Azelle Courtial
email: azelle.courtial@ign.fr

Guillaume Touya
email: guillaume.touya@ign.fr

Xiang Zhang

Keywords: Generalization, Evaluation, Constraints, Raster, Deep learning

Deep learning techniques have recently been experimented for map generalization. Although promising, these experiments raise new problems regarding the evaluation of the output images. Traditional map generalization evaluation cannot directly be applied to the results in a raster format. Additionally, the internal evaluation used by deep learning models is mostly based on the realism of images and the accuracy of pixels, and none of these criteria is sufficient to evaluate a generalization process. Finally, deep learning processes tend to hide the causal mechanisms and do not always guarantee a result that follows cartographic principles. In this article, we propose a method to adapt constraint-based evaluation to the images generated by deep learning models. We focus on the use case of mountain road generalization, and detail seven raster-based constraints, namely, clutter, coalescence reduction, smoothness, position preservation, road connectivity preservation, noise absence, and color realism constraints. These constraints can contribute to current studies on deep learning-based map generalization, as they can help guide the learning process, compare different models, validate these models, and identify remaining problems in the output images. They can also be used to assess the quality of training examples.

Introduction

Deep learning is now widely used and is successfully applied in various tasks in computer vision and natural language processing. The potential of this approach for map generalization has recently been demonstrated [START_REF] Touya | Is deep learning the new agent for map generalization?[END_REF][START_REF] Feng | Learning cartographic building generalization with deep convolutional neural networks[END_REF][START_REF] Du | Segmentation and sampling method for complex polyline generalization based on a generative adversarial network[END_REF].However, to bring the deep learning approach to map generalization to the next level, the question of how good the generalization results are must be addressed, whether it is during or after the generalization process.

Automated evaluation has multiple roles in the map generalization process due to the iterative nature of generalization [START_REF] Mackaness | Chapter 5 -Evaluation in the Map Generalisation Process[END_REF]: (i) to evaluate the results while tuning the process and its parameters, (ii) to control the iterations of the process to make sure the map is improved by the generalization algorithms, (iii) to assess the quality of the final output map. For all these different types of evaluation, a balance between the legibility of the map and the preservation of important structures (e.g., topology and spatial distribution) is targeted [START_REF] Zhang | Automated evaluation of generalized topographic maps[END_REF][START_REF] Zhang | Automated evaluation of building alignments in generalized maps[END_REF][START_REF] Stoter | Evaluation in Generalisation[END_REF], while respecting the fitness for use of the map Figure 1: Deep learning image generation failure, roads images are from our experiments, while other are from [START_REF] Zhu | Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks[END_REF].

(you do not want to disconnect your roads if your map are to be used to visualize trip itineraries). For most of these evaluation tasks, researchers found that the constraint-based approach provides a useful framework [START_REF] Beard | Constraints on rule formation[END_REF][START_REF] Bard | Quality Assessment of Cartographic Generalisation[END_REF][START_REF] Burghardt | Investigations on cartographic constraint formalisation[END_REF][START_REF] Stoter | Methodology for evaluating automated map generalization in commercial software[END_REF][START_REF] Touya | Social Welfare to Assess the Global Legibility of a Generalized Map[END_REF].

However, the use of deep learning implies some other challenges for the evaluation, mainly due to the use of raster (images) instead of vector format:

• The credibility, noise, and realism of the predicted images have to be evaluated. Indeed, deep learning generates a new image by choosing pixel values. The main constraint for the choice of the pixel values is that the prediction has to look like the target (i.e., ground truth) from a set of training examples. Figure 1 illustrates some unrealistic images of road map produced using deep learning, and some other failure cases by generative adversarial networks (GAN) [START_REF] Zhu | Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks[END_REF]. So there is no guarantee that the results will follow cartographic rules, represent the same geographic information or be a functioning map. A good neural network should produce maps that respect these objectives. Nevertheless, as most deep learning models do not explicitly incorporate these considerations, it is more reasonable to examine if and how the map objectives are satisfied with an independent evaluation process.

• Most deep learning techniques are raster-based and the raster format is a limitation for using the traditional map generalization evaluation methods. Although converting to a vector format is possible, it can complicate the whole end-to-end process. Sometimes it is even not suitable. For example, when the evaluation is used to guide the learning process during generalization, real-time conversion can be demanding.

The aim of this article is to explore how constraint-based evaluation could contribute to addressing these challenges.

To be specific, we focus on the evaluation of generalized raster images that should reflect the content of the map. To train the deep network for map generalization, our models take as input, images of the detailed map data, and try to predict images in the domain of generalized maps. The model predictions (i.e., generated images) have to look like a generalized map. Following previous research on deep neural network for road generalization in raster format.

The The following section is a literature review about map generalization evaluation and the challenges of deep learning approaches for map generalization. Then, we present our use case in 3 section. In 4 section, we propose seven raster-based constraints and their measurements for our use case. Finally, we present some concrete examples of the application of these constraints in our evaluation 5 section and some discussions about map generalization using deep learning quality management 6 section.

2 Literature Review

Deep Learning for Map Generalization

Deep learning methods are machine learning techniques mainly applied to images, point clouds, text, and graphs. Deep learning models are neural networks made of numerous abstraction layers. Each abstraction layer encodes the input until obtaining a prediction in the shape of the target, then the target and prediction are compared with a loss function.

Finally, the weights of the abstraction layer are backpropagated in order to minimize the loss. Thus, the deep neural network iteratively learns to reproduce the target from the examples, and probably for most similar input situations. [START_REF] Courtial | Generative Adversarial Networks to generalise Urban Area in Topographic Maps[END_REF]. Then, evaluation can be used to choose the most adapted configuration of a network or guide the post-processing (e.g., identifying a noise to eliminate). Finally, the main interest of an evaluation measure calculated using images is to guide the learning process, i.e., including a particular evaluation measure in the loss function of the neural network.

Deep learning techniques are new at dealing with cartographic information, and very few models are used for map generalization. Among the possible kinds of deep learning models, three have already been employed to resolve tasks in the field of map generalization: classification models, segmentation models, and image generation models. The next paragraphs detail the interest of these techniques for map generalization and the relative need for evaluation.

Classification models are models that classify a whole image into a category. They can be used to obtain information about some geographic objects and contribute to information enrichment before its generalization. This approach has been used to classify the building shapes [START_REF] Yan | Graph convolutional autoencoder model for the shape coding and cognition of buildings in maps[END_REF], and to detect some complex objects like highway interchanges [START_REF] Touya | Deep learning for enrichment of vector spatial databases: Application to highway interchange[END_REF]. In these examples, there is specific evaluation needed neither regarding the nature of the object (geographic information) nor regarding the nature of the task (generalization). The result of the classification is a class, and the evaluation consists in determining if the predicted classes correspond to the expected one, using a confusion matrix. However, in some cases the target classification is fuzzy and the relevance of global classification is more important than just the value of each element, e.g., for road section classification between the classes "selected at target scale" or "eliminated at target scale" [START_REF] Courtial | Can Graph Convolution Networks Learn Spatial Relations? Abstracts of the ICA[END_REF]. Here, the global reduction of edge density while maintaining the structure and connectivity of the road network is more important than the exact accuracy of the selected sections.

Segmentation models are used to classify the pixels that belong to a target object on an image. In map generalization, this method can be used to detect some objects and enrich geographic information as well as (or even better than) classification models [START_REF] Touya | Deep learning for enrichment of vector spatial databases: Application to highway interchange[END_REF]. But segmentation models can also be used to predict the shape of a generalized object from an image of the detailed shape. The potential of this technique for building [START_REF] Feng | Learning cartographic building generalization with deep convolutional neural networks[END_REF], roads [START_REF] Courtial | Exploring the Potential of Deep Learning Segmentation for Mountain Roads Generalisation[END_REF], and coastline [START_REF] Du | Segmentation and sampling method for complex polyline generalization based on a generative adversarial network[END_REF] generalization has been explored. The result of segmentation is a mask of the generalized objects. The common evaluation measure in the segmentation literature is the intersection over union. This measure compares the predicted mask with the target one. However, in map generalization, the target is one possible generalization and a different shape can represent a good generalization too. Moreover, this measure is highly affected by the position and size of map objects and very lightly affected by local changes. Consequently, a map object (road or building) that is well generalized but displaced of two pixels to the left (e.g., to avoid the overlap with a river) would be evaluated as a bad result, while a road with a small disconnection or bend coalescence will be evaluated as correct. The issue of evaluating map generalization is to ensure the information preservation and the legibility of the output rather to compare with a reference.

Generation and style transfer approaches aim to generate a new image from an input one. These approaches are based on the combination of a generator network that produces an image and a discriminator that evaluates if this image is false or if it is a real target image. The generator learns to fool the discriminator while the discriminator is trained to improve its classification accuracy. Such an adversarial strategy would eventually come to a balance where the generator outputs images that look realistic (i.e., they look like real maps when the target is a map), and the discriminator cannot tell which one is real. Various such networks have been proposed to produce images of a map from aerial images [START_REF] Isola | Image-to-image translation with conditional adversarial networks[END_REF], to produce an image of a map in the style of Google Maps from an image of OSM data in a raw style [START_REF] Kang | Transferring Multiscale Map Style Using Generative Adversarial Network[END_REF], and to produce an image of a generalized map from an image of detailed one [START_REF] Courtial | Generative Adversarial Networks to generalise Urban Area in Topographic Maps[END_REF]. For all these experiments, the evaluation is still challenging, as the network learns to reproduce a target domain that is not clearly defined. In the literature of generative model evaluation, one of the most important issues is the realism of the image. This question reflects the credibility of the output. In map generalization, however, information preservation and legibility are more important. The internal measures of the network are not sufficient to deal with both constraints simultaneously.

Map Generalization Evaluation

Map generalization evaluation is essential to control, assess, and validate the generalization process. In practice, map generalization is often evaluated visually by cartographers [START_REF] Stoter | Evaluation in Generalisation[END_REF]. Hence, automated evaluation is necessary, and developing evaluation criteria is still challenging [START_REF] Mackaness | Chapter 5 -Evaluation in the Map Generalisation Process[END_REF][START_REF] Zhang | Automated evaluation of building alignments in generalized maps[END_REF][START_REF] Stoter | Evaluation in Generalisation[END_REF].

Constraints aim to formalize map specifications [START_REF] Beard | Constraints on rule formation[END_REF]. Thus, they are essential for many tasks in cartography including map generalization evaluation [START_REF] Werder | Formalization of Spatial Constraints[END_REF][START_REF] Touya | Collaborative Generalisation: Formalisation of Generalisation Knowledge to Orchestrate Different Cartographic Generalisation Processes[END_REF]. Constraints are associated with an object and have a specific function. For example, the constraint "buildings must be greater than a certain size" aims at improving the legibility of buildings on the map. They can be classified according to the object they apply to [START_REF] Ruas | Modèle de généralisation de données géographiques à base de contraintes et d'autonomie[END_REF], their nature or function [START_REF] Galanda | Modelling constraints for polygon generalisation[END_REF], or both of them [START_REF] Burghardt | Investigations on cartographic constraint formalisation[END_REF]. Evaluation is usually based on at least some constraints that aim to ensure information preservation and some other constraints to improve the legibility of the map objects [START_REF] Zhang | Automated evaluation of generalized topographic maps[END_REF].

Evaluating a map or a generalization process from a set of constraints raises the following issues:

1. The identification of relevant constraints/desired characteristics of the map. Both the objectives (legibility and preservation) need to have similar weights in the evaluation to avoid caricatured results [START_REF] Zhang | Automated evaluation of generalized topographic maps[END_REF]. Blank images are legible but do not preserve any information, and on the contrary, the identity transformation is a perfect preservation but does not improve legibility at a smaller map scale.

2. The identification of the relevant level of analysis for each constraint. This is a key issue in constraint definition: a constraint on each road section is different from those on a complete road, or on the whole road network.

Ruas identifies three levels of constraints [START_REF] Ruas | Modèle de généralisation de données géographiques à base de contraintes et d'autonomie[END_REF]: micro that relate to an individual object of the database used to make the map; meso that relates to a group of objects; and macro that relates to a population (e.g., all roads of the map).

3. The definition of a measurement method. The complexity of the measure depends on the constraint. For instance, constraints on object size can be easily measured while others require advanced spatial analysis, e.g., alignment preservation [START_REF] Zhang | Automated evaluation of building alignments in generalized maps[END_REF] and line position accuracy [START_REF] Skopeliti | A methodology for the assessment of generalization quality[END_REF].

4. The computation of a level of satisfaction. The goal of this step is to determine if the result of the measure is acceptable. To do this, we can define an evaluation function [START_REF] Bard | Quality Assessment of Cartographic Generalisation[END_REF]. For example, for the constraint "the road position must be preserved," the measure could be the distance between two points from the previous and the new position, and the evaluation function would evaluate how the constraint is respected or violated given the measured distance.

5. The fusion of all the constraints into a synthetic value would permit the comparison of map generalization methods. However, generalization constraints often express conflicting ideas and cannot be totally satisfied at the same time: in a dense area, the legibility constraint demands information selection and/or object displacement which violates the preservation constraints. Consequently, several strategies can be used for this step [START_REF] Touya | Social Welfare to Assess the Global Legibility of a Generalized Map[END_REF]: minimize the value of worst constraint satisfaction, reduce the amount of critical measures, reduce the number of conflict areas, or increase the mean satisfying of all constraints, etc.

3 Use Case

Mountain Roads Generated by Deep Learning

In this article, we present how raster-based constraints can be defined, set up, and applied to the specific case of mountain roads. We focus on evaluating mountain road graphic generalization for a display at the 1:250,000 scale.

The goal is to be able to evaluate the images generated by three different deep learning models, i.e., CycleGAN [START_REF] Zhu | Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks[END_REF], Pix2Pix [START_REF] Isola | Image-to-image translation with conditional adversarial networks[END_REF], and U-Net [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] with a reference generalization (obtained by a combination of vector generalization algorithms and manual corrections). We choose to compare these three models as they have already been experimented for road map generalization, and the lack of evaluation methods make their comparison difficult.

To make the comparison as fair as possible, we train the networks with the same amount of training examples, and tiles constructed similarly. We based our experiments on images of roads in the Alps extracted from IGN (the French national mapping agency) maps at 1:250,000 (target) and 1:25,000 (input) scale [START_REF] Courtial | Alpinebends -a benchmark for deep learning-based generalisation[END_REF]. Our images (input and output) Road network Image represent 2.5 km2 in 256×256 pixels, so the resolution is around 10 m. The width of the road symbols on these images corresponds to road importance and varies between 1 and 6 pixels. The reference images represent the expected generalization.

The main challenge of this use case is the raster format of maps. In vector map generalization, road segments, faces, and networks are used to set up constraints. On the contrary, a raster is a grid of valued pixels, and this format does not permit to define the limit of each object. Consequently, the constraints should not hold on to the notion of object shape or limits, and other levels of analysis have to be explored, such as the pixel, a set of pixels, or the image. These levels are presented in Table 1. We can construct a pixel set using the following rule: two pixels that are 4-connected and have the same color should belong to the same pixel set. Consequently, a road map is composed of road and background pixel sets.

Choosing the Constraints

Most of our constraints are directly derived from the usual requirements of mountain road map generalization, and road map generalization [START_REF] Mustière | GALBE: Adaptive Generalisation -The need for an Adaptive Process for Automated Generalisation, an Example on Roads[END_REF][START_REF] Duchêne | Individual Road Generalisation in the 1997-2000 AGENT European project[END_REF]. We expect the mountain road generalization to reduce the coalescence in bend series and to smooth the road shape while preserving the position, connectivity, and structure of the network. Displacements are tolerated to avoid coalescence with other cartographic themes (rivers, for example), as well as bend removals sometimes, but the position has to be preserved as much as possible. We consequently decided to observe and measure the following elements:

1. The smoothness, the generalized shapes have to be smoother, less detailed than the initial ones;

2. The coalescence reduction, there should be no coalescence left;

3. The position preservation, the distances between input and generalized roads have to remain as small as possible;

4. The structure preservation, the road network structure and connectivity have to be preserved.

All these constraints are classical, and we only need to adapt the way they are measured. In addition, the observation of the example images of our use case validates our hypothesis that specific measures for deep learning results are required. Figure 3 describes an extract of our testing set. A visual evaluation of these images would guide the constraint definition and let us make the following observations. U-Net prediction is promising but artifacts and disconnections impact both the information preservation and the legibility of maps. Pix2pix images are good generalizations for most situations, but bend series and complex situations are often not well generalized, and coalescence sometimes remains. CycleGAN gives a better prediction, and most of the situations are correctly generalized; some cases are too smooth, and the image is sometimes blurred. Finally, the images have very few visible noise pixels in the form of regular small blue spots. We observe that some predictions are unrealistic, blurred, or noisy. Consequently, we decided to add some realism constraints that aim at checking that the model will produce images that look like real maps.

Validation Set

In this section, we present the images used to set up and validate the proposed constraint measures. We propose to constitute what we call the validation set with both "real data" (images extracted from our deep learning results presented above), and "fake data": hand-drawn images of interesting configurations of roads. For example, Table 2 lists interesting fake situations to construct by modifying the shape of a random road.

Proposed Constraints ans Their Measurements

In this section, we present all the proposed constraints, associated measures, validation, and satisfaction functions. Most of these constraints are extensions of the measures first proposed in [START_REF] Courtial | Constraint Based Evaluation of Generalized Images Generated by Deep Learning[END_REF]. We implemented all our measures in Python using the open-source libraries PIL and skimage. The code should run on any square image, but most measures are only relevant for images constituted of roads (pixels close to red) and backgrounds (pixel close to white). We choose to define a qualitative level of constraint satisfaction: "very good," "good," "neutral," "bad," "very bad." For our use case, we propose the definition of a threshold for each satisfaction level, based on the examination of the validation set of our experiment.

Clutter Reduction

Definition Clutter can be defined as too much and disorganized information in an image, and it can be seen as a proxy for image complexity. It has been shown that clutter makes the use of the images more difficult [START_REF] Rosenholtz | Measuring visual clutter[END_REF]. This is why we introduce a constraint to avoid that clutter raises during generalization. This constraint may be adapted for images of generalized maps in general, not only for images of generalized mountain roads.

≤ x < 10% 10% ≤ x < 20% 20% ≤ x < 30% 30% ≤ x < 60% x ≥ 60%
Measure Clutter measures are image-level measures that aim to evaluate the complexity of an image. Their adaptation for map legibility evaluation is still a question [START_REF] Dumont | Assessing the Variation of Visual Complexity in Multi-Scale Maps with Clutter Measures[END_REF]. We have tested four different measures for estimating the legibility of our road images:

• The edge density [START_REF] Rosenholtz | Measuring visual clutter[END_REF],

• The sub-band entropy [START_REF] Rosenholtz | Measuring visual clutter[END_REF], that is similar to JPEG compression, • The quad-tree method, measures the number of homogeneous square cells in the image [START_REF] Touya | Comparing Image-Based Methods for Assessing Visual Clutter in Generalized Maps[END_REF],

• The object segmentation-based measure [START_REF] Bravo | A scale invariant measure of clutter[END_REF].

Most of these measures represent a quantity; for more clarity, we propose to convert these quantities into a ratio using the formula 1

clutter_generalisation = |clutter initial -clutter prediction | clutter initial (1)
Validation and Limits Figure 4 presents the value of the edge density measure for test images (the images presented in Figure 3). This measure gives similar values for each image except the CycleGAN one. Indeed, these images include pixel noise that highly impacts the measure. This observation is similar for all tested clutter calculation methods, this is why we decide to use the edge density to measure clutter. The measure succeeds in detecting a complexity increase in the images from this method. However, if we look at the U-Net generalization, while a human could judge it as quite more complex than the others, the clutter measure only indicates a small difference.

Constraint Satisfaction

We propose the satisfaction values associated to the clutter measure presented in Table 3.

The thresholds were calibrated with our validation set of images (Figure 3).

Smoothness

Definition The smoothness constraint is a legibility measure that should constrain the generalized road to be smoother than the input road. The more the road is smooth, the fewer irregularities are visible, the more the shape is legible. This constraint is designed for images of generalized roads in general.

Measure To measure smoothness, we have to detect and measure the number of irregularities around the roads. We detect irregularities at the pixel level using mathematical morphology. We simplify the problem by transforming the image from colored (RGB) to a black and white map that represents the presence or absence of a road. A closing (dilation followed by erosion) of road pixels on the image should permit, with an adapted size to fill in the concave irregularities of the road shape. Then, the deletion of the road shape (irregularpixels = closingpixels -roadpixels) isolates the irregular pixels near the road as illustrated in Figure 5. Then we also propose a measure at an image level: the ratio of the initial roughness that is removed by generalization [START_REF] Feng | Learning cartographic building generalization with deep convolutional neural networks[END_REF].

smoothness = irregular initial -irregular prediction irregular initial (2)
Parameters The size of the closing determines the size of the biggest irregularity that should be smoothed by generalization; if we choose a threshold too large, concave portions that are not irregularities might be wrongly filled. This measure seems to work the best at the resolution and scale of our use case with a closing value of three pixels (30 m). Our experiment has shown that closing with less than three pixels detects very few irregularities while closing with more pixels fills the bends and tiny regular background areas.

Validation and Limits Figure 6 represents the distribution and some example values of the smoothness measure. This measure is globally well correlated with our visual perception of smoothness.

However, we observed the following limitations: Table 4: Smoothness level of satisfaction (at image level).

Very good Good Neutral Bad Very Bad x > 60% 60% ≥ x > 40% 40% ≥ x > 20% 20% ≥ x > 0%

x ≤ 0%

• The accuracy of the irregularity detection has to be questioned; depending on the threshold so irregularities can be missed and/or some bends can be wrongly identified as smoothness problems.

• The measure depends on the resolution and scale of the road; other use cases might require different parameters.

• The measure is quite sensitive to the initial situation: roads with very concave shapes, even well generalized, will be evaluated as less smooth compared to straighter roads. Moreover, convex irregularities are not identified.

In most of the situations where the road is concave at the right, it is convex at the left (and conversely), but in some really rare cases where the road shape is large, the left and right sides of the shape are not correlated (illustrated in Figure 7). In these cases, the convex irregularities are not correctly measured: (a) is considered as not smooth, while (b) and (c) as similar.

• If the initial image has no irregularity the ratio measure cannot be computed.

Constraint Satisfaction

We propose the satisfaction values associated with the measure of smoothness constraint presented in Table 4.

Coalescence Reduction

Definition The coalescence constraint is a legibility constraint. Coalescence is the fact that the symbol of a line overlaps itself in the interior of a sharp bend [START_REF] Mustière | GALBE: Adaptive Generalisation -The need for an Adaptive Process for Automated Generalisation, an Example on Roads[END_REF]. The generalized image should contain no symbol coalescence. This constraint is designed for generalized mountain roads, where coalescence might occur.

Measure To measure the coalescence, we also use mathematical morphology operations to detect pixels that belong to a coalescent part of the road. This measure is also based on black and white images. The measure is also at the pixel level if we consider the map of the coalescent pixels and aggregated at the image level as we consider the ratio of pixels in an image. First, a dilation of roads (of n pixels) simulates the coalescence occurring when the road symbol is enlarged due to scale change; then an erosion of size n + w (the maximal width of roads) erases the parts that are not coalescent; and finally, a dilation of w pixels permits to evaluate the number of pixels in the coalescent areas. This process is illustrated in Figure 8. Finally, similarly to the smoothness constraint, we propose to measure the evolution of coalescence instead of the number of coalescent pixels using Eq. 3. Parameter In this process, the size n represents the maximal distance between two coalescent parts of the road from the map specifications. For our use case (at the 1:250k scale), a threshold of five pixels is the most adapted, and it corresponds to a distance of 50 m. This value allows the detection of most of the coalescent areas. However, it also detects some un-coalescent areas as presented in Figure 9, while a thinner threshold would miss many coalescent areas.

Coalescence = coalescent_pixel(initial) -coalescent_pixel(prediction) coalescent_pixel(initial) (3)
Validation and Limits Figure 10 presents the distribution and some examples of values for the coalescence reduction measure. This measure is more questionable than the others. First of all, it does not exactly measure the coalescent parts but the pixels that could be coalescent with a larger symbol. This definition implies that the overlapping parts of the coalescent road are identified as a unique road and not detected as coalescent. Nevertheless, the observed ratio value is most of the time consistent with our visual evaluation of coalescence. But frequently, high coalescence values are explained by two parallel roads that are separated by fewer pixels than the threshold (detected as coalescent by the measure), and this is not coalescence as these are different roads, but just symbol overlap. These situations are limited to 5% of the cases in our real test set.

Constraint Satisfaction

We propose the satisfaction values associated to the measure of coalescence constraint presented in Table 5. As there is an over-detection of coalescence, we cannot really consider that any coalescence is bad, so we compute satisfaction based on the reduction of coalescent areas.

Table 5: Coalescence level of satisfaction (at image level).

Very good Good Neutral Bad Very Bad x > 30% 30% ≥ x > 20% 20% ≥ x > 10% 10% ≥ x > 0%

x ≤ 0%

Position Preservation

Definition Then, this is one of the most basic constraints for map generalization, the generalized road has to be as much as possible at the same location as the initial one, which is as close as possible as the real position. This constraint is designed for an image of generalized roads in general.

Measure We measure this constraint using the intersection over union (IoU) of the road pixels of the initial and generalized images. Here, we also based our computation on transformed black and white images. The intersection between the road pixels in the initial image and the road pixels in the generalized image is computed and the area of the intersection is divided by the area of the union namely, the pixels that belong to generalized or the initial roads (Eq. 4). This is a classical measure to assess image segmentation and determine which parts of the road pixels are common in the initial image. Figure ?? represents the intersection and the union of a road and its generalization.

IoU = road_pixel(initial) ∩ road_pixel(prediction) road_pixel(initial) ∪ road_pixel(prediction) (4)
Parameters We propose to use a buffer of size n around roads instead of just the road pixels. This threshold would permit to model the area where the roads are allowed to move or be distorted without considering the position as too far. This threshold can be very large because the displacement offset can be large for mountain roads at this scale (1:250k).

In our image set, the position is always visually preserved. However, the measured position accuracy is most of the time under 50%, so introducing this buffer with a size of 20 pixels (200 m) better reflects this preservation and the variation between two generalization methods: some cause more displacement than others.

Validation and Limits Figure 12 represents the distribution and some examples of values for the position measure. Despite a big dissimilarity of IoU values in our image set, we observe that all real images visually respect the position accuracy, even those with a threshold of IoU very low. Moreover, the measure is not regular: with a similar displacement, the IoU of two large objects will be more important than the one of two small objects. Another limit of this method is that it permits quantifying the displacement, but it does not identify if the displacement is relevant. Road displacements and distortions are permitted when they tend to increase legibility. Finally, the road position accuracy is a minor constraint: it can be seen as a soft constraint whose satisfaction does not need to be maximized.

Constraint Satisfaction

We propose the satisfaction values associated with the measure of position constraint presented in Table 6. Very good Good Neutral Bad Very Bad x > 30% 20 < x < 30 % 10 < x < 20% 5 < x < 10%

x < 5%

Road Connectivity Preservation

Definition Then, we think that more than the position, the road network structure preservation is a key point for road maps. The connection between roads permits the creation and following an itinerary that is the most important function of such maps. This constraint is adapted for images of generalized roads in general.

Measure For the connectivity preservation constraint, we measure connections and disconnections using the number of sets of contiguous road pixels and contiguous background pixels. Here, we also based our calculation on transformed black and white images. Figure 13 illustrates how these numbers can reveal changes in the structure. A disconnection will reduce the number of background pixel sets and increase the number of road pixel sets. This measure also detects problems such as the creation of a loop, which deteriorates both the correctness and legibility of the information. Equation 5shows how we combine these numbers of contiguous parts to measure continuity preservation. (x, y) is a pair of a detailed and a generalized image, nroad is the number of road pixel sets in the image, and nback is the number of background pixel sets.

structure_alteration = |nback(x) -nback(y)| + |nroad(x) -nroad(y)| nback(x) + nroad(x) (5)
Figure 13: Different cases of disconnections or new connections in road images.

x < 10% 10% ≤ x < 40% 40% ≤ x < 70% 70% ≤ x < 100% ≥ 100%
Parameters We choose to use the 4-connectivity to construct the sets of pixels; pixels of the same value connected by diagonal only, do not belong to the same group of pixels. We also include a threshold for set size: the groups composed of one or two pixels are not considered, consequently the appearance or disappearance of such group is not measured.

Validation and Limits Figure 14 gives the distribution and some examples of values for the connectivity measure, each pair represents an initial situation and the corresponding generalization. This measure is quite consistent with our observation but it seems to overvalue structure changes. This problem is explained by the fact that all changes have equal weight in the calculation, even the addition of very small sets or the disconnection of one pixel, which are not visible at the target scale. Then, this measure is sensitive to border effects (illustrated in the last column of Figure 13). When a road has bends at the border of an image, these bends can be connected or disconnected by a small displacement, without a real change in the structure.

Constraint Satisfaction

We propose the satisfaction values associated to the road connectivity constraint presented in Table 7.

Color Realism

Definition The next two constraints are not adapted from conventional vector-based constraints but are specific to images generated by deep learning models. They aim at the quantification of errors that do not exist in traditional map generalization. The goal of the color constraint is to make sure that the generated images do not convey any unexpected information. In a map, a different color represents a different object. This constraint is adpated for all images of map generated using deep learning.

Measure We propose to measure if there is color noise: in our use case, it means pixels that are not roads (red) or background (white). We have decided to use an implementation of the CIEDE2000 distance between colors [START_REF] Sharma | The CIEDE2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations[END_REF] because of its consistency with human color perception. Then, we counted the number of pixels that are visibly too distant from red or white.

Parameters We choose to consider a pixel as white or red when the distance with pure red (255,0,0) or pure white (255,255,255) is higher than 9.

Validation and Limits

Very good

Good Neutral Bad Very Bad x < 2.5% 2.5 ≤ x < 5% 5 ≤ x < 7.5% 7.5 ≤ x < 10% ≥ 10%

Constraint Satisfaction

The satisfaction values of this constraint are computed with the threshold distances presented in Table 8.

Noise Absence

Definition We also found that some deep learning models can generate some noise pixels: e.g., isolated road pixels that are not roads. This is due to the fact that the loss function that is optimized in the training process is global and might not be fully optimized after all the iterations of the training. This constraint is designed for all images of maps generated using deep learning, not just generalized mountain roads.

Measure We just count the number of sets of contiguous road pixels that are too small to be real roads. Similarly to the road connectivity preservation constraint, we use the 4-connectivity to define sets of contiguous pixels.

Parameters The threshold for noise detection depends on the image resolution. For our use case, we have fixed a size below 6 pixels.

Validation and Limits Figure 16 presents the distribution and some examples of values for this measure. This measure is quite satisfying as it permits us to distinguish noise in both the fake images that we created with noise in our validation set, and in the images generated by a U-Net model, which often contain noise pixels.

Constraint Satisfaction

The satisfaction values of the noise constraint are computed with the thresholds presented in Table 9.

How to Use the Constraints in Deep Learning Map Generalization

In this section, we present how our evaluation method can be used in three examples: comparing deep learning models, identifying and solving the quality issues of a deep learning model, and validating the results of a model. Our

Table 9: Noise level of satisfaction (at image level).

Very good Good Neutral Bad Very Bad x = 0

x < 3 3 ≤ x < 6 6 ≤ x < 9 ≥ 9 observations are based on the evaluation of the results obtained with U-net, cycleGAN, and pix2pix models, trained and tested with default parameters on the same image set from the Alps.

Comparing Models

In this section, we propose to explain how the evaluation can be used to compare different deep learning models or similar models used with different parameters. A good generalization solution is a balance between several criteria, so it is not easy to determine if a solution is better than another. The constraints could contribute to evaluating which network configuration gives better results in an ambiguous case where it is visually hard to determine a preference. For comparison purposes, a good combination of different measures is needed to give a synthetic measure of the global quality of a tile image. For this combination, several methods are possible, but the method has to preserve and highlight the order and dissimilarity of quality. Moreover, the mean and worst values are interesting for this task. Instead of the worst value, we prefer the last decile value that excludes exceptionally bad values.

For our use case, the mean value allows a thinner comparison of the interest of CycleGan and pix2pix methods, which have quite visually similar results. Table ?? shows the mean and last decile values for the constraint measures and the associated level of satisfaction, for all the images from our test set. This table shows that unsupervised learning (CycleGAN) preserves more information while supervised learning (pix2pix) learns that distortions are tolerated. On the other hand, the comparison of the worst value permits highlighting the irregularities of the methods. The comparison shows that the reference images are sometimes less well generalized than deep learning predictions. This is due to defects in our data with roads displayed in the reference generalized image, but not in the initial data. Then, this analysis helped us in the identification of these badly generalized images, and excluding them from the example set would improve the learning.

Finally, this comparison allows the identification of parameters or models that are more appropriate for several road configurations. It could contribute to improving results in most cases by using the best model for each configuration. Thus, the output generalization of the study area would be a partition of tiles generalized by different deep learning models. The determination of the most appropriate methods for each configuration relies on a robust evaluation that permits to compare methods.

Identifying and Solving the Quality Issues of a Model

In this section, we explain how a model can be used to detect the quality issues of a given deep learning model and then eventually improve the model. On one hand, the constraints at the image level must provide a comprehensive indicator of quality for the constrained property, which allows the identification of a quality issue. On the other hand, constraints at the pixel level (e.g., coalescence and smoothness) permit the identification of conflicts and situations that are badly generalized within each tile. If a characteristic is often badly generalized, it means that the model is not adapted to learn to reproduce this characteristic. For example, disconnections are important errors in map generalization, but they affect very few pixels, so they are not penalized in the learning process. Indeed, the learning process is based on the minimization of a loss function, and the default loss seems not to be adapted to reproduce structure preservation. Since this default has been identified, it can be solved by the exploration of adapted loss functions. For example, by modifying the weight of loss components and increasing the part that guarantees, the resemblance of the input and prediction images, or by including geometric consistency in the loss computation [START_REF] Fu | Geometry-Consistent Generative Adversarial Networks for One-Sided Unsupervised Domain Mapping[END_REF]. Another improvement could be to add the constraint violation measure to the loss function. Therefore, the model learns to minimize the sum of the loss function and the constraint violation (e.g., the connection preservation measure).

If a situation or a kind of situation is often badly generalized, it might reflect an imbalance in the training dataset. For example, in our experiments, the cases with two close parallel roads are under-represented. Here, the solution is to change the training set either by adding images that represent similar situations or by modifying example weights to give more impact in the optimization to the examples that are less represented than others. The calculation of the constraint satisfaction with the pair of images initial/reference from the training set instead of prediction/initial would permit the detection of such situations.

Validation of a Model

To validate a model, we propose to verify that predictions from this model give results that are under a defined threshold of satisfaction for each constraint. The definition of the threshold of validation is quite similar to the categorization for identifying quality issues. We can consider that the images with a "good" or "very good" value are valid, and then a method with a majority of valid images is valid [START_REF] Touya | Social Welfare to Assess the Global Legibility of a Generalized Map[END_REF]. These thresholds are defined by examination of the test data and especially the constraint satisfaction values obtained with the reference images, which should be good by definition and should be evaluated as "good" or "very good" for most situations and constraints.

Discussion

6.1 Do We Need a Meso Level of Analysis?

In this section, we will discuss the interest of using a meso level of analysis. First, the use of a sliding window over the image can create a new sub-image or a patch that can be better adapted for the calculation of some constraints. This process can permit locating problems in the image for constraints that are only measurable at the image level. With this method, we can also identify the images that suffer from border or size effects: indeed these images would have a remarkable distribution of a constraint value in a patch. For example, the connectivity constraint of an image that suffers from border effects would have a high value for a patch at the border but not for a patch in the middle of the tile.

Then, we believe it could be useful to define a meso constraint that synthesizes the satisfaction of several pixellevel constraints [START_REF] Ruas | Modèle de généralisation de données géographiques à base de contraintes et d'autonomie[END_REF] for pixels contained in a meso object. For example, instead of measuring the ratio of the pixels that are not smooth, we could define that an image is smooth if all or most of its pixels are not irregular. This definition first designed for the agent models is also adapted to guide the deep learning processes where the loss is a statistical summary of the pixels or patches values.

6.2 Is it Possible to Automatically Evaluate a Generalization with Images Only?

Image representation allows a visual evaluation of generalization. The automation of this evaluation is a complex combination of multiple constraints. In this article, we describe how some constraints can be adapted for a calculation based on raster data. However, we were unable to prove that all existing constraints can be converted by this method. Especially, the relation preservation constraints seem to be challenging as the raster format does not permit to encode object boundaries and spatial relations.

Moreover, the evaluation of a map must relate to the map function. The automatic evaluation of map efficiency is still challenging for both raster and vector data, as it relates to human perception. A good value of evaluation is a necessary condition for the final map to be a functional map but not a sufficient one.

Finally, the proposed measure still suffers from multiple limitations. We think the raster evaluation can be useful at a stage where vector conversion is not ideal. However, the vectors remain more accurate and practical and permit a comparison with classical generalization approaches or validation.

6.3 Can These Methods be Adapted to a Multi-Theme Map?

Then, a challenge is to adapt this constraint-based evaluation to maps with several themes, not just roads. In these maps, the spatial relations are even more important and concern several different objects, e.g., buildings are aligned along a road, or a road is crossing a forest. And the preservation of these relations is an important point in information preservation as they contribute to the global structure and understanding of the map [START_REF] Mackaness | The Importance of Modelling Pattern and Structure in Automated Map Generalisation[END_REF] and help to determine map location.

Moreover, the raster representation makes conflict detection harder as each pixel can have only one value even if there is a coalescence of several elements at this place. Finally, the identification of groups of objects can be done using a distance criterion using buffers, but the identification with similarity criteria could be more difficult. In Table ??, we propose to estimate the difficulty of the raster adaptation of constraints from [START_REF] Stoter | Methodology for evaluating automated map generalization in commercial software[END_REF]. We identify four levels of adaptability of measure:

• Directly measured in the image: ++ ;

• May require image processing: + ;

• May require a new paradigm: -;

• No possible adaptation/may require vectorization: -.

Conclusion

Deep learning techniques are still recent but promising and their development and integration into the map generalization process would be conditioned to the existence of common and adapted evaluation methods. This work shows the applicability and the interest of constraint-based evaluation on images generated by deep learning.

Measuring constraint current values based on raster data can require image processing (segmentation, mathematical morphology statistical analysis, etc.) or a new level of analysis (e.g., considering a group of pixels instead of an object). But sometimes, it seems not possible to measure a constraint value with an image. For the use case of mountain roads, we demonstrate that four of the most important constraints can be adapted. We note that the measures for our test set are often consistent with our visual evaluation. To go further in the validation of these constraints, it would be useful to couple this constraint-based quantitative evaluation with a user survey to make sure that these measures do reflect the human perception of a "good map."

Figure 2 :

 2 Figure 2: Steps where evaluation is required in a deep learning process.

Figure 2

 2 Figure 2 presents the main steps where an evaluation would be required in the setup of a deep learning process. First of all, the evaluation of the training examples at the training set preparation step would permit to make sure the examples are good enough, or to filter examples of lower quality; and the quality of the deep learning model is mostly conditioned to the training set quality[START_REF] Courtial | Generative Adversarial Networks to generalise Urban Area in Topographic Maps[END_REF]. Then, evaluation can be used to choose the most adapted configuration of a network or guide the post-processing (e.g., identifying a noise to eliminate). Finally, the main interest of an evaluation measure calculated using images is to guide the learning process, i.e., including a particular evaluation measure in the loss function of the neural network.

Figure 3 :

 3 Figure 3: Examples of image in the test set.

Figure 4 :

 4 Figure 4: Distribution and examples of the value for edge-density clutter measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Figure 5 :

 5 Figure 5: Effect of the smoothing operation on a road and its generalization.

Figure 6 :

 6 Figure 6: Distribution and examples of values for the smoothness measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Figure 7 :

 7 Figure 7: Different kinds of pixels detected as irregular: a) concave only irregularity, b) convex only irregularity, c) symmetric irregularity.

Figure 8 :

 8 Figure 8: Steps for detecting coalescent pixels, from left to right: initial image, dilation, erosion, dilation.

Figure 9 :

 9 Figure 9: Effect of the coalescence detection on a road and its generalization. a) example with a few overdetection. b) example with an overdetection of coalescence in the generalized image.

Figure 10 :

 10 Figure 10: Distribution and examples of values for the coalescence reduction measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Figure 11 :

 11 Figure 11: Intersection and union of roads and their generalization.

Figure 12 :

 12 Figure 12: Distribution and examples of values for the position measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Figure 14 :

 14 Figure 14: Distribution and examples of values for the connectivity alteration measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

  Figure 15 represents distribution and examples of values for the color measure. This measure has permitted quantifying the observed noise of CycleGAN images and verifying that images from other methods do not suffer from this problem.

Figure 15 :

 15 Figure 15: Distribution and examples of values for the colour measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Figure 16 :

 16 Figure 16: Distribution and examples of values for the noise measure. Each point represents an image from our testing set, the corresponding level of satisfaction of the constraint is indicated in orange.

Table 1 :

 1 Table of possible levels of analysis for raster and vector data.

	Level of analyse	Vector data	Raster data
	Micro	Roads	Pixel value
		Road network faces Set of background pixel
	Meso	Stroke
		Other set of road	Set of road pixel
	Macro

Table 2 :

 2 Alterations applied to create fake test images.

	Legibility alteration	Information alteration
	Enlarged bends	Appearance of a road
	Enlarged road width	Noise in the image
	Small displacement	Big structure alteration
	Loop creation	Continuity alteration
	Blurred road	Continuity alteration at border
	Only background in the image Only roads

Table 3 :

 3 Clutter level of satisfaction derived from the measure.

	Very good	Good	Neutral	Bad	Very Bad
	0%

Table 6 :

 6

Position's level of satisfaction (at image level).

Table 7 :

 7 Road connectivity's level of satisfaction.

	Very good	Good	Neutral	Bad	Very Bad

Table 8 :

 8 Color realism level of satisfaction (at image level).

Table 10 :

 10 Table of value for comparing models.

	Method	Constraint	Reference CycleGAN	Pix2pix	U-Net
	Mean value	Smoothness	0,85	0,92	0,82	-0,23
	Satisfaction		Very good Very good Very Good Very Bad
	Worst value		0,54	0,80	0,50	-3,37
	Satisfaction		Good	Very good	Good	Very bad
	Mean value	Coalescence	0,47	0,62	0,63	-0,02
	Satisfaction		Very good Very good Very Good Very bad
	Worst value		-0,20	0,38	0,42	-0,91
	Satisfaction		Very bad	Very good	Good	Very bad
	Mean value	Position	0,19	0,63	0,22	0,23
	Satisfaction		Neutral	Very good	Good	Good
	Worst value		0,08	0,54	0,13	0,16
	Satisfaction		Bad	Very good	Neutral	Neutral
	Mean value	Connectivity	0,32	0,39	0,74	2,65
	Satisfaction		Good	Good	Bad	Very bad
	Worst value		0,45	1,07	1,46	3,91
	Satisfaction		Neutral	Bad	Very bad	Very bad
	Mean value	Color	0	0,67	0,61	0
	Satisfaction		Very good	Good	Very good Very good
	Worst value		0	0,96	1,01	0
	Satisfaction		Very good	Bad	Good	Very good
	Mean value	Noise	0,19	0,25	5,06	12,75
	Satisfaction		Good	Good	Good	Very bad
	Worst value		1	1	10,5	20
	Satisfaction		Very good Very good	Very bad	Very bad
	Majority vote		Very good Very good	Good	Very bad

Table 11 :

 11 Table of possible constraint conversion to raster.

	Id	Type	Geometry	Property	Adaptation
	1-1	Minimal dimension	polygon	area	++
	1-4	Minimal dimension	polygon	area of any hole in a polygon	+
	1-5	Minimal dimension	line/polygon	length of an edge/line	--
	1-13 Shape	polygon	squareness	-
	1-18 Orientation	any	general orientation	--
	2-1	Minimal dimensions any -any	minimal distance	+
	2-3	Orientation	line/polygon -line/polygon orientation	--
	2-9	Topology	any -any	topological consistency	+
	3-4	Topology	line and polygon	intersection	-
	3-5	Topology	line and polygon	connectivity	-
	3-7	Topology	polygon	spatial distribution	--
	3-8	Topology	polygon	spatial distribution	--

We also presented the interest of a new type of constraint designed for evaluating images generated using deep learning. We propose two examples of such constraints. Then, this article has demonstrated the interest in the constraint-based approach for comparing, improving, and validating deep learning models. Moreover, proposed measures are indicators of a prediction's quality and they can be used as a term of the loss function that the deep neural network aims to minimize.

However, the seven proposed constraints for the case of roads generalization are only partly sufficient for the validation of deep learning models, and the limitations remain important (size and border effects). Future research is necessary to improve the proposed constraints, develop new ones, and try to connect raster-based and vector-based constraint evaluation.

Data and Code availability Data and code presented in the submitted work are available in the DeepMapGen repository, https://github.com/ umrlastig/ DeepMapGen.

Informed Consent This research involved no human participants.

Ethical Approval This research involved no human participants.

Conflict of Interest

The authors declare no competing interests.


Benjamin Boissi Ère

Hadrien Benoit

Renaud Ansart

Herv É Neau

Daniel Gauthier

Gilles Flamant

Mehrdji H Émati

Hydrodynamics of Dense Fluidized Beds for Application in Concentrated Solar Energy Conversion

In the frame of the call for projects of the European Commission which aims to find alternative HTF in order to extend working temperature and to decrease environmental impact of standard Heat Transfer Fluid (HTF) used in concentrating solar power(CSP) plants, we proposed to use Dense Particle Suspensions -DPS-fluidized with air (approximately 50% of solid) in tubes as new HTF. DPS will enable operating temperature over 1 000 ˚C which corresponds to the sintering temperatures of the solid against 560 ˚C for the most efficient molten salts, thus increasing the plant efficiency and decreasing the cost per kWh produced, have no lower limitation of temperature and are riskless. A cold mockup of receiver using DPS has been built for the preliminary study of the concept. The operation of the mockup has shown the possibility to ensure a regular and adaptable upward flow of solid in the range 10 to 65 kg • h -1 per tube. This paper compares the experimental results of the cold mockup running with the predictions of a multi-fluid approach 3D numerical code.

INTRODUCTION

In the field of high solar concentration industry, the reduction of the cost per kWh produced needs technical breakthroughs and innovations. Among the different solar concentration technologies: parabolic, fresnel, cylindrical-parabolic and central receiver system (CRS), the last one seems to be the most efficient [START_REF] Ortega | Central receiver system solar power plant using molten salt as heat transfer fluid[END_REF]. For central receiver, the increase in HTF operating temperature belongs to the most waited improvements because it will enable more efficient power cycles.

CRS mainly uses molten salts as HTF. Current molten salts have a melting point around 200 ˚C, making mandatory the use of auxiliary energy source during nongeneration periods, and a maximal operating temperature of about 560 ˚C, which is too low to enable a supercritical cycle [START_REF] Singer | Assessment of solar power tower driven ultrasupercritical steam cycles applying tubular central receivers with varied heat transfer media[END_REF].

The proposed concept, concerns the HTF used in central receiver systems [START_REF] Flamant | Dispositif collecteur d' Énergie solaire[END_REF]. To overcome the two previous drawbacks and to decrease the environmental impact of standard HTF, we suggest the use of dense gas-particle suspensions as an alternative heat recovery and storage fluid. These suspensions do not have lower temperature limit and the maximum operating temperature corresponds to the sintering temperature of the solid which is higher than 1000 ˚C. Moreover, as their volume composition is roughly 50% of solids and 50% of gas, their density and their volumetric heat capacity approximate that of a liquid.

The objective is to make these suspensions circulate into the solar absorber made of vertical pipes. The particles may then be used as an heat storage medium after being heated and stored in a storage vessel as it is currently done in CRS working with molten salts and as cannot be done by other fluids such as gas or steam.

In the frame of the "'CSP2" project, supported by the seventh Framework Program of the european commission (FP7), whose aim is the construction of a 150 kW th solar pilot plant in the Odeillo solar furnace in France, a cold laboratory scaled mockup has been built at the Chemical Engineering Laboratory of Toulouse (LGC). This mockup must allow to understand and determine the operating conditions enabling stable and adjustable upward and downward solid flows [START_REF] Boissiere | Étude hydrodynamique d'un nouveau concept de r écepteur solaire à suspension dense de particules[END_REF]. For this kind of fluid, the heat transfer efficiency is strongly linked to the particle agitation and the renewal rate at the wall, both unreachable with the available equipment. In order to access the local behavior of particles in an exchanger tube, we have undertaken 3D numerical simulations of this flow on a similar geometry with a multi-fluid approach code. In this paper, is presented the results of this work and the comparison with experiment.

EXPERIMENTAL STUDY

Experimental device: cold mockup

The mockup was designed to be tested under cold conditions. A detailed scheme is given The minimum fluidizing and bubbling velocities and their corresponding solid volume fraction presented in Table 1 were determined experimentally.

(kg • m -3 ) (µm) (µm) = d50 d90 -d10 (mm • s -1 ) (mm • s -1 )
Values 3210 63.9 78.2 0.93 5.0 7.0 0.575 0.59

Metrology

The flow properties experimentally accessible are the static pressure of air and the gas flow rate going through each tube. Differential pressure controllers placed every 20 cm along the pipe allow to determine the average porosity of suspension. Indeed, when the suspension is fluidized (air velocity higher than minimum fluidizing velocity), the pressure loss is equal to the solid weight divided by the tube section.

∆P = mp • g S = (1 -αg) • ρp • V • g S =⇒ αg = 1 - ∆P ρp • g • h (1)
The air flow rate going through each tube is determined with an helium tracking device.

Helium is injected at a sufficient height of the tube to be sure to recover the totality at the top. The flow of air coming from the bin dilutes the helium. The air flow rate is given by the concentration of helium measured at the top of the tube through the Equation ( 2).

Fair = 1 [He] -1 • FHe (2)
The measurements were realized on a reference test case of 50 kg • h -1 of solid per tube and a superficial air velocity of 15 mm • s -1 for the fluidization of the bin.

A preliminary experimental study (4) has demonstrated , by determination of the fluidizing curve with decreasing air velocity, that the pressure loss sustained by air for the fluidization of this powder is about 130 mbars • m -1 . At the minimum fluidization state, the volume proportion of solid is about 0.43. This work has also shown that the same pressure profiles and the equal repartition of air between the tubes result in the same solid flow rate in each tube.

NUMERICAL SIMULATION

Mathematical model

Three dimensional numerical simulations are carried out using an Eulerian n-fluid modeling approach for poly-dispersed fluid-particle flows implemented in NEPTUNE CFD software which is developed and implemented by IMFT (Institut de M écanique des Fluides de Toulouse). This software is a multiphase flow code developed in the framework of the NEPTUNE project, financially supported by CEA (Commissariat à l' Énergie Atomique), EDF ( Électricit é de France), IRSN (Institut de Radioprotection et de S ûret é Nuc éaire) and AREVA-NP. The multiphase eulerian approach is derived from a joint fluid-particle PDF equation allowing to derive the transport equations for the particle velocity's moment. In the proposed modeling approach, transport equations (mass, momentum and fluctuant kinetic energy) are solved for each phase and coupled through inter-phase transfer terms.The momentum transfer between the phases is modeled using the drag law of Wen & Yu limited by Ergun equation for the dense flows [START_REF] Gobin | Fluid dynamic numerical simulations of a gas phase polymerization reactor[END_REF]. The collisional particle stress tensor is derived in the frame of the kinetic theory of granular media [START_REF] Boelle | Second-order prediction of the prediction of the particle-phase stress tensor of inelastic spheres in simple shear dense suspensions[END_REF]. The turbulence modeling is achieved by the k -ε model extended to the multiphase flows (accounting for additional source terms due to the inter-phase interactions). For the dispersed phase, a coupled transport equation system is solved on particle fluctuating kinetic energy and fluid-particle fluctuating covariance (q 2 p -q f p ). The effect of the particle-particle contact force in the very dense zone of the flow are taken into account in the particle stress tensor by the additional frictional stress tensor [START_REF] Srivastava | Analysis of a frictional-kinetic model for gasparticle flow[END_REF].

The mesh

The symmetry of the flow in each tube of the cold mockup allowed us to simplify the mesh geometry, thus limiting the calculation cost. The mesh includes one tube with the same dimensions as in the mockup and a bin with the same height.

The mesh (Figure 3) contains 1,650,000 hexahedra, based on O-grid technique with approximately ∆ r = 1.2 mm and ∆ z = 1.5 mm. The chamber has a horizontal section area of 0.02 m 2 , a height of 40 cm and is equipped with a lateral solid entrance and a regulation valve at the top. The tube is plunged in the chamber. Its length is 2 m and it is set 10 cm above the bottom.

Figure 3: Mesh: the bin with one dipped tube.

Boundary conditions and phase properties

The geometry is composed of two inlets: the fluidization grid, which is a wall for the particles and where an air flow rate of 1.3 kg • h -1 (15 mm.s -1 ) is imposed and the lateral injection of solid in the bin with a solid flow rate of 50 kg • h -1 at a 50 % porosity.

There are also two outlets: the top of the tube is a free outlet for air and particles at atmospheric pressure, and an outlet placed at the top of the bin imitates the regulation valve. The pressure regulation valve is modeled as a pressure loss implemented on the cells of the valve conduct.

The pressure loss coefficient is defined as a function of the pressure in the chamber sky to imitate the behavior of the actual system, which uses a Proportional-Integral-Derivative (PID) controller.

The wall boundary conditions in the tube are defined as friction for the gas. For the particles, the influence of this boundary condition is studied at the section Boundary conditions and phase properties. The particles are taken monodisperse with a diameter of 64 µm.

The calculations were run on 128 cores. The solid mass flow rate of the study case is 50 kg.h -1 . The radial profiles presented are averaged over a section of the tube. The calculation of the average value begins when the transient regime is over. The steady state is reached when the weight of solid in the system becomes constant, in other words, when the solid flow rate exiting the tube is equal to the feeding solid flow rate.

THE PRELIMINARY STUDY

Effects of numerical parameters

Before running the simulation on the complete geometry, we studied the effect of different numerical parameters on a simplified column geometry of 50 cm high and 36 mm of inner diameter.

The study of the mesh refinement with a fixed number of 30 width cells allowed us to determine the optimal mesh size for convergence. By comparing various cells heights from 1 to 5 mm, we determined that the cell height had to be 1.5 mm. With bigger cells the bubbles are dissolved and the flow becomes homogeneous, and using smaller ones only increase the number of cells with no difference in the results. As observed by Agrawal [START_REF] Agrawal | The role of mesoscales structures in rapid gas-solid flows[END_REF], the unresolved structures have a strong effect on macroscopic behaviour.

Several particle wall boundary conditions were tested. According to comparisons with PEPT measurements in dense fluidized bed, Fede et al. ( 9) have shown that free-slip boundary condition over estimates the downward mean particles velocity near wall region and that no-slip conditions improve that velocity profile. The Table 2 presents the averaged volumetric fraction of solid determined with the pressure profile measured at the wall depending on the wall boundary condition imposed for particles. The friction condition represented by a Johnson and Jackson boundary conditions [START_REF] Johnson | Frictional-collisional constitutive relations for granular materials, with applications to plane shearing[END_REF] gives the closest results to the experiment.

As observed on Figure 2, the SiC particles are strongly non-spherical and present many angular shapes. This non-sphericity is considered by the correction of the drag coefficient through the coefficients C Shape and f Shape as described by Loth [START_REF] Loth | Drag of non-spherical solid particles of regular and non regular shape[END_REF]. With the morphological analysis of the particles, the coefficient C Shape and f Shape have been estimated to 5.3 and 1.15 respectively. This corresponds to a strong non-sphericity of the particles. The results of the 3 cases tested reported Table 3, shows that the solid volumetric fraction obtained with the strongly non-spherical shape is the closest to the experimental results.

Influence of gas compressibility

In order to study the effect of gas compressibility, the solid front ascending velocity have been studied in the case of no supply of solid, with a totally closed pressure regulation valve, and for a fluidizing air velocity of the bin of 15 mm • s -1 . Table 4 summarizes the experimental and numerical results of solid front velocity ascension for the incompressible case (ρ gas = cst) and the case said "compressible" (the gas density follows the ideal gas law).

Table 4: Parameters for numerical simulation for the simulation 1.

Incompressible Compressible Experiment

Solid front velocity 0.28 m.s -1 0.13 m.s -1 0.08 m.s This dynamic test shows the necessity to consider the gas compressibility in the si/mulations. For the following of the study, we have considered the wall-particle boundary condition of Johnson and Jackson [START_REF] Johnson | Frictional-collisional constitutive relations for granular materials, with applications to plane shearing[END_REF], a strong non-sphericity for the solid phase, and a gas density following the ideal gas law.

RESULTS

Comparison on pressure profile and gas flow rate

In this section, the radial profiles are presented by circumference and time averaging.

Figure 4(a) shows the gas pressure drop is linear along the tube for experimental measurements, 3D numerical simulation and for Bubble-Emulsion model prediction (4). These profiles are typical of pressure losses through fluidized beds. The numerical prediction of the multi-fluid approach overestimates the pressure losses. This means that the volume fraction of solid is also overestimated. This difference between experiments and simulations can be attributed to the approximations that were done for the simulations: the particles were considered mono-disperse and spherical whereas the powder range of granular distribution goes from a few micrometers to more than a hundred and they have irregular, jagged shapes. The account of the poly-dipersed particles distribution needs to consider smaller particles which would require an even finer mesh. An alternative could be to use sub-grid model which take account of the effects on smaller scales [START_REF] Parmentier | A functional subgrid drift model for filtered drag prediction in dense fluidized bed[END_REF] without decreasing the mesh dimensions and keep a reasonable CPU time. Meanwhile, the prediction of pressure profile of the 1D Bubble-Emulsion model modified to take account of the solid transport by bubble also presented 4(a) shows good agreement with experiment. This good agreement is mainly due to the hypothesis of an emulsion under the minimum fluidizing state.

The radial profiles of the vertical solid mass flux of the Figure 4(b) shows that there is a solid recirculation near the wall region. Indeed, the solid mass flux is positive at the center of the tube and negative near the wall. This recirculation of solid is a favorable flow property for the process efficiency because it enhances the renewal rate of particles at the wall and thus the thermal transfer.

Figure 4(c) shows that the solid is fluidized everywhere in the tube even near the wall region, because the slip velocity is higher than the interstitial minimum fluidizing velocity. The slip velocity increases with height of tube because. Indeed, the gas expansion with the decrease of pressure results in an increase of gas velocity.

Figure 4(d) shows that solid volume fraction is higher close to the wall. Solid volume fraction is lower at the base of the tube than at its top. It can be explain by the expansion of air resulting in higher slip velocities when rising in the tube.

CONCLUSION

The object of this paper was the experimental and numerical study of a new concept of solar receiver using DPS as new HTF, circulating in vertical narrow tubes.

The construction of a cold mockup has shown the possibility to have a constant and adaptable upward flow of solid, equally distributed between the tubes in parallel. The analysis of the numerical parameters in a simple geometry has led to the following conclusions. The optimal cells for mesh are 1.2 mm wide and 1.5 mm high. A correction of the drag must be applied to take the particles shapes into account. The particles wall boundary conditions that best represents reality is a friction condition.

The simulation of the actual process was done step by step in a complex geometry. It was necessary to take the air density variation into account, which was done by the application of a perfect gas model. The process needs to be controlled, this is why a regulation was encoded. The analysis of the radial profile of particles vertical velocities shows that there is a solid recirculation in the tube. This recirculation observed qualitatively through the particle agitation will be an essential parameter for the thermal exchange.

Careful analysis of the results of these numerical simulations have to be led to determine the renewal rate of particles law near the wall. We will also study the effect of solid flow rate on these flow properties. In the frame of the european project "CSP2", these numerical predictions will be compared to local measurements obtained by PEPT method [START_REF] Chan | Particle motion in the CFB riser with special emphasis on PEPT-imaging of the bottom section[END_REF].

Figure 1 .

 1 It is a cold mockup of solar receiver made of two passes, one with vertical up-flow of solid and the other with downward flow of solid. Each pass is composed of two parallel tubes of 34 mm inside diameter made of transparent PVC. The bins are 400 mm height and have a 200x400 mm base. The following only deals with the upward part of the solid flow. The bottom bin is fluidized with 3 Nm 3 • h -1 of air and is fed with solid by a hopper in the range 10 kg • h -1 to 130 kg • h -1 . The freeboard pressurization of the bed by the regulation valve connected to a PID controller results in the rising of the solid in the tubes, until it reaches the top bin which is under atmospheric pressure. The solid is silicon carbide mainly for its thermal properties, availability and low cost. The chosen particle size offers a good fluidization quality for very low air fluidization velocities.

Figure 1 :

 1 Figure 1: Overall design of solar receiver mockup.

Figure 2 :

 2 Figure 2: SEM photographs of SiC.

  Vertical component of slip velocity. Solid volume fraction.

Figure 4 :

 4 Figure 4: (a) Vertical pressure profile at the wall. ( ): Bubble-Emulsion prediction; ( ): Multifluid approach prediction; ( ): Experiment, (b)(c)(d) Radial profiles of vertical solid velocity, vertical slip velocity and solid volume fraction at different heights. (•): z = 0.5 m; (+): z = 1.0 m ; (×): z = 1.5 m.

Table 1 :

 1 Properties of SiC.

	Properties	ρp	d32	d50	Span	U mf	U mb	ε mf	ε mb

Table 2 :

 2 Influence of the wall boundary condition on the overall volume fraction of solid.

	Wall boundary condition Free-slip No-slip Friction Experiment
	Solid Volume Fraction	0.495	0.513 0.479	0.43

Table 3 :

 3 Influence of the particle shape on the overall volume fraction of solid with free-slip condition.

	Particle shape	Sphere Strong non-sphericity Experiment
	Solid Volume Fraction	0.495	0.476	0.43

ACKNOWLEDGMENTS

This work was developed in the frame of the CSP2 European project. Authors acknowledge the European Commission for co-funding the "CSP" Project ( 14) -Concentrated Solar Power in Particles -(FP7, Project N ˚282 932). The simulations were carried out using the super-calculator Hyperion, to which we have access thanks to CALMIP (Calcul en Midi-Pyr én ées), within the framework of the project P11032, and the authors are grateful to CALMIP for this.