Method for determining one or more models of the mechanical behavior of a turbine blade, and associated device

Abstract

Method for determining one or more mechanical behavior models of a turbine blade, and associated device. Such a method comprises the following steps: a simplified numerical simulation step, during which, for each stress field within a set of stress fields, a simplified mechanical response of the turbine blade subjected to the considered stress field is determined; a partitioning step, during which said set is subdivided into K distinct stress field groups based on said simplified mechanical responses; a pre-characterization step, during which an expected performance criterion (G) is determined for a dictionary of mechanical behavior models comprising K models, taking into account said partitioning; and the determination of the mechanical behavior model(s), taking into account said performance criterion. Figure to be published with the abbreviation: Figure 5

Description

Title of the invention: Method for determining one or more mechanical behavior models of a turbine blade, and associated device Technical field

[0001] In general, the invention relates to the field of numerical simulation of the mechanical behavior of mechanical parts, for example their deformation and aging. It relates in particular to the determination of mechanical behavior models for a turbine blade subjected to different stress fields, for example to different temperature fields. These mechanical behavior models synthetically represent the expected mechanical behavior for the blade in question. These mechanical behavior models, associated with a stress field classifier trained by artificial intelligence methods, make it possible to usefully accelerate aging studies by numerical simulation. Aging is understood to mean the evolution of the state of the mechanical part over time, including fatigue damage. TECHNOLOGICAL BACKGROUND

[0002] Numerical simulation by finite elements of the deformations undergone by a mechanical part subjected to different stresses has made significant progress in recent years. Such simulations make it possible in particular to predict how a given mechanical part will age, for a long period of use (for example several years), which could not have been the subject of direct tests or measurements. More generally, these simulations make the design and prototyping phase of a mechanical system faster and more efficient.

[0003] But certain situations remain difficult to simulate, because they require a very significant amount of calculation time.

[0004] This is the case in particular for the characterization of aging, in particular the fatigue resistance of a blade of a turbomachine turbine, for example a high-pressure turbine located just downstream of the combustion chamber. In this case, a high and generally inhomogeneous temperature is exerted on the blades of the turbine, in addition to the pressure field and the centrifugal force experienced by these blades.

[0005] For a given temperature field, exerted on this blade, simulating the mechanical response (or the mechanical behavior) of the blade, for several cycles corresponding to several flights of an aircraft, requires very significant computing resources.

[0006] Furthermore, 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 an entire statistical distribution of temperature fields.

[0007] With a classic finite element numerical simulation, the calculation time or the calculation resources (number of processors or calculation clusters to be made to work in parallel) necessary for such a characterization then become prohibitive.

[0008] To reduce the computational time required to perform such simulations, an approach based on clustering different possible inputs (different possible stresses), and a classification to recommend a reduced model (for example a reduced model of mechanical behavior) was proposed in the article “Localized Discrète Empirical Interpolation Method” by B. Peherstorfer et al., SIAM J. Sci. Comput. (2014). More recently, an approach based on such an order reduction of the mechanical problem and on a temperature field classifier was proposed in the following article: “Model order reduction assisted by deep neural networks (ROM-net)” by T. Daniel et al., Adv.Model. and Simul. in Eng. Sci. (2020) 7:16.

[0009] According to this approach, instead of determining the mechanical deformation at each node of a mesh of the blade to be characterized, with a number of degrees of freedom (scalars) which would therefore be equal to at least three times the number of nodes of the mesh, the mechanical response of the blade is studied on the basis of a simplified mechanical behavior model, Mj, parameterized by a smaller number of degrees of freedom.

[0010] Such a mechanical behavior model Mj describes, for example, the deformation of the turbine blade considered in the form of a weighted sum of deformation modes, typical of the mechanical problem studied (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 in question, etc.). The degrees of freedom, or in other words the parameters of this mechanical behavior model, then correspond, in this weighted sum, to the coefficients which weight the contributions of these different deformation modes (or, in other words, of these different elementary mechanical modes).

[0011] The mechanical response of the turbine blade, subjected to a given temperature field T, is then determined over time for several flight cycles, by numerical simulation, on the basis of this mechanical behavior model Mj (during step Ssim of [Fig.l]). Thanks to the simplified nature of this model, it is possible carry out a simulation corresponding to a significant duration of use, while maintaining limited calculation time.

[0012] Such a mechanical behavior model Mj is sometimes called a “reduced model” in this technical field. It is representative of the mechanical response of the turbine blade (in particular its deformation) when the blade is subjected to a given type of stress (for example to a given type of temperature field T). It is therefore in some way a typical response of the turbine blade.

[0013] According to the approach proposed in the article by T. Daniel et al. mentioned above, before carrying out the numerical simulation S sim in question, on the basis of the reduced model Mj, this reduced model Mj is chosen, in a dictionary of reduced models, according to the temperature field T exerted on the turbine blade 1. The reduced model Mj is selected, by a temperature field classifier 2, from among the different reduced models Mb ...Mj, ..., MK of this dictionary ([Fig.l]). The reduced model Mj which is selected by the classifier 2 is, among these reduced models Mi , .. .Mj, ..., Mk, the one which is the most representative of a complete mechanical response expected for the turbine blade, 1, when it is subjected to the temperature field T in question. Classifier 2, which includes for example a neural network, is trained by artificial intelligence techniques during a training step S5 ([Fig.2]).

[0014] The method described in the article in question comprises the following steps: • S0 (not shown in [Fig.2]): creation of a training database of possible temperature fields Tb . . .,1), .. .,Tn, • S1: simplified numerical simulation step, during which, for each temperature field Ti, .. .,1), .. .,Tn considered, a simplified mechanical response u; of the turbine blade 1, subjected to the temperature field T; in question, is determined, • S20: partitioning stage (or “clustering” in English), during which: • we group these different temperature fields Tb ... ,1), ... ,Tn into K groups Ci,.. .Cj,.. .CK each grouping, among these temperature fields, those which have similar simplified mechanical responses u;, and • for each group Cj, one or more typical temperature fields 1)* (sometimes called “snapshot”, i.e. “overview”, in this technical field) are identified in the group, the simplified mechanical responses of which are representative of all the simplified mechanical responses of the group Cj considered, • S3: high-fidelity numerical simulation step, during which, for each typical temperature field Tj* previously identified, a high-fidelity mechanical response is determined for the turbine blade subjected to the typical temperature field T)* considered, then • S4: step of determining the mechanical behavior models Mi, .. .Mj, ..., Mk, from said high-fidelity mechanical responses U^F. by a model order reduction technique.

[0015] The possible temperature fields Tb .. .,T;, .. .,Tn created in step S0 are representative of temperature fields likely to be exerted on the turbine blade in question, in the turbine considered. The characteristics of these temperature fields can come from measurements, or from fluid mechanics simulations, or from prior knowledge concerning the temperature fields usually encountered in such a situation. These temperature fields are at least partially random and differ from each other. These n temperature fields, which are in a way training fields, are used to train the classifier and to construct the clusters and their associated reduced models, with a view to being used for new temperature fields subsequently, in the use phase.

[0016] During the simplified numerical simulation step SI, for each temperature field, the mechanical response of the turbine blade is determined, over time, for a turbine stress cycle corresponding to the typical duration of a complete flight of an aircraft, powered by a turbomachine incorporating this turbine.

[0017] Such a numerical simulation is not necessarily sufficient to determine the long-term evolution of the mechanical characteristics of the blade (which would require a simulation over many successive cycles), nor its lifetime. But it already makes it possible to identify the type of mechanical behavior of the blade, when it is subjected to the temperature field T; in question. Due to the simplified nature of this numerical simulation, the mechanical responses determined at this stage are called “simplified” mechanical responses, while those determined at stage S3 are called “high fidelity” mechanical responses.The mechanical behavior models determined in step S4 are, in a way, simplified models, not because they come from a simplified numerical simulation (on the contrary, they are determined from the high-fidelity mechanical responses), but because they are obtained by model order reduction, from the high-fidelity mechanical responses.

[0018] And during the high-fidelity numerical simulation step S3, for each typical temperature field T;* (for each “snapshot”), we determine the mechanical response of the turbine blade (the "high fidelity" response), over time, for many turbine loading cycles, instead of doing so for only one or a few cycles (in order to more precisely characterize the long-term evolution and aging, in particular fatigue damage of the blade).

[0019] The different mechanical response models Mb ..., Mj, ..., MK determined in step S4 are each associated with one of the K clusters Cb ..., Cj, ..., CK formed during the partitioning step S20. Thus, the mechanical behavior model Mi is the model associated with the cluster Cb II is representative of an average mechanical behavior of the turbine blade, for the set of temperature fields T; belonging to this cluster Cb Similarly, the mechanical behavior model Mj is the behavior model associated with the temperature fields belonging to the cluster Cj.

[0020] Each mechanical behavior model Mb..., Mj, ..., MK is determined, by model order reduction, from the high-fidelity mechanical responses u^F corresponding to the typical temperature fields T;* of the cluster Cb..., Cj, ..., CK considered. Thus, the mechanical behavior model Mi is determined (in step S4) from the high-fidelity mechanical responses u^F obtained for the typical temperature fields T;* of the cluster Cb for example.

[0021] In step S5, the training data used to train the temperature field classifier 2 comprises the set of temperature fields Tb.. .1),... Tn, each with a label which is the number of the cluster to which the temperature field T; considered belongs. For example, if the field Ti belongs to the cluster C3, k(1) = 3, and the pair (Tb 3), where the label 3 identifies the cluster to which Tb belongs, is one of the training data used in step S5.

[0022] Once the classifier 2 has been trained, to characterize the aging of the blade 1 when it is subjected to a given temperature field T, this field T is provided as input to the classifier ([Fig.l]). As output, the classifier 2 provides the number j of the cluster Q that it associates with this temperature field T, which makes it possible to know which of the mechanical behavior models Mb..., Mj, ..., MK is most suitable for studying the mechanical response to this temperature field T (in this case the model Mj, corresponding to the cluster Cj).

[0023] In practice, the cluster that classifier 2 associates with this temperature field T is that of clusters 1 to K for which the simplified mechanical responses u; are the closest (i.e.: the most similar) to a mechanical response that can be expected for this temperature field T.

[0024] Once the mechanical behavior model best suited to the field T has been identified by the classifier 2, the mechanical response of the blade 1 is determined, over time, for a significant duration of use (for example for several hundreds of flight cycles), by numerical simulation based on the reduced model of mechanical behavior in question (step Ssim of [Fig.l]).

[0025] The tool, formed by the trained classifier 2 and the set of associated mechanical behavior models Mb..., Mj, ..., MK, is called ROM-net in the article by T. Daniel et al. mentioned above. It makes it possible to reduce the calculation time required to characterize the lifetime of a turbine blade, which may be subjected to many different temperature fields.

[0026] But the accuracy of the fatigue life predictions obtained with such a ROM-net, and the speed with which they are obtained (i.e.: the speed of execution of the Ssim step) depend on the parameters of the ROM-net. They depend for example on the number K of clusters used to subdivide the set of temperature fields Tb .. .T;,.. .,Tn. And before having built the ROM-net, it is difficult to know if the values ​​chosen for these parameters will lead to a ROM-net that is both accurate and fast.

[0027] Determining such a ROM-net, on the basis of ROM-net parameters fixed at arbitrary values, can then lead to performances which are not optimal, or which do not correspond to desired criteria in terms of precision or calculation time. SUMMARY

[0028] In this context, a method is proposed for determining one or more mechanical behavior models of a turbine blade, adapted to a set of stress fields to be exerted on said blade, the method comprising the following steps: • simplified numerical simulation step, during which, for each stress field of said assembly, a simplified mechanical response of the turbine blade subjected to the stress field considered is determined, • partitioning step, during which K groups of distinct stress fields are identified in the set of stress fields, each group being characterized by one or a few typical stress fields, the simplified mechanical responses of which are representative of the set of simplified mechanical responses of the group considered, • high-fidelity numerical simulation step, during which, for each typical stress field previously identified, a high-fidelity mechanical response is determined for the turbine blade subjected to the typical stress field considered, then • step of determining the mechanical behavior models associated with each group, from said high-fidelity mechanical responses, by model order reduction, • the method further comprising, before the high-fidelity digital simulation step, a pre-characterization step during which one or more performance criteria, expected for the mechanical behavior model(s) taking into account said partitioning, are determined.

[0029] The performance criterion(s) are for example determined as a function of individual projection errors calculated for different stress fields of said assembly, each individual projection error corresponding to a projection error between: • the simplified mechanical response associated with the stress field considered, and • a reduced mechanical response, associated with the group to which the stress field considered belongs, the reduced mechanical response having been determined by order reduction from the simplified mechanical responses associated with the typical stress fields of said group.

[0030] When building a ROM-net, it is the high-fidelity numerical simulation step that requires the most calculations. It is therefore particularly interesting to have a first estimate of the final performances expected for the ROM-net, for the set of parameters considered, before carrying out the high-fidelity numerical simulation step.

[0031] This avoids carrying out this step, which is computationally intensive, for one or more ROM-net parameters which would in fact be poorly adapted (because they would lead, for example, to insufficient precision).

[0032] The parameter(s) in question comprise the number K of clusters used to subdivide the set of stress fields. They may also comprise a number N of elementary mechanical modes composing each of the reduced mechanical responses in question (number N which is intended to also be the number of elementary mechanical modes composing each of the mechanical behavior models determined in step S4). These parameters may also comprise the number ns of typical temperature fields identified for each cluster (i.e.: the number of “snapshots” per cluster), for which high-fidelity mechanical responses are determined).

[0033] The partitioning step and the pre-characterization step can also be executed several times for different values ​​of the parameters in question, so as to identify optimal values ​​of these parameters, or at least values ​​making it possible to satisfy one or more performance conditions, before executing the high-precision digital simulation step, which is then executed for this well-adapted set of values.

[0034] The high-fidelity digital simulation step, which is demanding in terms of computation time, can thus be executed only once, directly for well-adapted values ​​of the ROM-net parameters.

[0035] It is interesting to execute the partitioning and pre-characterization step several times because this makes it possible to choose the values ​​of these parameters as best as possible while requiring only a few additional calculations.

[0036] Indeed, a partitioning operation is in itself very inexpensive. And moreover, these different partitionings can be carried out on the basis of the same set of simplified mechanical responses, initially obtained during the simplified numerical simulation step. In other words, this simplified simulation step does not have to be executed again to carry out a new partitioning. And the similarities (or dissimilarities) between simplified mechanical responses, on the basis of which the partitioning is carried out, do not have to be recalculated at each partitioning: they can be reused during these different partitioning calculations.

[0037] The method presented above therefore makes it possible to substantially improve the performance of the ROM-net finally obtained, at a very low additional cost in terms of calculations.

[0038] It may be noted that at least one of the performance criteria mentioned above provides an indication concerning a precision (representation precision) that can be expected from the ROM-net, for the parameter values ​​considered. Indeed, this or these performance criteria are determined as a function of projection errors between: • on the one hand, the mechanical responses obtained during the simplified numerical simulation step, for different stress fields, and, • on the other hand, reduced mechanical responses analogous to the mechanical behavior models which will finally be produced after step S4, and which prefigure what these mechanical behavior models will be (although less precise than these mechanical behavior models, because determined from the results of simplified simulation, and not from the results of high-fidelity simulation).

[0039] The simplified numerical simulation step is simplified in the sense that it requires fewer calculations than the high-fidelity numerical simulation step, for example because it is a simulation corresponding to a shorter duration of use of the turbine than for the high-fidelity numerical simulation (corresponding for example to one or a few flight cycles, instead of a large number of flight cycles), or carried out with a less fine time step or mesh than for the high-fidelity numerical simulation.

[0040] The stress fields in question may, for example, be temperature fields exerted on the turbine blade. But, as will appear to man of the trade, this method can also be applied to other types of stresses exerted on the turbine blade, such as pressure exerted on the blade by a fluid flowing in the turbine, for example.

[0041] The method which has just been presented is implemented by an electronic device comprising at least one memory and a processor capable of executing logical operations. The device in question can be implemented in the form of a computer, or a calculator whose constituents 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 computing resources (for example remote computing resources, provided by a "cloud").

[0042] 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 according to all technically possible combinations: • each stress field is a temperature field exerted on said turbine blade; • the partitioning step is executed several times, for different values ​​of the number of groups K, and after each execution of the partitioning step, the pre-characterization step is executed, so as to pre-characterize the partitioning with K groups obtained previously; • each reduced mechanical response is composed of a number N of elementary mechanical modes; • N is less than a number of nodes of a turbine blade mesh; • after each execution of the partitioning step, the pre-characterization step is executed several times successively for different values ​​of the number N of elementary mechanical modes composing the reduced mechanical responses; • the method comprises a parameter selection step during which, for each pair of values ​​grouping together one of said values ​​of the number K and one of said values ​​of the number N, it is tested whether the performance criteria, determined for this pair of values ​​of K and N, satisfy one or more pre-established performance conditions, and the pair or pairs of values ​​for which the test is positive are selected, this or these pairs of values ​​constituting admissible pairs of values; • when at least one pair of admissible values ​​has been selected: • during the high-fidelity numerical simulation step, the high-fidelity mechanical responses are determined for the typical stress fields corresponding to the partitioning for which the value of K is that of said pair of admissible values, and • during the step of determining the mechanical behavior model(s), the determined model(s) are each composed of a number of elementary mechanical modes equal to the value of N given by said pair of admissible values; during the parameter selection step, a pair of optimal values ​​satisfying one or more pre-established optimization conditions is further selected from among the pairs of admissible values ​​of the numbers K and N, the steps of high-fidelity numerical simulation and determination of the mechanical behavior model(s) being executed for this pair of optimal values ​​of the numbers K and N; the pair of values ​​of K and N satisfying the said optimization conditions is the pair grouping: that of the admissible values ​​of N which is minimal, and N being minimized, that of the admissible values ​​of K which is the largest; when no pair of admissible values ​​of the numbers K and N could be selected during the parameter selection step, then, during the step of determining the mechanical behavior model(s), a single, global mechanical behavior model is determined for all of said stress fields; the performance criterion(s) determined during the pre-characterization step include an average projection error, equal to an average of said individual projection errors; said or one of said performance conditions is that said average projection error is less than a given maximum permissible error; the performance criterion(s) determined during the pre-characterization step include a partitioning gain which is determined as a function of an average, for different stress fields of said set, of the ratio between: a projection error associated with an overall reduced mechanical response on the one hand, and the individual projection error obtained for the stress field considered on the other hand, the overall reduced mechanical response being the same for the different stress fields considered, and being determined by model order reduction from the simplified mechanical responses corresponding to all the typical stress fields identified during the partitioning step; said or one of said performance conditions is that the partitioning gain is greater than a given limit gain; this makes it possible to ensure that the model dictionary is more efficient than a single global model; • the limit gain is a function of the number K; • the limit gain is an increasing function of K; • if no pair of admissible values ​​of the numbers K and N could be selected during the parameter selection step, then a user is asked, via a human-machine interface, to modify said maximum admissible error or said limit gain; • during the partitioning step, the typical stress fields are identified on the basis of a dissimilarity between the simplified mechanical responses, the dissimilarity between two simplified mechanical responses being calculated according to a normalized scalar product equal to the scalar product of the two mechanical responses considered, divided by a norm of a first of these two responses and by a norm of a second of these two responses; • the said dissimilarity is all the greater the smaller the normalized scalar product in question; • said dissimilarity is equal to or proportional to the square root of 1 minus the square of said normalized scalar product;

[0043] The present technology also relates to an electronic device comprising at least one processor and a memory, programmed to execute the method which has been presented above. The optional characteristics, presented above in terms of method, can also be applied to this device.

[0044] The present technology and its various applications will be better understood by reading the following description and examining the accompanying figures. BRIEF DESCRIPTION OF THE FIGURES

[0045] The figures are presented for illustration purposes and in no way limitative.

[0046] [Fig.l] [Fig.l] schematically represents an identification operation, by a temperature field classifier, of a mechanical behavior model adapted to a given temperature field T, exerted on a turbine blade.

[0047] [Fig.2] [Fig.2] schematically represents steps of a method of construction of a prior art ROM-net.

[0048] [Fig.3] [Fig.3] schematically represents steps of a method for constructing a ROM-net implementing the teachings of the invention.

[0049] [Fig.4] [Fig.4] schematically represents one of the stages of [Fig.3], more in detail.

[0050] [Fig.5] [Fig.5] shows values ​​of an average projection error, or, in other words, of an average representation error, evaluated for different partitionings of a set of temperature fields.

[0051] [Fig.6] [Fig.6] shows values ​​of a partitioning gain, evaluated for these different partitionings. DETAILED DESCRIPTION

[0052] [Fig. 3] schematically represents steps of a method for determining one or more mechanical behavior models of a turbine blade 1 (i.e. a turbine blade), well adapted to a set E of stress fields to be exerted on this blade. In the embodiment described here, these stress fields are temperature fields Tb..., T;,..., Tn exerted on the turbine blade 1 (at different points of the blade).

[0053] As explained in the “technological background” section, these mechanical behavior models are simplified in the sense that they are parameterized by a lower number of degrees of freedom than for a conventional numerical simulation by finite elements (in which the number of degrees of freedom would be for example equal to 3 times the number of nodes of the mesh of the blade 1). Although simplified, they are constructed in such a way as to be as representative as possible of the mechanical responses likely to be obtained when the blade 1 is subjected to a temperature field typical for this turbine. These mechanical behavior models, associated with a trained temperature field classifier, form a “ROM-net” which makes it possible to considerably accelerate the numerical study of the aging of such a turbine blade.

[0054] The method of [Fig. 3] has several points in common with the method for determining a ROM-net shown in [Fig. 2] and which was described above in the section concerning the technological background. The method of [Fig. 3] thus comprises the following steps (see Figures 3 and 4): • S0 (not shown): creation of a training database of possible temperature fields .. .,Tn, • SI: simplified digital simulation, • S20: partitioning and identification of typical temperature fields T; * • S3: high-fidelity digital simulation, • S4: determination of one or more mechanical behavior models, • S5: training of the temperature field classifier.

[0055] These steps are identical, or at least similar to steps S1 to S5 of [Fig.2],

[0056] But compared to the method of [Fig.2], the method described here comprises a pre-characterization step S24, executed after the partitioning step S20 and before the high-fidelity numerical simulation step S3 (see [Fig.4]).

[0057] During this pre-characterization step, a first estimate of performance is determined, expected for the mechanical behavior model(s), taking into account in particular the partitioning carried out in the previous step S20.

[0058] Thanks to this first estimate, it is possible to choose a partitioning, and more generally well-adapted calculation parameters before executing the high-fidelity digital simulation step S3.

[0059] It is interesting to estimate these performances beforehand, before executing step S3, because this step involves large calculations, and it would therefore be very expensive in terms of calculation to have to execute it several times, until obtaining a suitable set of mechanical behavior models.

[0060] Here, on the contrary, we anticipate the performances which should be obtained for such or such parameterization, and we then determine the mechanical behavior model(s) for parameters chosen as a function of these expected performances.

[0061] The different stages of [Fig.3] are now described in more detail, one after the other.

[0062] These different steps are executed on the basis of a set E of temperature fields which includes n temperature fields Tb..., Ti5..., Tn.

[0063] Each of these temperature fields brings together temperature values ​​exerted at different nodes of a mesh of the turbine blade. These are, for example, temperature values ​​exerted on the surface of this blade, due to the presence of a hot (and turbulent) fluid flowing in the turbine.

[0064] SI step: simplified numerical simulation.

[0065] During this step, for each temperature field Tb..., T;,..., Tn, a simplified mechanical response ub..., u;,..., un is determined, which corresponds to a mechanical response of the blade 1 subjected to the temperature field considered (which is possibly variable over time) for a duration corresponding to the typical duration of a complete flight of an aircraft.

[0066] Each of these simplified mechanical responses ub..., ui5..., un is a field defined over the entire turbine blade 1. It may correspond, for example, to the displacement field, to the Von Mises stresses, to the cumulative plastic deformation, or to any indicator of a state of local damage to the material. Each of these responses u; = u(T;) is stored in the form of a vector containing the values ​​of the field at the integration points or at the nodes of the finite element mesh of blade 1 (according to the definition of u). This vector includes N components (N being for example the number of nodes of the mesh).

[0067] Step S2: Determination of parameters for high-fidelity numerical simulation; partitioning.

[0068] This step, executed after step SI, is shown in more detail in [Fig.4].

[0069] It comprises a first step S21, during which several steps are executed both the partitioning step S20 and the pre-characterization step S24. This makes it possible to obtain a mapping of the a priori expected performances for the ROM-net, for different values ​​of parameters K and N. In this case, K is the number of clusters used to subdivide the set of temperature fields in step S20. And N is a number of elementary mechanical modes which compose reduced mechanical responses Vb.. .Vj,.. .VK occurring during the pre-characterization step S24, reduced mechanical responses which prefigure the mechanical behavior models Mb.. .Mj,..,MK which will be finally obtained.

[0070] Step S2 then comprises a selection step S22, during which it is determined whether the values ​​of the parameters K and N, tested in step S21, make it possible to satisfy certain performance conditions, and, in this case, an optimal pair of values ​​(Ko, No) is selected from these values. This selection step is carried out on the basis of the performances previously evaluated in step S21.

[0071] During step S21 (mapping step), partitioning step S20 is executed several times successively, for values ​​of K ranging from 2 to Kmax. Kmax is a maximum number of clusters, predetermined (for example set by a user), for example between 5 and 20.

[0072] And for each execution of step S20, the pre-characterization step S24 is executed several times successively, for different values ​​of N ranging for example from 1 to Nmax, the value of K remaining equal to the current value used during the last execution of the partitioning step S20.

[0073] This makes it possible to obtain a first estimate of the performances which could be obtained for a dictionary of reduced models formed of K mechanical behavior models Mb..., Mj, .. .MK, each of these models being composed of N elementary mechanical modes.

[0074] For these reduced models Mb..., Mj, .. ,MK to be of interest in terms of calculation acceleration (acceleration of the Ssim step, compared to a conventional finite element numerical simulation based directly on the mesh of blade 1), it is desirable in practice that the number N of modes (i.e.: the number N of degrees of freedom of each model Mj) be of the order of, or less than N1 / 3- Also, the Nmax value can be set for example to CN173 where C is a coefficient between 0 and 1, for example equal to 0.5 (indeed, there would be no point in pre-characterizing the performance of a ROM-net for a very large number of modes N, which we know will be of no interest in practice).

[0075] In step S20, the partitioning is carried out as explained below, for the current values ​​of K and N, to be tested.

[0076] The partitioning is carried out here on the basis of only a part of the temperature fields T; of the set E. This makes it possible to keep the other temperature fields (not used in step S20) in order to then test (in step S24), independently, the performances corresponding to this partitioning.

[0077] The part El of the set of temperature fields E, used to carry out the partitioning, here groups the fields Tn +i to Tn, with n'=n / 2. While the part E2, used in step S24 to evaluate the performance of this partitioning, groups the fields Ti to Tn •. More generally, each of these two parts groups together approximately half of the temperature fields of the set E (for example, plus or minus 20%). Since each temperature field is at least partially random, it can also be considered that each of these two parts groups together fields selected randomly from the set E (i.e., distributed randomly in the space of the temperature fields).

[0078] During step S20, K groups Ci,... Q,... ,CK of temperature fields are identified, in the set of temperature fields (E), each of these groups being characterized by one or several typical stress fields T;*, the simplified mechanical responses of which are representative of the set of simplified mechanical responses u; of the group Q considered.

[0079] These typical mechanical responses Tj* are sometimes called “snapshots” (i.e. “overviews”) in this technical field. In this case, for each group Cj, a number ns of typical temperature fields Tj* (belonging to this group) are identified in step S20.

[0080] For each of these typical temperature fields Tj*, a high-fidelity mechanical response will be determined in step S3. The quantity of calculations to be performed in step S3 therefore depends directly on the total number of typical temperature fields Tj* selected during the partitioning step, a number which is equal to K.ns (ns snapshots per cluster, and K clusters). In practice, to limit the quantity of calculations to be performed in step S3, the product K.ns is limited to a maximum value nSmax: K.ns -nSmax. nSmax can for example be between 10 and 40.

[0081] Furthermore, so that each mechanical behavior model Mj is as representative as possible of the mechanical responses of the corresponding cluster Q, it is desirable to determine the model from several high-fidelity mechanical responses (which are calculated for each of the typical temperature fields T;*), and therefore to maximize ns. Moreover, according to the model order reduction method used in step S4, the number N of modes making up each mechanical behavior model can be limited to the value of ns (i.e. N - ns), which, again, encourages using a relatively large value of ns. Also, to maximize ns while taking into account the constraints concerning the volume of calculations to be carried out, we choose here ns = nSmav / KJ, during each execution of step S20.

[0082] Step S20 is executed here using the so-called “k-medoids” algorithm, in this case in a 2-step hierarchical version. The “k-medoids” algorithm is thus executed a first time to identify the K clusters. Then, for each cluster, it is executed again to identify ns sub-clusters.

[0083] In the k-medoids algorithm, each cluster is represented by one of its members, called a medoid (unlike a cluster barycenter, the medoid is always an element of the database). It can be seen as the most central database element within the cluster. Thus, each of the K.ns sub-clusters provides a medoid, which is one of the typical temperature fields T)* in question. The ns medoids of the sub-clusters of cluster number j (i.e.: the ns typical temperature fields T;* of this cluster) will be used to construct the mechanical behavior model Mj of the dictionary.

[0084] The version of the k-medoids algorithm used may for example be its version called “PAM” (for “Partitioning Around Mediods”) described in detail in one or other of the following references: • L. Kaufmann and P. Rousseeuw. “Clustering by means of medoids. Data Analysis based on the Ll-Norm and Related Methods”, pages 405-416, 01 1987; • Chapter 2 in L. Kaufman, PJR Leonard Kaufman, and PJ Rousseeuw. “Finding Groups in Data: An Introduction to Cluster Analysis”. A Wiley-Interscience publication. Wiley, 1990.

[0085] In other embodiments, however, another version of this algorithm or another type of partitioning algorithm could be employed.

[0086] In any case, the partitioning step is carried out on the basis of distances ôü' between temperature fields: within each sub-cluster, the typical temperature fields T)* are identified as being the most central, in the sense of this distance ôir between temperature fields.

[0087] The distance ôir corresponds to a distance between the temperature field T; and the temperature field Tr. Here, this distance is representative of a dissimilarity between: the simplified mechanical response u; obtained for the temperature field T; on the one hand, and the simplified mechanical response u; • obtained for the temperature field Tr on the other hand. In this case, this dissimilarity is evaluated in accordance with the formula below:

[0088] _ / “y mW

[0089] This distance does not depend on the intensity of the fields u;, u; • but only on their shapes (it is linked to a notion of angle between mechanical responses). Indeed, if U; • is equal to u; to within a multiplicative coefficient, then the distance ôü- is zero. It is therefore well suited for the construction of a ROM-net, since it makes it possible to determine whether two different temperature fields will lead to similar mechanical behaviors (i.e.: of the same shape), or not. Moreover, in practice, a partitioning carried out on the basis of this distance leads to projection errors (in other words, representation errors) that are lower than with another type of distance such as the Euclidean distance between the fields u; and u; •.

[0090] However, in other embodiments, the distance used to perform the partitioning could be a distance other than that defined above. It could thus be the Euclidean distance, the so-called “L1” distance (also called “city-block” distance, or “Manhattan” distance), or a distance in the sense of Grassmann such as that used in the article by T. Daniel et al. mentioned above in the “technological background” section.

[0091] In practice, the distances ôü- corresponding to the different pairs of temperature fields (1), Tr) are calculated only once and stored in memory, before executing step S2. They are then used several times, during each execution of the partitioning step S20.

[0092] As explained above, the medoids of the ns sub-clusters of each of the K clusters, i.e. the typical temperature fields T;* associated respectively with these different sub-clusters, are identified from the temperature fields Tn + b .. .,Tn of the first part El of the set E.

[0093] Once this is done, we determine, for each temperature field Tb.. .1),... ,T n • of the second part E2, the number K( i) of the cluster to which the temperature field T; considered belongs (because this is useful later during step S24). To do this, we identify, among the medoids of the K clusters (without considering the medoids of the sub-clusters), the medoid which is closest to 1), in the sense of the distance ôir, and we then considers that the temperature field T; belongs to the same cluster as the corresponding temperature field.

[0094] As already indicated, the pre-characterization step S24 is then executed several times, for example for N ranging from 1 to Nmax.

[0095] Step S24 here comprises the following operations: • For each cluster Cb..., Q,..., CK, determination of a reduced mechanical response Vj characteristic of the cluster Cj considered, by model order reduction, from the simplified mechanical responses u; associated with the typical stress fields T)* of this cluster; • determination of a global reduced mechanical response, VG, by model order reduction, from the simplified mechanical responses u; corresponding to the typical stress fields T;* belonging to all the clusters (without being limited to a single cluster); • for each temperature field T; with i=l..n', determination of an individual projection error q which is a projection error between: • the simplified mechanical response u; corresponding to this temperature field 1), and • the reduced mechanical response Vj of the cluster Q to which the temperature field T in question belongs (i.e. for j = K(i)); • for each temperature field T; with i=l..n', determination of an error qGji which is a projection error between: • the simplified mechanical response u; corresponding to this temperature field T;, and • the reduced overall LV mechanical response. • determination of expected performance criteria for the ROM-net, for the values ​​of K, N, ns corresponding to this execution of step S24, these performance criteria being determined on the basis of the individual projection errors q; and the errors r|G>i.

[0096] The model order reduction technique used to determine the reduced mechanical responses Vb..., Vj,..., VK, and to determine the overall reduced mechanical response VG is a linear dimension reduction technique, for example of the POD type (according to the English acronym for "Proper Orthogonal Decomposition", i.e. decomposition orthogonal to the eigenvalues), or of the "POD snapshot" type.

[0097] Each reduced mechanical response Vb..., Vj,..., VK, as well as the global reduced mechanical response VG, here takes the form of a matrix containing N rows and N columns and such that VjV / = denotes the identity matrix of order N. The columns of

[0098]

[0099]

[0100]

[0101]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107]

[0108]

[0109] This matrix corresponds to the N elementary mechanical modes which make up the reduced mechanical response Vj. By projection error, we mean a difference, in this case a normalized difference between: the response u; considered on the one hand, and the orthogonal projection of this response onto the reduced mechanical response Vj or VG on the other hand. The individual projection error ip is then expressed as follows, here: ------------ avec I K L And, likewise: In the embodiment described herein, the performance criteria mentioned above include an average error h as well as a partitioning gain G. The average error is equal to the mean, in this case the arithmetic mean of the individual errors q;, for the different temperature fields Tb..., T;,.. .Tn ■ : As for the partitioning gain G, it is equal to the average of a ratio qGi / p between the projection error "without partitioning" qGi, and the individual projection error "with partitioning" rp. This arithmetic average is also calculated for the set of temperature fields Tb..., T;,.. .Tn -, i.e.: G = g{k, N, The gain G provides an estimate of the gain in precision that can be expected from a ROM-net based on a dictionary grouping K mechanical behavior models (and whose parameters are K, N and ns), compared to the use of a single global reduced model, identical whatever the temperature field considered. Figure 5 shows values ​​obtained for the average projection error 7, as a function of K and for different values ​​of N. As can be seen in this figure, the larger K and N are, the smaller this error is. These values ​​are obtained for N = 2000, for Nmax = 5 and nSmax = 20 (for this example, intended to illustrate the interest of the process, the value of N remains quite small; in practice, it can have a much larger value, to simulate as realistically as possible the fatigue resistance of a turbine blade). For each curve plotted in this figure, the dotted part corresponds to configurations for which the amount of calculation that would be necessary in step S3 is considered too large, because it does not respect the condition ns - nSmax / K (which results in the exclusion of certain values ​​of N, because N is at most equal to ns, and therefore at most equal to nSmax / K = 20 / K, here).

[0110] [Fig.6] shows values ​​obtained for the partitioning gain G, under the same conditions as for [Fig.5]. Here too, for each curve plotted in this figure, the dotted part corresponds to configurations for which the amount of computation that would be necessary in step S3 is considered too large. As can be seen in this figure, the partitioning gain is all the greater as the number of clusters K is large. On the other hand, the gain G does not increase monotonically with the number of modes N, which may seem surprising at first sight and underlines the interest of carrying out this mapping step before selecting the ROM-net parameters. Here, for example, it is maximal for N=3 (and not for N=5).

[0111] The selection step S22 is executed, after step S21, on the basis of the expected performances and G determined in step S21.

[0112] During step S22, for the different pairs of values ​​that can be envisaged for K and N (values ​​that go from 2 to Kmax for K, and from 1 to Nmax for N), it is tested whether the pair of values ​​considered makes it possible to satisfy certain pre-established performance conditions. The pair or pairs of values ​​for which this test is positive constitute admissible pairs of values.

[0113] Here, these performance conditions to be satisfied are: • that the average error W is less than the maximum admissible error q* (relevance condition), and • that the partitioning gain G is greater than a limit gain g (profitability condition).

[0114] The maximum permissible error q* can, for example, be between 0 and 0.5. In the case of [Fig.5], for example, it is 0.35.

[0115] As for the limit gain g, it represents a threshold above which we consider that a dictionary with several mechanical behavior models is more interesting than a single global reduced model (due to the gain in precision provided by the dictionary), despite the complexity brought by the use of such a dictionary. This limit gain can for example be greater than or equal to 2.

[0116] Here, the limit gain g is in fact a function which depends on K, in this case which increases with K. This makes it possible to take into account the fact that the attribution error, by classifier 2, increases when K increases (when the number of clusters is very large, the attribution errors, or in other words the classification errors increase). For the use of a dictionary with K models to be profitable, it is therefore necessary that the gain in precision is substantial when K is high, to compensate for these possible attribution errors.

[0117] To take into account these possible allocation errors, the limit gain g(K) is determined here in accordance with the following formula:

[0118] / ,3 #=W=........S)""'" • where p(K) is the probability of an exact assignment by the temperature field classifier, when the number of clusters is K, • where G* is a gain threshold, for which the use of a dictionary (rather than the use of a global model) is considered profitable, the attribution by the classifier being assumed to be perfect (without error); G* is for example greater than or equal to 2 (in the case of [Fig.6], it is equal to 2), and where • E is a value strictly between 0 and 1, representing an average ratio between the projection error on a global reduced model, and the projection error on a model in the dictionary which would not be the right one (due to an attribution error; As an example, here, E is equal to 0.75.

[0119] As for the values ​​of p(K), we have p(1)=1, and p(K) which decreases with K. As an example, we can take p(6)=0.8 (value taken from the article by T. Daniel et al. mentioned above), and p(Kmax)=1 / Kmax.

[0120] The curve g(K) obtained for these example values ​​is represented in dashes in [Fig.6].

[0121] For the numerical example corresponding to Figs. 5 and 6, five pairs of admissible values ​​for K and N (for which the ROM-net is expected to be both relevant and profitable) are identified. In these figures, the points corresponding to these pairs of admissible values ​​are surrounded by a circle. These are the following pairs: (K=4, N=3) and (K=4, N=4), (K=5, N=3) and (K=5, N=4) and (K=6, N=3).

[0122] It will be noted that other performance conditions than those indicated above could be used. Thus, in other embodiments, only the relevance condition, relating to 7, could be taken into account, for example. Or, the performance condition could relate to a combination of 7 and G (such as the sum G+1 / 7), to be compared to a threshold value. Furthermore, instead of calculating an average of the individual projection errors, and testing whether it is less than a threshold (to test whether the corresponding parameters are admissible), one could calculate an average of the inverses of the individual projection errors (in a way, an average of the individual projection accuracies), and test whether this average is greater than a given accuracy threshold.

[0123] In any case, if only one pair of admissible values ​​has been identified for the parameters K and N, it is this pair which is retained to then execute steps S3 and S4.

[0124] And if several pairs of admissible values ​​are identified (as is the case for the example in Figures 5 and 6), a pair of optimal values ​​(Ko, No) satisfying one or more optimization conditions is selected from these pairs. It is then this pair of optimal values ​​which is then used to execute steps S3 and S4.

[0125] Here, the pair of optimal values ​​is selected so as to minimize N (here N=3), and then, among the pairs for which N is minimal, so as to maximize K (here K=6). The pair of optimal values ​​is therefore (K=6, N=3), here.

[0126] Choosing the smallest admissible value of N is very interesting from the point of view of the speed of the ROM-net which will finally be obtained (at the end of the process of [Fig.3]). Indeed, as the mechanical behavior models will then each be composed of a small number of elementary mechanical modes, the long-term simulations carried out during the Ssim step (once in the ROM-net usage phase) will require only a few calculations.

[0127] Then choosing the largest admissible value of K (N remaining minimal) makes it possible to maximize the precision of the ROM-net (see [Fig.5]).

[0128] Alternatively, other optimization conditions could be used. For example, if the speed of the ROM-net is less crucial, and one wishes to favor optimal precision, one could select, among the pairs of admissible values, the one for which K is the largest, and then, K being maximized, the one for which N is minimal (which is slightly different from the optimization criterion given above). Or, one could directly select the pair for which the average error 7 is the smallest. One could also choose the pair of values ​​of K and N so as to be as close as possible to a bend that the curve presents or G(K) (because, beyond the bend in question, an increase in K only allows a modest gain in precision, by "bend" one designates for example the point of the curve where the radius of curvature is the strongest).

[0129] The selection step S22 may further comprise a step of interaction with a user, via a human-machine interface (such as a touch screen, for example). During this interaction step, if no pair of admissible values ​​could be identified for the parameters K and N of the ROM-net, the user is offered to modify the values ​​of q* and / or G* or E, for example in the direction of a lower performance requirement.

[0130] During this interaction step, the results of step S21 can be communicated to the user, via the human-machine interface (for example in the form of the graphs of figures 5 and 6, displayed on the interface), to allow this user to check the performances likely to be obtained in the end before launching the high-fidelity digital simulation.

[0131] Steps S3 and S4: high-fidelity digital simulation and determination of the mechanical behavior model(s).

[0132] When at least one pair of admissible values ​​has been identified in step S2 for the parameters K and N, the method continues with the determination of a ROM-net with K mechanical behavior models.

[0133] In this case, step S3 is executed on the basis of the partitioning corresponding to the value of K retained at the end of step S2. This is for example the optimal value Ko mentioned above. We then determine, for the different typical temperature fields Tj* identified during the partitioning corresponding to the value Ko, the high-fidelity mechanical responses u^F corresponding to these fields (which are in a way “high-fidelity snapshots”).

[0134] Then, during step S4, for each of the Ko clusters, the mechanical behavior model Mj of the cluster Q considered is determined (step S41 of FIG. 3), by model order reduction, from the high-fidelity mechanical responses U^F of this cluster, the model in question being composed of a number of elementary mechanical modes equal to the optimal value No. The model order reduction method used is the same as that used to determine the reduced mechanical responses Vj.

[0135] During step S5, the temperature field classifier 2 is trained on the basis of the partitioning corresponding to the optimal value Ko in question (or, failing that, corresponding to the only admissible value of K that could be identified in step S2).

[0136] When no pair of admissible values ​​could be identified in step S2 for the parameters K and N, the method continues by determining a single global mechanical behavior model Mo.

[0137] In this case, after step S3, this global model is determined by model order reduction, from all the high-fidelity mechanical responses u^F determined in step S3, without taking into account their belonging to this or that cluster (step S42 of FIG. 3). The high-fidelity mechanical responses uVF in question are determined for the typical temperature fields T)* identified during one of the partitionings carried out (and tested) in step S2. This partitioning, from which the global mechanical behavior model Mo is determined, can be selected, from among the different partitionings carried out, as explained below.

[0138] As projection errors qGi on a global reduced response VG have been determined for each partitioning and for each value of N considered, during the executions of step S24, it is then possible to calculate, for each pair of values ​​of K and N, a global average error ^g, equal to the average of the projection errors riGi for this case is - I1 1 V / ? • We can then choose to determine ' nG\K,N,ns) = -.L^Gi p the global model Mo for the pair of values ​​of K and N leading to the smallest value of the global average error ^g, or, for example, for the pair of values ​​of K and N for which N is the smallest (to promote speed, once in the use phase), associated with the value of K giving, for said value of N, the lowest average projection error.

[0139] Different variants can be made to the method which has just been described, in addition to those already mentioned above.

[0140] For example, instead of carrying out the mapping step S21, then the selection step S22, one could test the performance conditions mentioned above after each execution of step S24, as and when, and stop the iterations of steps S20 and S24 as soon as these conditions are met, that is to say as soon as a pair of admissible values ​​could be identified for K and N.

[0141] Furthermore, the constraints related to the computation time of step S3 (which are translated here by ns=nSmax / K) could be taken into account in different ways. And instead of executing steps S20 and S24 for values ​​of the parameters K, N and ns chosen directly so as to satisfy these computation time constraints of step S3, one could execute steps S20 and S24 for a wider range of values, and then select a triplet of values ​​of K, N and ns on the basis of performance conditions relating to the relevance and profitability of the ROM-net (as regards K and N), but also on its speed of obtaining (as regards ns).

Claims

1. Claims Computer-implemented method for characterizing the aging of a turbine blade (1) comprising a determination of one (Mo) or more mechanical behavior models (Ml, Mj, MK) of the turbine blade (1), adapted to a set (E) of stress fields (Tl, Ti, Tn) to be exerted on said blade (1), the method comprising the following steps: - (SI) simplified numerical simulation step, during which, for each stress field (Tl, Ti, Tn) of said assembly (E), a simplified mechanical response (ui) of the turbine blade (1) subjected to the stress field (Ti) considered is determined, - (S20) partitioning step, during which we identify K groups (Cl, Cj, CK) of distinct stress fields in the set of stress fields (E), each group (Cj) being characterized by one or a few typical stress fields (Ti*), whose simplified mechanical responses are representative of the set of simplified mechanical responses (ui) of the group (Cj) considered, - (S3) high-fidelity digital simulation step, at during which, for each typical stress field (Ti*) previously identified, a high-fidelity mechanical response (w^F) is determined, for the turbine blade subjected to the typical stress field (Ti*) considered, then - (S4) step of determining the model(s) of mechanical behavior (Mo, Ml, Mj, MK), from said high-fidelity mechanical responses (tt^), by model order reduction, - the method further comprising, before the high-fidelity digital simulation step (S3), a pre-characterization step (S24) during which one or more performance criteria (h, G), expected for the mechanical behavior model(s) (Mo, Ml, Mj, MK) taking into account said partitioning, are determined, the performance criterion(s) (h, G) being determined in function of individual projection errors (pi) calculated for different stress fields (Tl, Ti, Tn') of said set (E), each individual projection error (pi) corresponding to a projection error between: • the simplified mechanical response (ui) associated with the stress field (Ti) considered, and • a reduced mechanical response (Vj), associated with the group (Cj) to which the stress field (Ti) considered belongs, the reduced mechanical response (Vj) having been determined by model order reduction from the simplified mechanical responses (ui) associated with the typical stress fields (Ti*) of said group (Cj).

2. Method according to claim 1, in which each stress field is a temperature field (Tl, Ti, Tn) exerted on the turbine blade (1).

3. Method according to claim 1 or 2, in which - the partitioning step (S20) is executed several times, for different values (2, Kmax) of the number of groups K, and in which - after each execution of the partitioning step, the pre-characterization step (S24) is executed, so as to pre-characterize the partitioning into K groups (Cl, Cj, CK) obtained previously.

4. Method according to one of claims 1 to 3, in which, - each reduced mechanical response (Vj) is composed of a number N of elementary mechanical modes, and in which - after each execution of the partitioning step (S20), the pre-characterization step (S24) is executed several times successively for different values (1, Nmax) of the number N of elementary mechanical modes composing the reduced mechanical responses.

5. Method according to claims 3 and 4, • comprising a parameter selection step (S22) during which, for each pair of values grouping one of said values (2, Kmax) of the number K and one of said values (1, Nmax) of the number N, it is tested whether the performance criteria ('1, G), determined for this pair of values of K and N, satisfy one or more pre-established performance conditions, and the pair or pairs of values for which the test is positive are selected, this or these pairs of values constituting pairs of admissible values, and in which • when at least one pair of admissible values has been selected: • during the high-fidelity numerical simulation step (S3), the high-fidelity mechanical responses (U^) are determined for the typical stress fields (T;*) corresponding to the partitioning for which the value of K is that of said pair of admissible values, and • during step (S4) of determining the mechanical behavior model(s), the (Mo) or models (Mb Mj, MK) determined are each composed of a number of elementary mechanical modes equal to the value of N given by said pair of admissible values.;

6. Method according to claim 5 wherein, during the parameter selection step (S22), a pair of optimal values (Ko,No) satisfying one or more pre-established optimization conditions is further selected from among the pairs of admissible values of the numbers K and N, the steps of high-fidelity digital simulation (S3) and determination of the mechanical behavior model(s) (S4) being executed for this pair of optimal values of the numbers K and N.

7. Method according to claim 6 in which the pair of values of K and N satisfying said optimization conditions is the pair grouping: that of the admissible values of N which is minimal, and N being minimized, that of the admissible values of K which is the largest.

8. A method according to claim 5 wherein, when no pair of admissible values of the numbers K and N could be selected during of the parameter selection step (S22), then, during the step (S4) of determining the mechanical behavior model(s), a single global mechanical behavior model (Mo) is determined, for the set (E) of said stress fields.

9. Method according to one of claims 1 to 8 in which the performance criterion(s) determined during the pre-characterization step (S24) comprise an average projection error (h), equal to an average of said individual projection errors Oli)-

10. A method according to claim 9, taken in dependence on one of claims 5 to 8, wherein said or one of said performance conditions is that said average projection error (H) is less than a given maximum permissible error (q*).

11. Method according to one of claims 1 to 10 in which - the performance criterion(s) determined during the pre-characterization step (S24) comprise a partitioning gain (G) which is determined as a function of an average, for different stress fields of said set (E), of the ratio between: a projection error (qGi) associated with a global reduced mechanical response (VG) on the one hand, and the individual projection error (qi) obtained for the stress field (Ti) considered on the other hand, - the global reduced mechanical response (VG) being the same regardless of the stress field (Tj) considered, and having been determined by model order reduction from the simplified mechanical responses (ui) corresponding to the set of typical stress fields (T;*) identified during the partitioning step.

12. A method according to claim 11, taken in dependence on one of claims 5 to 8, wherein said or one of said performance conditions is that the partitioning gain (G) is greater than a given limit gain (g).

13. Method according to one of claims 1 to 12 wherein, during the partitioning step (S20), the typical stress fields (T;*) are identified on the basis of a dissimilarity (ôü ) between simplified mechanical responses, the dissimilarity (ôir) between two simplified mechanical responses (u;, u; ) being calculated as a function of a normalized scalar product equal to the scalar product of the two mechanical responses considered (iii, u; ), divided by a norm of a first (uj) of these two responses and by a norm of a second (u; ) of these two responses.

14. Electronic device comprising at least one processor and one memory, programmed to execute the method according to any one of the preceding claims.

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