Vladyslav Sydorov
email: vladyslav.sydorov@inria.frkarteekalahari

Karteek Alahari
email: karteek.alahari@inria.fr

Cordelia Schmid
email: cordelia.schmid@inria.fr

Focused Attention for Action Recognition

published or not. The documents may come

Introduction

The arrival of diverse, large-scale datasets [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF][START_REF] Goyal | The "Something Something" video database for learning and evaluating visual common sense[END_REF][START_REF] Caba Heilbron | Ac-tivityNet: A large-scale video benchmark for human activity understanding[END_REF][START_REF] Gunnar | Hollywood in homes: Crowdsourcing data collection for activity understanding[END_REF] has paved the way for the success of large 3D CNN architectures [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF][START_REF] Wang | Non-local neural networks[END_REF] for video-based action recognition. These architectures operate on batches of consecutive video frames and are characterized by extensive 3D convolutions over space and time.

The inputs to these computationally-heavy models have a large temporal dimension and, to constrain the already large number of parameters to reasonable limits, the spatial extent is relatively limited. This means the original video has to be downscaled, which causes some of the finer details to be lost. Consequently the loss of the details may negatively affect fine-grained recognition abilities of the network. Additionally, the annotations for videos in aforementioned datasets simply indicate that an action is happening at some point of time, without spatial localization. Under such conditions, one cannot fully exploit the receptive field of the network, as irrelevant information will inevitably be included along with useful video features.

We propose a general method that alleviates this problem of detail degradation and helps to further adapt the model to the video recognition task. First, a measure of saliency is defined to compare relative "importance" of video regions to solving the task. The measure of saliency is then used to extract focused representations in a weakly-supervised way, which are processed by an extended network to improve recognition performance. c 2019. The copyright of this document resides with its authors. It may be distributed unchanged freely in print or electronic forms. The network is provided with the relevant crop of the video, and can process it at a higher resolution.

Consider the example shown in Figure 1(a). Here the actions occurring are "Washing something with a towel" and "Tidying up a table" and the important regions of the image are the table, the towel and the person interacting with them. The relevant part of the video occupies only a small region of the input frame, while a large fraction of the frame is occupied by the background and is of lesser relevance to recognizing the action.

We utilize a saliency measure to source focused inputs from the original video data, which contain a more relevant portion of the video, as shown in Figure 1(b). The input dimensionality of the model remains fixed, but the tighter crop around the salient region allows the network to compute more expressive statistics of the relevant portion of the video and analyze it in finer detail.

In this paper we present a framework to leverage the focused attention, and show how providing the model with an additional focused view of the inputs allows it to better adapt to the wide variety of possible videos. Specifically, we obtain network-driven saliency estimations by back-propagating through a pretrained network. This allows us to define an attention procedure, through which we find the video region that encloses the most salient area. With the help of this attention procedure we extract these salient regions, and provide them to the network as an additional modality. This effective use of attention allows us to improve action recognition performance.

Related Work

Action Recognition: Video action recognition has evolved considerably since the introduction of deep networks. Starting from the highly successful two-stream networks [START_REF] Simonyan | Two-stream convolutional networks for action recognition in videos[END_REF], a selection of approaches emerged that focuses on bringing together spatial and temporal information. Some of them model feature evolution with RNNs [START_REF] Li | VideoLSTM convolves, attends and flows for action recognition[END_REF][START_REF] Ng | Beyond short snippets: Deep networks for video classification[END_REF], others try to find a robust technique for sampling the video frames [START_REF] Lin | Temporal shift module for efficient video understanding[END_REF][START_REF] Wang | Temporal segment networks: Towards good practices for deep action recognition[END_REF][START_REF] Wang | Temporal segment networks for action recognition in videos[END_REF][START_REF] Zhou | Temporal relational reasoning in videos[END_REF].

The temporal aspect of video recognition is more directly captured by 3D convolutional networks [START_REF] Tran | Learning spatiotemporal features with 3d convolutional networks[END_REF][START_REF] Varol | Long-term temporal convolutions for action recognition[END_REF]. Recently, the I3D family of deep networks [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF] has managed to model long-range temporal dependencies by employing extensive 3D convolutions. These networks have achieved superior performance compared to previous attempts. Our approach is generally applicable to any CNN. We utilize it in conjunction with the I3D model in this paper.

Attention for video recognition: This problem has been pursued in various formsreinforcement learning for sampling frames [START_REF] Yeung | End-to-end learning of action detection from frame glimpses in videos[END_REF], guided pooling of temporal [START_REF] Girdhar | Attentional pooling for action recognition[END_REF][START_REF] Long | Attention clusters: Purely attention based local feature integration for video classification[END_REF] and spatial [START_REF] Sharma | Action recognition using visual attention[END_REF] features.

Self-attention approaches, initially described for language modeling [START_REF] Vaswani | Attention Is All You Need[END_REF], have proven especially useful. A drop-in extension to I3D via non-local pooling [START_REF] Wang | Non-local neural networks[END_REF] performs selfattention within the spatiotemporal space of the CNN input. Video action transformers [START_REF] Girdhar | Video Action Transformer Network[END_REF] modify this approach to consider only the relationships between human regions and rest of the input volume. In the same spirit, Sun et al. study pairwise interactions between 1 × 1 convolutional features [START_REF] Sun | Actor-centric relation network[END_REF]. Long-term feature banks [START_REF] Chao-Yuan | Long-Term Feature Banks for Detailed Video Understanding[END_REF] consider the temporal aspect by accumulating temporal features and applying attention pooling between accumulated and individual frame features. Self-attention has also been utilized to model long-term interactions between memory states of a 3D LSTM [START_REF] Wang | Eidetic 3D LSTM: A Model for Video Prediction and Beyond[END_REF].

An important difference between the methods described above and our approach is the way inputs are treated. Contemporary approaches focus on discovering relationships between video features while staying within the bounds of an input RGB volume. In contrast, we follow the attention cues to the original videos, and provide more informative inputs. Essentially, we provide a different perspective on how attention can be utilized.

Processing videos in different modalities: Videos are commonly processed by CNNs in different modalities. Originally, optical flow has been used almost universally as an additional stream [START_REF] Simonyan | Two-stream convolutional networks for action recognition in videos[END_REF], then pose features have been considered for this purpose [START_REF] Choutas | Potion: Pose motion representation for action recognition[END_REF][START_REF] Zolfaghari | Chained multi-stream networks exploiting pose, motion, and appearance for action classification and detection[END_REF]. Recently SlowFast has looked at RGB features via two branches at different temporal speeds [START_REF] Feichtenhofer | SlowFast Networks for Video Recognition[END_REF].

We consider "focused" inputs as a similar yet distinct modality of the input data, as they provide additional information about finer details to the network.

Saliency/unsupervised methods: A lot of work has been done on understanding CNN behavior and visual explanation for decisions they make. Erhan et al. [START_REF] Dumitru Erhan | Visualizing higher-layer features of a deep network[END_REF] seek to visualize filters by maximizing activations. Simonyan et al. [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF] acquire image-specific saliency maps via a single back-propagation pass and use them to obtain segmentation masks. Guided backpropagation further improves the quality of saliency maps [START_REF] Tobias Springenberg | Striving for simplicity: The all convolutional net[END_REF][START_REF] Matthew | Visualizing and understanding convolutional networks[END_REF]. Zhou et al. [START_REF] Zhou | Learning Deep Features for Discriminative Localization[END_REF]generate class activation maps to identify discriminative image regions, while GradCAM [START_REF] Ramprasaath | Grad-CAM: Visual Explanations From Deep Networks via Gradient-Based Localization[END_REF] extends this technique to provide high-quality visualization with strong localization capabilities.

While we are not aiming to produce class maps, the unifying idea of utilizing the network itself to produce saliency is at the core of our approach. We leverage the gradients to identify the salient regions in the video in a network-specific way.

Saliency methods have also been used to guide recognition. Sudhakaran et al. [START_REF] Sudhakaran | Attention is All We Need: Nailing Down Object-centric Attention for Egocentric Activity Recognition[END_REF] employ Imagenet-pretrained network to obtain CAM maps [START_REF] Zhou | Learning Deep Features for Discriminative Localization[END_REF], which are then sed to rescale video features, in a form of soft spatial attention. In contrast, we use task-specific saliency and hard attention.

Hard Attention: In the work of Jaderberg et al. [START_REF] Jaderberg | Spatial transformer networks[END_REF], spatial transformers act as a differentiable attention mechanism that aid image classification, their network learns to crop patches from the input image. Concurrently to our work, Katharopoulos et al. [START_REF] Katharopoulos | Processing Megapixel Images with Deep Attention-Sampling Models[END_REF] utilize learnable attention to efficiently sample informative patches from high-resolution images, thus reducing computation time and memory footprint.

3 Focusing attention

Motivation

The state-of-the-art approach to the problem of action recognition in videos is to process the video with a complex CNN based model to answer the question of whether a certain action is present in it. We define a video as an RGB volume of dimension T × H ×W × 3, where T is the temporal dimension, H and W are height and width of the frames respectively. When solving an action recognition task, we assume that the video contains certain spatiotemporal regions, which can be leveraged to recognize the action occurring in it. In other word, this RGB video contains some relevant sub-regions and some portion of irrelevant background data.

Current state-of-the-art models assume fixed size inputs. These CNN networks are also heavy, with a big receptive field in the temporal dimension T , which implies that the spatial dimensions H and W are limited due to the memory constraints. For example, I3D as described in [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF][START_REF] Feichtenhofer | SlowFast Networks for Video Recognition[END_REF][START_REF] Chao-Yuan | Long-Term Feature Banks for Detailed Video Understanding[END_REF] has inputs of size 224 × 224 only.

Fitting the video into these dimensions involves scaling and cropping operations, during which some of the finer details are inevitably lost. A common tactic used both in video and image recognition [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF][START_REF] Simonyan | Very Deep Convolutional Networks for Large-Scale Image Recognition[END_REF] is to crop the center region from the input. This accounts for the common bias of recorded media to contain the concept of interest in the center, but is in the essence a heuristic operation -there is no guarantee that important features are in fact contained in the center. Interestingly, a lot of recent video action datasets focus on close-up [START_REF] Goyal | The "Something Something" video database for learning and evaluating visual common sense[END_REF][START_REF] Monfort | Moments in time dataset: one million videos for event understanding[END_REF] or egocentric [START_REF] Gunnar | Charades-ego: A large-scale dataset of paired third and first person videos[END_REF] videos, both settings in which this problem is less likely to occur.

We state that an attention operation will be helpful for the purpose of extracting useful data from the video inputs and fully exploiting the receptive field of the network. Moreover, we think that an attention operation should be tied to the network itself, instead of depending on externally sourced cues like human gaze data or additional object detectors.

Saliency based attention

To be able to discern relevant regions versus background we need to define an "importance" or "saliency" metric. "Saliency" has multiple connotations and in general terms can be thought of as a measure of where would a person or a model "look" to make a decision regarding the input.

Let us consider an input space R T ×H×W ×3 + of RGB videos. An action recognition saliency over video V may be defined as a function S(V ) : R T ×H×W ×3 + → R T ×H×W + , which assigns high values to regions which are important for making a prediction and low values to others. With this spatiotemporal saliency metric we can measure the relative importance of each region. A saliency function should necessarily be dependent on the network parameters. The choice of saliency function will be discussed in section 4.1.

Processing a video with a CNN involves preprocessing steps to make it compatible with the CNN architecture. The video is transformed with a heuristic spatial transform:

V in = F(V, θ ) : R T ×H×W ×3 → R T ×H in ×W in ×3 , (1)
which commonly consists of individual crop and resize transforms, and where θ defines the parameters of these transforms. The parameters of these transforms do not take the contents into account and to address this we propose to employ a saliency based attention transform

A to obtain an input, focused on "important" regions:

V s in = A(V, S(V )), (2)
which produces a video volume V s in ∈ R T ×H in ×W in ×3 , similar to the V in obtained with heuristic operation (1), but prioritizes maximization of saliency score within.

We have described a general approach so far, which boils down to the usage of networkdependent attention for the purpose of providing a CNN with focused inputs. To define the approach precisely, we must choose a saliency function S and an attention transform A. In the next section, we present these details.

Attention for Action Recognition 4.1 Saliency function

We aim to efficiently locate regions important for action recognition in a class-independent way. Accordingly, we obtain saliency by taking a derivative with respect to the inputs. The technique consists in essence of a single back-propagation pass through the network, which allows for on-the-fly extraction of focused inputs during training. It was originally employed in a class-specific way, to visualize a network's notion of object class [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF].

The choice of the function being back-propagated is important -we experiment with using the loss function. The intuition for utilizing the loss function as a guidance is the following: loss minimization is the way of training networks to extract useful signal from the video, hence regions that affect the loss function the most contribute to the network predictions the most.

We train our model with binary cross-entropy loss per-class:

L CE (y n , ŷn ) = y n log ŷn + (1 -y n ) log(1 -ŷn ), (3)
where y n ∈ {0, 1} are ground truth labels and ŷn ∈ [0, 1] are class predictions. It is then possible to obtain saliency during training by back-propagating L CE . The downside of minimizing the training loss is that it requires knowledge of the ground truth labels y n , which are not available for the evaluation step. To account for this limitation, we choose to obtain saliency predictions by back-propagating the negative entropy over class predictions:

L E ( ŷn ) = ŷn log ŷn . (4)
In trained networks, low entropy of class outputs can be thought of as a mode, where the network is certain of the prediction being made. In accordance with the definition of saliency as a decision making mechanism, it would make sense to use L E to guide the saliency. We find that after pretraining, saliency maps obtained with both L E and L CE are similar, thus we can use L E as a proxy for L CE .

After propagating the cross-entropy/entropy back towards the video, we obtain the Jacobian J ∈ R T ×H in ×W in ×3 of the same output dimensionality as the RGB input. To get a positive per-pixel saliency magnitude we apply the L2-norm at each pixel location as follows:

s ti j = (J ti j1 , J ti j2 , J ti j3 ) , (5)
where J ti jc are indexed values of a Jacobian, thus obtaining

S(V in ) = s, s ∈ R T ×H in ×W in + .

Attention function

We obtain our saliency measure directly from the network, i.e., we compute it with respect to and of the same dimensionality as the network inputs V in . We assume that saliency is an inherent property of the video itself and is preserved by the operation F. This assumption allows us to reason about the quantity S(V ) by observing S(V in ) . We search a volume of dimension T × B h × B w that maximizes the sum of saliency scores inside. Concretely we first reduce the temporal dimensions via max-pooling s i j = max t=1...T (s ti j ). Next, we select a fixed size bounding box B = (B t , B l , B h , B w ), by finding B t , B l as follows:

B t , B l = argmax B t ,B l B t +B h ∑ i=B t B l +B w ∑ i=B l s i j , (6)
where B t , B l are the coordinates of the top left corner and B h , B w are the height and the width of the box respectively. The coordinates of the box are fixed across all time steps T . We utilize the parameters θ of the original transformation F to obtain box coordinates

B o = (B o t , B o l , B o h , B o w )
with respect to the original video V and consider that B o would also capture the maximum saliency in S(V ), if we had access to it directly. We crop the volume enclosed by B o from the original video and scale it to H in × W in . During the training phase, additional data augmentation functions can be applied (e.g., random crops, left-right flip, photometric adjustments). When accessing original video we reapply these augmentations. This procedure allows for efficient extraction of focused regions during training and evaluation stages.

Processing V in and V s in inputs

The attention operation as described above provides a different, if similar, facet of the input video to the model. Even assuming that the saliency measure we obtain is perfect, there is always a possibility that some context information is lost. Additionally, if the video crop V in is already a perfect representation then there is little need to apply a transformation. Another important issue is that the model has been trained by only applying the heuristic F and if we were to change the spatial transformation without further adjustments, the newly introduced domain shift might hurt the performance. In lieu of these considerations, we think it is reasonable to keep both the original transformed data and the focused inputs and process them together. This can be thought of as another take on the approach of processing the video stream in different input modalities, which is a reasonable concept, when applied to video understanding [START_REF] Feichtenhofer | SlowFast Networks for Video Recognition[END_REF][START_REF] Zhang | Two-Stream Oriented Video Super-Resolution for Action Recognition[END_REF].

When training the full model, we first train the network with the usual transform F until convergence, we call the result a Base model. We explore two ways of processing the inputs together: temporal concatenation and late fusion, which allows us to use the same architecture for both modalities.

Temporal concatenation: we extend the input by concatenating the original input V in and the focused region V s in along the temporal axis. This step is possible because the modalities of the inputs are similar. We finetune the Base model on temporally extended inputs.

Late fusion: we process the two modalities V in and V s in via separate networks, both initialized from Base, and aggregate the output scores via mean-pooling. When finetuning, we keep the branch that processes the original input V in fixed.

In other words, once the network training procedure reaches the performance limit, determined by the provided data, we allow further performance increase by accounting for important regions via extended inputs.

Experiments

We utilize a standard 3D CNN architecture [START_REF] Wang | Videos as space-time region graphs[END_REF], where the base model is ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF], inflated into a network with 3D convolutions in an I3D [START_REF] Carreira | Quo vadis, action recognition? A new model and the kinetics dataset[END_REF] fashion.

Data and setup

For our experiments we use the Charades dataset [START_REF] Gunnar | Hollywood in homes: Crowdsourcing data collection for activity understanding[END_REF], containing 9 848 videos across 157 action classes. In each video, a person can perform one or more actions. We evaluate the video classification task, which involves recognizing all the actions in blea video, without temporal localization. We train and evaluate on the publicly available subset of Charades following standard protocol [START_REF] Wang | Videos as space-time region graphs[END_REF][START_REF] Chao-Yuan | Long-Term Feature Banks for Detailed Video Understanding[END_REF]. RGB frames are extracted at the rate of 8FPS.

Network Details

We use a model finetuned from Kinetics to test our method. We call it Charades-Base.

Kinetics-Base: As the base model we utilize ResNet50-I3D [START_REF] Gunnar | PyVideoResearch[END_REF], which is publicly available along with the source code and pretrained weights. This model has been trained on Kinetics from scratch for 300 epochs. It achieves 64.01% (83.70%) top-1 (top-5) accuracy on the Kinetics validation set. The base network accepts inputs T in × H in ×W in × 3, representing T in RGB frames, where H in = W in = 224. At the last layer of the network, the logit outputs are of dimension T out × N class . For T in = 64, the corresponding T out = 7.

Charades-Base: We adopt the Kinetics base model for Charades and introduce a different loss function. Specifically, a cross-entropy loss (L CE ) over network predictions Ŷ is used, to accommodate for the multilabel nature of the Charades dataset. In the last layer of the network we accordingly replace the softmax operation with a sigmoid. The Charades dataset has temporal annotations Y frame for every video frame along with per-video labels Y videowe incorporate this by minimizing a sum of two loss components:

• Video loss: L CE (Y video , M( Ŷ ))
, where M : T out × N class → N class is a mean-pooling operation across the temporal dimension of the network outputs.

• Frame loss: L CE (Y frame ,U( Ŷ )), where U : T out × N class → T in × N class is a bilinear interpolation operation that upsamples the logit outputs to match the input temporal dimension.

Implementation

Saliency. We explore several bounding box sizes for the Charades validation set and find that for our network with expected spatial inputs H in = W in = 224, B h , B w = 128 performs the best. These zoomed-in boxes enclose roughly 30% of the original area.

Training. First, we finetune the baseline model from Kinetics to the Charades dataset without using focused inputs. We then apply the proposed approach to further improve the recognition performance.

Charades-Base: We initialize the model with Kinetics-Base weights. We cover temporal extents T in of 16, 32 and 64. We use mini-batch SGD with momentum set to 0.9 and batch sizes are set accordingly to 20, 16, 12 for T in = 16, 32, 64 respectively. Training lasts for 50 epochs, with an initial learning rate 0.375 that is decreased by a factor of 10 at epochs 15 and 40.

Multi-modality models: From the Charades-Base model we tune several attention and ablation models for additional 25 epochs, in both temporal concatenation and late fusion versions. The initial learning rate is 0.02, the batch size is 8, and the learning rate is dropped by a factor of 10 at epoch 15. We finetune BN layers, small batch size notwithstanding. At T in = 64 GPU memory requirements for late fusion and temporal concatenation are 24GB and 32GB respectively. We thus train late fusion models on 2 Titan X GPUs and temporal concatenation models on 2 Tesla P100s.

The Repeat model is tuned from Charades-Base, but no attention step is executed, instead the V in inputs are repeated in the temporal dimension. The Random crop and Center crop experiments perform attention step without utilizing saliency, i.e., during training a random or a centered B h × B w cuboid respectively is cropped from the input volume. When training Attention models, a choice of saliency function can be made at both training and evaluation stages. To denote this choice, we utilize a two-index notation L, where the first index corresponds to the saliency function used during training, the second to the function used during evaluation. We train L E / E , L CE / CE and L CE / E Attention models. We emphasize that the L CE / CE experiment only serves as a reference for what attention guided by training loss can achieve, since utilizing L CE during evaluation involves accessing ground truth data.

During training, a left-right flip is applied with p = 0.5, while the transform F amounts to resizing the video to 256 × 256 and randomly cropping a 224 × 224 cuboid.

Evaluation. During evaluation, F is composed of resizing the video to 256 × 256 and then obtaining a 224 × 224 center crop. At test time we sample 10 clips per video and combine the predictions using max pooling, following prior work [START_REF] Wang | Videos as space-time region graphs[END_REF][START_REF] Chao-Yuan | Long-Term Feature Banks for Detailed Video Understanding[END_REF].

Results

In this section we experimentally evaluate the models on the Charades action recognition task. We show that our baseline model performs on par with the state-of-the art. Most importantly, we show that, when trained on videos augmented by focused attention, the performance of the model improves over the baseline. Finally, we perform several ablation studies.

Baseline model

In Table 1 we show the performance of our Charades-Base baseline at different temporal extents T in as well as other approaches. We include neural network approaches that were state-of-the-art for Charades before appearance of I3D [START_REF] Gunnar A Sigurdsson | Asynchronous temporal fields for action recognition[END_REF][START_REF] Wang | Videos as space-time region graphs[END_REF][START_REF] Zhou | Temporal relational reasoning in videos[END_REF] and show that I3D outperforms them. Next, we consider a baseline I3D network from a recent publication [START_REF] Wang | Videos as space-time region graphs[END_REF] and demonstrate our network performing at the same level, despite being pretrained on only In Table 2 we present the experimental results of the proposed attention approach. Charades-Base and Repeat are the two baseline experiments, where Charades-Base corresponds to the baseline performance of the model without using additional modality and Repeat allows us to reason about the stacked model performance in the absence of the attention transform A.

We see that Random crop and Center crop strategies do not provide any useful signal to the network, in fact randomly cropping the volume results in a lower performance. This indicates that the attention transform A should be guided by a reasonable, input-dependent saliency function.

The Attention models L CE / CE , L E / E and L CE / E outperform the baselines and allow us to confirm the usefulness of the saliency-guided attention step. We see that L CE and L E saliencies are very similar, as evidenced by a very small difference in performances of L CE / CE , L E / E and L CE / E . We also perform ablation studies "L CE / CE (upscaled)" and "L CE / E (upscaled)", for which we do not query the original video for the higher resolution inputs and instead simply upscale the focused video. These experiments show lower performance than L CE / CE and L CE / E , proving that querying the original video is important.

Conclusion

In this paper we introduce a new approach to leverage attention for action recognition. Instead of being constrained within the bounds of input data dimensionality, we take a step further and leverage the original video data. We demonstrate that the preprocessing steps of a video action pipeline have a notable effect on the quality of results. Consequently, we allow the network-dependent saliency measure to guide the preprocessing operation to select regions of the data that are more useful for the recognition task. Crucially, these regions can be processed in higher detail, allowing to access additional information, which is not used by conventional attention approaches.

Figure 1 :

 1 Figure 1: Example of attention on Charades action recognition dataset. a) Saliency scores (displayed as a heatmap) are localized around the object, a box maximizing the saliency measure within is selected. b) The network is provided with the relevant crop of the video, and can process it at a higher resolution.

Table 1 :

 1 Baseline performance. We show the mean Average Precision (mAP%). Kinetics dataset. Recent works utilizing attention report higher performance on this task, but employ long-range temporal techniques such as spatiotemporal graphical models[START_REF] Wang | Videos as space-time region graphs[END_REF] or memory banks[START_REF] Chao-Yuan | Long-Term Feature Banks for Detailed Video Understanding[END_REF].

	Method	Model		Pretraining	mAP
	2-Stream [20]	VGG16		ImageNet	18.6
	Asyn-TF [20]	VGG16		ImageNet	22.4
	Multiscale TRN [42]	Inception		ImageNet	25.2
	I3D baseline (T in = 32) [33] ResNet50-I3D ImageNet+Kinetics 31.8
	Ours (T in = 16)	ResNet50-I3D	Kinetics	27.8
	Ours (T in = 32)	ResNet50-I3D	Kinetics	30.3
	Ours (T in = 64)	ResNet50-I3D	Kinetics	31.6
	6.2 Attention
	Method	Temp. concat. Late fusion
	Charades-Base		31.6	31.6
	Repeat		31.7	31.6
	Random crop		31.4	30.6
	Center crop		31.6	31.0
	Attention L CE / CE		33.2	32.9
	Attention L CE / CE (upscaled)	33.2	32.7
	Attention L CE / E		33.2	33.0
	Attention L CE / E (upscaled)	33.1	32.8
	Attention L E / E		33.3	33.1

Table 2 :

 2 Our performance at T in = 64, mAP%

Acknowledgements. This work was supported in part by the Indo-French project EVER-EST (no. 5302-1) funded by CEFIPRA, ERC advanced grant ALLEGRO, and a grant from ANR (AVENUE project ANR-18-CE23-0011). We gratefully acknowledge the support of NVIDIA with the donation of GPUs used for this work.


F Viprey

H Nouira

S Lavernhe

C Tournier

To improve the accuracy of manufactured mechanical parts, the geometric errors of a machine-tool should be evaluated and compensated in order to better master the deviations between the actual and nominal tool positioning (volumetric accuracy). Thus, a novel Multi-Feature Bar (MFB) for machine-tool geometric errors' identification was designed and manufactured. The MFB standard is made of Invar material. The proposed design of the MFB allows extracting three intrinsic parameters: one linear positioning and two straightness errors. The calibration of the MFB was performed on an accurate coordinate measuring machine (CMM) when applying the reversal technique, in order to separate the MFB's error forms from the motion errors of the CMM's mechanical guiding systems. Furthermore, an intercomparison was conducted between four National Metrology Institutes (LNE, PTB, CMI, UM) to evaluate the reliability of the proposed calibration methodology. Findings resulting from this intercomparison reveal dimensional stability of the MFB standard for geometric errors identification on CMM and machine-tool. Therefore, the use on machine-tool of the calibrated MFB, regardless of the harsh environment, guarantees its metrology traceability to the SI metre definition of few micrometres (<5 µm).

Inter-comparison on multi-feature bar calibration for determining machine-tool geometric errors 1 Introduction

During the manufacturing process, the inspection of produced parts directly in the machine-tool(s) represents a more and more frequent need in the industry.

The manufacture and measurement of these parts are usually performed through high precision machine tools (MT) which must be in line with to the SI metre definition published by the Bureau International des Poids et Mesures (BIPM).

The volumetric error in quasi-static conditions is defined in ISO 230-1:2012 [START_REF]Test code for machine tools -Part 1 : Geometric accuracy of machines operating under no-load or quasi-static conditions[END_REF] as a relative deviation between actual and ideal tool and workpiece positioning on machine-tool. This deviation is essentially generated by geometric errors [START_REF] Mekid | A review of machine tool accuracy enhancement through error compensation in serial and parallel kinematic machines[END_REF] (position and orientation errors and motion errors [1] [3]). Several other sources of the volumetric error such as thermal errors and loads can be identified by measuring the error motions [START_REF] Schwenke | Geometric error measurement and compensation of machines -An update[END_REF]. Therefore the traceability and minimisation of the volumetric error on MT represent new challenges for researchers of National Metrology Institutes (NMIs) of manufacturing laboratories and/or plants. In this context, NMIs develop and deliver standards and procedures suitable to assess and ensure the traceability of the measurement capability of in-process metrology. More particularly, this task consists in designing and manufacturing highly accurate multi-purpose material standards, which are robust against environmental influences and high mechanical stress, for the mapping of volumetric and task-specific measurement errors. The Laboratoire National de Métrologie et d'Essais (LNE, French NMI) has designed and manufactured a novel Multi-Feature Bar (MFB), depicted in Figure 1. The new design of the MFB machined in Invar due to its small / ratio equal to 7.7 10 -8 m/W and specific technical attributes enables the identification of geometric errors on machine-tool in a harsh environment [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF]. MFB consists of a repetition of a 3Dpattern in the  direction. Each pattern contains 7 features: 4 flat surfaces (vertical planes) and 3 cylinders (one vertical inner cylinder and two horizontal outer cylinders). The 3D pattern is displayed in Figure 2. Many measurements are carried out on each pattern. The processing of the measured data (48 probed points) allows extracting one point of interest Oi corresponding to the intersection of the 7 features previously mentioned. The expected measurements and the post-processing of the measured data (a total of 316 probed points) can be completed according to the steps detailed in [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF]. The steps are repeated as many times as necessary to cover the whole geometry of the MFB. Thus, the identified points of interest O i = (x i , y i , z i ) offer 3 intrinsic geometric parameters in the local frame of the R MFB : 1 linear positioning and 2 straightness errors contrary to the commercially available hole bar. R MFB is built using the measured data on the surfaces of the patterns [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF].

To ensure the traceability of measurement on machine-tool to the SI metre definition, the calibration of the MFB is carried out using an accurate CMM. In addition, the calibration is repeated 5 times in order to characterise its repeatability in a repeatability condition of measurement specified by the International Vocabulary of Metrology (VIM) [START_REF] Jcgm | International vocabulary of metrology -Basic and general concepts and associated terms-Bureau International des Poids et Mesures[END_REF]. In other words, the calibration was performed in a condition of measurement, out of a set of conditions that includes the same measurement procedure, same operator, same measuring system, same operating conditions and same location, and replicate measurements of the same or similar objects over a short period of time. Furthermore, the specified reproducibility condition of measurement in the VIM is a condition of measurement, out of a set of conditions that includes different locations, operators, measuring systems, and replicate measurements on the same or similar objects. Therefore, despite a clearly specified calibration protocol, it is necessary to evaluate the reliability of the proposed calibration methodology by an intercomparison.

Agreements enable the NMIs to demonstrate the international equivalence of their measurement standards and their measurement certificates. These agreements are governed by the Mutual Recognition Arrangement (MRA) of the Comité International des Poids et Mesures (CIPM) (CIPM MRA). The CIPM MRA [START_REF] Bipm | Reconnaissance mutuelle des étalons nationaux de mesure et des certificats d'étalonnage et de mesurage émis par les laboratoires nationaux de métrologie -Bureau International des Poids et Mesures, 2ème édition[END_REF] ensures the international recognition of the Calibration and Measurement Capabilities (CMC) of participating laboratories. CMCs, once approved, are published in the CIPM MRA database, which also contains other technical information. This approval is objectively established on the basis of mutual recognition criteria. Approval is based on implemented means such as:

-international comparisons of measurements, referred to as Key Comparisons; -international comparisons of additional measurements; -the establishment by LNMs of quality systems and demonstrations of their skills.

In this context of CMC, a European intercomparison of the MFB was conducted by LNE. This paper aims at comparing the calibration results of different NMIs to evaluate the reliability of the proposed calibration methodology. The paper is structured as follows: Section 2 presents the summary of the calibration procedure of the MFB; Section 3 deals with the European intercomparison and the description of the reproducibility condition; Section 4 is dedicated to the NMIs' results and comparisons in relation to reference values.

Calibration Procedure

First of all, the MFB is assembled on the work holder. This task aims at maintaining the MFB during the measurement and providing an isostatic settingup of the MFB. The calibration must be performed along any axis of CMM, but preferably along the axis presenting the smallest linear positioning error (i.e. E xx , E yy , E zz or xtx, yty, ztz). The achievement of the assembly as displayed in Figure 1 is imperative to perform the calibration in similar conditions as those in the machine tools.

During the tightening, it is necessary to make sure that the washer is in contact with the spheres (isostatic link) and the mounting flange. The washer does not move after tightening. The tightening torque applied to the four nuts must be equal to 1.5 Nm. The nuts should be gradually embedded (Figure 3). Once the assembly is completed, the MFB is put into place on the CMM for the calibration. For the calibration of the MFB, the reversal technique is applied in order to separate the motion errors of the used accurate CMM from the straightness errors of the MFB. The absolute length between each hole can be corrected by a substitution technique. The motion errors of CMM contain both systematic component and random component. The application of the reversal technique allows determining the systematic components. The random component is still mixed with the geometric errors of the MFB; this is the reason why the measurements are repeated at least 5 times to average out the zero mean random errors.

For the application of the reversal technique, the MFB is carefully aligned along the CMM axis with the smallest linear positioning error that was previously identified. The sine error due to the alignment defect along the entire length of MFB must be inferior to 0.1 mm. The calibration technique and expression of the linear positioning and straightness errors of the points of interest O i are specified in [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF]. For the calibration of the MFB, the following instructions should be observed:

-The MFB should be aligned along the selected axis with the smallest linear positioning error, -The calibration of the selected axis should be performed previously by any other internal standard, -The MFB should be aligned along the selected CMM, -The measurement should be carried out by using a stylus with a diameter of 6 mm and a maximum length of 50 mm, -The selected feedrate of motion during the deflection of the stylus shall not exceed 8 mm/s. Each operator was provided with the instructions for the calibration process and the probing programmes. In any case, the operator has to repeat the measurement five times. The total number of measurements is equal to ten measurements (including five direct measurements and five reversal measurements). The calibration procedure is based on the schema displayed in Figure 4 where A, B and C are external flat surfaces of the MFB.

European intercomparison

An intercomparison of measurements is performed on the LNE Material Standard: the MFB. This intercomparison between several organisations (as seen in Table 1 and Figure 5) ensures that the measurements of the MFB all produce results that are within an acceptable tolerance. The measurements are performed by four NMIs across Europe: LNE, PTB, CMI, UM. The reproducibility condition is then respected: different locations, operators, measuring systems, and replicate measurements of the same or similar objects (in this case the MFB) with the same measurement procedures.

The MFB and all the elements necessary for the fixation, such as the plate, the modular inspection equipment system, the screws and the torque wrench are inserted in the individual Transport Case. Therefore, the NMIs perform measurements for the calibration of the MFB with the same equipment, except the CMM.

Results and reference values

The extraction of the points of interest is performed by LNE from the probed points in the institutes by using the CMMs. The results of identified errors E xx MFB , E yx MFB , and E zx MFB are shown respectively in Figure 6, Figure 7 and Figure 8. The uncertainty depicted in Figure 6, Figure 7 and Figure 8, is the repeatability of measurement. Deviations between the results of participants especially for Exx HB (Figure 6), and Ezx HB (Figure 8) may be noticed.. In order to identify the cause of this deviation, various factors were investigated and are discussed below.

Impact of the MFB's weight and orientation

The MFB is fixed on a specific holder, built with modular inspection equipment systems, via isostatic assembly composed of 3 mechanical linkages (spherical joint, point surface joint and point curve joint). This type of assembly avoids any transmission of the mechanical and thermal deformation of the holder to the MFB. Three mechanical linkages are located on the points of minimum deflection that were analytically calculated (0.16 µm without tightening operations) [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF]. This value is compensated by reversal technique during the calibration of the MFB. Therefore, the impact of deflection is not the reason that could explain the difference between Ezx HB results. It can be concluded from Figure 8 and Table 1 that the MFB was deformed during intercomparison, and particularly between the first measurements by LNE and CMI. Moreover, modular inspection equipment system enables the orientation and location of the MFB along any direction in the MT workspace (Figure 5). The change in length of the MFB when standing vertically, due to its own weight, is equal to 52 nm and is neglected with respect to the magnitude of the identified errors [START_REF] Viprey | Novel multi-feature bar design for machine tools geometric errors identification[END_REF].

Impact of the tightening operation

The tightening operation generates a deviation equal to -4.7 µm on the deflection of the MFB coupled with a deviation on its length (-3.7 µm) with the variability of 2,7 µm identified by deflection and length measurement for 50 cycles of tightening and loosening. Despite careful attention and design of the clamping system, the latter introduces random errors during the calibration process. The use of a clamping system reducing all deformations due to tightening could be investigated in the future (e.g. using isostatic decoupling link).

Reference values

The reference values were calculated by using all the measurements. Indeed, the five measurements of each participant are used to calculate a mean value, named reference value of the identified errors E xx MFB , E yx MFB , and E zx MFB . The results are presented in Table 2.

Conclusions

This paper describes an intercomparison of the novel Multi-Feature Bar (MFB) calibration. The progress of this innovative geometry that ensures the identification of three parameters has been tested by NMIs and provides a high level of confidence regarding the calibration. Despite reference values with small associated standard uncertainties of reproducibility, improvements and studies aiming at an easier and faster calibration will be carried out on the basis of the feedback provided by NMIs. Further works will be conducted by the application of procedures from this specific set of key comparison data to provide a key comparison reference value (KCRV) and the associated uncertainty [START_REF] Cox | The evaluation of key comparison data[END_REF].

Figure 1 :

 1 Figure 1: Multi-Feature Bar.

Figure 2 :

 2 Figure 2: 3D pattern of the MFB.

Figure 3 :

 3 Figure 3: Fixture equipment of MFB.

Figure 4 :

 4 Figure 4: Calibration procedure of the MFB.

Table 1 :Figure 5 :

 15 Figure 5: LNE material standard during measurement in the institutes.

Figure 6 :

 6 Figure 6: Linear positioning error E xx MFB .

Figure 7 :

 7 Figure 7: Horizontal straightness error E yx MFB .

Figure 8 :

 8 Figure 8: Vertical straightness error Ezx MFB .

  for each participant are shown in Figure 9, Figure 10, and Figure 11. The Figure 11 depicts a variation of the geometry during the intercomparison between LNE and CMI.

Table 2 :

 2 Reference values of the MFB The deviations between the reference values of E xx MFB , E yx MFB , and E zx MFB , and the identified values of E xx MFB , E yx MFB , and E zx MFB

		E xx MFB		E yx MFB		E zx MFB
	O i mean (µm) u (µm) mean (µm) u (µm) mean (µm) u (µm)
	1	0	0	0	0	0	0
	2	-3,0	0,8	-1,3	0,4	-7,1	0,3
	3	-7,5	0,9	-1,3	0,4	-5,5	0,9
	4	-11,2	1,1	-1,8	0,4	-0,1	1,4
	5	-13,3	1,3	-4,4	0,5	5,0	1,6
	6	-16,9	1,3	-3,5	-0,3	4,7	1,7
	7	-19,7	1,2	-4,1	0,3	5,0	1,7
	8	-22,9	1,2	-5,5	0,3	13,1	1,4
	9	-26,4	1,8	-7,3	0,4	14,9	1,1
	10	-33,2	1,8	-9,3	0,3	19,0	0,9
	11	-35,1	0,9	-11,2	0,3	10,5	0,4
	12	-40,3	1,2	0	0	0	0

Acknowledgments

The authors sincerely thank the EMRP organisation. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union (IND62: JRP-TIM).


Hichem Benlaoukli

Sorin Olaru
email: sorin.olaru@supelec.frsilviu.niculescu@lss.supelec.fr

Silviu-Iulian Niculescu

CIFA 2008 Commande prédictive des systèmes avec l'entrée affectée par retard variable

Keywords: MPC, Commande prédictive, système incertain, système à retard, ensembles invariants

Cette article présente une méthodologie pour le calcul d'une loi de commande pour un système soumis à des contraintes de fonctionnement et affecté par un retard variable. Dans une première phase, on construit une loi de commande prédictive "classique" qui garantie l'existence d'un ensemble positif invariant robuste (RPI) non vide. Deuxièmement, on proposera deux algorithmes itératifs pour l'approximation de cet ensemble RPI. Ce problème peut être traité dans le cadre des systèmes affines par morceaux dû à la formulation explicite de la loi de commande obtenue via la programmation multiparamétrique.

I. Introduction

L'interaction entre un système réel et un processus physique avec des excitations exogènes peuvent être difficilement caractérisée comme instantané. Une des façons classiques de modéliser ces situations et phénomènes est par l'utilisation de retard. Les retards (variables, distribués ou non) décrivent le couplage entre les dynamiques, propagation et phénomène de transport, hérédité et compétition dans les dynamiques de population. Plusieurs exemples peuvent être trouvés dans [START_REF] Niculescu | Delay effects on stability. A robust control approach[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Michiels | Stability and stabilization of timedelay systems. An eigenvalue based approach[END_REF]. Il y'a un consensus dans la définition du retard comme un paramètre critique dans l'analyse des comportements dynamiques et des performances des systèmes.

Indépendamment des problèmes mathématiques liés à la représentation, les systèmes à retard sont connus par leurs problèmes de commande due à l'instabilité introduite par des actions de commandes différées. Une façon naturelle pour neutraliser l'effet du temps mort est de prédire l'évolution du système avec une attention particulière pour la sensibilité de prédiction pour modèles instables.

La commande prédictive à base de modèle (MPC -"Model Predictive Control") est une technique de commande qui consiste à résoudre un problème de commande optimale à temps fini sur un horizon glissant. Cette formulation temporelle permet la prise en charge des contraintes dès le stage de conception [START_REF] Normey-Rico | Control of Dead-time Processes[END_REF] et se révèle d'être très populaire pour les praticiens. Plusieurs stratégies ont été pro-posées afin de renforcer la stabilité du MPC, [START_REF] Maciejowski | Predictive Control with Constraints[END_REF], [START_REF] Goodwin | Constrained Control and Estimation[END_REF] avait comme ingrédients principaux, la fonction coût terminale et contraintes terminales invariantes [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. Malheureusement, considérer , les incertitudes introduites par l'existence d'un retard variable, mène à un problème d'optimisation minmax complexe. En plus, les garanties de stabilité nécessitent la description des contraintes terminales qui réduisent le domaine de faisabilité.

Dans ce papier on utilise la loi de commande prédictive obtenu en utilisant le modèle nominal. La loi de commande affine par morceaux correspondante transformera la dynamique en boucle fermée vers un système affine par morceaux. Quant au retard variable, il introduit un certain degré d'incertitude qui doit être analysé pour garantir le fonctionnement global. L'existence d'un ensemble invariant non vide peut être garantie sous certaines conditions et des procédures algorithmiques seront proposées dans ce sens.

Deux problèmes liés à la robustesse pour le calcul de la loi de commande seront traités en détail:

• réglage du MPC nominal en utilisant les arguments de l'optimalité inverse;

• caractérisation de l'ensemble positive invariant robuste maximal (MRPI). Notons que l'ensemble MRPI peut ne pas être déterminé avec un nombre fini d'itérations. Pour le deuxième point on propose la construction itérative par procédure duale expansive/contractive d'approximation garantie.

Le reste du papier est organisé comme suit: section 2 formule le problème de commande et définit le modèle qui sera utilisé dans le calcul de la commande MPC; section 3 traite la construction de la loi de commande explicite affine par morceaux et section 4 traite le problème des approximations pour l'ensemble positive invariant robuste maximal associé au système bouclé. La section 5 présente un exemple et les conclusions sont tirés dans la section 6.

Notations: Co {e 1 , . . . , e1} définis la combinaison convexe des éléments e 1 , . . . , e1. B 0 (ε) est la boule centrée à l'origine avec un rayon ε. Le symbole * dans la description matricielle est la transposé. Le reste des notations sont standards.

II. Formulation du problème

Considérons un système linéaire LTI suivant:

ẋ = A c x(t) + B c u(t -τ ) (1)
affecté par un retard variable. Il est supposé que la variation du retard est bornée:

τ ∈ [0, τ max ] (2)
Notons que les instants du temps discret

x k = x(t k ) = x(kT e ) (3)
où T e est le temps d'échantillonnage. Considérons:

d = ⌈τ /T e ⌉ ǫ = dT e - τ (4)
où τ est la valeur probable du retard (s'il n'y a pas d'information statistique sur la distribution du retard qui permettra d'extraire une valeur espérée, la valeur centrale de l'intervalle [0, τ max ] peut être choisie, i.e. τ = τ max /2). Maintenant, on peut décrire le modèle LTI discret affecté par le retard:

x k+1 = Ax k + Bu k-d -∆(u k-d -u k-d+1 ) (5)
A = e AcTe (6)
obtenue en assumant que la commande u est maintenue constante entre deux instants d'échantillonnages,

u(t) = u k , ∀t ∈ [t k , t k+1 ).
Ce modèle LTI discret (5) sera considéré par la suite comme modèle nominal.

x k+1 = Ax k + Bu k-v -∆(u k-v -u k-v+1 ) ∆ ∈ Ω x Ω x = Co {∆ 0 , ∆ 1 , . . . , ∆ n } v ∈ {0, 1, 2, . . . , h} (10)
où v c'est une variable entière qui décrit le retard en nombre de pas d'échantillonnage. La valeur maximale du retard est:

h = τ max T e (11)
et les réalisations extrêmes ∆ i sont obtenues en utilisant la forme canonique de Jordan pour la matrice A c dans (9), calculée pour les valeurs extrêmes de ǫ, à savoir 0 et τ max . Il peut être montré que le nombre minimum des sommets caractérisant l'incertitude polytopique pour le système original à temps variable est n + 1 où n est la dimension du système d'état [START_REF] Olaru | Predictive control for linear systems with delayed input subject to constraints[END_REF].

Utilisant la présentation de l'espace d'état étendu basée sur l'équation [START_REF] Maciejowski | Predictive Control with Constraints[END_REF], on obtient en utilisant (A, B, ∆), le modèle de prédiction nominal:

ξ k+1 = F ξ k + Ḡu k (12)
En effet en définissant:

ξ T k = x T k u T k-h u T k-h-1 . . . u T k-d u T k-d-1 . . . u T k-1 (13)
les matrices F , Ḡ sont données par les équations [START_REF] Larin | About the inverse problem of optimal control[END_REF][START_REF] Benlaoukli | Computation and bounding of robust invariant sets for uncertain systems[END_REF]. Pour le même vecteur d'état ξ k , on peut décrire la variation structurelle provoquée par la variation du retard par l'existence d'un modèle affecté par une incertitude paramétrique :

ξ k+1 = F ξ k + Gu k (F, G) ∈ Ω ξ Ω ξ = Co {(F 1 , G 1 ), (F 2 , G 2 ), . . . , (F s , G s )} (18)
En utilisant A, B et l'incertitude polytopique pour la matrice ∆, avec les réalisations extrêmes ∆ i , i = {1, . . . , n}, les matrices de transitions seront données par [START_REF] Furtmueller | Disturbance suppression for an industrial level control system with uncertain input delay and uncertain gain[END_REF](17).

Les variables δ i , i = 1, . . . , h sont telles que:

h i=1 δ i = 1; δ i ∈ {0, 1} (19)
Le nombre de réalisations extrêmes est s = h(n + 1) (due au fait qu'il y'a h combinaisons de variables logiques admissibles). Cependant, n-1 de ces réalisations, correspondant à ∆ = 0 sont, en fait, générées par les combinaisons des variables logiques des blocs envoisinant tel que la complexité globale du modèle polytopique est donnée par s = nh+1 réalisations extrêmes.

L'évolution du système satisfera les limitations physiques sur la commande et les restrictions sur la région accessible dans l'espace d'état. Leurs descriptions seront regroupées dans un ensemble d'inégalités linéaires:

Cξ k ≤ W (20)
En ce qui suit, l'objective de la commande est la régulation de l'état ξ k vers l'origine en satisfaisant les contraintes par l'approche de la commande optimale sur un horizon glissant. Le critère de performance est construit tel que la loi de commande obtenue garantie pour une région non vide au voisinage de zéro un comportement robuste en présence d'un retard variable.

III. Loi de commande explicite

Dans cette section le modèle de prédiction nominal [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] sera utilisé pour le calcul d'une loi de commande affine par morceaux. Chaque morceau sera associé à une sousensemble de contraintes saturées.

A. Commande prédictive

La stratégie de commande MPC pour le système à retard considéré ici, construit à chaque pas d'échantillonnage k la séquence de commande optimale:

k * u = {u k|k , . . . , u k+N -d-1|k } (21) F =                              A h-d 0 0 . . . 0 B -∆ ∆ d-2 0 . . . . . . 0 h -d            0 . . .
I m 0 . . . . . . d -2              . . . . . . . . . 0 . . . . . . 0 I m . . . . . . . . . 0 . . . I m 0                              ; (14)
Ḡ = 0 | 0 . . . . . . . . . | 0 0 | . . . . . . 0 I m T (15)
F =            A δ h (B -∆) δ h-1 (B -∆) + δ h ∆ δ h-2 (B -∆) + δ h-1 ∆ . . . . . . δ 1 (B -∆) + δ 2 ∆ 0 0 I m 0 . . . 0
           ; (16)
G = δ 1 ∆ 0 . . . . . . 0 I m (17)
pour l'index de performance

k * u = arg min {u k|k ,...,u k+N -d-1|k } { ξ k+N |k T P ξ k+N |k + N j=1 ξ k+j|k T Qξ k+j|k + N -d-1 j=0 u T k+j|k Ru k+j|k (22) tel que:        ξ k+j+1|k = F ξ k+j|k + Ḡu k+j|k Cξ k+j|k ≤ W ; j = 1, . . . , N -1 u k+i|k = 0, i = N -d, . . . , N -1 ξ k+N ∈ X N ; qui évalue
u k = k * u (1) = u k|k (23)
Utilisant les nouvelles mesures, le nouveau vecteur de commande est calculé par optimisation, ainsi une commande en boucle fermée est obtenue. La construction de la loi de commande prédictive sera influencée par le choix de l'horizon de prédiction N , les facteurs de pondérations sur la trajectoire de l'état Q = QT ≻ 0 et la commande R = RT ≻ 0. Pour la pénalisation de l'état terminal la matrice P est généralement construite tel que l'horizon de prédiction peut être étendu à l'infini par l'introduction du terme ξ k+N |k T P ξ k+N |k dans (22). Néanmoins ξ k+N |k doit satisfaire quelques conditions matérialisés par la contrainte terminale qui force cette prédiction d'atteindre un ensemble invariant prédéfini X N . Dans la sous section 3.3, le choix de l'indexe de performance sera discuté (en particulier les matrices Q, R et indirectement P ) tel que la loi de commande calculée présente certain degré de robustesse vis à via la variable de retard.

B. Programmation multiparamétrique

Après l'expression des prédictions comme fonctions de l'état présent et les futures commandes, le problème d'optimisation dans (22) peut être formalisé comme un problème quadratique multiparamétrique ( [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], [START_REF] Goodwin | Constrained Control and Estimation[END_REF], [START_REF] Dua | Multi-Parametric Model-Based Control: Theory and Applications[END_REF], [START_REF] Olaru | Avoiding constraints redundancy in predictive control optimization routines[END_REF])

k * u (ξ k ) = arg min ku 0.5k T u Hk u + k T u Gξ k subject to : A in k u ≤ b in + B in ξ k ( 24
)
où ξ k est le vecteur des paramètres. La fonction coût est convexe et l'optimum est unique si H ≻ 0, l'ensemble des contraintes linéaires décrivent des polyèdres paramétrés tel que l'optimum sous contraintes peut être décrit par la combinaisons de sommets paramétrés ( [START_REF] Olaru | Avoiding constraints redundancy in predictive control optimization routines[END_REF]). Des procédures géométriques universelles existe pour partitionner l'espace paramétré selon l'ensemble des contraintes actives et exprimer analytiquement la valeur optimum comme une fonction affine par morceaux de ξ k pour chaque combinaison. Globalement l'optimum k * u (ξ k ) est une fonction affine par morceaux sur une partition polyédrale de l'espace d'état.

De plus, la solution explicite pour la loi MPC peut être obtenue en retenant la première composante de k * u (ξ k ), ainsi la commande prédictive est exprimé en terme d'une loi de commande rétroaction affine par morceaux:

u k = K M P C i ξ + κ M P C i , with i s.t. x ∈ D i , (25)
pour D i , régions polyédrales dans ℜ n+hm . Remarque 1: Le modèle de prédiction est linéaire, l'origine est un point faisable (dans le cas générale placé à l'intérieur du domaine faisable) et représente un point d'équilibre pour le système [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Les problème (22), et (24), sont faisables et en plus de ça, l'optimum associé sera sans contraintes. La conséquence directe est que la loi de commande affine correspondant à la région D i0 contenant l'origine (0 ∈ D i0 ) n'a plus de partie affine (κ M P C i0 = 0) et la loi de commande locale correspond à la commande optimale sans contraintes (K M P C i0 = K LG ).

C. Réglage du MPC pour la robustesse

Considérons un problème de commande min-max sur un horizon infini pour le système polytopique (18):

min u k ,u k+1 ,u k+2 ... max F ∈Ω ξ J ∞ (26) où J ∞ = ∞ i=0 ξ T k+i Qξ k+i + u T k+i Ru k+i (27)
u k = Kξ k (28)
où Q > 0, R > 0 sont des matrices de pondération convenable a priori et K, le gain de retour jouant le rôle d'argument d'optimisation. Utilisons l'idée dans [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], et en notant

P = GS -1 et Y = KS, pour S ≥ I, l'LMI suivante peut être construite:     S SF T + Y T G T SQ 1/2 Y T R 1/2 F S + GY S 0 0 Q 1/2 S 0 GI 0 R 1/2 Y 0 0 GI     ≻ 0, (29)
La loi de commande stabilisante est donnée par K = Y S -1 où Y , S et le scalaire G sont les solutions du problème LMI (similaire avec la construction dans [START_REF] Kothare | Robust constrained model predictive control using linear matrix inequalities[END_REF]):

min G,S, Y G     S SF i T + Y T G T i SQ 1/2 Y T R 1/2 F i S + G i Y S 0 0 Q 1/2 S 0 GI 0 R 1/2 Y 0 0 GI     ≻ 0,
for all i = 0, . . . , s S ≥ I (30)

Remarque 2: La LMI dans (30) n'est pas dépendante de l'état et ainsi la loi de commande correspondante est un gain de retour fixe. La loi trouvée u k = Kx k représente une commande robuste stabilisante dans le cas sans contraintes.

La procédure de réglage des paramètres MPC est la suivante: étant donné les matrices F , Ḡ et Y, S de (30), les matrices Q ≥ 0 et R > 0 (et indirectement P ≥ 0) seront construite tel que la solution optimale du problème sans contraintes (22) soit:

k * u =      Y S -1 Y S -1 ( F + ḠY S -1 ) . . . Y S -1 ( F + ḠY S -1 ) N -1      ξ k (31)
La solution (non-unique) ( Q, R) doit satisfaire:

Q = P -F T P F + Y S -1 T ( R + ḠT P Ḡ)Y S -1 (32)
RY S -1 + ḠT P ḠY S -1 + ḠT P F = 0 (33)

Ce problème peut être résolu dans le cas général en employant la formulation LMI [START_REF] Larin | About the inverse problem of optimal control[END_REF]:

min α P -F T P F + Y S -1 T ( R + ḠT P Ḡ)Y S -1 ≻ 0 Z RY S -1 + B T P BY S -1 + B T P A I ≻ 0 Z ≺ αI, P ≻ 0 (34)
Théorème 1: La loi de commande MPC nominale, calculée en utilisant l'index de performance obtenu comme probleme d'optimalité inverse en tenant compte d'une loi de commande linéaire robuste dans le cas sans contraintes, est robustement stabilisante pour le système (18) malgré les contraintes dans une voisinage (nondegeneré) de l'origine V .

La démonstration est constructive et suit les arguments décrits dans cette section.

IV. ENSEMBLE INVARIANT ROBUSTE

A. RPI maximale

Le problème de synthèse étant résolu, on dispose d'une loi de commande supposé stabiliser le système à retard variable. La question est: quel est l'ensemble invariant pour le système en boucle fermée?

Par la suite on va utiliser le fait qu'une approximation peut être obtenu par la construction de l'ensemble invariant robuste (MRPI) pour un système affine par morceaux (PWA) affecté d'incertitudes. En effet, un système PWA (avec incertitudes polytopiques) est obtenu par l'association du système LTI (affecté par des incertitudes polytopiques) en boucle fermée avec la loi de commande affine par morceaux:

ξ k+1 = f P W A (ξ k ) = (F + GK M P C i )ξ k + κ M P C i for ξ k ∈ D i (F, G) ∈ Ω ξ Ω ξ = Co {(F 1 , G 1 ), (F 2 , G 2 ), . . . , (F s , G s )} (35) où D i sont des partitions polytopiques D = ∪ i D i .
Les dynamiques reliées à une réalisation extrême des incertitudes polytopiques du PWA seront décrites par:

ξ k+1 = f j P W Ai (ξ k ) = (F j + G j K M P C i )ξ k + κ M P C i for ξ k ∈ D i , j ∈ {0, 1, . . . , s} (36)
La description de l'ensemble MRPI pour un tel système PWA n'est pas immédiate parce que la détermination en temps fini de cet ensemble n'est pas garanti. Néanmoins, par le fait que la partition de l'espace d'état est donné par des régions polyédrales seront utilisées pour construire l'approximation appropriée.

Pour décrire cette construction géométrique, les opérateurs image et la préimage sur un ensemble Ψ ∈ ℜ n+hm définit comme:

Im fP W A (Ψ) = j ζ ∈ ℜ n+hm |∃ξ ∈ Ψ, s.t. ζ = (F j + G j K M P C i )ξ + κ M P C i for ξ ∈ D i ∩ Ψ} (37) P reIm fP W A (Ψ) = j {ξ ∈ D|∃ζ ∈ Ψ, s.t. ζ = (F j + G j K M P C i )ξ + κ M P C i for ξ ∈ D i } (38)
B. Procédure contractive L'idée dans [START_REF] Benlaoukli | Computation and bounding of robust invariant sets for uncertain systems[END_REF] est de soustraire de la partition d'état D = ∪ i D i qui définit le système PWA, ces régions pour lesquelles une des dynamiques extrême évoluera en dehors de D. C'est une procédure itérative tel que après chaque itération, l'ensemble D est modifié, ainsi les évolutions possibles sont à revérifier.

La complexité de cette procédure réside dans le fait que la soustraction d'ensembles convexes n'est pas convexe. En d'autre terme, si D est convexe, il n'y a pas de garanties qu'il restera convexe après une itération de la procédure contractive. Indirectement, cela nous ramène à la déduction que l'ensemble MRPI set peut être non convexe.

Procédure 1: Algorithme Contractif V 0 = D k = 0 while (condition de précision)

V k+1 = P reIm fP W A (Im fP W A (V k ) ∩ V k ) k = k + 1 C. Procédure expansive
Dans ce cas, au lieu d'exclure graduellement les régions en dehors de l'ensemble MRPI, en commence par un ensemble RPI et en ajoute les régions qui évolue dans un pas à l'intérieur de l'ensemble RPI. L'ensemble résultant est RPI et il est croisant d'une façon monotone (dans le sens d'inclusion) et il est limité par MRPI.

Un avantage important de la procédure expansive et que, les résultats intermédiaires sont robustement invariant et peuvent être considérés comme candidats pour l'approximation de l'ensemble MRPI.

Procédure 2: Algorithme Expansif

find σ > 0 s.t. E(σ) ⊂ D i0 V 0 = E(σ) k = 0
while (condition de précision)

V k+1 = P reIm fP W A (Im fP W A (D) ∩ V k ) k = k + 1

D. Approximation intérieure

Notons l'ensemble invariant maximal Ψ et les itérations obtenus avec les procédures expansive et contractive Ψ e i et Ψ c i respectivement. La procédure expansive Ψ e i ⊂ Ψ et la procédure contractive Ψ c i ⊃ Ψ ne possèdent pas de mesure de convergence vers l'ensemble invariant maximale. Néanmoins, pour l'utilisation des deux procédures on obtient une approximation intérieur pour l'ensemble MRPI:

Ψ e i ⊂ Ψ ⊂ Ψ c i ( 39
)
Considérons la distance de Hausdorff sur les polyèdres. La distance d H (Ψ c i , Ψ e i ) peut fournir une mesure de l'approximation de l'MRP donnée par Ψ e i et ainsi une condition de précision:

Ψ e i ⊂ Ψ ⊂ Ψ c i ⊂ Ψ e i ⊕ B 0 (d H (Ψ c i , Ψ e i )) (40)

V. Exemple

Considérons le procédé reporté dans [START_REF] Furtmueller | Disturbance suppression for an industrial level control system with uncertain input delay and uncertain gain[END_REF]. L'objectif est la commande du niveau de coulée dans une cuve, le schémabloc est présenté dans la figure 1. Le système continu à contrôler est considéré comme un double intégrateur avec un retard variable:

ẋt = 0 0 1 0 x t + 1 0 u t-τ , τ ∈ [0s, 0.3s] (41)
Le modèle discret est donné par:

x k+1 = 1 0 0.1 1 x k + 0.1 0.05 u k-i - ∆(u k-v -u k-v+1 ), with v ∈ {0, 1, 2} (42)
Dans un premier temps on calcul la matrice ∆. Due au fait, que dans la représentation originale, le vecteur d'état x k est à deux dimensions, l'incertitude polytopique sera:

∆ ∈ Co 0 0 , 0.05 0 , 0 0.013 (43)
Dans la représentation d'état étendue, le modèle nominal obtenu pour τ = 0.3 est:

ξ k+1 =       1 0 1 0 0 0.1 1 0.05 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0       ξ k +       0 0 0 0 1       u k (

VI. Conclusion

Une loi de commande prédictive a été calculée pour des systèmes à temps de retard variable.

Les contraintes sont prises en charge dès le stage de conception et l'approximation itérative de l'ensemble invariant positif maximal offre l'information sur la région de l'espace d'état où la commande est viable.

Fig. 1 .

 1 Fig. 1. Schéma général du procédé à controlé.

  compte, que le retard maximal est de 3 instants d'échantillonnages, on choisit un horizon de prédiction N = 5 afin d'avoir une solution explicite moins complexe (47 régions dans la partition de l'espace d'état, voir la figure2). Néanmoins, il est à mentionner, qu'en général, le domaine de faisabilité est d'autant plus grand si l'horizon de prédiction est grand.

Fig. 2 .

 2 Fig. 2. Projection de la partition de la solution explicite sur les deux premières composantes de l'espace d'état étendu.Le modèle polytopique dans la représentation d'état étendue qui englobe (utilisant 7 réalisations extrêmes) le système à retard variable, va nous permettre d'employer la procédure contractive pour l'approximation de l'ensemble invariant maximal. Dans la figure (3) l'approximation du MRPI est obtenue après 5 itérations.Finalement dans la figure (4) une simulation temporelle avec variation du retard est présentée (l'état initial à (0; -2)).

Fig. 5 .

 5 Fig. 5. Le signal de commande et la variation du retard.

Références

Fig. 3 .

 3 Fig. 3. La partition de la solution explicite et l'approximation de l'ensemble MRPI.

Fig. 4 .

 4 Fig. 4. Evolution temporelle de l'état du système.


Charly Euzenat
email: charly.euzenat@ens-paris-saclay.fr

Sylvain Lavernhe

Christophe Tournier

FVPM simulation of scratching induced by a spherical indenter

Keywords: Scratching, FVPM, Grinding, Multi-scale Modeling

come

FVPM simulation of scratching induced by a spherical indenter

Introduction

The performances of high added value parts are strongly linked with the ability to master the manufacturing process at all involved steps. In industry, finishing operations based on abrasion material removal play a major role in ensuring functional requirements and remain mandatory to improve surface characteristics after machining whether the surface is designed to provide optical, friction, or aerodynamic functions. The abrasion process is characterized by complex interactions between the abrasive tool and the machined surface in order to remove and repel matter. These interactions are generated by the path of sharp particles, inducing grooves over the surface [START_REF] Gahr | Preface[END_REF]. The global geometric shape as well as the local texture of the surface are affected by the process which, in turn, may influence the overall quality and limits the service lifespan of mechanical components. In these regards, manufacturers meet their requirements relying on empirical rules combined with an experienced based know-how. The contributions of models and simulation algorithms are not limited to improving the manufacturing process but also allow a better understanding of the abrasion mechanism itself.

The modeling of material removal mechanism has been investigated through several techniques. The chip thickness, the induced grinding force along with the thermal effects models have been developed. A review of these grinding models has been proposed by [START_REF] Tönshoff | Modelling and simulation of grinding processes[END_REF] and a multi-scale review of the experi-mental and simulation approaches has been proposed by [START_REF] Brinksmeier | Advances in modeling and simulation of grinding processes[END_REF]. Several ways of defects have been identified by [START_REF] Klocke | Analysis of defect mechanisms in polishing of tool steels[END_REF]. Three hypotheses have been made to describe the removal mechanisms. The flow hypothesis and the abrasion hypothesis are mechanical interactions inducing high local strain and stress to remove and repel matter. The mechanical defects induced by the abrasive particles fall into four main categories :

• Microploughing • Microfatigue • Microcutting • Microcracking
The plastic deformations encountered by the specimen, repels the material at the sides of the grooves and results in microploughing. The pile up model developed in [START_REF] Chen | Analysis and simulation of the grinding process. part II: Mechanics of grinding[END_REF] accounts for these plastic deformations in microploughing. Depending on the material characteristics, the interactions may although generate chips and cut the specimen instead of creating bulges. Microfatigue occurs when several grains paths are located in the same area until the material fails. Brittle materials like ceramics are more subjected to microcracking as a result of high stress induced by abrasive particles.

From a macroscopic and empirical point of view, several models exist in order to predict the surface obtained following a finishing process. One of the most common to induce the removal rate is the Preston model [START_REF] Preston | The theory and design of plate glass polishing machine[END_REF] (Eq. 1).

dz dt = K p .P.V (1)
This equation reflects the fact that the removal rate is related to both the contact pressure P and the relative velocity V. The Preston coefficient K p is the equivalent for the efficiency of the abrasive tool against process parameters. Several other laws derived from the Preston model have been proposed to improve the prediction relevance.

dz dt = K p .P α .V β (2)
Klocke [START_REF] Klocke | Analysis of defect mechanisms in polishing of tool steels[END_REF] has developed a general-purpose model for abrasion (Eq. 2) with two additional degrees of freedom in the equation. This model requires more involved identification procedure yet offers a wider range of possibilities while modeling various tool-workpiece pairs. Whatever the shape of the equations, the laws identified above are mainly used as a description of the interactions but are not yet implemented in simulations on their own.

Applying the concept of continuum mechanics to abrasion is under high investigation in the literature, especially allowed by recent development of computing capabilities. Still, the simulation of a complete abrasion process is out of reach within reasonable computational time, this is why test cases involving a single abrasive grain are often preferred to evaluate numerical methods. The scratching test procedure is used to characterize a material under the path of a spherical indenter and can be considered as the basic interaction involved during abrasion. This elementary damage procedure has been previously chosen by [START_REF] Leroch | Smooth particle hydrodynamics simulation of damage induced by a spherical indenter scratching a viscoplastic material[END_REF] for the validation of SPH simulation code. Single grain scratching and impact have been successfully simulated using both FEM and meshless numerical methods [START_REF] Takaffoli | Finite element analysis of single impacts of angular particles on ductile targets[END_REF][START_REF] Takaffoli | Material deformation and removal due to single particle impacts on ductile materials using smoothed particle hydrodynamics[END_REF].

Theory

During a simulated scratching, the material cohesion and behavior inside the specimen is numerically emulated through a set of equations. In the general case, analytical solution cannot be found, hence the use of numerical resolution methods. As a consequence, the discretization of the volume and the time range are required to compute approximations of the solution. Among all the numerical techniques available, one can divide them in two main categories. Mesh-based techniques features connectivity information between the interpolation points whereas meshless methods rely on a set of uncorrelated points. The abrasion process simulation includes large deformations so that a regeneration algorithm is often needed to handle element distortion. Concerning meshless methods, the particle distribution used to interpolate vector fields inside the specimen allows to take into account any amount of deformation whether small or large. That's why particle-based methods are a suitable choice for wear simulations. Finite Volume Particle Method (FVPM) is a meshless method which includes many of the attractive features of both particle methods, such as SPH, and conventional mesh-based Finite Volume Methods (FVM) [10]. In FVPM, the Sheppard function is used as the interpolating

W i = 1 x -x i ∞ ≤ h i 0 x -x i ∞ > h i (3)
ψ i = W i (x) k W k (x) (4)
U = i ψ i u i (5)
Any spatial field can be interpolated from the spatial scheme composed of overlapping particles, as shown in Figure 1. The kernel function W i defined by equation 3 allows to take into account the neighbor particles within a defined domain size h i . The equation 4, which sets the Shepard value ψ i for a given point, consists in a normalization of the kernel over all the considered particles. Equation 5gives the value U of the interest quantity as the summation over all the neighboring particles. This principle allows to retrieve the value at any given point in between particles. Since no connectivity information is embedded in the particle cloud, an additional cost is driven by the need to retrieve closest neighbors. Nevertheless spatial hashing algorithms allow to reduce the addressed complexity. The interpolation of the derivatives is carried out with the same spatial discretization.

ρ dC dt = ∇ . σ + ρg + f c (6)
The equation 6 accounts for the internal equilibrium that structures the solid in question and represents a local application of the fundamental principle of dynamics. The velocity field of the solid is denoted by C and σ represents the stress tensor. The solid is eventually subjected to gravity g and external forces f c . Solving this equation requires the evaluation of the divergence of constraints ∇ . σ. The speed of sound is given by the material's Young's modulus, E, and Poisson's ratio, ν, according to:

v = E 3ρ 0 (1 -2ν) (7)
The FVPM method is based on calculating the stress divergence through the flow that passes through the common surface between two adjacent interpolation zones (Eq. 8).

Ω i ∇ . σ = S i σ . dS = S i n . σ . dS = j (Γ i j -Γ ji ).σ i j (8) Γ i j = - m l=1       ∆S l δ + l δ - l       (9)
The integral of the divergence of constraints in the volume Ω i is calculated using the flow-divergence theorem and solved numerically in equation 9 based on neighboring volumes as shown in Figure 2. The variables δ + and δ -denote the kernel summation outside and inside the particle number i. The Figure 2 depicts the calculation of interaction vector Γ i j between particle i and j. The area is partitioned into surfaces of constant δ + and δ -and the intersection border is divided into 4 segments. Each segment corresponds to one term in the summation of equation 9. The superposition of rectangular supports is necessary to calculate the interactions between the particles. The choice of this geometry was made in order to accurately calculate the contact surfaces between the rectangular supports. More details on the implementation of this method can be found in [START_REF] Jahanbakhsh | Finite Volume Particle Method for 3-D Elasto-Plastic Solid Simulation[END_REF].

In order to link strain and stress, the Johnson Cook model is implemented. This model includes a non-linear hardening coefficient n and the effect of strain rate εp and temperature T * .

σ = A + Bε n p       1 + Cln       εp ε0             1 -T m * (10)
The failure of the material is modeled by the removal of particles whose plastic strain reaches a certain value ε f . This value is a function of the strain rate εp , the pressure p relative to the von Mises stress σ v and the temperature T * . The parametrization of this model relies on five material constants D i .

ε f =       D 1 + D 2 exp(D3 p σ v )             1 + D 4       εp ε0             [1 + D 5 T * ] (11)
In FVPM, like in SPH, computational nodes usually move with the material velocity, which is compatible with the Lagrangian form of the equation of motion. This enables the method to handle moving interface problems without issues of mesh deformation or tangling. Contrary to mesh-based methods, whose core functions rely on generating a rigidity matrix to be inverted, working with particles as interpolation points implies to compute internal forces exerted on each particle at each time step. The forces are correlated to accelerations which are integrated as time passes from one step to another. The propagation of acoustics waves in the specimen is part of the outcomes of the simulation. Thus, using this force based framework raises numerical instabilities against the chosen time step. The plasticity simulation is based on radial return stress evaluation.

CFL = vδt δx (12)
The CFL condition given in equation 12 ensures that the numerical integration scheme is stable. It is depends on the characteristic speed v in the material, the value of time steps δt and the particle spacing δx. CFL number is comprised between 0 and 1 which, in turn, bounds the time step to keep the simulation stable. A refinement of the particle spacing δ x will not only lead to an increase in the number of particles but also to a reduction in the time step.

The presented article is based on a FVPM implementation developed by [START_REF] Jahanbakhsh | Simulation of silt erosion using particle-based methods[END_REF] which is named SPHEROS. This simulation code features several solvers emulating fluid or solid behavior and relies on the open MPI [START_REF] Gabriel | Open MPI: Goals, concept, and design of a next generation MPI implementation[END_REF] message passing interface for parallel computations on clusters. The elasto-plastic deformations induced by a rigid spherical particle over the specimen surface have been evaluated to complete the erosion simulations resulting from the impact of silt particles on a surface [START_REF] Jahanbakhsh | Finite Volume Particle Method for 3-D Elasto-Plastic Solid Simulation[END_REF]. Residual stress repartition have been found coherent with [START_REF] Meguid | Three-dimensional dynamic finite element analysis of shot-peening induced residual stresses[END_REF]. The main difference between erosion and abrasion lies in the amount of plastic strain reach by the specimen, and which is much more important concerning abrasion.

Scratching test case

The scratching test consists in damaging a solid under the motion of an indenter. The indenter moves tangentially to the surface under a normal load which results in the creation of a scratch. The geometry of the groove associated with the required efforts characterize the ability of the solid to resist to local plastic deformations. The scratching simulation has been done on copper, since this material characteristics have been widely studied in the literature. The associated material properties and Johnson Cook material parameters are listed in tables 1, 2 and 3.

D 2 4.89 [-] D 3 -3.03 [-] D 4 0.014 [-] D 5 1.12 [-]
The specimen is composed of overlapping particles as shown in Figure 3. The repartition of particles is generated as an assembly of two sphere quarters and a half cylinder. The spheres are located at the edges and the cylinder represents the core of the specimen. This repartition consists in an assembly of several hulls whose distance to the center line of the specimen is the same and whose particle diameter is constant. An example of one hull is given in Figure 4. This repartition is suitable to generate regular variation of particles radius in the specimen. In this respect, the particles are always surrounded by others with a similar size. This avoid the instabilities generated in the interface due to large variation of particle size in the same area. The spacing length is refined in the areas of interest to balance relevance against computing time while the largest particles aim at extended the size of the simulation without penalizing computational complexity. The finest particles have a diameter 15 times smaller than the indenter in the presented test case.

The indenter is modeled by a single spherical particle. The indenter size in this simulation has a 400 µm diameter. The interaction between the specimen and the spherical particle modeling the indenter consists in a penalty force directed from the center of the indenter toward each of the particle in contact. Although the Coulomb model for friction is available in the code, the friction not been taken into account to stay coherent with [START_REF] Leroch | Smooth particle hydrodynamics simulation of damage induced by a spherical indenter scratching a viscoplastic material[END_REF]. In the simulation, the indenter speed is 100 m.s -1 whereas experimental speed is 10 mm.s -1 . The high speed chosen here does not reflect reality, but reduces the simulation time [START_REF] Ducobu | On the introduction of adaptive mass scaling in a finite element model of ti6al4v orthogonal cutting[END_REF]. The C coefficient of the equation 10 has been set to zero so that the indenter velocity has no influence on the specimen behavior. This allows to use a relatively high speed of scratching. This choice has been imposed by the fact that the FVPM model simulates the material strain and stress based on a dynamic res- olution of the equations. The kinematics of the indenter consists in a vertical descent until the chosen penetration depth is reached. Then the indenter slides tangentially to the surface over a distance defined 10 times longer than its radius.

The simulations have been run in a computational cluster. The main characteristics of the test case are listed in table 4.

Results and discussion

Several simulations have been run with different penetration depth h s from 5 µm to 60 µm. A screen shot of the simulation is displayed in Figure 5. In Figure 6 the surface has been extracted from the points cloud using a level set of an implicit function. As shown in Figure 6 bulges are present at the side of the indentation.

The evolution of the normal load L and tangential effort F has been plotted against scratching distance D in Figure 7. The first part of the curve corresponds to the indentation. At this point, normal effort is maximum and tangential effort is still null. Then, as the indenter begins to slide over the surface, tangential resistance is opposed to the motion and leads to a stationary state. Oscillations are observed from these curves and are due to the periodical removal of damage particles during the simulation. For each indentation depth, the corresponding normal load L and tangential effort T have been collected in the simulated data and plotted in Figures 8 and9. In addition, the numerical results obtained have been compared with the article of Leroch [START_REF] Leroch | Smooth particle hydrodynamics simulation of damage induced by a spherical indenter scratching a viscoplastic material[END_REF] in these two figures. Orders of magnitude of the proposed simulations are consistent with the literature. Normal load simulated with the FVPM model tends to over estimate the real value whereas tangential efforts are coherent. For small indentation depth, the normal load is higher than both experimental and simulated literature results. For medium loads the values fit the literature data. While the depth reaches the highest values, the uncertainty increases because of the high damages generated in the specimen which leads to particles being periodically removed. The fact that friction coefficient has not been taken into account may explain the tangential effort tendency to be under the other values. The particle distribution used with the FVPM model, which is coarser than [START_REF] Leroch | Smooth particle hydrodynamics simulation of damage induced by a spherical indenter scratching a viscoplastic material[END_REF], may have an influence in the over estimation of normal load.

Conclusion

The elasto-plastic FVPM model implemented in SPHEROS have been assessed for the problem of scratching a specimen under the tangential motion of an indenter. The chosen simulations conditions aimed at replicating the main raw interactions encountered during the abrasion process.

The simulation kinematics have been imposed and the efforts have been observed afterward. Various penetration depth have been tested. The topography is coherent with experimen- tal observations found in literature. FVPM appears to be an interesting alternative to molecular dynamic model or SPH algorithms. Nevertheless numerical instabilities have been encountered while large plastic strain was reached. The plastic simulation algorithm based on radial return stress evaluation may not to be sufficiently stable for large deformations.

This issue need to be considered before investigating the implementation of a several particles grinding process. The simulation framework is suitable for taking into account abrasive tool wear during the process. An increased computing time is expected against the problem complexity, which will need to be addressed.

The primitive geometry for the indenter could be model by an elasto-plastic model in order to simulate abrasive wear which is a common problem while industrializing mechanical part.

  2h

Figure 1 :

 1 Figure 1: rectangular support kernels

ΔSFigure 2 :

 2 Figure2: Interactions between particles i and j[START_REF] Jahanbakhsh | Finite Volume Particle Method for 3-D Elasto-Plastic Solid Simulation[END_REF]

Figure 3 :Figure 4 :

 34 Figure 3: Specimen particle repartition

Figure 5 :

 5 Figure 5: Simulation result for h s 60 µm

Figure 6 :

 6 Figure 6: Extracted topography heights [µm]

Figure 7 :

 7 Figure 7: Scratching efforts for a 60 µm depth

Figure 8 :

 8 Figure 8: Normal load exerted on the indenter

Figure 9 :

 9 Figure 9: Tangential load exerted on the indenter

Table 1 :

 1 Material properties for copper

	Material property	Value	Unit
	Density, ρ	8960	[kg.m -3 ]
	Young's modulus, E	124	[GPa]
	Poisson's ratio, ν	0.34	[-]

Table 2 :

 2 Johnson-Cook parameters for copper

	Parameter name	Value	Unit
	Initial yield stress, A	90	[MPa]
	Hardening coefficient, B	292	[MPa]
	Hardening exponent n	0.31	[-]

Table 3 :

 3 Johnson-Cook damage parameters for copper

	Parameter name	Value	Unit
	D 1	0.54	[-]

Table 4 :

 4 Simulation characteristics

	Number of cores	64
	Number of particles	21143
	Simulation time	5 h 28 min

Preprint submitted to ElsevierSeptember 19, 2019