An error indicator-based adaptive reduced order model for nonlinear structural mechanics -- application to high-pressure turbine blades

The industrial application motivating this work is the fatigue computation of aircraft engines' high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes {the classical unenriched proper orthogonal decomposition method} fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the {error indicator} becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving 5 million degrees of freedom, where the whole procedure is computed in parallel with distributed memory

Nonintrusive approximation of parametrized limits of matrix power algorithms -- application to matrix inverses and log-determinants

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones

Parametrized non intrusive space-time approximation for explicit dynamic fem applications

In the following work, a benchmark of different non-intrusive model reduction approaches is performed on an explicit dynamic contact 3D-problem. The main purpose of this work is to evaluate the stability of the reduced model with respect to time along with the precision of these approaches with respect to the true solutions of interest. These solutions are the prediction of displacement and velocity fields. The precision of these approaches is also evaluated with respect to the evolution of some materials parameters. Six parameters vary in this study and we would like to predict the whole transient fast dynamic impact response with respect to each parameters. To this end, several models are trained : Proper Orthogonal Decomposition (POD) and Deep convolutional Neural Network (DcNN), in addition, a vectorized version of Interpolation in Grassman Manifolds is proposed. The benchmark performed illustrate that using DcNN’s allows to achieve the best precision and stability in predicting physical fields

Mathematical study of the sensitivity of the POD method (Proper orthogonal decomposition)

Dans cette thèse, nous nous sommes intéressés à l’étude mathématique de la sensibilité paramétrique de la méthode de réduction de modèles par projection connue sous le nom de POD pour Proper Orthogonal Decomposition. Dans beaucoup d’applications de la mécanique des fluides,la base de projection (base POD) calculée à un paramètre caractéristique fixe du problème de Navier-Stokes, est utilisée à la suite pour construire des modèles d’ordre réduit ROM-POD pour d’autres valeurs du paramètre caractéristique. Alors, la prédiction du comportement de ce ROM-POD vis-à-vis du problème initial est devenue cruciale. Pour cela, nous avons discuté cette problématique d’un point de vue mathématique. Nous avons établi des résultats mathématiques de sensibilité paramétrique des erreurs induites par application de la méthode ROM-POD. Plus précisément, notre approche est basée sur l’établissement d’estimations a priori de ces erreurs paramétriques, en utilisant les méthodes énergétiques classiques. Nos résultats sont démontrés pour les deux problèmes de type Burgers et Navier-Stokes. Des validations numériques de ces résultats mathématiques ont été faites uniquement pour le problème de type Burgers.In this thesis, we are interested in the mathematical study of the parametric sensitivity of the reduced order model method known as the POD method (proper orthogonal decomposition). In several works applied to fluid mechanics, the POD modes are computed once and for all in association with a fixed parameter that characterize the equations of the fluid mechanics : Navier-Stokes system. Then, these modes are used in order to compute reduced order models (ROM) associated to these equations, for different parameter values. So, one needs a tool for predicting the behavior of the reduced order model with respect to the complete problem, when the parameter’s value is changing. We have discussed this problem from a mathematical point of vue. In fact, we have established mathematical results on the parametric sensitivity of the errors induced by applying the ROM-POD method. More precisely, our work is based on developing a priori estimations of these parametric errors, by using classical techniques of energy estimation.Our results are proved for the two problems of Burgers and Navier-Stokes. Numerical validations are established only in the case of the Burgers equation

Etude mathématique de la sensibilité POD (Proper orthogonal decomposition)

In this thesis, we are interested in the mathematical study of the parametric sensitivity of the reduced order model method known as the POD method (proper orthogonal decomposition). In several works applied to fluid mechanics, the POD modes are computed once and for all in association with a fixed parameter that characterize the equations of the fluid mechanics : Navier-Stokes system. Then, these modes are used in order to compute reduced order models (ROM) associated to these equations, for different parameter values. So, one needs a tool for predicting the behavior of the reduced order model with respect to the complete problem, when the parameter’s value is changing. We have discussed this problem from a mathematical point of vue. In fact, we have established mathematical results on the parametric sensitivity of the errors induced by applying the ROM-POD method. More precisely, our work is based on developing a priori estimations of these parametric errors, by using classical techniques of energy estimation.Our results are proved for the two problems of Burgers and Navier-Stokes. Numerical validations are established only in the case of the Burgers equation.Dans cette thèse, nous nous sommes intéressés à l’étude mathématique de la sensibilité paramétrique de la méthode de réduction de modèles par projection connue sous le nom de POD pour Proper Orthogonal Decomposition. Dans beaucoup d’applications de la mécanique des fluides,la base de projection (base POD) calculée à un paramètre caractéristique fixe du problème de Navier-Stokes, est utilisée à la suite pour construire des modèles d’ordre réduit ROM-POD pour d’autres valeurs du paramètre caractéristique. Alors, la prédiction du comportement de ce ROM-POD vis-à-vis du problème initial est devenue cruciale. Pour cela, nous avons discuté cette problématique d’un point de vue mathématique. Nous avons établi des résultats mathématiques de sensibilité paramétrique des erreurs induites par application de la méthode ROM-POD. Plus précisément, notre approche est basée sur l’établissement d’estimations a priori de ces erreurs paramétriques, en utilisant les méthodes énergétiques classiques. Nos résultats sont démontrés pour les deux problèmes de type Burgers et Navier-Stokes. Des validations numériques de ces résultats mathématiques ont été faites uniquement pour le problème de type Burgers

Sur la sensibilité paramétrique de l'Hyper-réduction en Dynamique des structures

International audienceDans cette communication, nous présentons un résultat mathématique sur la sensibilité paramétrique de la technique d'Hyper-réduction de modèles, par une base réduite obtenue grâce à la méthode de la décomposition orthogonale aux valeurs propres (POD) et un domaine d'intégration réduit (RID) associé à cette base. Nous nous intéressons plus particulièrement au cadre des équations hyperboliques de la dynamique des structures avec une loi de comportement visco-élastique. Nous présenterons au cours de la conférence une comparaison avec des simulations numériques pour la configuration d'un cas académique de dynamique d'une plaque d'acier

A novel Gappy reduced order method to capture non-parameterized geometrical variation in fluid dynamics problems

In this work, we propose a new Gappy reduced order method to fill the gap within an incomplete turbulent and incompressible data field in such a way to satisfy the physical and topological changes of the fluid flow after a non-parameterized geometrical variation in the fluid domain 1. A single baseline simulation is assumed to be performed prior geometrical variations. The proposed method is an enhancement of the Gappy-POD method proposed by Everson and Sirovich in 1995, in the case where the given set of empirical eigenfunctions is not sufficient and is not interpolant for the recovering of the modal coefficients for each Gappy snapshot by a least squares procedure. This happens when the available data cannot be written as an interpolation of the baseline POD modes. This is typically the case when we introduce non-parameterized geometrical modifications in the fluid domain. Here, after the baseline simulation, additional solutions of the incompressible Navier-Stokes equations are solely performed over a restricted fluid domain, that contains the geometrical modifications. These local Large Eddy Similations that we will call hybrid simulations are performed by using the immersed boundary technique, where the latter is a fluid boundary and is defined by the baseline velocity field. Then, we propose to repair the POD modes using a local modification of the baseline POD modes in the restricted fluid domain. The modal coefficients of the least squares optimization of the Gappy-POD technique are now well recovered thanks to these updated modes, i.e. the residual of the Gappy-POD technique in the restricted fluid domain is now equal to zero. Furthermore, we will propose a physical correction of the latter enhanced Gappy-POD modal coefficients thanks to a Galerkin projection of the full Navier-Stokes equations upon the new compression modes of the available data. This repairing procedure of the global velocity reconstruction by the physical constraint was tested on a 3D semi-industrial test case of a typical aeronautical injection system. The speed-up relative to this new technique is equal to 100, which allows us to perform an exploration of two new designs of the aeronautical injection system

Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of 2.2 ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC

Data-Targeted Prior Distribution for Variational AutoEncoder

International audienceBayesian methods were studied in this paper using deep neural networks. We are interested in variational autoencoders, where an encoder approaches the true posterior and the decoder approaches the direct probability. Specifically, we applied these autoencoders for unsteady and compressible fluid flows in aircraft engines. We used inferential methods to compute a sharp approximation of the posterior probability of these parameters with the transient dynamics of the training velocity fields and to generate plausible velocity fields. An important application is the initialization of transient numerical simulations of unsteady fluid flows and large eddy simulations in fluid dynamics. It is known by the Bayes theorem that the choice of the prior distribution is very important for the computation of the posterior probability, proportional to the product of likelihood with the prior probability. Hence, we propose a new inference model based on a new prior defined by the density estimate with the realizations of the kernel proper orthogonal decomposition coefficients of the available training data. We numerically show that this inference model improves the results obtained with the usual standard normal prior distribution. This inference model was constructed using a new algorithm improving the convergence of the parametric optimization of the encoder probability distribution that approaches the posterior. This latter probability distribution is data-targeted, similarly to the prior distribution. This new generative approach can also be seen as an improvement of the kernel proper orthogonal decomposition method, for which we do not usually have a robust technique for expressing the pre-image in the input physical space of the stochastic reduced field in the feature high-dimensional space with a kernel inner product

A priori compression of convolutional neural networks for wave simulators

Convolutional neural networks are now seeing widespread use in a variety of fields, including imageclassification, facial and object recognition, medical imaging analysis, and many more. In addition, there areapplications such as physics-informed simulators in which accurate forecasts in real time with a minimal lag arerequired. The present neural network designs include millions of parameters, which makes it difficult to installsuch complex models on devices that have limited memory. Compression techniques might be able to resolvethese issues by decreasing the size of CNN models that are created by reducing the number of parametersthat contribute to the complexity of the models. We propose a compressed tensor format of convolutionallayer, a priori, before the training of the neural network. 3-way kernels or 2-way kernels in convolutionallayers are replaced by one-way fiters. The overfitting phenomena will be reduced also. The time neededto make predictions or time required for training using the original Convolutional Neural Networks modelwould be cut significantly if there were fewer parameters to deal with. In this paper we present a methodof a priori compressing convolutional neural networks for finite element (FE) predictions of physical data.Afterwards we validate our a priori compressed models on physical data from a FE model solving a 2D waveequation. We show that the proposed convolutional compression technique achieves equivalent performance inthe prediction error as classical convolutional layers with fewer trainable parameters(around 20%) and lowermemory footprint