Unsupervised Learning of Automotive 3D Crash Simulations using LSTMs

1 INTRODUCTION . Data driven virtual product design is nowadays an essential tool in the automotive industry saving time and resources during the development process . For a new car model , numerical crash simulations are performed where design parameters are changed to study their effects on physical and functional properties of the car such as firewall intrusion , weight , or cost ( Fang et al. , 2017 ) . Since one simulation run takes a couple of hours on a compute cluster , running a large number of simulation is not feasible . Therefore , a system that is able to use a limited dataset and predict new simulations would make the development process faster and more efficient . The rise of deep neural networks ( DNNs ) in recent years encourages further research and industrial usages . Besides manifold research for autonomous driving , it is natural for the automotive industry to seek and evaluate the possible applications of DNNs also in the product design stages . As an example , we investigate car crash tests , in which for example the plate thickness of certain parts strongly influences the bending behavior of structural beams and as a result also the intrusion of the firewall into the passenger compartment . Here , numerical crash simulations for different variations of such thicknesses are used as a dataset for learning . The aim is to design a system based on a DNN architecture that learns the crash behavior and would be able to imitate the crash dynamics . Car crash simulations are based on a mathematical model of the plastic deformations and other physical and mechanical effects . They are defined on a computing mesh of currently up to three million points and up to a hundred time steps are stored . Each data instance is a simulation run—of pre-selected parts and/or time steps—that is very high dimensional . Working with this data directly exasperates any machine learning ( ML ) method , but a transformation of this data presented in IzaTeran & Garcke ( 2019 ) allows to obtain a new representation that uses only a small number of coefficients to represent the high resolution numerical solutions . The transformed representation is employed here to compress the mesh geometries to feature sets suitable for neural networks , while avoiding to directly handle geometries in the machine learning method . This way , a network designed for video prediction and embedding based on a long short-term memory ( LSTM ) based architecture ( Srivastava et al. , 2015 ) can be adapted for mesh data . Since LSTM is a recurrent neural network that allows to exhibit temporal dynamic behavior with feedback connections , it is a natural choice for learning the 3D sequences . The aim is that the network learns the observed crash behavior including translation , rotation , or deformation of the parts in the model . Since the contribution of this paper is using DNNs for analyzing car crash data , the related works are categorized into a group of publications in which DNNs are extended for 3D graphics and one that concerns the use of ML techniques for analyzing car crash simulations . For the latter , one typically uses different embedding techniques to obtain a low dimensional representation for the intrinsic underlying data space and to cluster simulations with similar characteristics together ( Bohn et al. , 2013 ; Diez , 2018 ; Garcke & Iza-Teran , 2015 ; Iza-Teran & Garcke , 2019 ; Le Guennec et al. , 2018 ) . The majority of publications about 3D DNN tried to extend CNN for 3D space and focus on description learning and shape correspondence , also known as geometric deep learning , ( Bronstein et al. , 2017 ; Masci et al. , 2015 ; Monti et al. , 2017 ; Boscaini et al. , 2015 ; 2016 ; Litany et al. , 2017 ; Halimi et al. , 2018 ; Maturana & Scherer , 2015 ; Su et al. , 2015 ; Wang et al. , 2017 ) and some developed CNN filters for unorganized point clouds ( Qi et al. , 2017a ; b ) . The very active research is so far very compute resource consuming and there is no extension of ConvLSTM for 3D space to our knowledge , but for prediction one would need an LSTM ( or GAN ) approach . However , a couple of very recent works introduce new feature sets and architectures for mesh embedding using autoencoders and LSTM ( Tan et al. , 2018b ; Qiao et al. , 2018 ; Tan et al. , 2018a ) . The feature representation is using local shape deformations obtained by solving an optimization problem at each node and a global optimization for compensating for rotations . They have shown that after training the network , a sequences of 3D shapes as an animation can be generated by doing operations in the latent space . The bidirectional LSTM architecture is shown to outperform autoeconders ( Tan et al. , 2018a ) . An LSTM based learning network has also been proposed in Qiao et al . ( 2018 ) , where the obtained feature representation is then taken as the temporal data to be feed into a CNN that takes the features and represents them in a lower dimensional latent space . This information is subsequently feed into the LSTM module . 2 MESH DATA AND ITS REPRESENTATION . Data collection is a bottleneck for deep learning applications . If the training data is not diverse enough , the network would neither be able to properly learn the intrinsic data space nor be able to return reasonable output for before unseen data . We focus on surface mesh data from numerical simulations of car crashes , where in the industrial setting the car model is divided into several physcical car parts . As a simple car model we use a Chevrolet C2500 pick-up truck , a model with around 60,000 nodes from the National Crash Analysis Center1 . The data stems from numerical crash simulation2 of a frontal crash for random variations of the plate thickness of nine structural components , a setup similar to Bohn et al . ( 2013 ) . The thickness variations result in different deformation behavior . Figure 1 shows a snapshot of the crash simulation for the truck model . The geometries of the car model and parts are available in a regular mesh format and the correspondence of vertices between different simulations and over the time of the simulation are known by their node id . Therefore , instead of working with the meshes one can simply work with vertices and treat them like organized point clouds , while being able to recover the mesh and connectivity at any time . Now , the features for training the network are obtained from these point clouds and the network outputs a feature vector that is later post-processed to a point cloud or mesh . Instead of working directly with 3D surface meshes , a set of features is extracted for training the network . Such a feature set should be able to represent the dynamics of the crash efficiently . In IzaTeran & Garcke ( 2019 ) a compact representation for deforming shapes has been presented for the car crash case . The approach is based on the property that the Laplace Beltrami Operator ( LBO ) on a surface is invariant to isometric deformations . That is , the LBO is the same if a surface mesh is de- 1from NCAC http : //web.archive.org/web/ * /www.ncac.gwu.edu/vml/models.html 2computed with LS-DYNA http : //www.lstc.com/products/ls-dyna formed in such a way that it is neither stretched nor teared apart , which is the case for the considered simulations , which additionally all start from the same geometry . Consequently the eigenvectors , which form an orthogonal basis , do not change under an isometric transformation and can be used to represent mesh functions such as the deformations ( as three functions , one each for x , y , z ) . The representation is obtained by projecting mesh functions onto the common orthogonal basis to compute so-called spectral coefficients . It turns out that most of the variations of the deformed shapes are concentrated in a small number of coefficients . Therefore an efficient representation is obtained using few spectral coefficients . As shown by Brezis & Gómez-Castro ( 2017 ) using suitable assumptions , for the L2-approximation of functions controlled in the H1-Sobolevnorm the orthonormal basis stemming from the Laplace operator provides an optimal approximation in a certain sense ; this result can be extended to the LBO and functions in the Sobolev space H2,2 . Furthermore , the spectral representation can be understood as a mesh surface analogy to the Fourier decomposition of signals . This representation is introduced here for training LSTMs and allows to bypass the complexity of dealing with large meshes directly . To formalise , in Bn×m we collect the first m unit eigenvectors of the LBO , where n is the number of points in the considered 3D shape . The spectral coefficients C3×m at one time step are obtained by C3×m = R3×nBn×m , ( 1 ) where R3×n contains the x , y , z coordinates of the n points from the considered 3D shape . Four B matrices are required for four distinct parts in the dataset , which are four distinct geometries . Recovering the 3D shapes from their spectral coefficients is possible by R ′ 3×n = C3×mB T n×m , ( 2 ) since Bn×m is orthonormal . R′ is an approximation of R and by choosing larger m , the approximation error gets smaller . In other words , the more eigenvectors are used in Bn×m , the more details are saved in the spectral representation . Figure 2 visualizes the localization and histogram of average error for reconstructing four selected parts from their spectral coefficients . The averaging is done over 205 × 10 samples for each part independently . Note that the bounding box of the entire data is [ 2750 , 4500 ] × [ −600 , 600 ] × [ 250 , 650 ] , determined over simulation time . Comparing the histogram with the tight bounding box ’ s dimensions shows that the reconstruction error is not very high for m = 40 , while the part with blue color coding seems to have more overall error . Note that the error is localized mostly toward the front of the parts , this area goes through very large deformations during the crash . Note that with m = 100 the observed maximal observed error would go down from 40 to 20 . 3 MODEL DESCRIPTION . For frame prediction in video and to obtain latent representations a number of quite interesting applications in robotics and computer vision have been developed . Different DNN architectures have been investigated very extensively . Srivastava et al . ( 2015 ) was one of the first proposals of an unsupervised LSTM based representation learning method for video . The logic behind the choice is that the same operation must be applied at each step to propagate dynamics to the next step . This implies that the underlying dynamics of the dataset remains the same , i.e . the same dynamics acting on any state , at anytime , should produce the next state ( Srivastava et al. , 2015 ) . Others extended this work by adding convolutional operations in LSTM , introducing new recurrent networks , or using model based architectures ( Xingjian et al. , 2015 ; Finn et al. , 2016 ; Kalchbrenner et al. , 2017 ; Lotter et al. , 2016 ; Sharma et al. , 2015 ) . Car crash simulation data can basically be considered a sequence of 3D geometries . In analogy to the video processing case , one can see each geometry in the sequence as a video frame . Inspired from the work on video prediction , we use a similar architecture for the case of car crash representation learning . We choose a two branch LSTM architecture with reconstruction and prediction decoders from Srivastava et al . ( 2015 ) . The encoder LSTM maps an input sequence of k time steps into a fixed length vector representation . This representation is decoded using a decoder LSTM to perform the reconstruction of the input sequence of size k. Receiving a few steps l , l < k , from the beginning of the sequence , the other decoder LSTM predicts future sequences of length k − l , see Figure 3 , where we here use l = k/2 for simplicity . The architecture is in Srivastava et al . ( 2015 ) shown to have a better performance for predicting future frames compared to other approaches . The design of the prediction encoder-decoder LSTM is same as that of the autoencoder LSTM , except that it is like a supervised method in which the decoder LSTM predicts future behavior that comes after the input sequence while the encoders hidden state will capture information about the representation of the input sequence . In comparing to an autoencoder LSTM that receives the entire k time steps of the simulations and reconstruct them again , the prediction encoder-decoder LSTM receives just the first l steps and predicts the remaining time steps . Therefore , the input sequence acts as seed points for generating the rest of the sequence . This compositional architecture is supposed to address the shortcomings that one has to confront by the use of an autoencoder or encoder-predictor alone . Namely , on the one hand an autoencoder LSTM suffers from a bias by memorizing the inputs , which is not sufficient for predicting future frames . On the other hand , an encoder-decoder LSTM suffers from the bias to save information mostly about the last few frames since these carry more information for predicting the future frames , but then the representation at the end of the encoder will ignore large parts of the input . In the compositional architecture , the model is trained to also predict all of the input sequence , therefore it can not just store information about the last few frames ( Srivastava et al. , 2015 ) . It is worth to mention that the dataset contains the translation , rotation , and deformation of car parts during the crash . Therefore the network would learn the entire degrees of freedom of the crash dynamics and would be able to imitate the crash by the prediction decoder after receiving the initial time steps as seeds .

Summary: The paper presents an LSTM model to predict the evolution for 3D models for crash tests. The core model is a video prediction model (Srivastava et al. 2015). Instead of using the original 3D geometry, the authors propose to predict a feature representation based on prior work (Brezis & Gomez-Castro 2017, Bohn et al. 2013, Teran & Garcke 2019).