Complexity

Volume 2018, Article ID 3791543, 10 pages

https://doi.org/10.1155/2018/3791543

## Computer Vision with Error Estimation for Reduced Order Modeling of Macroscopic Mechanical Tests

^{1}Centre des Matériaux, Mines ParisTech PSL Research University, Evry 91003, France^{2}Safran Analytics, rue des Jeunes Bois, Châteaufort, CS 80112, 78772 Magny les Hameaux Cedex, France^{3}CEMEF, Mines ParisTech PSL Research University, CS 10207, 06904 Sophia Antipolis Cedex, France

Correspondence should be addressed to David Ryckelynck; rf.hcetsirap-senim@kcnylekcyr.divad

Received 29 June 2018; Accepted 13 November 2018; Published 2 December 2018

Academic Editor: Francisco J. Montáns

Copyright © 2018 Franck Nguyen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

In biomechanics, computer vision and mechanical testing have been coupled to obtain patient-specific simulation approaches, as proposed in [1]. At the same time, with the growth of industry 4.0, imaging techniques are more and more widespread in factories. When combined with artificial neural networks, digital images enable the classification of products to obtain the best possible process, as proposed in [2] for olive batches classification in oil extracting process or as shown in [3] for composite materials manufacturing. Nowadays, we have the possibility of extending these methods to the classification of mechanical parts produced in industry, in order to develop part-specific decision approaches. For mechanical parts, the quality of manufacturing processes has a direct influence on the ultimate mechanical properties of the manufactured parts. For example, the way the fracture is initiated in a specimen often reveals defects in the material whose origin can be tracked back to the manufacturing process [4]. In general, the numerical computation of mechanical stresses in a given manufactured part allows the predictive evaluation of the link between the ultimate mechanical properties of this part and the manufacturing process. The reader can find an example of how to optimize a process for curing composite parts in [5] according to this paradigm. The mechanical modeling of manufactured parts has for purpose to verify if defects induced by a manufacturing process are tolerable, if an observed part must be rejected, or if the manufacturing process must be improved.

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 [6] for remaining useful life predictions of engineering systems. We show that the strength of the proposed hybrid modeling is its ability to incorporate an error estimator related to the modeling chain with computer vision and convolutional neural networks (CNN) [7, 8].

As explained in [9], computer vision with deep convolutional neural networks has achieved state-of-the-art performance on standard recognition datasets and tasks. In this paper, we explore the capabilities of a CNN as a recommender system for the mechanical modeling of structures submitted to various loading. The proposed hybrid modeling couples a noncentered principal component analysis (PCA) and a CNN, in order to preserve an accurate description of spatial information.

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 [10]. In hyperreduced order models these two reduced spaces are different [11]. Then, because we consider projection of physics-based equations, hyperreduced order models preserve the physical parameters involved in the high dimensional equations. The reduced spaces involved in the hyperreduced modeling are spanned by empirical modes extracted from simulation data, by using the proper orthogonal decomposition [10] of known finite element predictions. This procedure is similar to a noncentred principal component analysis (PCA). Hence, the proposed modeling via computer vision exploits both simulation data and observational data, which are, respectively, finite element predictions and digital images of mechanical tests.

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 [12, 13], in order to preserve small reduced-bases of empirical modes. Moreover, cluster-based reduced order modeling (CROM) has been proposed in [14] to define a small subset of critical data to learn an efficient (CROM) with a sparse approximation space [15]. In this paper, a dictionary of hyperreduced order models is generated by considering clusters of possible loading environments in the observed mechanical tests. Then, the identification of the hyperreduced order model is done via recognition of a class of mechanical loads by a convolutional neural network. In practice, each item of the dictionary is not directly a hyperreduced order model. In order to face a possible variability on the geometry of the observed structures, it is more robust to define an item of the dictionary as a set of finite element solutions for various ideal geometries and for a given class of mechanical loads. Hence, the proposed workflow is robust enough to face geometrical defects in the observed mechanical parts. We assume that the mesh for the finite element modeling of the parts is obtained by using image-meshing techniques, as proposed in [16], of segmented 3D digital images obtained by X-ray computed tomography. We refer the reader to [17, 18] for more details on finite element modeling of 3D images obtained by X-ray computed tomography. An example of X-ray computed tomography applied to manufactured parts can be found in [19]. The ideal geometries involved in the proposed workflow were obtained by using computer-aided-design (CAD). Such CAD models are usually used to find an optimal design of the manufactured parts, with parametric finite element simulations [20].

#### 2. Materials and Methods

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 [21]. The parameters aim to describe all the possible mechanical problems, in advance, before the observation of a realization of one situation. This approach defines a tensor for the description of all possible displacements. The order of this tensor is the number of scalar parameters involved in the parametric equations, plus one. For instance, if a single parameter is introduced then we need two indices i and j to have access to the value of the degree of freedom of a finite element model for the value of the parameter. When* D* parameters are introduced, we need* D+1* indices: i, , …, to have access to a scalar value in the tensor containing all the possible displacements. This tensor formalism aims to introduce a sampling procedure of the parameter space in order to get an estimation of the reduced approximation, or a reduced basis, for the displacements. For instance, this sampling procedure can be achieved by the proper generalized decomposition (PGD) [21] or the Tensor Train decomposition [22, 23]. In this paper, the loading environment is depicted by an image of 3968 × 2976 pixels. Then the parameter space dimension is around 12 millions (the number of pixels) for the description of all possible loading environments. Hence the tensor formalism for model-order reduction would require the decomposition of a tensor of order 12 millions. To our knowledge, no tensor decomposition method has been applied to such a huge tensor order. A purely tensor approach seems to be unaffordable. In this paper, we do not pretend to model all the possible solutions of mechanical equations related to a huge parameter space. We do not follow the usual paradigm of low rank approximations. The proposed image-based modeling aims to exploit available data for fast approximate predictions with fast error estimation.

The workflow of the proposed modeling via computer vision is shown in Figure 1. Four kinds of inputs are required:(i)a 2D digital image of the part in the test machine, this image is denoted by ;(ii)a database where are saved all simulation data, ordered with respect to a cluster index and the index of the available ideal mesh in the list , these are the meshes used when generating the simulation data by the finite element method;(iii)a 3D voxel image of the part alone, we assumed that this 3D image is obtained by X-ray computed tomography;(iv)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.