Abdesslam Benzinou
email: benzinou@enib.fr

Sébastien Carbini

Kamal Nasreddine

Romain Elleboode

Kélig Mahé

Discriminating stocks of striped red mullet (Mullus surmuletus) in the Northwest European seas using three automatic shape classification methods

Keywords: Striped red mullet, otoliths, stock identification, year identification, shape analysis, Fourier descriptors, Principal Component Analysis (PCA), shape Geodesics

Stock identification is of primarily importance for population structure assessment of economically important species. This study investigates stocks of striped red mullet using three automatic methods of stock identification based on otolith shape and growth marks. Otolith shape is known to be a promising approach for stock identification but interpreting patterns of variance is a difficult problem. In this study, images in reflected and transmitted light were acquired from 800 otoliths sampled in the Northwest European seas from South Bay of Biscay to North Sea. The growth marks are pointed out manually by an expert. The external shape of otoliths were automatically extracted by computer vision process and then three automatic classification methods were compared, two classical state-of-the-art methods based on Fourier descriptors and Principal Component Analysis (PCA), and a recently proposed method based on shape Geodesics.

Introduction

Striped red mullet (Mullus surmuletus) occurs along the coast of Europe from the South of Norway [Wheeler, 1978] and the North of the Scotland [START_REF] Gordon | The fish populations of the west of scotland shelf. Part II[END_REF] to Gibraltar, also along the northern part of West Africa to Dakar, in the Mediterranean and Black Seas. Striped red mullet has been extensively studied in terms of quantity in the Mediterranean Sea and some studies were carried out in the Bay of Biscay [START_REF] Desbrosses | Contribution à la biologie du rouget-barbet en atlantique nord[END_REF][START_REF] Desbrosses | Contribution à la connaissance de la biologie du rouget barbet en atlantique nord (iii) mullus barbatus (rond) surmuletus fage mode septentrional fage[END_REF][START_REF] N'da | Sexual cycle and seasonal changes in the ovary of the red mullet, mullus surmuletus, from the southern coast of brittany[END_REF]] that correspond to oldest exploitation areas in the Atlantic Ocean. Within the Atlantic Ocean, there are two main areas where this species is caught in this region: Bay of Biscay and in the Eastern English Channel. This species has been initially exploited by the Spanish fleets along their coast to the Bay of Biscay. Initially considered as a valuable by catch [START_REF] Marchal | A comparative analysis of metiers and catch profiles for some french demersal and pelagic fleets[END_REF], the development of striped red mullet exploitation and a strong increase in landings along the English Channel and the southern North Sea by French, English and Dutch fleets have been observed since the 1990's. The strong increase of catches is essentially due to French trawlers and supplemented by the Netherlands and United Kingdom fleets which are carried out in the Eastern Channel and the south of North Sea [START_REF] Mahé | Le rouget barbet de roche mullus surmuletus (L. 1758) en manche orientale et mer du nord[END_REF]. This could be attributed to an expansion of its migration distribution, abundance of this species coupled by the decline of traditionally targeted species in these areas and the sea-water warming trend [START_REF] Ices | Report of the working group on assessment of new MoU species (WGNEW)[END_REF][START_REF] Marchal | A comparative analysis of metiers and catch profiles for some french demersal and pelagic fleets[END_REF][START_REF] Poulard | The impact of climate change on the fish community structure of the eastern continental shelf of the bay of biscay[END_REF]. Reports indicate a steady increase in East English Channel landings reaching ten times recorded landings in 1990 [START_REF] Carpentier | Channel habitat atlas for marine resource management (charm phase ii). INTERREG 3a Programme[END_REF][START_REF] Marchal | A comparative analysis of metiers and catch profiles for some french demersal and pelagic fleets[END_REF]. Striped red mullet is still considered as a non-quota species in the Northeast Atlantic region and evaluation of the level of stock exploitation has only started since 7 years [START_REF] Ices | Report of the working group on assessment of new MoU species (WGNEW)[END_REF].

Stock identification and spatial structure information provide a basis for understanding fish population dynamics and provides reliable resource assessment for fishery management [START_REF] Reiss | Genetic popula-tion structure of marine fish: mismatch between biological and fisheries management units[END_REF]. Each stock may have unique demographic properties and responses or rebuilding strategies to exploitation. Biological attributes and productivity of species may be affected if the stock structure and fisheries management are not well considered [START_REF] Smith | Loss of genetic diversity due to fishing pressure[END_REF].

There are a variety of techniques for stock identification such as genetics and morphometry studies. Genetic studies have been carried out in the Mediterranean Sea [Apostolidis et al., 2009;[START_REF] Galarza | Patterns of genetic differentiation between two co-occurring demersal species: the red mullet (mullus barbatus) and the striped red mullet (mullus surmuletus)[END_REF]Mamuris et al., 1998a,b]. In the Gulf of Pagasitikos (Greece sea), the analyses of three molucar markers revealed that this is a panmictic population [Apostolidis et al., 2009]. However, on the level of the Mediterranean basin, the siculo-Tunisian Strait seems to be the transition zone between the Mediterranean's eastern and western populations [START_REF] Galarza | Patterns of genetic differentiation between two co-occurring demersal species: the red mullet (mullus barbatus) and the striped red mullet (mullus surmuletus)[END_REF]. A sharp genetic division was detected when comparing striped red mullet originating from the Atlantic Ocean and from Mediterranean Sea. Among all available techniques, otolith shape has been proven to be relevant feature for species and/or stock discrimination issues [Begg and Brown, 2000;[START_REF] Burke | Otolith shape analysis: its application for discriminating between stocks of irish sea and celtic sea herring (clupea harengus) in the irish sea[END_REF][START_REF] Campana | Stock discrimination using otolith shape analysis[END_REF][START_REF] Stransky | Geographic variation of golden redfish (sebastes marinus) and deep-sea redfish (s. mentella) in the north atlantic based on otolith shape analysis[END_REF]Stransky et al., 2008b]. Otolith shape reflects the growth pattern of the fish as well as being markedly species specific. As a result, otolith shape can be used to differentiate stocks of the same species. Another relevant feature for stock identification is the growth law as growth is highly correlated to the environmental conditions and is thus stock specific.

In the present study, the stock identification was investigated with two methods based either on otolith shape or on growth marks (and both information). Images in reflected and transmitted light were acquired from 800 otoliths sampled in the Northwest European seas from South Bay of Biscay to North Sea. Growth marks have been pointed out manually by an expert.

External shapes were extracted by computer vision process and then three automatic classification methods were compared, two classical state-of-theart methods based on Fourier descriptors, Principal Component Analysis (PCA), and a recently proposed method [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF] based on shape geodesics. {Figure .1 goes here } The otoliths were selected from the routine surveys on board the RV "Thalassa" and RV "Gwen-Drez" conducted by the Ifremer Institute (France) and from fisheries markets. Fish were caught by otter trawl, bottom pair trawl and set gillnets. Both sagittal otoliths were removed and cleaned before drying and storing in paper envelope. One otolith per fish was examined using a light microscope connected to a video camera and a dedicated imageanalysis system TNPC (digital processing for calcified structures) developed by Ifremer, ENIB and Noesis society. For the year discrimination issue, the test is carried out on dataset

Materials and methods

Otolith datasets

(3) and dataset (4) separately. As dataset (4) is composed of randomized classes, the classification performances on this dataset should be close to those of a theoretical random classifier (i.e. 50%). The difference in performances between dataset (3) and dataset (4) will give an idea of the validity of the results.

Shape-based stock identification

The shape-based classification process can be decomposed in three main steps (Figure .2). First, the otolith contour is extracted as described in next section ( § 2.2.1) using an automatic threshold. Three approaches to extract reduced-dimension feature vectors from the contours were considered: Fourier Transform (FT), Principal Component Analysis (PCA) and a technique issued from shape geodesics [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF]. The discriminative power of each approach is evaluated using its own distance matrix as input for a classifier. In other words, for a query input the feature vector is considered as the distance matrix calculated between this individual and the training individuals. Here, we investigate the performances of two widely used classifiers: (1) the K-Nearest Neighbors (KNN) classifier with the "leave-one-out" heuristic and (2) the Support Vector Machine (SVM) classifier [Vapnik, 1995] with two randomly-selected sub-samples, one of them is used to build the SVM-model which is tested on the other.

{Figure .2 goes here }

Automatic contour extraction

The otolith image is acquired using two imaging modalities: by transmitted light or by reflected light. These two modalities could give additional information. To extract the otolith outline, a mixed image is built in order to integrate information available in both modalities (Figure .3). This mixed image is a mean between the transmitted light image and the negative of the reflected light image. Image contours are detected as local maximum of the image gradient, approximated using a Sobel filtering. The resulting contours are filled by a morphological closing operation and filtered to retain the largest connected component which corresponds to the edge of the otolith. The advantage of mixing both image modalities is illustrated on example given by figure .3. The mixed image gives more details about the contour especially on the region of the excisura major.

{Figure .3 goes here }

The resulting contour is then sampled into 300 points which describe adequately the otolith shape.

Fourier descriptors

Shape can be described using complex Fourier descriptors [START_REF] Granlund | Fourier preprocessing for hand print character recognition[END_REF] or using elliptic Fourier descriptors [START_REF] Kuhl | Elliptic fourier features of a closed contour[END_REF]. For otolith shape analyses, both techniques have been extensively used and proved to be efficient [Duarte-Neto et al., 2008a;[START_REF] Kristoffersen | Population structure of anchovy engraulis encrasicolus L. in the mediterranean sea inferred from multiple methods[END_REF][START_REF] Mérigot | Characterization of local populations of the common sole Solea solea (Pisces, Soleidae) in the NW mediterranean through otolith morphometrics and shape analysis[END_REF]Stransky et al., 2008a[START_REF] Cardinale | Effects of sex, stock, and environment on the shape of known-age atlantic cod (gadus morhua) otoliths[END_REF][START_REF] Galley | Combined methods of otolith shape analysis improve identification of spawning areas of Atlantic cod[END_REF][START_REF] Robertson | Shape analysis and ageing of orange roughy otoliths from the south tasman rise[END_REF][START_REF] Schulz-Mirbach | Differences in otolith morphologies between surface-and cave-dwelling populations of Poecilia mexicana (teleostei, poeciliidae) reflect adaptations to life in an extreme habitat[END_REF][START_REF] Smith | Regional differences in otolith morphology of the deep slope red snapper etelis carbunculus[END_REF]Torres et al., 2000]. In our previous work [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF] we have showed that for red mullet otoliths, classification results are still similar by using these two methods. Elliptic Fourier descriptors are more appropriate than complex Fourier descriptors when otolith contours are composed of series of ellipse arcs (as for Trachurus mediterraneus otoliths for example). Hence, for striped red mullet otoliths we have chosen to use the complex descriptors which can be implemented more efficiently.

With a view to achieving translation, rotation and scaling invariance, the first descriptor is aborted and the selected descriptors are scaled with respect to the first non zero coefficient resulting in the so-called normalized Fourier descriptors. The distance between two shapes is computed as the Euclidean distance between the associated vectors of the normalized Fourier descriptors.

Principal Component Analysis (PCA)

Principal Component Analysis (PCA) was first introduced by Pearson in [START_REF] Pearson | On lines and planes of closest fit to systems of points in space[END_REF] as a mathematical tool that transforms data linearly correlated to uncorrelated variables called principal components. PCA is extensively used in fisheries research for otolith shape analyses, in particular for otolith stock identification. Usually, PCA is applied on Fourier coefficients in order to assess differences in otolith shape [Duarte-Neto et al., 2008b;[START_REF] Mérigot | Characterization of local populations of the common sole Solea solea (Pisces, Soleidae) in the NW mediterranean through otolith morphometrics and shape analysis[END_REF][START_REF] Schulz-Mirbach | Differences in otolith morphologies between surface-and cave-dwelling populations of Poecilia mexicana (teleostei, poeciliidae) reflect adaptations to life in an extreme habitat[END_REF]. PCA can also be applied on morphometric variables [Torres et al., 2000], on a binary low resolution image of the contour [Bermejo and Monegal, 2007] or for standardizing the otolith contour orientation [START_REF] Piera | Otolith shape feature extraction oriented to automatic classification with open distributed data[END_REF]. However, PCA is not invariant to affine transformations, it is applied for pattern recognition when the coordinates of input vectors can be ordered. In face recognition for example, eyes and lips centers are manually selected, then the images are rotated, in order to make the line connecting eye centers horizontal, and resized to make the distances between the centers of the eyes equal. The PCA is carried out on data vectors formed by cropped part of images. In the case of calcified structures, it is not always obvious to order the data vector coordinates on the basis of clearly defined landmarks.

A normalization procedure should then be applied to the raw contours to be invariant with respect to translation, rotation and scaling, so that the normalized shape is the result of the fish history, independently of acquisition settings.

The translation invariance is obtained simply by subtracting the coordinates of the mass center to the coordinates of all points. Scale invariance is also simply obtained by dividing each point of the contour in polar coordinates by the mean radius. For rotation normalization, a first solution could be to align shapes according to the main axis. This axis can be defined by the two farthest points of the shape or by minimizing the covariance using a PCA like in [START_REF] Piera | Otolith shape feature extraction oriented to automatic classification with open distributed data[END_REF]. However, on striped red mullet otolith, the main axis does not correspond to a meaningful biological feature. Instead, we propose to normalize shapes according to the center of excisura major.

The corresponding point of the excisura major can be detected automatically after subtraction of the original otolith shape from the corresponding filled shape. Then, each shape is aligned according to the axis that passes through this point and the mass center of the otolith contour (Figure .4).

{Figure .4 goes here } After contours normalization, PCA is applied on a matrix where each of the rows represents a different contour and the columns represent the information about the contours: the Cartesian normalized coordinates and the local curvature are all put together in a row, one after the other.

To compute a distance between all contours in a given dataset, we proceed with a "leave-one-out" heuristics. One after another, each contour C i of the dataset is left out and PCA is computed on the remaining contours.

Then contour C i is projected into the eigenspace generated by the eigenvectors. Finally the distances between the projected contour and each of the other projected contours of the dataset are computed as Euclidean distances in the eigenspace.

The geodesics approach

A potential drawback of Fourier and PCA approaches comes from the implicit global (spatial) characterization of the shape. Each descriptor holds information about all points of the shape as it is calculated using all points.

Therefore, local (spatial) discriminant shape signatures, such as shape discontinuities or landmarks, may not be well exploited by such a global characterization [START_REF] Parisi-Baradad | Otolith shape contour analysis using affine transformation invariant wavelet transforms and curvature scale space representation[END_REF]. In contrast, a Geodesic approach was recently proposed [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF] to take advantage of local shape features while ensuring invariance to geometric transformations (e.g. translation, rotation and scaling). In this approach, we have defined distance between shapes as a deformation cost stated as a matching issue, i.e. determining the optimal matching between two otolith contours with respect to a similarity measure.

The distance d(Γ 1 , Γ 2 ) between two shapes Γ 1 and Γ 2 is stated as the minimum, over all mapping functions Ψ, of the similarity measure, E D (Γ 1 , φ(Γ )) between the reference shape Γ 1 and the mapped shape φ(Γ 2 ).

d(Γ 1 , Γ 2 ) = min φ∈Ψ E D (Γ 1 , φ(Γ 2 )) (1)
As the important biological information is considered in the shape of contour and not in its size, the shapes are parameterized in function of the normalized curvilinear abscissa s which has a value between 0 and 1 independently of the original contour length. A robust criterion is introduced in order to improve the robustness of the proposed distance to outliers coming from biological interindividual variabilities. The principle is supported by the use of a function that adjusts weight ω in order to penalize the data points of high variation compared to other points.

Given two shapes locally characterized by the angle θ(s) between the tangent to the curve and the horizontal axis, the distance between two contours is defined by:

d(θ 1 (s), θ 2 (s)) = 2 inf φ arccos s φ s (s) cos ω(r(s))r(s) 2 ds ( 2
)
where φ s = dφ ds and r(s) = θ 1 (s)θ 2 (φ(s)). The term φ s (s) allows to avoid torsion and stretching along the curve. The weight function ω is issued from the robust estimator of Leclerc [START_REF] Black | On the unification of line processes, outlier rejection, and robust statistics with applications in early vision[END_REF];

ω(r(s)) = 2 σ 2 exp( -r 2 (s) σ 2 )
where σ is the standard deviation of data errors r(s).

Formally, the numerical computation of d(Γ 1 , Γ 2 ) is solved by using a dynamic programming technique (refer to [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF] for more details).

Growth marks based stock identification

The growth-based classification process consists of three main steps (Figure .2). First, an expert manually points out the growth marks on the otolith image (Figure .5). This step can be done using TNPC software (www.tnpc.fr) in parallel with the image acquisition step; it is not a contradiction with the automatic process of classification. Then distance between the growth laws of two otoliths is computed using the Euclidean distances between growth vectors. In case of two different aged otoliths, distance is computed using only the growth marks available on both otoliths. For example, in figure .5 this distance is computed using the three growth marks on each otolith.

Given two growth vectors

G 1 = {G 1j } j=1•••N 1 and G 2 = {G 2j } j=1•••N 2 ,
the growth distance is considered as the Euclidean distance:

d Growth = Ng j=1 (G 2j -G 1j ) 2
(3)

where N g = min{N 1 , N 2 } is the number of growth marks available in both vectors.

Finally, all distances between otoliths are computed leading to a distance matrix used as input for an SVM classifier. The feature vector is considered as the distance calculated between the query input and all training individuals.

{Figure .5 goes here }

Results

Here performances are evaluated in terms of correct classification rates.

We have started experiments with the hypothesis that the six stocks (NS, EEC, WEC, CS, NBB and SBB) are considered as individual separated stocks with specific characteristics of shapes.

Compared to KNN, SVM classifier performs slightly better in terms of correct classification rate (from 30% to 32.7% on dataset (1)) but at the cost of increasing dramatically the standard deviation of the performances between classes (from 10.9 to 15.2 on dataset (1)). Thus, as KNN classifier results in stable performances across the classes, it has been chosen

for shape-based classification. In contrast, applying KNN for growth-based stock identification gives a correct classification rate of 25.5% whereas SVM gives higher correct classification rate (35.4%) for the same dataset (dataset (1)). Hence, SVM has been chosen for growth-based classification.

The correct classification rates remain high with respect to the random classification but these rates show that the hypothesis of separated stocks should be aborted. The six stocks are then grouped into three stocks leading to a correct classification rate of 67%. Grouped stocks have in the first hypothesis close shape characteristics and could not be really distinguished easily. Classification errors could be due to genetic factors, migration among others. A rate of 100% could then not be reached with the presence of all these factors on the otolith shape. In comparisons to other stock identification methods, otolith shape is a promising approach but interpreting patterns of variance can be difficult [START_REF] Cadrin | Stock identification methods: Applications in Fishery science[END_REF].

In the following, geographical zones are ordered in the tables according to their positions (from north (NS) to south (SBB)); thus neighbor classes are also neighbor geographical zones.

Dataset (1)

Results on dataset (1) are given in tables .1-.3. Geodesic approach reaches 30% of correct classification (Table .3) while this rate is 19.7% for Fourier approach ( As in [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF], we have tested stock identification with both growth and shape information in order to improve classification per-formances. The mean correct classification rate is then increased to reach 49.4% (Table .5).

{Table .5 goes here }

Dataset (2)

On dataset (2), Fourier approach reaches 16.4% of mean correct classification (Table .6), PCA approach reaches 19% of correct classification (Table .7) while Geodesic approach reaches 24.9% (Table .8). These scores are also better than a random classification that would theoretically reach 14.3% (for seven classes).

{Table .6 goes here } {Table .7 goes here } {Table .8 goes here }

Datasets (3) and (4)

Regarding the year discrimination issue on dataset (3), the mean classification rate of the Fourier approach (56%, Table .9) is too close to the theoretical mean classification rate of a random classifier (50% for two classes).

Thus the classical Fourier approach fails on this specific year discrimination issue. The mean classification rate on the random dataset (4) (43%, Table .10) is lower but quite close to the theoretical mean classification rate of a random classifier (50% for two classes), it shows that with this approach, two arbitrary sets of the same stock and same year have no significant shape differences.

Regarding PCA and Geodesic approaches, the mean classification rate on dataset (3) (60%, Table .9) is higher than the mean classification rate on the random dataset (4) (49.5%, Table .10). This shows that the otolith morphology varies over two consecutive years and that this difference in shape is higher than between two arbitrary groups of the same year and same stock.

{Table .9 goes here } {Table .10 goes here }

Discussion

Comparison of the three shape-based approaches

Performances of the three shape-based approaches are compared in table .11. On both dataset (1) and dataset (2), the Geodesic approach exhibits highest performances followed by PCA approach and Fourier approach last.

Regarding the stock discrimination issue on dataset (1) (Tables .1, .2 and .3), the three methods show that the population of striped red mullet can be geographically divided in three zones:

• The Bay of Biscay (NBB+SBB)

• A mixing zone composed of the Celtic Sea and the Western English Channel (CS+WEC)

• A northern zone composed of the Eastern English Channel and the

North Sea (EEC+NS)

To further the "three zones" hypothesis, we have tested the classification when the otoliths were grouped in three classes corresponding to the three zones. The results of this classification using the geodesic approach is shown in table .12 below. It clearly validates the hypothesis as the obtained mean correct classification rate reaches 54.3% and the error scores are higher between two neighbors zones than between two unconnected zones. Finally, this rate raises to 67.31% when the SVM classifier is used with geodesic distances coupled with the growth information (Table .13).

Regarding the year discrimination issue, classical Fourier approach fails while PCA approach shows a small difference in shape and Geodesic approach exhibits the highest difference (Table .11). Thus Geodesic approach seems the most appropriate method for this task.

{Table .11 goes here } {Table .12 goes here } {Table .13 goes here }

Relevance of the shape and growth information

In this study three different approaches have been compared for shapebased stock identification, two state-of-the-art methods (Fourier and PCA)

that have been extensively used in marine research on different species, and a recent method (Geodesic) that proved to give very good performances on different shapes [START_REF] Nasreddine | Variational shape matching for shape classification and retrieval[END_REF] and in particular on otolith shapes [START_REF] Nasreddine | Shape geodesics for the classification of calcified structures: beyond fourier shape descriptors[END_REF]. Although these three methods result in high correct classification rates on several problems, they give quite low correct classification rate for the particular cases tested in this study. It tends to prove that otolith shape is not relevant for the particular case of striped red mullet if we consider the six stocks separately. The growthbased stock identification results are not so far from the shape-based stock identification results. This study shows that both information are influenced by different living conditions and different environments and can serve as stock identifier. This identification is not very high as otolith shape is highly due to the genetics. This result tends to prove that the genetic information is quite homogeneously spread across all geographical zones in the north

  Striped red mullet otoliths were extracted from fish randomly sampled from the southern bay of Biscay to the North sea. The study area was divided into six geographic sectors: the NS (North Sea ; ICES Division IVab), the EEC (Eastern English Channel ; ICES Division VIId), the WEC (Western English Channel ; ICES Division VIIe), the CS (Celtic Sea ; ICES Division VIIh), the NBB (North Bay of Biscay ; ICES Division VIIIa) and the SBB (South Bay of Biscay ; ICES Division VIIIb) (Figure .1). All sampling were collected from September to December 2009 except the EEC otoliths which were collected from October-November 2007 and 2008.

  Images of whole otoliths have been acquired using both transmitted and reflected lights. From 800 otoliths coming from six different stocks of striped red mullet (Figure .1), four different image datasets will be considered: Dataset (1) : 600 otoliths sampled from six different stocks (100 otoliths per stock): • NS: North Sea (IVab) -2009 • EEC08: Eastern English Channel (VIId) -2008 • WEC: Western English Channel (VIIe) -2009 • CS: Celtic Sea (VIIh) -2009 • NBB: North Bay of Biscay (VIIIa) -2009 • SBB: South Bay of Biscay (VIIIb) -2009 Dataset (2) : 700 otoliths: the 600 otoliths of dataset (1) with 100 other otoliths from Eastern English Channel but of a different year: • EEC07: Eastern English Channel (VIId) -2007 Dataset (3) : 200 otoliths: those from Eastern English Channel (VIId) over the two consecutive years 2007 and 2008: • EEC07: Eastern English Channel (VIId) -2007 • EEC08: Eastern English Channel (VIId) -2008 Dataset (4) : 200 otoliths from North Sea (IVab) from the same year 2009 randomly divided in 2 classes: • NS09a: North Sea (IVab) -2009 a • NS09b: North Sea (IVab) -2009 b These datasets illustrate two different types of applications of otolith shape classification: stock discrimination (datasets (1) and (2)) and year discrimination (datasets (3) and (4)). Both issues are quite hard for current state-of-the-art computer vision techniques because the external shapes of the considered otoliths exhibit very few differences.

Figure . 1 :

 1 Figure .1: Map of the stocks of striped red mullet involved in this study.

Figure . 2 :

 2 Figure .2: Shape-based and growth-based classification general schemes.

Figure . 3 :

 3 Figure .3: Contour extraction using transmitted light image (left), reflected light image (middle) and resulting mixed image (right). Note that the contour extracted using the mixed image is more efficient.

Figure . 4 :Figure . 5 :

 45 Figure .4: Contour extraction and normalization. Left: contour before normalization, right: contour after rotation normalization. In this figure we show the main axis passing through the mass center and the excisura major center.

Table .

 .

	1) and 25% for PCA (Table .2). These scores are
	better than a random classification that would theoretically reach 16.7% (for
	six classes).
	{Table .1 goes here }
	{Table .2 goes here }
	{Table .3 goes here }
	In table .4, classification results are given when the growth information
	is used for stock identification. The mean correct classification obtained by
	SVM reaches 35.4%.
	{Table .4 goes here }

Table .

 .

	1: Confusion matrix (in %) for the Fourier approach on dataset (1) achieved by
	KNN classifier. Mean correct classification rate: 19.7%.
		Fourier approach on Dataset (1)
				Actual Class
	Estimated Class NS EEC08 WEC CS NBB SBB
	NS	18	20	11	18	18
	EEC08	21	28	25	17	6
	WEC	8	19	12	16	7
	CS	21	12	18	13	11
	NBB	16	9	14	16	23
	SBB	16	12	20	20	35

Table . 7

 . Table .5: Confusion matrix resulting from an SVM classifier on geodesic distances coupled with growth distances (dataset (1)). Mean correct classification rate: 49.4 %. Table .6: Confusion matrix (in %) for the Fourier approach on dataset (2) achieved by KNN classifier. Mean correct classification rate: 16.4%. : Confusion matrix (in %) for the PCA approach on dataset (2) achieved by KNN classifier. Mean correct classification rate: 19%.

	Table .3: Confusion matrix (in %) for the Geodesic approach on dataset (1) achieved by
	KNN classifier. Mean correct classification rate: 30%. Geodesic approach on Dataset (1) Growth and Geodesic-based approach on Dataset (1) PCA approach on Dataset (2)
	Estimated class Estimated Class NS EEC07 EEC08 WEC CS NBB SBB Actual class Actual Class Actual Class NS EEC08 WEC CS NBB SBB Estimated Class NS EEC08 WEC CS NBB SBB NS 15 20 11 8 5 NS 43.75 12.00 2.44 12.25 5.00 3.57 NS 20 10 11 17 14 8 7 11 EEC08 28 44 17 23 5 EEC08 31.25 66.00 21.95 18.36 0.00 0.00 EEC07 16 15 17 8 14 16 14 5 WEC 9 9 22 11 7 WEC 12.50 16.00 60.98 4.08 0.00 25 EEC08 12 15 24 14 16 8 7 9 CS 24 15 24 32 15 CS 8.33 6.00 9.76 44.89 20.00 10.71 WEC 12 16 14 22 14 16 13 13 NBB 10 5 16 13 27 NBB 0.00 0.00 2.44 20.41 45.00 25.00 CS 19 12 16 14 15 11 9 22 SBB 14 7 10 13 41 SBB 4.17 0.00 2.44 0.00 30.00 35.71 NBB 13 19 9 10 14 15 28 40 SBB 8 13 9 15 13 26 22
	Table .2: Confusion matrix (in %) for the PCA approach on dataset (1) achieved by KNN Table .4: Confusion matrix resulting from an SVM classifier on growth distances (dataset
	classifier. Mean correct classification rate: 25%. (1)). Mean correct classification rate: 35.4 %. Fourier approach on Dataset (2)
	PCA approach on Dataset (1) Growth-based approach on Dataset (1) Actual Class
	Actual Class Actual class Estimated Class NS EEC07 EEC08 WEC CS NBB SBB Estimated
	Estimated Class NS EEC08 WEC CS NBB SBB class NS EEC08 WEC CS NBB SBB NS 15 10 22 7 18 13	11
	NS NS EEC07	29 20.34 15 42.49 19 13	15 12 16.58	5.62	19 23	10 3.84 14	11.13 11	11
	EEC08 EEC08 EEC08	12.30 17	18 50.38 16 31	16 24 13.12	2.36	21 18	10 5.56 17	16.28 7	11
	WEC WEC WEC	12.14	6	14	19.26 17	13	26 41.35 5.13 14	11 7	21 8.54 14	13.58 5	11
	CS CS CS	41.7	20	17	2.74 14	21	15 43.71 8.74 8	20 17	11 3.33 7	0.29 12	11
	NBB NBB NBB	11.47 16	15	8.79 14	11	12 8 30.07	13 3.66 26.47 19.54 21 12 15 20	22
	SBB SBB SBB	12.02 11	7	20.12 10	11	16 12 12.00	16 1.39 11.12 43.34 27 16 15 32	23

Table . 8

 . : Confusion matrix (in %) for the Geodesic approach on dataset (2) achieved by KNN classifier. Mean correct classification rate: 24.9%.Table.13: Classification results (in %) on dataset (1) with the Growth and Geodesic-based approach when the otoliths were grouped in three classes according to their geographical zones. Mean correct classification rate: 67.31% (SVM classifier).Growth and Geodesic-based approach on Dataset (1) with otoliths grouped by zones

					Actual Class
	Estimated Class Northern zone Mixing zone			Bay of Biscay
	Northern zone		74.30	26.76				8.61
	Mixing zone	Geodesic approach on Dataset (2) 22.31 58.25		22.00
	Bay of Biscay		3.39	Actual Class 14.99			69.39
	Estimated Class NS EEC07 EEC08 WEC CS NBB SBB
	NS	10	13	16	8	7	2	10
	EEC07	23	32	22	27	28	19	13
	EEC08	23	15	36	13	17	6	5
	WEC	5	3	5	15	9	4	7
	CS	18	13	13	16	24	10	11
	NBB	9	13	3	12	6	23	20
	SBB	12	11	5	9	9	36	34

west European seas.

This study has proven that by coupling both information (shape and growth patterns), stock discrimination becomes more efficient. These two information are independent and multivariate analysis, including them with other independent information (chemical concentrations, . . . ), should be investigated for stock identification.

The observations above lead to two hypothesis on the striped red mullet:

• some adults move from one zone to another,

• some larvae or juveniles perform migration during growth.

Acknowledgement

The authors want to thank the European commission for providing the financial support of this work through the NESPMAN project, all scientists and crew on board the RV "Thalassa" and the RV "Gwen Drez" for their help with sample collection.


A Lebée

G Foret

Efficacité en raideur et contraintes de renforts périodiques pour FPSO Stiffness and stress efficiency of periodic reinforcing pads for FPSO's hull

Keywords: Renforts Composites, Homogénéisation des structures Composite reinforcement, Homogenization of structures 1

Les coques de FPSO (Floating Production Storage and Offloading units) subissent une corrosion accélérée à tel point qu'il est nécessaire de les renforcer à l'aide the plaques composites. Lorsqu'il faut couvrir de grandes surfaces ces renforts suivent un motif périodique et ne sont pas connectés entre eux. Ainsi dans ce travail, nous appliquons les concepts de l'homogénéisation périodique à une modélisation éléments finis de la coque et du renfort pour estimer l'efficacité mécanique du dispositif. On montre que pour la plupart des configurations envisagées, les renforts ne sont pas suffisants pour baisser le niveau de contraintes dans la coque. Il est donc préconisé de les connecter entre-eux.

FPSO units store crude oil extracted from subsea wells. They must stay close to the offshore platform between two offloadings to crude oil tankers. Otherwise the production stops and generates large financial losses. It turns out that an accelerated corrosion is observed in many of these vessels : from an original averaged thickness of 17mm for the primary hull, it can drop to 15mm. It becomes then mandatory to restore the original thickness by means of reinforcement devices. Most reinforcement techniques rely on additional steel sheets which are welded directly on the hull. However, this requires to fully empty the FPSO and ensure that no flammable gas is remaining. Hence, new reinforcement techniques are currently under consideration. One of them is offered by the company "Coldpad ® " and is under investigation here. It is based on the use of composites and polymers directly bonded to the hull to be reinforced.

Configuration of a single pad

For a rather localized damage, a single reinforcement pad is bonded to one side of the hull (Figure 1). The in-plane geometry of a single pad is part of other design studies. However, the standard cross section of the reinforcement is as follows. Starting from the hull, there is a rather thick layer of epoxy adhesive in order to accommodate the surface irregularities. Then, there is a thick layer of fiber reinforced composite to which is bonded a thin protection plate. Because epoxy resins are very Adhesive Composite reinforcement Protection plate Peripheral seal Damaged steel substrate Fig. 1.: Typical cross section of a single pad sensitive to seawater (swelling), an elastomeric seal goes all around the pad and separates polymer materials from chemical ingression. In addition, the pads will protect the hull from further corrosion. The set-up of the reinforcement is as follows. The pads are prefabricated (composite, protection and seal) and the hull surface is prepared. Then the pads are bonded on-site using infusion techniques. Composites are used in order to reduce the weight of the pads which are carried to the FPSO's damaged area.

Configuration with large damaged areas

There are also large damaged areas. Typical cases are in the water ballast tanks or at the FPSO deck (Figure 3). In this case, it is not possible to have a single pad and a periodic tiling of pads is under consideration (Figure 6). In the present study, the pads are not connected together. Connecting the pads would increase labor. A design without connection would be preferred. Without reinforcement, the hull is mostly loaded in its plane (membrane stress). Hence, its design consists mostly in satisfying plastic yielding criterion and out-of plane buckling load. When adding the reinforcing pads, the adhesive layer sustains mostly transverse shear close to the free edges and transfers longitudinal strains to the composite reinforcement and protection plate (shear lag effect : Figure 2). Thus we expect a decrease of the strain inside the damaged hull and an increase of the bending stiffness of the reinforced hull. Let us point out right away that the reinforcement is added on only one face of the hull. The neutral axis of the hull is shifted in the reinforced area and the real effect of the pad is likely to be more complex than this very simple description.

Simple description of the reinforcing effect

The approach chosen in this study

There are many questions which needs to be answered regarding this new technology. The design of single pads is already part of other studies. They focus on the optimization of the edges geometry in order to limit stress concentrations. A large campaign of experiments was also performed in order to assess the strength of the pads. The present study gives a first approach for assessing the behavior of a periodic assembly of pads which covers a large damaged area. Hence, we want to focus on the capability of the pads to actually reinforce the hull in terms of equivalent stiffness and stress inside the hull in the linear regime. Especially, we want to identify potential geometries which work without connecting the pads together. Thus, non-linear effects (like buckling), stress concentrations and creep will be intentionally kept aside for future studies.

2.

Assessment of the reinforcement pads through periodic plate homogenization 2.1. Configuration of the hull to be reinforced

The structural organization of a FPSO is globally as follows. The primary structure is constituted of a succession of transverse web frames from the prow to the poop (Figure 3). They are positioned every five meters in the longitudinal direction. The hull is also reinforced in the longitudinal direction with "T" stiffeners which are about 450 mm height spaced about 1 m in the cross direction.

In this study we consider the case where the damaged area is at the deck of the FPSO. Hence the pads are bonded directly on the hull and tiled according to the stiffeners. For handling reasons, the maximum size of a pad is about 1 m × 2 m. Between two web frames there will be about 3 to 5 pads. It is not possible to model several pads including the web frames with refined enough FE simulation. Thus, in the following, we assume that a representative periodic pattern includes only the hull, the pads and the stiffeners but not the web frames.

Periodic plate homogenization in few words

Thanks to this assumption, the assembly of pads is periodic. Thus, homogenization appears as suitable tool for studying the mechanical behavior of the reinforced hull. Homogenization consists in isolating a unit-cell which generates the whole pattern and performing all relevant computations on this unitcell. The main advantage of this method is that it is not necessary to simulate the whole reinforced area of the hull which reduces significantly the computational burden. In addition, all load cases are treated in the same procedure. It is then possible to try a large number of geometries for the pads.

In the present case, we will homogenize the periodic assembly of hull+stiffeners and reinforcement pads as a simple equivalent plate. At the macroscopic level, a Kirchhoff-Love plate model will be derived. Let us recall that the membrane stress N αβ and the bending moment M αβ are the usual generalized stresses for Kirchhoff-Love plates. They work respectively with the associated strain variables : E ∼ , the membrane strain and K ∼ the curvature. The homogenization procedure consists in finding the equivalent plate stiffness A ∼ ∼ , B ∼ ∼ , D ∼ ∼ , which links the strains E ∼ and K ∼ to the stresses N ∼ and M ∼ .

In addition to that, at the microscopic scale, it is possible to relocalize the stress generated by (E ∼ , K ∼ ) (or (N ∼ , M ∼ )) inside the unit-cell. All these aspects are extensively detailed in [? ? ? ]. The key point is to find the solution of auxiliary problems which apply the macroscopic plate loadings "on average" on the unit-cell. Finally, those auxiliary problems will be applied to the non-reinforced hull and to the reinforced hull and the corresponding local fields and equivalent stiffnesses will be compared.

Stress and stiffness reinforcement rates

In order to estimate the efficiency of the pads we define stress and stiffness rates. Because the homogenization procedure will return six localization fields corresponding to (E ∼ , K ∼ ) or (N ∼ , M ∼ ), we need to specify how the hull is actually loaded.

Choice of the macroscopic load

The loading which mostly sets the design of the FPSO is the weight of the hydrocarbon payload.

Whether the vessel is empty of full, this load generates global bending in the longitudinal direction. This means that the bottom of the FPSO is mostly under membrane traction in the longitudinal direction and that this membrane traction is force driven (Figure 4). Hence, let us define N 0 11 the macroscopic membrane traction inside the original assembly of hull and stiffeners (without reinforcement). Now that one of the six plate elementary loadings is set, there remains to fix the other five loadings. The answer to this question is actually not straightforward. Assuming there are no web frames, the stiffened hull is free to contract in the cross in-plane direction (in-plane Poisson's effect). The same remark holds for the three curvatures (M 11 , M 22 , M 12 ) and the in-plane shear (N 12 ). Hence without web frames, it seems consistent to let free the other plate loadings :

N 11 = N 0 11 , N 22 = 0, N 12 = 0, M 11 = 0, M 22 = 0 and M 12 = 0 (Eq. 1)
We call this macroscopic load configuration "uni-axial traction" (or "load case N").

On the contrary, the web frames restrain to some extent the plate kinematics (except the longitudinal strain E 11 ). Hence we suggest a "dual" macroscopic load configuration defined as "pure axial-strain" (or "load case E") :

N 11 = N 0 11 , E 22 = 0, E 12 = 0, K 11 = 0, K 22 = 0 and K 12 = 0 (Eq. 2)
Because the web frames are very close together (every 5m), it is expected that the actual macroscopic load of the plate is closer to uni-axial strain than to uni-axial traction. However, both cases are studied here to be exhaustive.

Definition of reinforcement rates

In this section we define the reinforcement rates for both macroscopic loads.

We define the stress reinforcement rate as the ratio :

τ σ = σ r /σ 0 (Eq. 3)
where, σ 0 is Von Mises stress in the non-reinforced stiffened hull and σ r is the maximum Von Mises stress with reinforcement. When this rate is lower than 1, the hull is reinforced. We define also the stiffness reinforcement rate as :

τ A = A r -A 0 A 0,s (Eq. 4)
where A r is the apparent stiffness of the reinforced and stiffened hull for the given macroscopic load, A 0 is the apparent stiffness of the non-reinforced and stiffened hull. Finally A 0,s is the apparent stiffness of the hull without stiffeners. We choose this stiffness as reference because the we are concerned with the reinforcement of the hull only. The stiffeners are simulated only because their size is comparable to the size of the pads which forces us to include them in the simulations.

3.

Application to the reinforced hull

Detailed description of the unit-cell

The hull is made of steel. The Young modulus and the Poisson's ratio are fixed in the whole study to : E h = 210 GPa and ν h = 0.3. The thickness is also fixed to t h = 15 mm. It will be discretized using 3D linear brick elements with reduced integration. The size of the stiffeners varies in the whole FPSO. It is fixed here to a typical configuration : The web height is h w = 450 mm and its thickness t w = 13 mm. The flange total width is h f = 75 mm and its thickness t f = 29 mm. The spacing between two stiffeners is set to H = 925 mm. They are made of the same steel as the hull. In terms of finite elements we choose to model them as shells.

Since the elastomeric seal stiffness is much smaller than the adhesive, it is not modeled at all. In addition, the protection plate overlap over the seal is also not modeled in order to simplify the geometry of the finite element model (Figure 1 and5).

The adhesive layer is one of the critical part of the design. It is assumed isotropic with a constant Poisson's modulus ν a = 0.4. Its thickness will vary : t a ∈ [5 mm, 20 mm] as well as its Young modulus :

E a ∈ [650 MPa, 8000 MPa].
It is discretized with 3D linear elements with reduced integration. The composite layer will be made of epoxy resin reinforced with glass or carbon fibers. Here we assume uni-directional layup in the longitudinal direction. For the simulations, the composite constitutive law is transversely isotropic. The elastic moduli are derived through the mixture law between moduli of the epoxy matrix (E m ∈ [2 GPa, 4 GPa] and ν m = 0.4), the fibers moduli and the volume fraction of fibers ρ f ∈ [0.3, 0.5] (? ]). It is discretized with 3D linear elements with reduced integration. The thickness of the composite reinforcement is t r ∈ [5 mm, 20 mm].

The protection plate is assumed to be made of stainless steel. Its thickness is t i ∈ [1 mm, 6 mm]. Even if it is well known for having a non-linear constitutive behavior, here we assume that it has a linear and isotropic behavior. The moduli are set to : E i = 190 GPa and ν i = 0.3. A typical cross section of the edge of a pad modeled with finite elements is given in Figure 5. For fabrication reasons, there is a gap between the composite and the seal which is filled with adhesive. This gap width is w a = 20 mm. Finally, several in-plane configurations are under consideration. They are introduced in Figure 6. The typical length of a pad is L = 2000 mm. The width of the seal is w g = 20 mm.

First design -worst case

In this section, we present the results for the case of rectangular pads which were first considered as potential geometries (see Figure 6). It enables us to give the main features of the effect of the pads.

Parameters

We choose the following set of parameters : Adhesive thickness : t a =5 mm, Stainless steel thickness : t i =1 mm, Adhesive Young modulus : E a =8000 Mpa, Composite thickness : t r =20 mm, Composite longitudinal Young modulus : E l =117 GPa The composite reinforcement is quite thick with material characteristics leading to highest moduli. The adhesive thickness is as thin as possible with a very high modulus. The stainless steel thickness is the thinnest one considered for the design. This first choice of parameters is meant to be as favorable as possible for the efficiency of the pads : thin adhesive layer and thick composite.

Stiffness reinforcement

The following stiffness reinforcement rates are found :

-uni-axial strain : τ E A =0.467 -uni-axial traction : τ N A =0.457 Hence the average stiffness of the hull is increased around 45% compared to the damaged situation. With this configuration the pad achieves the reinforcement in terms of stiffness.

Stress reinforcement

On Figure 7 is displayed the stress rate field for an assembly of four unit-cells in the uni-axial strain loading case. In most areas, the rate is below 1 which confirms the reinforcing effect of the pads. However, the stress rate jumps just between the pads to rather high values which are : -uni-axial strain : τ E σ = 2.59 -uni-axial traction : τ N σ = 2.27 This is more clearly observed on the right of Figure 7 where only the hull is displayed. A hinge appears along the line which separates two pads. This hinge is generated by two phenomenon. The first one comes from the fact that actually the hull is not reinforced in this region. Then one expect the stress being higher than in the reinforced area. However, one would expect the stress having values close to 1 (the rate without reinforcement). In the present case, the stress rate jumps to values twice larger. When looking more closely at the deformed shape, it appears that the hull is locally bent which generates large local stress. Because the pads are bonded only on one side of the hull, the neutral axis is shifted between reinforced and non-reinforced areas. This eccentricity couples membrane loading and bending. Since the reinforced area is much thicker than the non-reinforced hinge, all the curvature is concentrated in the non-reinforced hinge. This folding phenomenon critically affects the efficiency of the pads and we seek better geometries in the following. However, let us point out also that in the reinforced area, the stress rate is globally below 84% which illustrates the reinforcing potentialities of the device.

Comparison of in-plane geometries

As already observed, the rectangular shape of the pads leads to local bending between the pads. In this section we explore the influence the the in-plane geometry of the pads in order to identify possibly interesting shapes. We consider the shapes given in Figure 6. Basically we tried to limit the number edges in the cross direction and circumvent the possibilities of folding by not aligning edges. In addition, these geometries must be producible. In order to compare quantitatively the geometries, we perform a numerical campaign following a factorial experiment. Only parameters related to the thicknesses of the layers and the constitutive behavior are investigated. The values are chosen as the extreme values which can be reached for the design of the pads and where given in Section 3.1.

Stiffness reinforcement

The stiffness reinforcement rate for the uni-axial strain load case is plotted in Figure 8 for each simulation and each geometry. For clarity, the simulations were sorted with respect to the rate of the Rectangle geometry. The stiffness rate covers a rather broad range from 5% to 50%. In the present study the target is about 10%. The reinforcement rate does not depend strongly on the geometry (all the geometries follow the same trend. Finally, it comes out that the Rectangle geometry is the most efficient in terms of stiffness whereas the Diamond family shows lower rates, especially the Triple Skew Diamond Geometry. A simple explanation for this is that the latter is the geometry which presents the longest perimeter. Hence its reinforcing surface is smaller and the length of free edges for the pads is larger. However, this does not seem to dramatically decrease the stiffness reinforcement rate.

Stress reinforcement rate

The stress rate of all geometries is given for each simulation on Figure 9. Thick lines show the uniaxial strain load case whereas the thin lines give the corresponding rate for the uni-axial traction to be exhaustive. Again the simulations are sorted with respect to the stiffness reinforcement rate of the Rectangle geometry. This means that most compliant configurations are located on the left of the figure whereas stiffest configurations are located on the right. The most remarkable observation is that the geometry dramatically affects the stress rate, especially for large stiffness reinforcement. The second observation is that most of the geometries increase the stress inside the hull. More precisely, it appears that the different geometries have different global trends (increasing or not from left to right) but follow similar local patterns depending on the change of parameters between each simulation. We conclude that the local patterns are the consequence of fine tuning of parameters which will be investigated later on, whereas the global trend shows the influence of the in-plane geometry and is discussed here.

Considering the Rectangle geometry, the stress rate increases when the stiffness rate increases (the stress rate is larger for simulations on the right of the figure). This means that increasing the stiffness and decreasing the stress are antagonistic demands. This is because of the hinge phenomenon already described : the more the hull is reinforced, the thicker is the pad, the more local is the hinge, the larger is the bending stress. We conclude that the Rectangle geometry will never lead to a suitable design. The same remark holds for the Triangle and Diamond geometries even if the effect is smaller. The only geometries for which the stress rate does not clearly increase with the stiffness are the Skew Diamonds and the Triple Skew diamonds. Actually these are also the only one which are close to reinforcing the hull (stress rate below 1).

Conclusion regarding the in-plane geometry

Performing the same simulations for different in-plane geometries points out that they do not affect significantly the stiffness reinforcement rate. This design criteria is affected by other parameters which will be investigated in the following. However, the geometry critically affects the stress rate. The hinge phenomenon is present in almost all situations and leads to an antagonistic demand between increasing the stiffness and decreasing the stress inside the hull. Finally, only the Triple Skew Diamond seems to have potentialities. Consequently, we select this geometry in the following for optimizing the design and investigating the influence of the remaining parameters. Let us already point out that this geometry is much more complex than the one of the rectangle.

Investigation of the Triple Skew Diamond Geometry

In order to have a comprehensive description of the possibilities of this geometry, we performed the same factorial experiment as in the previous section to which we added two values for the length of the pad : L = 2000 mm and L = 2500 mm. We investigate longer pads than firstly considered since it will turn out to be a critical parameter. 8).

Overview of results

On Figure 10 is given the stress reinforcement rate as function of the stiffness reinforcement rate for all simulations and for both load cases. Clearly, the most restrictive load case is the uni-axial strain and the uni-axial traction leads to much lower stress rate. Contrary to the other geometries, here the stress rate decreases globally with larger stiffness reinforcement rate which is the expected reinforcement effect. However, the achieved stress reinforcement are rather low : 0.944 for the best case in uni-axial strain.

Parameters influence

A simple procedure was applied to bring out the influence of the parameters. Considering one parameter, for each of its values investigated we compute the average all simulations (the other parameters varying). The average is performed for the two rates. Because we have a factorial experiment, for each value of a single parameter, the average is performed on exactly the same set of remaining parameters.

The results are given on Figure 11. The graphs are organized in three rows corresponding to both reinforcement rates. The columns correspond to each investigated parameters. Globally, there is a clear distinction between the load cases only for the stress reinforcement rates (the gap being 7% on average) whereas results are almost identical with the stiffness rate. This is because the uni-axial strain enforces more kinematic constraints on the unit-cell than the uni-axial traction.

Let us consider the stiffness reinforcement rate. The stainless steel thickness t i and the composite thickness t r have a clear positive effect on the stiffness reinforcement. In addition high tensile characteristics for the composite and high adhesive modulus are also favorable to the stiffness rate. These influences are quite expectable. It is noticeable that the length of the pad L and the thickness of the adhesive have less influence on the stiffness rate. This tends to indicate that the shear lag effect (Figure 2) is not so large.

Because the stress reinforcement rate is highly correlated to the stiffness reinforcement rate (cf. Figure 10) the same influences of the parameters would be expected on the stress rate. However, this is not exactly the case. The positive influence of the tensile characteristics of the composite and of the composite thickness on the stress rate is confirmed. Similarly, the adhesive thickness does not seem to critically affect the stress rate. However, the influence of the stainless steel thickness and the adhesive modulus is less clear and the influence of the length of the pad is stronger than for the stiffness rate. The stainless steel thickness has a favorable effect on the stress rate because it increases the stiffness of the hull. However, it has a negative effect because it increases the bending stiffness of the pad. When the pad bending stiffness increases, it localizes the curvature along the hinges coming from the gap between pads. This effect is stronger than with the composite layer. This is because the composite layer is strongly anisotropic : the very compliant transverse shear stiffness of this layer accommodates more easily the hinge. Hence it seems more preferable to increase the thickness of the composite layer than the thickness of the stainless steel. The influence of the adhesive modulus is also ambiguous. High moduli should decrease the stress rate because they increase the stiffness rate. However, the opposite trend is observed for the uni-axial load case. Further analysis of the results shows that the best reinforcing rates are achieved for E a = 3000 MPa which shows that there is probably an optimum value to be found. Finally, when the pads length increases, the hinge are more oriented in the longitudinal directions. This might explain the larger influence of this parameter.

Design recommendations

In the light of the present study, it possible to express some recommendations for the design of reinforcing pads not connected together. The first point is that the geometry critically affects the stress reinforcement rate. Because the pads are not connected, there is a shift of neutral axis which generate local hinges between pads. Exploring potential geometries shows that the Triple Skew Diamond one mitigate enough this phenomenon. One cannot exclude other potential geometries which might be more efficient. However, they must be suitable for fabrication.

One question was to check if the rather large thickness of the adhesive layer would generate a large shear lag effect and preclude any reinforcement. This effect is present, however it is not dominating the behavior of the pad (for the length considered here). Thus, having rather thick and compliant adhesive layer does not compromise the strengthening effect of the pad. On the contrary it might be beneficial since it accommodates the hinge phenomenon. Having t a = 20 mm and E a = 3000 MPa seems a good compromise. Finally, from the parameters influence study it appears that it is preferable to reinforce with composite than with stainless steel because the transverse shear compliance of the composite accommodates better the hinge phenomenon. In addition, the composite should have the highest tensile properties in order to maximize its reinforcing effect.

Conclusion

The present work shows the application of homogenization techniques in the linear elastic regime, in order to assess the efficiency of periodically tiled reinforcing pads. The following conclusion are drawn :

-The pads are able to restore the original stiffness of the hull.

-The pads efficiency in terms of stress reduction is very low. This is because the pads shift the neutral axis of the hull and generate local hinges. Hence, even if it were possible to decrease the stress inside the hull, this would generate a very large increase of the local stiffness (see Figure 10) and would seriously perturb the global behavior of the hull. In addition, we focused on the longitudinal load for the sake of simplicity but there are other load directions which will require investigation and for which the Triple Skew Diamond geometry will not be suitable. Consequently, it is highly recommended to connect mechanically the pads together in order to achieve a continuous reinforcement.

  Introduction 1.1. The Coldpad reinforcement technique

  Fig. 2.: Schematic view of the reinforcing effect of the pads.

Fig

  Fig. 3.: Structural detail of the FPSO unit bottom

  Fig. 5.: Cross section modeled with FE

  Fig.6.: The several tilings explored in this study. In red, the unit-cell. In green, the quarter of unit-cell actually simulated.

Fig. 7 .

 7 Fig. 7.: Global view of the reinforced hull with rectangular pads. The hull is under uni-axial strain in Direction x and the contour plot is the stress rate. 4 unit-cells are displayed.

  Fig. 8.: Stiffness rate inside the hull for each simulation and each geometry. The simulations are sorted from left to right with respect to the stiffness rate of the Rectangle configuration.

Fig

  Fig. 11.: Sensitivity analysis for the Triple Skew Diamond factorial experiment. Blue markers are for the uni-axial strain and brown markers for the uni-axial traction load cases.

  Stress rate inside the hull for each simulation and each geometry. The simulations are sorted from left to right with respect to the stiffness rate of the Rectangle configuration (like Figure

			Rectangle E	Rectangle N	Narrow Rectangle E	Narrow Rectangle N
			Triangle E	Triangle N	Diamond E	Diamond N
		2.60	Skew Diamond E	Skew Diamond N	Triple Skew Diamond E	Triple Skew Diamond N
		2.40
		2.20
	Stress reinforcement rate	1.40 1.60 1.80 2.00
		1.20
		1.00
		0.80
			11 3 1 15 9 12 7 5 4 13 27 31 2 10 28 16 19 23 32 25 17 29 20 21 26 8 14 18 24 6 30 22
				Simulation #
	Fig. 9.:

  Fig. 10.: Stress reinforcement rate vs. Stiffness reinforcement rate for the Triple Skew Diamond factorial experiment

		1.05
		1.00
	Stress reinforcement rate	0.90 0.95
		0.85	Load Case E
			Load Case N
		0.80
		0.00	0.10	0.20	0.30	0.40	0.50	0.60
					Stiffness reinforcement rate