Efficient planning of peen-forming patterns via artificial neural networks
Wassime Siguerdidjane, Farbod Khameneifar, Frédérick P. Gosselin
EEfficient planning of peen-forming patterns via artificial neuralnetworks
Wassime Siguerdidjane a , Farbod Khameneifar a,b, ∗ , Fr´ed´erick P. Gosselin a,b a Department of Mechanical Engineering, Polytechnique Montr´eal, Canada b Aluminium Research Centre REGAL, Canada
Abstract
Robust automation of the shot peen forming process demands a closed-loop feedback inwhich a suitable treatment pattern needs to be found in real-time for each treatment it-eration. In this work, we present a method for finding the peen-forming patterns, basedon a neural network (NN), which learns the nonlinear function that relates a given targetshape (input) to its optimal peening pattern (output), from data generated by finite elementsimulations. The trained NN yields patterns with an average binary accuracy of 98.8% withrespect to the ground truth in microseconds.
Keywords: shot peen forming, neural network, finite element method, bilayer mechanics,inverse problem
1. Introduction
The performance of artificial neural networks has been proven on visual pattern recog-nition tasks [1]. Here, we demonstrate that the inverse problem of shot peen forming,consisting in finding a suitable treatment, can be formulated as a visual pattern recognitionproblem, and accurately and efficiently solved using a neural network.Peen forming is the process of shaping thin metal parts by bombarding them with astream of shot at high velocity. Treating certain regions of a part gives rise to local cur-vature, without expensive geometry-specific tools, such as dies and matrices. This processis frequently used in the aerospace industry to correct distortion in machined parts and toform large panels such as wing skins [2].The final deformed shape of the part depends on the treated regions, referred to as thepeening pattern. To select the peening pattern, today’s industry mostly relies on human in-tuition with operators trained in this craft with a qualitative understanding of the mechanicsthrough trial and error. This is time-consuming, inaccurate and expensive.Predictive models are crucial to predict the outcome of a planned peening pattern. Differ-ent modelling approaches have been proposed for peen forming [3, 4, 5]. Notably, Faucheux ∗ Corresponding author
Email address: [email protected] (Farbod Khameneifar)
Preprint submitted to Manufacturing Letters August 19, 2020 a r X i v : . [ phy s i c s . c o m p - ph ] A ug t al. [6] proposed the bilayer framework. Fig. 1 depicts how the cumulative effect of mul-tiple shot impacts induce surface compressive residual stresses, which in turn result in thelocal expansion of the treated shallow portion of the plate. Combining the treated expandedlayer with the unaffected bottom layer forms a curved bilayer. Thermal expansion efficientlymodels the effect of the treatment, and is readily available in most finite element software.Adjusting the thermal expansion coefficient and the thickness of the active layer controlsthe intensity of the process. Figure 1: Representing the mechanics of peen forming as a bilayer plate problem: (a) Initially flat plate; (b)Visualization of the local effect of an impact; (c) Multiple impacts expand the upper portion of the plate,while the rest of the plate is unaffected, causing a geometrical incompatibility; (d) Final deformed plateuniformly treated. Inspired by Ref. [7].
The shape of real peen-formed parts still deviates from the model’s predicted shape,mainly due to unknown initial residual stresses in parts [7]. Since the measurement of resid-ual stresses is either expensive or destructive [8], a potential solution is to adjust the modelusing feedback during the process. The long-term objective of our research is to maximizethe robustness of automated peen-forming by incorporating modelling and 3D shape mea-surement in the control loop for adaptive adjustment of the model with the feedback fromshape measurement during peen forming.The bilayer framework efficiently simulates the deformed shape when applying a peeningpattern, i.e., the direct problem. However, the inverse problem can only be solved throughiterative optimization. The difficulty is that the direct problem should be solved in everyiteration of the optimization, which makes it inefficient to solve the inverse problem. Forclosed-loop process control, the inverse problem needs to be solved for each treatment itera-tion. The development of a more efficient method for real-time process planning is thereforenecessary.Here we demonstrate that with the bilayer framework, predicting the peening treatmentto form a flat plate into a target shape can be formulated as a pattern recognition prob-lem, solvable with a trained neural network with sufficient accuracy, while only requiring acomputation time of the order of microseconds.
2. Theoretical Background on Neural Networks
Neural Networks (NNs) are nonlinear function approximators optimized through gradient-based methods to fit a given dataset. NNs learn to predict the most probable output giventhe training data provided. They learn from known references (ground truths). The con-ventional training framework requires the dataset to be separated into a training set to2ptimize the networks parameters, a validation set to validate and tune the networks hyper-parameters, such as the networks depth and learning rate, and a test set isolated to evaluatethe networks performance on unseen data. This training framework ensures that the modelis not overfitting the training data [9].
3. Methodology
We propose to train a NN on data generated with the Finite Element Method (FEM)based on the bilayer framework [6]. The objective of this NN is to predict the requiredpeening pattern, when given a map of the average Gaussian curvature of each element.Fig. 2 shows the data generation pipeline. 60,000 examples were generated. 40,000examples were used for training the NN, while 10,000 were used for validation. The remaining10,000 examples from the dataset served to test the final performance of the model.Since symmetry conditions are applied during the simulation, only a quarter of the plateis considered. The program generates the training patterns by initially creating randommaze images of the size of a quarter of the plate (Fig. 2a) using the depth-first searchalgorithm [10] (adapted from [11]). The A ∗ search algorithm [10] (adapted from [12]) thenfinds a path within the maze to navigate the workpiece from one randomly selected point toanother, hence creating a random continuous path (Fig. 2b). The path is then widened byrecursively padding it with one pixel on each side. This operation ensures that the generatedpattern is larger than a single pixel, which corresponds to the resolution of the mesh. Thiswork assumes that the peen forming patterns considered would be experimentally enforcedthrough the use of shot peening masks, and therefore are not limited by the diameter ofthe nozzle. All pixels on the generated path are assigned a value of 1 representing therequirement of treatment (blue pixels in Fig. 2c). Fig. 2d presents the final patternimage after being symmetrized to cover the full plate. The generation of these randompatterns offers a diversity of continuous shapes that span the surface of the plate, to representarbitrary reachable peen forming cases.The open-source FEM software CalculiX [13] then computes the deformed shape (Fig.2f) of the initially flat plate with a static analysis considering geometric nonlinearities. Thepattern is applied on the plate at a single time to represent the use of a mask. The bilayermodel considers a real plate treated in the same way to be held flat during the peen formingprocess, and released to observe the deformation induced by the treatment.The square plate of size L × L × h (32 mm ×
32 mm × h active = 0 . α , and thepassive bottom layer of thickness h passive = 0 . T = 1 ◦ C is applied. Increasing the α value of the treated elementsapplies the load on the plate. This simulates a bi-axial expansion (cid:15) = αT in the active layerof the treated elements. This study considered 1 . − ≤ α ≤ . − ◦ C − . As in Ref. [6],the dimensionless bending moment generated by the expansion of the active layer is definedas Γ = 6 (1 + ν ) (cid:15) h active h passive L h (1)3here ν = 0 .
33 is the Poisson coefficient of the material of the plate. The process-orientedinterpretation of Γ defines the dimensionless peening intensity.Since the peen forming effect is predominantly bending, curvature is selected as an ef-ficient representation of the plate deformation rather than nodal deflection. The Gaussiancurvature is computed at each node of the mesh with the Gauss-Bonnet theorem imple-mented in the python library PyMesh [14], and then averaged for each element. This shapesthe data as two images of the same size, encoding respectively the target shape (Fig. 2f) andits corresponding pattern (Fig. 2d). The neural network takes as input the target curvaturemap and outputs the most probable peening pattern to reach it. The output pixel valuescorrespond to the probability that an element should be treated, based on the statisticaldistribution of training data. Figure 2: Data generation pipeline: (a) Random maze image where each pixel represents an elementof the FEM; (b) Continuous path in the maze found by the A ∗ search algorithm; (c) Resulting peeningpattern after path padding operations used to widen the peening area. This pattern covers a quarter of theplate, since symmetry conditions are leveraged in the FEM simulation; (d) The final pattern image usedas ground truth, after being symmetrized to cover the full plate; (e) Initially flat mesh of a quarter of theplate; (f) Deformed mesh resulting from the FEM bilayer simulation using the generated peening pattern;(g) Gaussian curvature image. Although a convolutional neural network [15] would have been an alternative, this workdemonstrates that the simpler multilayer perceptron (MLP) is adequate for the current task.The number of elements in the plate mesh defines the width of the proposed six-layer MLP[16]. The images in the dataset are flattened into vectors to be passed to the MLP. Themodel is implemented with Keras 2.2.4 [17] and Tensorflow 1.12.0 [18] as backend. BatchNormalization [19] is applied to every layer. A grid search led to the model hyper-parameters4 igure 3: Learning curve and hyper-parameters. selection (Fig. 3). The Adam optimizer [20] executed the training. Backpropagation al-gorithm [21] minimized the mean squared error (MSE) loss function between the predictedpatterns and the known reference patterns, until the training loss and validation loss con-verged (Fig. 3). The MSE loss function was chosen empirically by comparing its performancewith other loss functions such as cross-entropy.
4. Results and Discussion
The data generation took 20 hours, and the model training took 15 minutes. The predic-tion time of the model only takes 45 microseconds. This last value is crucial when consideringthe application of this model to a real-time process. The development and the evaluation ofthe model were conducted on a laptop computer with an Intel Core i7-6700HQ CPU, with16GB of RAM and under the Ubuntu 18.04 operating system. Graphics Processing Unit(GPU) acceleration was not leveraged in this study.The presented results were computed on a test set of 10,000 examples simulated withΓ = 20. Fig. 4a shows that the binary accuracy evaluated on the test set averages 98.8%.This accuracy metric is widely used for image classification, and in this context it representsthe percentage of elements correctly classified. Fig. 4b shows that the majority of the testexamples exhibit a MSE close to zero.The Hausdorff distance [22] computed between the mesh nodes of the deformed plateunder the peening pattern produced by the NN and the reference peening pattern measuresthe deviation of the resulting shape from the target shape. Both meshes are defined inthe same simulation reference frame, with the centre node at the origin. Normalizing thisdistance by the diagonal of the mesh bounding box offers a dimensionless metric proportionalto the size of the deformed plate. Fig. 4c shows the distribution of the normalized Hausdorffdistance for the test set. It further demonstrates the performance of the model, with anaverage of 0.02%. Fig. 4d presents the normalized confusion matrix computed on the testset. It shows that the percentages of false positives and false negatives are small. Theseresults are complemented with a satisfactory F1-score of 0.981.Fig. 4e shows a sample of the best and worst cases of patterns predicted by the NN ascompared to ground truth for a given input Gaussian curvature.5he relationship between the applied peening intensity and the simulated curvature of theplate is non-linear. For a uniformly peened square plate, at low intensity ( Γ <
20) it adopts aspherical deformation, whereas at high intensity ( Γ >
20) it adopts a cylindrical deformation[6]. We thus check how the ability of the NN to predict the right pattern is affected by thepeening intensity. To do so, we train the NN four times with four datasets obtained at fourpeening intensities. During this process, the hyper-parameters and network architectureremain unchanged. Fig. 4f represents the distribution of binary accuracy for various peeningintensities (i.e., different values of dimensionless bending moment Γ ). Overall, the accuracydecreases slightly with the increase in the peening intensity, but remains high even for verylarge intensities. All instances of the neural network converged to a low loss value, withoutexhibiting signs of overfitting. Figure 4: Distribution of (a) binary accuracy and (b) mean squared error over the test set; (c) Distributionof Hausdorff distance in the test set, normalized by the mesh bounding box diagonal. The fact that (a),(b)and (c) show high accuracy, low MSE and low Hausdorff distance for all test set examples, validates thegeneralization of the model beyond the training set; (d) The normalized confusion matrix demonstratesthe presence of small percentages of false negatives and false positives; (e) A random selection of the bestand worst examples from the test set, presenting the ground truth reference (left), the predicted pattern(middle), and the input Gaussian curvature image (right); (f) With the increase of intensity, the plates areproven to deform in a cylindrical mode, rather than spherical [6]. The violin plot presents the distributionof the test set binary accuracy for various dimensionless peening intensities, with highlighted mean values.We observe a slight increase in the accuracy standard deviation in the cylindrical domain. . Conclusion and outlook The main contribution of this work is the demonstration that the inverse problem ofpeen forming can be represented as a visual pattern recognition problem, by encoding thegeometry of the target shape in terms of Gaussian curvature. This approach solves theproblem via a neural network, a proven technique for pattern recognition tasks.The limitations of this model lie in the need for a large dataset that represents the variouspossible target shapes that can be achieved using a given treatment intensity. Proving thegeneralization of such model on any possible shape is an open problem.This work constitutes the first step towards the long-term goal of developing closed-looppeen forming by incorporating neural networks and the feedback provided by the geometricinspection data from a 3D scanner to compensate for deviations caused by unknown residualstresses in peen formed parts. The data-driven nature and differentiability of NNs are thecentral features that demonstrate their potential to adapt to new examples, therefore, correcttheir prediction errors based on new data.Since our developed method makes use of the bilayer framework, it would easily beadaptable to a broad range of problems which can also be simulated with bilayers such asbiological growth [23, 24], active materials [25] or solvent absorption [26].
Acknowledgements
This work was funded by Aerosphere Inc., the Fonds de recherche du Qubec Nature ettechnologies and the ministre de l’conomie, de la Science et de l’Innovation [Funding refer-ence number LU-210888].c (cid:13)
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