorable to the subsequent growth of a pure subset S _l . Indeed, thanks to this distance, this subset will be able to grow a minimum in size before a basic temperature field, belonging to another cluster C _{k ',} is found in its convex envelope E _l .

More generally, this distance limits the risk that the convex hull El of the subset Sl does not include, in the space of temperature fields, part of another Cluster Ck', that this part contains a temperature field basic or not. In other words, for the example of the , choosing in the cluster C1 seeds, which are far from the crosses representing the basic temperature fields of the cluster C2, makes it possible to avoid that a subset, resulting from one of these seeds, includes a part of the domain corresponding to the cluster C2, domain which is represented in white in the figure, whether this part contains a cross or not.

Furthermore, in each cluster C _k , the seeds are chosen so as not to be too isolated from the other basic temperature fields of this same cluster, so that the seed is conducive to subsequent growth of a subset. More precisely, each seed is chosen so that at least one other base temperature field of the same cluster C _k is located at a “distance” δ _ij from the seed which is less than a given threshold δ ₂ (otherwise said, which presents, with the seed considered, a dissimilarity δ _ij of mechanical response lower than the threshold δ ₂ in question).

• tous les champs de température de base T_i, ayant un voisin Ti’ qui appartient à un autre cluster et qui, par rapport à T_i, est situé à une distance δ_{i,i ’}inférieure au seuil δ₁, et
• tous les champs de température de base T_iqui n’ont aucun voisin T_i’’appartenant au même cluster et situé à une distance δ_{i,i ’ ’}de T_iinférieure au seuil δ_2.

To obtain seeds which are not too isolated with respect to the other basic temperature fields of their cluster C _k , and which do not have close neighbors belonging to a foreign cluster C _{k '} , one starts by selecting, under the form of a list F _k , all the basic temperature fields which satisfy these two criteria, in the cluster C _k considered. This list is obtained by discarding (by removing), from the complete list gathering all the basic temperature fields of the cluster C _k considered:

• all the basic temperature fields T _i , having a neighbor Ti' which belongs to another cluster and which, with respect to T _i , is located at a distance δ _{i,i '} lower than the threshold δ ₁ , and
• all the basic temperature fields T _i which have no neighbor T _i'' belonging to the same cluster and located at a distance δ _{i,i ' '} from T _i lower than the threshold δ _2.

The thresholds δ ₁ and δ ₂ are determined here as being equal respectively to ε ₁ .δ _ref and ε ₂ .δ _ref where δ _ref is a distance representative of a dimension of the cluster C _k considered (the values of the thresholds used can therefore vary, from one cluster to another). The coefficients ε ₁ and ε ₂ are less than 1 (and positive). For example, ε ₁ can be between 0.01 and 0.05, and ε ₂ can be between 0.1 and 0.3 (this means that we also select seeds that are relatively "isolated" This is useful, because data augmentation also aims to generate data in areas that lack it, i.e. which are poorly represented by the initial data).

• un champ de température de base du cluster C_k, considéré comme étant le représentant de ce cluster, et
• le champ de température de base, étranger au cluster C_k, et qui est le plus proche du représentant de C_k.

δ _ref is for example determined as being the distance between:

• a basic temperature field of the cluster C _k , considered to be the representative of this cluster, and
• the basic temperature field, foreign to the cluster C _k , and which is closest to the representative of C _k .

The representative of the cluster C _k is for example the basic temperature field whose mechanical response is closest to an average of the mechanical responses obtained for the various basic temperature fields of this cluster.

Once the list F _k of temperature fields has been obtained which, in the cluster C _k , are neither too isolated nor too close to the boundaries, a given number of seeds is selected from this list. This number is for example set beforehand by a user.

The seeds can be selected, in the list F _k , so as to be evenly distributed in the cluster.

Here the seeds can be selected as follows, from the list F _k . The first seed SD ₁ selected is constituted by the basic temperature field which is the representative of the cluster C _k considered. The other seeds are selected, one after the other, so that each new seed selected is constituted by the basic temperature field which, in the list F _k , is the farthest from the seed which is the closer to him. The second seed SD ₂ , for example, is constituted by the basic temperature field of the list F _k which is furthest from the first seed SD ₁ . And the third seed SD ₃ is constituted by the basic temperature field of the list F _k which is furthest either from the first seed SD ₁ , or from the second seed SD ₂ , depending on whether the seed closest to SD ₃ is SD ₁ or SD ₂ .

Thus, if L _k denotes the current list of selected seeds, the next seed added to this list is the basic temperature field T _j whose index j is given by the following formula: where δ is the distance between temperature fields mentioned above (also called dissimilarity of mechanical response).

Selecting the seeds in this way makes it possible to cover the whole of the cluster C _k in a homogeneous way, and therefore, then, to create pure subsets which are also distributed in a homogeneous way in the cluster. This is useful because pure subsets distributed in an inhomogeneous way would lead, after creation of additional training data within these subsets, to an over-representation of certain areas of the cluster, which is not optimal for learning the classifier.

Step A1 1 : growth of pure subsets.

Once the seeds SD _l , l=1..M have been selected, for each seed SD _l , a corresponding subset S _l is grown from the seed in question.

• A110 : ajout de l’un champ de température de base T_iau sous-ensemble S_l, le champ de température ajouté ayant la même réponse-type M_kque les autres champs de température du sous-ensemble S_l(autrement dit, le champ de température ajouté appartient au même cluster C_kque les autres éléments de S_l), puis
• A111 : test de pureté, pour le sous-ensemble S_lainsi augmenté.

Initially, the subset S _l contains only this seed SD _l . It is then grown by performing the following steps ( ):

• A110: addition of a basic temperature field T _i to the subset S _l , the added temperature field having the same standard response M _k as the other temperature fields of the subset S _l (in other words, the added temperature field belongs to the same cluster C _k as the other elements of S _l ), then
• A111: purity test, for the subset S _l thus increased.

If step A111 shows that the subset S _l , thus increased by another basic temperature field, is still pure, step A110 is executed again, then step A111, and so on until until step A111 shows that the subset _S1 is no longer pure. The iterations of steps A110 and A111 are then stopped, and the last base temperature field to have been added to the subset is removed from this subset _S1 , which makes it possible to make it pure again.

The growth of the subset S _l is therefore carried out step by step. More precisely, during the addition step A110, one adds to the subset S _l the basic temperature field T _i which is closest to the seed SD _l (in the sense of the distances δ _ij ), among the basic temperature fields of the cluster C _k which have not yet been selected (ie: which are not already part of the subset S _l ).

It will be noted that the subsets S _l are thus made to grow step by step, on the basis of a particular metric δ _ij which represents the dissimilarity between the mechanical responses associated with the temperature fields, and not on the basis of 'a usual distance (eg Euclidean) between the temperature fields themselves. This makes it possible to grow each subset with a good probability that it remains pure (since the purity criterion in question concerns the mechanical responses associated with the temperature fields). Otherwise formulated, growing the subsets S _l on the basis of the same metric δ _ij as that initially used to identify the different clusters C ₁ , C ₂ , … C _P (ie: the different classes) of temperature fields allows to optimize the probability that the subset remains pure, during its growth. This therefore makes it possible to obtain pure subsets of larger size. This precaution is welcome in practice, because even by making the pure subsets grow in this way, their maximum size remains generally small (they typically contain only a few temperature fields, of the order of 5 to 10, on average, for the examples presented here).

Similarly, the fact of selecting the seeds beforehand (at step A10) on the basis of this same metric δ _ij , initially used to identify the different clusters C ₁ , C ₂ , … C _P , makes these seeds good candidates for further growth of pure subsets.

Purity test

During the purity test step A111, it is checked, for the subset S _l to be tested, that no base temperature field T _{i '} , associated with a standard response M _{k '} different from the response- type M _k common to the basic temperature fields T _i of the subset S _l considered, is located inside the convex hull E _l of this subset S _l .

In other words, it is verified that no base temperature field, belonging to another cluster C _{k '} than the cluster C _k of the subset to be tested, is located inside the convex hull E _l of the subset S _l . When this condition is verified, the subset in question is considered pure.

This test amounts to checking, for each basic temperature field T _i' belonging to another cluster C _{k ' ,} that this temperature field T _i' cannot be expressed in the form of a linear combination of the fields of base temperature Ti belonging to the subset S _l in question, this linear combination having positive coefficients and normalized to 1.

The criterion to be tested is therefore relatively consuming, in terms of computation time, and this even if the number of degrees of freedom of the basic temperature fields T _i is relatively small, compared to the high resolution temperature fields T _H, initial _i (thanks to the prior step of selecting variables S0).

Also, to reduce the calculation time necessary to carry out the A111 purity test, the purity of each subset S _l is tested by projection, that is to say on the basis of temperature subfields , obtained by projection of the basic temperature fields T _i on a subspace comprising a number of degrees of freedom lower than the number Q of nodes of the mesh on which the basic temperature fields T _i are defined.

As indicated during the description of the variable selection step S0, each base temperature field T _i groups together a set of Q temperature values T _i ¹ ,….,T _i ^Q , which are the temperature values of this field at the various nodes of the mesh in question (which, in this case, is an already simplified mesh, compared to the initial high-resolution mesh which comprises Q _H nodes).

The subspace, on which the basic temperature fields T _i are projected, is a space of dimension Q _R , that is to say comprising a number of scalar degrees of freedom Q _R , lower than Q (in practice , it can be a subspace of very reduced dimensions, for example with only 5 dimensions instead of 87 or 42445 dimensions). This subspace corresponds here to the whole of the fields of temperature which give values of temperature in a number Q _R of nodes of the mesh only, instead of giving these values in all the nodes of the mesh on which are defined the fields of base temperature T _i .

Here, each temperature subfield therefore gives temperature values at the level of different nodes of a restricted group of nodes, comprising Q _R nodes only (for example 5 nodes only) instead of Q nodes. These nodes are chosen from among the nodes of the (reduced) mesh on which the basic temperature fields T _i are defined, that is to say from among the Q nodes of the group of relevant nodes S determined in step S0. In this case, they are chosen here randomly, from among the Q nodes of the group of relevant nodes S .

Each temperature subfield is then obtained, from the corresponding base temperature field T _i , by keeping, among the temperature values T _i ¹ ,….,T _i ^Q given by Ti, a restricted number of these temperature values, which are those at the level of the nodes of the restricted group in question.

Note, however, that other ways of projecting the temperature fields T_Iare conceivable. Any projection on a subspace ofR ^Qis conceivable. This subspace can for example correspond to the Q_Rfirst components of the canonical basis ofR ^Q, or to the Q_Rfirst left singular vectors in the singular value decomposition (SVD) of the collection of temperature vectors T_I ¹,….,T_I ^Q at the nodes of the reduced mesh.

Anyway, performing the A111 purity test mentioned above, not on the basis of the basic temperature fields T _i , but on the basis of the subfields , which is less bulky, makes it possible to accelerate this purity test step.

In addition, surprisingly, when the result of this purity test by projection is positive, this validates the purity of the subset S _l to be tested as reliably (as regards the validation of purity) as if the test had been carried out on the basis of the complete base temperature fields.

• si l’enveloppe convexe de l’ensemble regroupant lessous-champs de température , qui sont les projetés des champs de température de base T_iappartenant à un sous-ensemble S_l, ne contient aucunsous-champ de température qui soit associé à un champ de température de base T_i’d’un autre cluster que celui du sous-ensemble S_l,
• alors, l’enveloppe convexe E_ldu sous-ensemble S_lne contient aucun champs de température de base T_{i ’}d’un autre cluster que celui du sous-ensemble S_l.

Indeed, a mathematical demonstration shows that:

• if the convex hull of the unit grouping together the sub -fields of temperature , which are the projections of the basic temperature fields T _i belonging to a subset S _l , does not contain any temperature subfield which is associated with a basic temperature field T _i' of another cluster than that of the subset S _l ,
• then, the convex hull E _l of the subset S _l does not contain any basic temperature field T _{i '} of another cluster than that of the subset S _l .

That is to say, in summary: if the projection of the subset S _l is pure, then the subset S _l itself is also pure.

The demonstration of this property is not included here for the sake of brevity, and because it is not of direct interest for the implementation of the invention.

Note, however, that the converse of this proposition is generally not true. Indeed, if the projection of the subset S _l is not pure, nothing can generally be deduced therefrom concerning the purity or not of the subset S _l itself.

Thus, the project of the subset S _l may very well not be pure, whereas the subset itself is pure. Indeed, it is well understood that a projection, which constitutes in a way a crushing according to certain directions, can very well bring inside a subset of the elements belonging to another cluster, and which, before projection , were outside this subset. In other words, a projection can very well cause the purity to be lost.

• on exécute une étape de test par projection A112 ;
• lorsque le résultat du test par projection A112 est positif (c’est-à-dire, lorsque le projeté sous-ensemble S_lest pur), on détermine que le sous-ensemble S_lest pur ;
• en revanche, lorsque le résultat du test par projection A112 est négatif, on exécute à ce test par projection, mais en réalisant la projection sur la base d’un autre sous-espace (d’un autre groupe restreint de nœuds, sélectionnés aléatoirement).

To take into account these two aspects (namely: the purity of the projection implies the purity of the subset S _l itself; and the non-purity of the projection implies an uncertainty as to the purity of the subset S _l ), the A111 purity test step is performed as follows ( ):

• a projection test step A112 is executed;
• when the result of the test by projection A112 is positive (that is to say, when the projected subset _S1 is pure), it is determined that the subset _S1 is pure;
• on the other hand, when the result of the test by projection A112 is negative, one executes this test by projection, but by carrying out the projection on the basis of another subspace (of another restricted group of nodes, selected randomly) .

As long as the result of the projection test A112 is negative, and as long as the number it of iterations of this test is less than an iteration limit number, n _lim , step A112 is executed again.

If the result of the test by projection A112 finally becomes positive, during these iterations, it is determined that the subset S _l is pure.

On the other hand, if at the end of the number n _lim of iterations of the test by projection (each time with a projection on a different subset), this test has still not given a positive result, it is determined that the sub- set S _l to be tested is not pure. In reality, we do not know if it is pure or not. But, not having been able to ensure that it is pure, this subset is eliminated as a precaution, and no additional temperature field will be determined within it.

In practice, the A112 projection test can be performed as explained now.

During the projection test A112, it is determined, for the subset S _l whose purity is to be tested, whether the convex hull of the unit grouping together the sub -fields of temperature , which are the projections of the basic temperature fields T _i of the subset S _l , does not contain any temperature subfield which is associated with a foreign basic temperature field T _i' , of another cluster C _{k '} than the cluster C _k to which the subset S _l to be tested belongs.

This amounts to testing that no temperature subfield cannot be written as a linear combination of the temperature subfields of the subset S _l , assigned positive coefficients and normalized to 1. To test this criterion, one proceeds as follows.

For each temperature subfield , corresponding to one of the basic temperature fields T _i of the subset S _l , a vector π _i having Q _R +1 components is considered. The Q _R first components of the vector π _i are the Q _R temperature values given by the temperature sub-field . The last component of this vector is equal to 1.

We then form a matrix A, of dimension (Q _R +1)×n, by grouping in the form of a matrix the n column-vectors π _i corresponding to the n temperature sub-fields that comprises the subset S _l whose purity is to be tested.

For each temperature subfield belonging to another cluster C _k' than that of the subset S _l to be tested, a vector π _{i ' is constructed in the same way.}

To test if the temperature subfield is quite different from any linear combination of the temperature subfields of the subset S _l to be tested, assigned positive coefficients and normalized to 1, it is then tested whether the condition below is verified: where w is a vector grouping together n positive coefficients and where ε is a fixed tolerance level, comprised for example between 0.5% and 5%.

From a numerical point of view, to test this condition, we carry out, for the term on the left, a minimization of the least-squares type (with positive coefficients). The coefficients grouped in the vector w are not normalized to one (removing this constraint facilitates the minimization procedure in question). But the component equal to 1, added at the end of each vector π, makes it possible to carry out this test as if we were testing a linear combination with positive coefficients and normalized to 1.

Example digital

The results obtained using this method of determining learning data are briefly presented below for an example of implementation.

This example corresponds to the part represented on the , for which the high resolution mesh includes QH=42445 nodes. Furthermore, N=600 high resolution temperature fields TH,1 ,…, TH,N are produced initially, during step 10. And the number P of clusters (ie: the number P of response-types Mk) , is equal to 4, here.

As already indicated, in this example, the variable selection step S0 makes it possible, in the high resolution mesh, to select Q=87 relevant nodes, which makes it possible to simplify the high resolution temperature fields, in the form of basic temperature fields T₁ ,…, T_NOT, still numbering 600.

Then, during the A0 data augmentation step, about 60 pure subsets were identified (on average) in each cluster C ₁ ,…C _P of temperature fields. Each pure subset thus identified comprises on average 5 elements (5 basic temperature fields). During the purity test, by projection (step A112), the subspaces on which the temperature fields are projected are 5-dimensional subspaces (in other words, 5 nodes, instead of 87, for the projection purity test).

In this example, 5400 additional temperature fields T' _i , each matched with their label, were then obtained by convex linear combination within these pure subsets.

The set of training data obtained (basic + additional) then includes 6000 labeled temperature fields, used for training a temperature classifier.

• avec et sans augmentation de données (avec/sans DA), et
• avec une sélection de variables :
  • soit par la technique de sélection décrite plus haut, par maximisation de la pertinence avec minimisation de la redondance (FS),
  • soit par analyse en composantes principales (PCA).

The classification accuracy obtained after training was then tested for several types of classifiers:

• with and without data augmentation (with/without DA), and
• with a selection of variables:
  • either by the selection technique described above, by maximizing relevance with minimizing redundancy (FS),
  • or by principal component analysis (PCA).

These results are collated in the table below. Type of classifier used Dimension reduction Accuracy,
With AD Accuracy,
Without AD multilayer perceptron FS 87.0% 81.0% BCP 86.5% 81.5% Quadratic discriminant analysis FS 77.5% 70.5% BCP 76.0% 70.0% Gaussian naive Bayes FS 39.5% 34.5% BCP 38.5% 31.5%

These results show that the additional training data, as determined here, does indeed improve classification accuracy, and is therefore physically relevant.

These results also show that the method of selection of variables (FS) presented above is reliable (in addition to being fast to execute), in the sense that it effectively makes it possible to extract the most relevant information from the fields of high resolution temperature.

Electronic devices

The present technology also relates to an electronic device comprising at least one processor and one memory, programmed to execute the variable selection method and/or the data augmentation method described above. More generally, this electronic device can be programmed to execute the entire method for determining learning data, and for training the classifier of the .

The device in question can be made in the form of a computer, or a computer whose components are located in the same place, for example in the same room or in the same computer case. It can also be implemented in the form of a delocalized computer system using, via a computer network, remote, distributed computer resources (for example remote computing resources, provided by a “cloud”). .

The present technology also relates to an electronic device, comprising at least a processor and a memory, comprising a classifier, that is to say programmed to perform a classification function as described above, and in which the classifier has been trained based on the basic and/or additional training data presented above. This classifier is specific (and, as seen above, particularly precise) because of the particular training it has undergone. From a concrete point of view, the specificities of this classifier translate into the particular values of the coefficients (synaptic coefficients, for example) which parameterize the classifier, as obtained after training on the basis of the basic learning data and /or additional.

• une phase préalable d’entrainement comportant :
  • une détermination de données d’apprentissage de base et de données d’apprentissage supplémentaires, comme expliqué plus haut,
  • l’entrainement d’un classificateur de champs de température sur la base desdites données d’apprentissage de base et desdites données d’apprentissage supplémentaires, et
• une phase d’utilisation comprenant les étapes suivantes :
  • identification de l’une des réponses-types M₁, …, M_Pdudit élément d’aube par le classificateur de champs de température recevant en entrée ledit champ de température T_R, et
  • simulation numérique de la réponse mécanique dudit élément d’aube, soumis audit champ de température T_R, conformément au modèle de comportement mécanique simplifié défini par la réponse-type précédemment identifiée par le classificateur.

The present technology also relates to an electronic device, comprising at least one processor and one memory, programmed to implement a method for digital simulation of the mechanical response of a turbine blade element, subjected to a temperature field T _R given, the simulation method comprising:

• a preliminary training phase comprising:
  • a determination of basic training data and additional training data, as explained above,
  • training a temperature field classifier based on said basic training data and said additional training data, and
• a use phase comprising the following steps:
  • identification of one of the standard responses M ₁ , …, M _P of said blade element by the temperature field classifier receiving said temperature field T _R as input, and
  • numerical simulation of the mechanical response of said blade element, subjected to said temperature field T _R , in accordance with the simplified mechanical behavior model defined by the standard response previously identified by the classifier.

Claims (12)

Method for determining basic learning data for training a stress field classifier (2'),

• in which high-resolution training data gathers high-resolution stress fields (T_H,1, T_H,i , T_H,N) exerted on a turbine blade element (1'; 1) and, for each of these fields, a label (k) associated with the stress field considered, each high-resolution stress field (T_H,i) giving values of said constraint (T¹ _H,i, T^I _H,i, Tⁱ _H,i) at different nodes of a high resolution mesh of said blade element,
• the method comprising a step of selecting (S0) a restricted number (Q) of nodes of the high-resolution mesh, the selected nodes forming a group of relevant nodes ( S ),
• the basic training data gathering basic stress fields (T₁, T_I, T_NOT) which are obtained by restricting the high resolution stress fields (T_H,1, T_H,i , T_H,N) said group of relevant nodes,
• and wherein at least a portion of said relevant nodes are each selected based on an approximate value ( ) mutual redundancy information (I _{j,j '} ) between the node (j) considered and any one of the other nodes (j') of said mesh, said approximate value being determined on the basis of a distance (d _{j,j '} ) space between the two nodes considered.

Method according to the preceding claim, in which the said stress exerted on the turbine blade element (1'; 1) is a temperature exerted on this element, the high resolution stress fields (T_{H, 1}, T_H,i , T_H,N) and the basic stress fields (T₁, T_I, T_NOT) being respectively high resolution temperature fields and basic temperature fields. Method according to one of the preceding claims, in which the mutual redundancy information (I_{not a word '}) between the two nodes (j, j’) considered is representative of a level of information redundancy between:

• the values of said constraint (T^I _{H, 1}, T^I _H,i, T^I _{H, NOT}) at the first (j) of the two nodes, as given by the set of high-resolution stress fields (T_H,1, T_H,i , T_H,N), And
• the values of said constraint (Tⁱ _{H, 1}, T^{I '} _H,i, T^{I '} _{H, NOT}) at the second (j’) of the two nodes, as given by the set of high resolution stress fields (T_H,1, T_H,i , T_H,N).

Method according to one of the preceding claims comprising a calibration step (S11) which comprises the following steps:

• (S12) for each pair of nodes (j, j') of a subset grouping part of the nodes of the high resolution mesh:
  • determination of a reference value ( ) mutual redundancy information (I_{not a word '}) between the two nodes (j, j') considered, by statistical calculation on the basis of the values of said constraint (T^I _{H, 1}, T^I _H,i, T^I _{H, NOT}, Tⁱ _{H, 1}, T^{I '} _H,i, T^{I '} _{H, NOT}), at these two nodes, as given by the different high-resolution stress fields (T_H,1, T_H,i , T_H,N), And
  • calculation of the spatial distance (d _{j,j '} ) separating the two nodes (j, j') considered
• (S13) fitting a curve ( ), so that it is representative of a link between said reference values ( ) mutual redundancy information, and the corresponding spatial distances (d _{j,j '} ),
• and in which the approximate value ( ) of the mutual redundancy information between two nodes (i,i') of the network is the value given by said curve ( ), for the spatial distance (d _{j,j '} ) separating the two nodes considered.

Method according to one of the preceding claims, in which, for each high-resolution stress field (T_{H, 1}, T_H,i , T_H,N), the label (k) associated with the high resolution stress field (T_H,i) considered identifies a standard response (M_k), representative of an expected mechanical response for said blade element (1′; 1) in response to said high-resolution stress field (T_H,i), among different standard responses (M₁, M_k, M_P) of said turbine blade element. Method according to the preceding claim, in which each relevant node (j) is further selected on the basis of a degree of relevance ( ) of the value of said stress at the node (j) considered, with respect to the mechanical behavior of said blade element, said degree of relevance ( ) being representative of a level of dependence between, on the one hand, the label (k) associated with the high resolution stress field (T_H,i) considered, and, on the other hand, the value of said constraint (T^I _H,i) at the node (j) considered for this stress field. Method according to the preceding claim, in which:

• the group of relevant nodes (S) initially contains a first node, which is the node (j) of the high-resolution mesh with the highest degree of relevance ( ),
• we then execute an addition step, during which we add an additional node to the group of relevant nodes (S), the additional node being selected, among the nodes of said mesh which are not yet part of said group, so as to maximize a quantity equal: to the degree of relevance ( ) of the considered node, minus an average level of redundancy between this node and the nodes already present in the group of relevant nodes (S),
• the addition step then being executed again several times.

Method according to the preceding claim, in which the average level of redundancy, between the node considered and the nodes already present in the group of relevant nodes (S), is equal to the arithmetic mean of the approximate values ( ) mutual redundancy information (I_{I ,j} ) between said node (i) and the nodes (j) already present in the group of relevant nodes (S), the additional node, added to the group of relevant nodes (S), being the node that maximizes the following quantity: where i is the index identifying the node considered, in the high resolution mesh, and where is the number of nodes contained in the group of relevant nodes (S) during the current iteration of the add step. Method according to one of claims 5 to 8, further comprising a step of increasing data (A0) during which additional training data is determined from said basic training data, each constraint field basic (T₁, T_I, T_NOT) with a label (k) of the high resolution stress field (T_H,1, T_H,i , T_H,N) corresponding, the step of increasing data comprising the following steps:

• (A1) form at least one pure subset (S _l ), which groups together basic stress fields (T _i ) associated with the same standard response (M _k ) of the turbine blade element, and which has a convex hull (E _l ) containing no basic stress field (T _{i '} ) which is associated with another standard response (M _{k '} ) of said element,
• (A2) for at least one of said pure subsets (S _l ), determination of one or more additional training data, each additional training data comprising an additional stress field (T' _i ), obtained by linear combination of basic stress fields (T _i ) of the pure subset (S _l ) considered, assigned positive weighting coefficients and normalized to 1, the additional stress field (T' _i ) being accompanied by a label (k) which is the same as for the basic stress fields (T _i ) of said combination.

Numerical simulation method of the mechanical response of a turbine blade element subjected to a stress field (T_R) given including:

• a preliminary training phase comprising:
  • a determination of basic training data according to one of claims 1 to 8, or a determination of basic training data and additional training data according to claim 9,
  • training (20) a stress field classifier based on said basic training data, or on the basis of said basic training data and said additional training data,
• a use phase comprising the following steps:
  • identification of one of the typical responses (M ₁ , M _k , M _P ) of said blade element by the stress field classifier receiving said stress field (T _R ) as input,
  • digital simulation of the mechanical response of said blade element, subjected to said stress field (T _R ), in accordance with the simplified mechanical behavior model defined by the standard response previously identified by the classifier.

Electronic device comprising at least a processor and a memory, programmed to determine basic training data, or to determine basic training data and additional training data according to the method of any of the claims 1 to 9. Electronic device comprising at least one processor and one memory, programmed to execute the digital simulation method of claim 10.