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In this paper, computer vision enables recommending a reduced order model for fast stress prediction according to various possible loading environments. This approach is applied on a macroscopic part by using a digital image of a mechanical test. We propose a hybrid approach that simultaneously exploits a data-driven model and a physics-based model, in mechanics of materials. During a machine learning stage, a classification of possible reduced order models is obtained through a clustering of loading environments by using simulation data. The recognition of the suitable reduced order model is performed via a convolutional neural network (CNN) applied to a digital image of the mechanical test. The CNN recommend a convenient mechanical model available in a dictionary of reduced order models. The output of the convolutional neural network being a model, an error estimator, is proposed to assess the accuracy of this output. This article details simple algorithmic choices that allowed a realistic mechanical modeling via computer vision.

In biomechanics, computer vision and mechanical testing have been coupled to obtain patient-specific simulation approaches, as proposed in [

In this paper, we restrict our attention to the stress prediction in a part under an observed loading environment by a digital image, while including all its geometrical defects. We propose a hybrid approach that simultaneously exploits a data-driven model and a physics-based model, in mechanics of materials. The reader can find a review on hybrid modeling in [

As explained in [

Image processing for computer vision is usually very fast. It does not make sense to couple computer vision with numerical simulations of mechanical stresses that take hours of computation. Hence, we couple computer vision with reduced order modeling of structures, in order to get fast mechanical predictions of stresses.

A reduced order model is a surrogate model obtained by the projection of high dimensional equations on a reduced space, and it also involves a reduced approximation space for the variables of the high dimensional problem. When they are the same, the surrogate model is a Galerkin reduced order model [

In general, the empirical modes obtained by noncentred PCA are very sensitive to the loading environment imposed when computing the simulation data. If the variety of loading conditions considered to calculate simulation data is too wide, the number of empirical modes becomes too large. They can no longer reduce the numerical complexity of the mechanical balance equations. Clustering methods have been applied for model-order reduction in [

Prior to the stress prediction, a reduced order model is setup for the projection of the mechanical balance equations. Here the reduced order model concerns the displacement in the observed mechanical part. Usually, for the computation of reduced approximations in nonlinear problems, we formally consider all possible situations in a given parameter space [

The workflow of the proposed modeling via computer vision is shown in Figure

a 2D digital image of the part in the test machine, this image is denoted by

a database where are saved all simulation data, ordered with respect to a cluster index

a 3D voxel image of the part alone, we assumed that this 3D image is obtained by X-ray computed tomography;

a measurement of the load magnitude at the end of the mechanical test and a measurement of the displacement at one fixed point of on the part, at the end of the test.

Workflow of the reduced order modeling via computer vision. The inputs are on the top of the figure; the outputs are the stress prediction

The CNN network aims to recognize the index

The stress predicted by the hyperreduction is obtained via constitutive equations in the framework of elasticity. It depends on the spatial position in the part, denoted by

The following property holds: if the finite element solution is unique, if the solution of the hyperreduced equation is unique, and if the reduced basis is exactly reproducing the finite element solution, the hyperreduced solution is exact. Hence the following expression holds:

In hybrid hyperreduced order models proposed in [

As shown in Figure

The clustering of loading environments in the mechanical tests is performed by using the simulation data mentioned in the workflow in Figure ^{5} and can reasonably be up to 10^{7} in industrial applications. In very high-dimension spaces, all the data are “far away” from the centre. Hence, a feature extraction is required in the framework of mechanical modeling, prior to clustering the simulation data. Several tensor decomposition methods are available in the literature for feature extraction. For instance, the k-PCA has been coupled to the proper generalized decomposition method in [

The CNN architecture chosen for this work is based on the layer composition initially described in [

The image goes through a set of convolutional layers followed by max-pooling layers. The convolution and pooling stack is repeated 3 times. The rectified linear unit (ReLu) activation function [

We consider a very simple mechanical test on a part in order to check its manufacturing process via the stress distribution in the part and the related response of the part submitted to various loading environments on the top of it. An image of the experimental setup is shown on Figure

Experimental setup. A mechanical load of 159.9 N is applied on the top of the part in white. A displacement is measured at a fixed point.

The region of interest shown in red in Figure

The red, blue, and yellow regions have been submitted to variations of the Young modulus in order to enhance the space spanned by the simulation data. The region in red is the region of interest selected by the designer of the experimental setup. The mesh shown here is the mesh G_{4}. It has 5. 10^{5} degrees of freedom.

Prior to starting the manufacturing process, the experimental setup has been designed by using finite element simulations on four ideal geometries (M=4). The material of the part is elastic, for each mesh

For each geometry, three local variations of the Young modulus of -20% have been simulated by the finite element model, in order to enhance the space spanned by using the simulation data. The regions affected by these variations are shown in Figure

An example of the remapping of a vertical displacement field onto the bounding box is shown on Figure ^{5} dofs. The first feature mode (the first column of

On the left, examples of vertical displacements in the bounding box superposed to the related finite element prediction, for the mesh G_{1} on the top and the mesh G_{4} on the bottom, with a colour scale related to the displacement magnitudes. On the right, the first feature mode

We have arbitrary chosen K = 4 centroids to cluster the mechanical loadings. The L points

L=18 points

About 250 high resolution digital images for each class of loading have been generated before starting any mechanical experiment. Examples of such images are shown on Figure

Sample images for each of the four loading environment classes.

The CNN has been implemented with Keras library in TensorFlow. The train/test set was built following a 90/10 ratio upon the 1000 digital images of loading environments. The volume of training data was artificially increased by using a synthetic data augmentation strategy. Each “original” image was transformed using a combination of random sheer, zoom, and horizontal flip values. The sheer was limited to a maximum of 10% and the zoom to a maximum of 30%.

The training was performed by optimizing a multinomial cross entropy loss using a minibatch gradient descend approach with the RMSprop adaptive learning rate method; batch size is set to 32 and the model is trained on 120K steps (60 epochs).

The performance of the CNN is assessed on the test set; a top-1 error value of 1.9% was achieved. This great performance is explained by the easiness of the classification task due to no large variations between input images being observed, since they are all related to the same experimental setup.

The modeling via computer vision has been applied to a mechanical test. The mesh _{4}, so ^{5} dof in

The digital image of the mechanical part in the experimental setup is shown in Figure

Digital image

The reduced mesh

A transparency effect has been added to the full mesh

A finite element simulation takes 45 min. The stress prediction by the hyperreduced order model shown in Figure

von Mises stress,

The exact error on the average stress in the region of interest is 0.1%. The map of the error estimator is shown in Figure

Error estimator

The proposed reduced order modeling is related to very huge parameter space of dimension twelve millions, mainly due to the input image of the loading environment. The accuracy of the stress prediction is satisfactory to assess the quality of the process with mechanical considerations, even if the region of interest is hidden by the experimental setup. It is also reasonably fast in order to be inserted into a manufacturing process, aiming for part-specific decisions.

The output of the proposed workflow has a high spatial resolution. This is achieved by coupling a PCA, a clustering and a convolutional neural network. A local error estimator aims to indicate the discrepancy between the output and the stress that a finite element simulation would give corresponding to the loading environment recognized by the convolutional neural network. But this error indicator does not evaluate recognition errors, neither error on the mechanical behaviour of the observed material. So, the hyperreduced order model may not be the best that the available data could give.

In this paper, the inputs of the reduced order model are nonparametric loading conditions. They are defined solely by images of the loading environment. Since digital colour images are third order tensors, we expect a possible generalization of the workflow to more complex thermomechanical loading environments and more complex variations for the geometry of the observed parts.

Here, no Big-Data is required to train the proposed reduced order modeling via computer vision. The training starts with a nonsupervised machine learning by using a noncentred PCA and a clustering procedure, both on simulation data. Then, the CNN is trained on digital images by supervised machine learning upon the classes defined by the clustering procedure. Obviously, this approach can be implemented with larger sets of data.

The data used to support the findings of this study are available from the corresponding author upon request.

One of the authors is a Safran employee.

The experimental setup and the computer used for this publication have been funded by Mines ParisTech PSL Research University.