Driss El Khoukhi
email: driss.el-khoukhi@ensam.eu

Franck Morel

Nicolas Saintier

Daniel Bellett

Pierre Osmond

Viet- Duc Le

Jérôme Adrien

Experimental investigation of the size effect in High Cycle Fatigue: role of the defect population in cast aluminium alloys

Keywords: HCF, size effect, microstructural heterogeneities, cast aluminium alloys, fatigue scatter

Cast Al-Si alloys have been widely used in automotive applications with regard to their low density and excellent thermal conductivity. Many components made of these alloys are subjected to cyclic loads which can lead to fatigue failure. Furthermore, for these materials the well know size effect in fatigue, whereby the fatigue strength is reduced when the size is increased, can be significant and need to be properly evaluated. This paper analyses the role of casting defects on the fatigue strength's size effect sensitivity. A uniaxial fatigue testing campaign (R=0.1) has been conducted using two cast aluminium alloys, fabricated by different casting processes (gravity die casting and lost foam casting), associated with the T7 heat treatment, and with different degrees of porosity. The fatigue response of different specimens (smooth and notched) with different stressed volumes has been investigated. The first part of this article is dedicated to the experimental characterization of the size effect in both alloys via the concept of the Highly Stressed Volume. The second part investigates the effect of the Highly Stressed Volume on the critical defect size via Kitagawa-Takahashi diagrams. The results show that the magnitude of the size effect and the experimental scatter are strongly linked to the characteristics of the defect population present in the alloy. It is revealed that the alloy B, with a high density of pore and a population of defects with relatively large size, shows non-significant size effect and less scatter in fatigue strength. In comparison, alloy A that exhibits a low density of pore and a population of defects of relatively small size manifests significant size effect and high scatter in fatigue strength.

Introduction 1.1 Context, introduction and objectives

The fatigue data transferability from laboratory specimens to real components or structures is very often difficult to handle because of the high number of parameters potentially affecting the fatigue strength. Besides the loading mode, the microstructural heterogeneities, the stress gradient and the size of the loaded volume can show significant effects on the fatigue strength.

In particular, it is generally accepted that the fatigue strength of certain materials decreases with an increase in the volume [START_REF] Kelly | Effect of Specimen Size and Preparation on the Fatigue Strength of a Plain Carbon Steel Tested in Rotating Bending and in Torsion[END_REF][START_REF] Carpinteri | Size effect in S-N curves: A fractal approach to finite-life fatigue strength[END_REF]. This is referred to as either the "volume effect", the "size effect" or the "scale effect" and is often simply explained by an increase in the probability of encountering a large material defect in the fatigue active volume. The effect is influenced by factors related to the distribution of defects in the material and the impact of the manufacturing process on the homogeneity of the material.

Even though the experimental observations of the size and the gradient effects concern the macroscopic fatigue response, a few studies [START_REF] Flaceliere | Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects[END_REF][START_REF] Saintier | Non-local energy based fatigue life calculation method under multiaxial variable amplitude loadings[END_REF][START_REF] Banvillet | A volumetric energy based high cycle multiaxial fatigue citerion[END_REF][START_REF] Hidalgo | Fatigue life estimation of cast aluminium alloys considering the effect of porosity on initiation and propagation phases[END_REF] try to show how microscopic features like microroughness or microdefects can affect the global stress gradient and size effects. The idea is, from some statistical distributions of surface or volume defects, to derive the macroscopic fatigue strength distribution. When doing so, three main difficulties arise. First, the mechanisms of crack initiation must be known to identify which of the microstructural features are responsible for fatigue damage and hence which statistics must be described. The task is all the more tough when several damage mechanisms are active [START_REF] Tenkamp | Defect-correlated fatigue assessment of A356-T6 aluminum cast alloy using computed tomography based Kitagawa-Takahashi diagrams[END_REF][START_REF] Ammar | Effect of casting imperfections on the fatigue life of 319-F and A356-T6 Al-Si casting alloys[END_REF]. Second, the experimental description of the statistics must be carried out by using a relevant representative volume. The question is here to know what is the minimum size of the fatigue representative volume (FRV) and how this FRV is affected by the microstructure features distribution. Third, when the main fatigue damage mechanisms have been identified, the relation between the microstructural heterogeneities characteristics (size, shape …) and the fatigue strength levels must be estimated. The Kitagawa diagram is for instance largely used in fatigue to relate the fatigue strength to the defect size.

The work presented in this paper aims at getting a better understanding of how the highly stressed volume and the defect population affect the macroscopic fatigue response. It focuses on cast Al-Si alloys, widely used in the automotive industry. In order to manufacture engine components, the Groupe PSA indeed uses two foundry processes that result in components containing Microstructural Heterogeneities (MH), principally shrinkage pores and oxides, with different characteristics (size, shape and spatial distributions). These components are subjected to cyclic mechanical loads (in the High Cycle Fatigue regime) that can result in the appearance of cracks and thus lead to the failure of the structure. The effect of the MH on the fatigue behaviour has been well documented in references [13, 16 -19].

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A typical engineering design practice [START_REF] Jan | Prediction of the low cycle fatigue regime of the S-N curve with application to an aluminium alloy[END_REF] involves the identification of a fatigue criterion, used to size and validate real components, based on tests carried out on laboratory sized specimens, generally under simple loading modes (e.g. tension, bending, and torsion). However, real component loading conditions are often different to those applied to laboratory test specimens, particularly in terms of the size of the critical zone (i.e. size effect) and the homogeneity of the stress state in this zone (i.e. stress gradient effect). Hence, the industrial objective of this work is to develop predictive tools to ensure the passage from an elementary volume to the structure, taking in account the size effect and the stress gradient effect. In this work, the effect of microstructural heterogeneities, principally shrinkage porosity and oxides, on the statistical size effect in cast Al-Si alloys is investigated.

An example of fatigue design approach taking account the size effect is the Highly Stressed Volume (HSV) approach who was first introduced by Kuguel [START_REF] Kuguel | A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume[END_REF] in 1961, and then employed by Sonsino and Fischer [START_REF] Sonsino | Local Assessment Concepts for the Structural Durability of Complex Loaded Components[END_REF], and Lin and Lee [START_REF] Lin | Effects of highly stressed volume on fatigue strength of austempered ductile irons[END_REF]. The terminology Vn% is used to define the volume of material that is subjected to at least n% of maximum principal stress (σn%=n%×σmax). This volume of material is assumed to have an increased probability of fatigue crack initiation. The percent is assumed to be 95% by Kuguel [START_REF] Kuguel | A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume[END_REF] and 90% by Sonsino and Fisher [START_REF] Sonsino | Local Assessment Concepts for the Structural Durability of Complex Loaded Components[END_REF]. The concept has been verified for metallic components with good success using the V90% volume [START_REF] Menan | The Stress-Strength Interference Method Applied to Fatigue Design: The Independence of the Random Variables[END_REF]. For short-fiber reinforced specimens and components, a value of V80% proved to be more appropriate than V90% [START_REF] Sonsino | Fatigue design of highly loaded short-glass-fibre reinforced polyamide parts in engine compartments[END_REF]. The effect of HSV on the fatigue behaviour has been well documented in references [START_REF] Kuguel | A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume[END_REF][START_REF] Sonsino | Local Assessment Concepts for the Structural Durability of Complex Loaded Components[END_REF][START_REF] Lin | Effects of highly stressed volume on fatigue strength of austempered ductile irons[END_REF].

Due to its significance in structural reliability, certain design standards require the size effect to be taken into consideration. As an example, for the design fatigue curves of the ASME B&PV Code [START_REF] Kuguel | A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume[END_REF][START_REF] Sonsino | Local Assessment Concepts for the Structural Durability of Complex Loaded Components[END_REF][START_REF] Lin | Effects of highly stressed volume on fatigue strength of austempered ductile irons[END_REF], the factors of 2 (stress) and 20 (cycles) are adjustment factors applied to the small-specimen data to obtain reasonable estimates of the lives of actual reactor components. These factors were intended to account for data scatter (including material variability) and differences in surface condition and size between the test specimens and actual components.

Two Al-Si alloys with different porosity distributions are chosen in order to understand how the characteristics of the defect populations affect the statistical size effect. Computed Tomography analyses are conducted in order to characterize the porosity in terms of size, shape and spatial distributions in these alloys. Different specimen shapes, including both smooth and notched specimens are machined from these alloys, in order to obtain geometries with different Highly Stressed Volumes. High cycle fatigue tests are conducted to investigate the size effect on the fatigue strength using the HSV concept. Finally, fractographic analyses are conducted to investigate the relationship between the critical defects and the fatigue strength via the Kitagawa-Takahashi diagram. Statistical analyses of fatigue data are also presented.

It is important to stress that 140 samples have been tested in order to get a good approximation of the statistical distribution with a high level of confidence.

Materials and experimental conditions

Mechanical and Material Properties

To study the influence of casting defects on the statistical size effect, two primary cast aluminium alloys, referred to as alloys A [AlSi7Cu05Mg03-T7] and B [AlSi7Mg03-T7], have been used (see Figure 1). These alloys were fabricated by different casting processes (alloy A: gravity die-casting and Alloy B: lost foam casting), and subject to the T7 heat treatment. These processes result in different porosity populations (i.e. volume fraction, defect size…) and different mechanical properties [see table 1 and table 2]. Previous works concerning the characterization of these materials have been done by Le [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF] and Koutiri [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF]. Table 1 summarizes the mechanical and microstructural properties of these alloys [START_REF] Ammar | Effect of casting imperfections on the fatigue life of 319-F and A356-T6 Al-Si casting alloys[END_REF][START_REF] Chantier | A probabilistic approach to predict the very high-cycle fatigue behaviour of spheroidal graphite cast iron structures[END_REF]. From Table 1, we conclude that the materials do not show a large difference in terms of micro-hardness, yield stress, ultimate strength and Young Modulus. Nevertheless, in terms of the SDAS parameter and Elongation, a significant difference is observed. However, it is well known that the uniaxial fatigue resistance in the HCF regime is mainly controlled by the defect size [13, 16, 28 -30]. Therefore, in this work, it is assumed that the SDAS and Elongation are not the main parameters controlling the fatigue response. In order to characterize the defect size distribution in these alloys, Computed-Tomography (CT) analyses were undertaken. CT scans were done by the MATEIS laboratory at INSA Lyon with resolution of 8 µm/voxel. The AVIZO software was then used in order to analyse the raw data. The alloys were characterised in terms of size and shape distributions of their micro-shrinkage porosity. Also, the distance between each individual pore was determined. An inspection volume of 363 mm 3 was used. Figure 2 illustrates the defect population as imaged by microtomography on two notched fatigue specimens (see section 2. 4 for more details on specimen geometry).

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Porosity Analyses

J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF 5 2500µm 2500µm c) d) a) b)
It can be clearly seen that alloy B has considerably higher density of large pores when compared to alloy A.

With the CT scans, only pores in the scanned volume can be observed. However, oxides are also potential sites of crack initiation. Nevertheless, as the experimental results show, the dominant mechanism of failure is the one that starts from a pore. Therefore, to understand the effect of microstructural heterogeneities on the fatigue size effect, it seems sufficient to understand the effect of the porosity on the fatigue size effect.

Figure 3 shows the defect size distributions for the two alloys in terms of the equivalent Murakami parameter [START_REF] Murakami | Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions[END_REF], of the defect. The relationship between pore volume obtained by tomography and its equivalent 𝐴𝑟𝑒𝑎 𝑒𝑞 square root of the projected area is given in Eq. 1. This relationship is obtained by assuming a spherical pore shape.

(1)

𝐴𝑟𝑒𝑎 𝑒𝑞 = 𝜋 1/6 ( 3𝑉 4 ) 1/3
The use of this approximation makes it possible to compare the sizes obtained by the CT scans and those measured on the fatigue failure surfaces. In order to characterize the shape of the defects, the sphericity parameter is defined in equation Eq. 2.

(

) 𝑆 = 𝜋 1/3 (6𝑉) 2/3 𝐴 2
where A is the surface area of the pore and V is the pore volume. This parameter compares the shape of a pore to a sphere. Only the perfect sphere will have a sphericity of 1. Figure 4 shows the relationship between pore size and sphericity for both alloys. It can be seen that the largest pores have the lowest sphericity and a complex shape. This tendency has also been observed in the previous works [START_REF] Ammar | Effect of casting imperfections on the fatigue life of 319-F and A356-T6 Al-Si casting alloys[END_REF][START_REF] Ai | Probabilistic modeling of fatigue life distribution and size effect of components with random defects[END_REF][START_REF] Zhu | Evaluation of size effect on strain-controlled fatigue behavior of a quench and tempered rotor steel: Experimental and numerical study[END_REF]. According to Buffiere classification [START_REF] Ai | Probabilistic modeling of fatigue life distribution and size effect of components with random defects[END_REF][START_REF] Zhu | Evaluation of size effect on strain-controlled fatigue behavior of a quench and tempered rotor steel: Experimental and numerical study[END_REF], this repartition tend to indicate the presence of shrinkage porosity and gas pores. This last type of pores seems especially present in the alloy B.

In order to characterize the spatial distribution of the pores in the scanned volume, the data obtained from CT scans were used to evaluate the distance to its nearest neighbor is established using the fact that the distance between the centers of mass of two pores P and Q, with Cartesian coordinates respectively (p1, p2, p3) and (q1, q2, q3), is:

J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF 7 (3) 𝑑 𝑇 (𝑃,𝑄) = (𝑝 1 -𝑞 1 ) 2 + (𝑝 2 -𝑞 2 ) 2 + (𝑝 3 -𝑞 3 ) 2
The nearest-neighbor distribution is the distribution of distances from each pore P to its nearest neighbor in the scanned volume. The nearest neighbor distance is defined by Eq. 4 (see [START_REF] Schlömer | Farthest-point optimized point sets with maximized minimum distance[END_REF], [START_REF] Garb | Statistical analysis of micropore size distributions in Al-Si castings evaluated by X-ray computed tomography[END_REF] for more details). ( 4)

𝑑 𝑃 = min 𝑄 ∈ 𝑋\{𝑃} 𝑑 𝑇 (𝑃,𝑄)
Where X is the scanned volume. The characteristics of this distribution are summarized in Table 2 where the coefficient of variation is defined as the ratio of the standard deviation to the mean. Table 2 shows that the alloys have different characteristics in terms of size, shape and nearest-neighbor distance.

We can summarized the data contained in the table as follows:

-Defects size : o The mean of the defects size contained in ally B is 1.5 times higher than the mean of those contained in alloy A. o Either the coefficient of variation and the standard deviation for alloy A are lower than for the alloy B -Nearest-neighbor distance :

o The mean of the nearest-neighbor distribution for alloy A is two times higher than the mean of the alloy B. o The coefficients of variation of the nearest neighbor distributions of these alloys are almost the same. -Defect shape :

o The alloy A presents more regular shaped pores compared to those in the alloy B. The characteristics of the defect size and nearest-neighbor distributions is likely to control the magnitude of size effect and scatter in these alloys.

Experimental conditions of the fatigue tests

All fatigue tests were performed under uniaxial tensile loads with a stress ratio R = 0.1 using a Rumul Testronic resonant testing machine at approximately 100 Hz. The stopping criterion was a drop in the resonance frequency of 1 Hz, corresponding to the presence of a fatigue crack of approximately 2 mm on surface. All tests were performed at room temperature, in air. The tests were conducted following the staircase method with a maximum life of 2x10 6 cycles. The run-out specimens were retested until failure at higher load levels in order to identify the critical defect that initiates a crack. A step of 5 MPa stress amplitude was used in the staircase procedure.

Specimens geometry

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To study the effect of stressed volume on the fatigue strength of alloys A and B, different fatigue specimen geometries were defined, corresponding to different loaded volumes. These are shown in Figure 5 and are referred to as: "V1-Small Volume", "V2-Reference Volume", "V3-Large Volume" and "VN-Notched specimen".

VN V1 V2 V3

• D = 7 mm

• d = 5.6 mm • r = 1.2 mm • Kt=1.78 VN COUPE D-D VN-1 V1 V2 V3 • 𝐷 = 7𝑚 𝑚 , • 𝑑 = 5,6𝑚 𝑚 , • 𝑟 = 1,2𝑚 𝑚 • Kt=1.78 VN-1 COUPE D-D

Estimation of the Fatigue Active Volume "FAV"

To illustrate the difference between the four specimen shapes in terms of loaded volume, the concept of the Fatigue Active Volume "FAV" is used in an empirical manner.

In the following sections, the FAV is estimated using the combination of two criteria:  Damage mechanism. In the literature, much works [START_REF] Ammar | Effect of casting imperfections on the fatigue life of 319-F and A356-T6 Al-Si casting alloys[END_REF][START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF][START_REF] Sonsino | Fatigue design of highly loaded short-glass-fibre reinforced polyamide parts in engine compartments[END_REF][START_REF] Ai | Probabilistic modelling of notch fatigue and size effect of components using highly stressed volume approach[END_REF][START_REF] Hirano | Study on Consideration of Size Effects on Design Fatigue Curve[END_REF][START_REF] Houria | Influence of casting defect and SDAS on the multiaxial fatigue behaviour of A356-T6 alloy including mean stress effect[END_REF][START_REF] Ben Ahmed | The impact of defect morphology, defect size, and SDAS on the HCF response of A356-T6 alloy[END_REF] have shown that the fatigue failure usually occurs at defect or pores located on the surface or in a sub-surface layer. In the experimental work undertaken in this work, the crack initiation sites are located in a subsurface of 500 µm for alloy A and 650 µm for alloy B. Figure 6 shows a subsurface pore with the depth 't' in the crack initiation zone for alloy B. Figure 7 shows the cumulative distributions distance from the surface for all pores observed in the crack initiation zones in these alloys. value is based on the results for the V1 specimen, which has an hourglass shape (Figure 9). Most of the fatigue cracks in these specimens do not occur in the centre of the specimen where the cross-sectional area is the smallest and the stress is the highest. This is because the cast aluminium materials investigated are defect containing materials. Figure 9 shows the case of a specimen in which the fatigue crack is located approximately 5.82 mm from the middle of the specimen. Figure 8 shows the normalised stress at the crack location as a function of the distance from the centre of the specimen. From this figure, it can be seen that almost all of the specimens have failed at a stress level that is greater than 80% of the maximal stress. The combination of these two criteria is used to define the Highly Stressed Volume corresponding to 0.8𝜎 𝐼𝑚𝑎𝑥 located in a subsurface layer of thickness 500 µm for alloy A and 650 µm for alloy B. In the following section, this definition is referred to as the FAV. For each specimen geometry, an estimation of the Highly Stressed Volume has been done using finite element computation with a linear elastic analysis. Figure 10 shows FAVs identified for each case. The use of the notch for the fatigue specimens VN has been designed to get a very small active volume.

J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF 10 VN-1 V2 V3 V1           VN-1 V2 V3 V1           VN-1 V2 V3 V1             VN V-1 V-2 V-3
In the following sections, we will use the estimated FAV A for alloy A and FAV B for alloy B. The FAV corresponding to each specimen shape, as well as stress concentration factors and number of specimens tested for each configuration, are presented in Table 3. We point out also that the FAV of the sample V3 for alloy B is not indicated in table 3 since it was not tested for this alloy. The stress concentration factors Kt were estimated as follows:

𝐾 𝑡 = 𝜎 𝑚𝑎𝑥 𝜎 𝑛𝑜𝑚 Where,
being the maximum elastic normal stress in the axial direction at the notch tip, and is the 𝜎 𝑚𝑎𝑥 𝜎 𝑛𝑜𝑚 nominal elastic stress based on the net section.

Fatigue SN curves and effect of FAV

High cycle Fatigue S-N Curves

High-cycle fatigue tests have been performed on specimens with different geometries taken from two Al-Si cast aluminium alloys. S-N curves have been represented to characterize the statistical size effect in these alloys. For the sake of simplicity, only the results corresponding to the standard volume V2 and the small volume V1 are shown for both alloys. Fig. 10 shows the S-N curves for these volumes: A-V1, A-V2, B-V1 and B-V2. The average S-N curves have been fitted to the experimental data using a power law based on the least square method. From Figure 11, it can be seen that there is:  A large difference between the A-V1 and the A-V2 data. This indicates a significant size effect in alloy A,  For alloy B, the data for B-V1 and B-V2 are almost superimposed. Hence, the size effect is almost negligible for alloy B,  Less scatter in terms of fatigue strength is observed for alloy B data compared to alloy A data,  The mean fatigue resistance for alloy B is lower than that of alloy A. This difference can be associated to the difference in terms of Cu cotenant and the critical defects size between the studied alloys. Work is underway which aims to evaluate the effect of Cu on the fatigue strength.

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From these remarks, it can be concluded that alloy A shows a strong statistical size effect as compared to alloy B, at least for the range of investigated volumes.

Statistical size effect

In order to illustrate the influence of the size effect on the fatigue strength, the product of the nominal stress amplitude and stress concentration factor K t is used. That is, for the notched specimens the local stress 𝜎 𝐷 amplitude at the notch-tip is used. In references [START_REF] Serrano-Munoz | Location, location & size: defects close to surfaces dominate fatigue crack initiation[END_REF][START_REF] Buffière | Experimental study of porosity and its relation to fatigue mechanisms of model Al-Si7-Mg0.3 cast Al alloys[END_REF], it was shown that there is a good correlation between the fatigue strength expressed as and the highly stressed volume. In this work, the correlation will be made 𝐾 𝑡 𝜎 𝐷 in terms of the Fatigue Active Volume. Note that the notched specimens were designed to ensure minimal local plasticity at the notch-tip. At the highest loads experimentally applied for 5 specimens the von Mises stress exceeds the yield strength by 4%. Hence, the effect of cyclic plasticity is considered negligible. Let us also notice that the estimation of the fatigue limit of each specimen is assessed using the following formula proposed in [START_REF] Lanning | On the use of critical distance theories for the prediction of the high cycle fatigue limit stress in notched Ti-6Al-4V[END_REF].

(5)

𝜎 𝐷 = 𝜎 𝑛 -1 + (𝜎 𝑛 -𝜎 𝑛 -1 ) × 𝑁 𝑓 2 × 10 6
where is the interpolated fatigue limit at 2 millions of cycles.

is the stress amplitude level of the block 𝜎 𝐷 𝜎 𝑛 -1 prior to the block where failure occurs. For the specimens that fail at the first level ( ) of stress we supposed 𝜎 𝑛 the existence of a fictive level (

).

is the maximum stress level of the final block of cycles during which 𝜎 𝑛 -1 𝜎 𝑛 failure occurs, and Nf is the number of cycles to failure in the final loading block. The main objective here is to get the fatigue limit of each specimen and not for the whole batch. Figure 12 shows the effect of the fatigue active volume (FAV) on the local fatigue strength amplitude for 𝐾 𝑡 𝜎 𝐷 50% probability of failure for a stress ratio of R=0.1 at 2×10 6 cycles. For both alloys, the local fatigue strength amplitude decreases with increasing FAV, then stabilizes for each alloy at a specific FAV. The notched specimens show the highest strengths in terms of the local stress amplitude.

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Following these observations, it can be concluded that:

-the alloy A has a more pronounced volume effect than alloy B. This is probably due to the characteristics of the defect population (i.e. pore size, distance between large pores, etc.…).

-for alloy A, the fatigue strength stabilizes from a specific volume of 375 mm 3 . This could be considered as the Fatigue Representative Elementary Volume (FREV).

-for alloy B, the fatigue strength stabilizes rapidly at a specific volume of 110 mm 3 , which can be considered as the FREV. Figure 13 shows the same data as Figure 12 but in terms of the nominal stress. It can be seen that for alloy B the presence of the notch in the B-VN specimens dramatically decreases the fatigue resistance as expected. However, for alloy A, the introduction of the notch caused the increase of the fatigue strength of the specimen A-VN compared to the specimen A-V2. This seems to indicate that for alloy A the volume effect is dominant compared to the notch effect. For four notched specimens of alloy B, the fatigue failure surfaces show two initiation sites (Figure 15). This reflects the higher density of large pores in alloy B even with a small Fatigue Active Volume. The size of the critical defects for the batch B-VN is summarized in Table 4. For alloy A, which has a lower pore density, other crack initiation mechanisms were observed such as initiation from oxides, Persistent Sleep Band (PSB) and even the combination of the two mechanisms (see Figure 16, Figure 17 and Figure 18). However, the dominant fatigue damage mechanism is crack initiation and growth from pores. Note that these damage mechanisms are fully consistent with those observed by Le [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF] on specimen V2 under a load ratio R=-1. The statistics of crack initiation sites in these alloys are summarized in Table 5.

A-V1 29 Figure 19 shows the evolution of the critical defect sizes expressed in terms of ( ) as a function of the 𝐴𝑟𝑒𝑎 fatigue active volume for both alloys. The mean value of critical defect size in alloy B are larger than 300 µm. On the other hand, for alloy A, the mean value of critical defect size are less than 210 µm. The two curves show the same tendencies. For a small FAV, the average critical defect size is low. It starts to increase and then stabilizes for a FAV of approximately 110 mm 3 for alloy B and a FAV 375 mm 3 for alloy A. The average of critical defect size is 4 times for the notched specimens A-VN when compared to the stabilized volume. Nevertheless, the alloy B does not show the same level of increase. The difference in the average of critical defect size between the notched specimen B-VN and the stabilized volume is only 50 µm.

The postulate made by Weibull [START_REF] Weibull | The phenomenon of rupture in solids[END_REF], i.e. the size effect is explained by an increase in the probability of encountering a large defect in large volume, seems to apply in alloy A but not in alloy B. The size effect in alloy A is very noticeable in contrast to what happens in alloy B. The following sections give further details on the quantification of the magnitude of size effect in these alloys.

First quantitative approach of size effect

In this section, we aim to quantify the magnitude of size effect present in the studied alloys. The idea is to measure the difference between the means of the batches V1 and V2 in terms of defects and stresses. In statistics, Cohen's d is a parameter that quantifies the difference between the means of two populations divided by their standard deviation as a measure to represent the magnitude of mean difference between them. Cohen [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF] was the first to introduce this parameter. However, its accuracy relies on two assumptions, the Normality of the distribution and the Homogeneity of variances. In practice, two groups of samples do not necessarily have the same standard deviation. Therefore, to correct the assumption on the Homogeneity of variance, Hedges [START_REF] Hedges | Estimation of effect size from a series of independent experiments[END_REF] proposed to use the pooled standard deviation (Eq. 6). Then, the corrected Cohen's d is defined as the difference between two means divided by the pooled standard deviation (Eq. 7).

J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF 17 (6) 𝑑 = 𝜇 1 -𝜇 2 𝑠 𝑝
Where, µ 1 and µ 2 are the mean values of the populations.

The pooled standard deviation for the non-equal standard deviation is defined as:

(7) 𝑆 𝑝 = (𝑛 1 -1)𝑠 2 1 + (𝑛 2 -1)𝑠 2 2 𝑛 1 + 𝑛 2 -2
Where n 1 and n 2 are the sizes of the populations and s 1 and s 2 are their standard deviations. In Cohen's terminology, an effect with a small magnitude is an effect that you can see only through a careful study (in terms of number of samples, it takes a relatively large number of samples to reveal it). While, an effect with large magnitude is an effect which is big enough that you may be able to see with the naked eye (otherwise, with a relatively small number of samples you can reveal it). The Table 6 contains descriptors for magnitudes of 'd' from 0.01 to 2.0, as initially suggested by Cohen and expanded by Sawilowsky in 2009 [START_REF] Sawilowsky | New Effect Size Rules of Thumb[END_REF]. Using the experimental data, in the following section we quantify the size effect in terms of defect size and fatigue strength in the studied alloys by calculating the Cohen's d of the two populations of the small volume V1 and the standard V2 [Table 7]. We choose those volumes in order to avoid the notch effect. From these results in Table 8, the Alloy A present a large size effect in terms of defects and fatigue strength. In the other hand, alloy B presents a very small size effect in terms of defects and small size effect in terms of fatigue strength.

Statistical Analysis of the critical defect and fatigue strength distributions

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The aim of this section is to determine which typical statistical distribution (Lognormal, Weibull or Gumbel) is best suited to model the experimental data (i.e. both the fatigue strength distribution and the critical defect size distribution). Table 9 shows the evaluated probability functions where f is the probability density function (PDF), and F is the cumulative density function (CDF).

In addition, indicates the standard normal cumulative function.

ɸ(𝑥) = 1 2𝜋 ∫ 𝑥 -∞ 𝑒 - 𝑡 2 2 𝑑𝑡
Log-normal 1 𝜎𝑠 2𝜋 exp ( - (ln(𝜎) -µ) 2 2 𝑠 2 ) ɸ( ln (𝜎) -µ 𝑠 ) Weibull 𝛾 𝛼 ( 𝜎 𝛼 ) 1 -𝛾 𝑒𝑥𝑝 [ -( 𝜎 𝛼 ) 𝛾 ] 1 -𝑒𝑥𝑝 [ -( 𝜎 𝛼 ) 𝛾 ] Gumbel 1 𝛽 𝑒𝑥𝑝 ( -𝜎 -𝜗 𝛽 ) 𝑒𝑥𝑝 [ -𝑒𝑥𝑝 ( -𝜎 -𝜗 𝛽 ) ] 𝑒𝑥𝑝 [ -𝑒𝑥𝑝 ( -𝜎 -𝜗 𝛽 ) ]
where µ is the mean and σ is the Standard deviation of logarithmic values for the lognormal distribution. α is the scale parameter and γ is the shape parameter for the Weibull distribution. υ is the location parameter and β is the scale parameter for Gumbel law.

Statistical tests are also used; the Anderson Darling test (AD) combined with its corresponding P-value and the log-likelihood criterion. The reason for choosing the AD test is that in terms of fatigue we are typically interested in the distributions tails, to which the AD test gives greater importance.

To summarize the use of these criteria: -The best distribution is the one that has the smallest AD value.

-If the corresponding P-value of the AD test is less than , corresponding to the 5% significance level in 𝛼 this case, then the distribution does not show a good-fit to the data.

-The best distribution is the one that has the highest Log-likelihood value. Following these criteria, the distributions have been ranked and then the distribution with the best overall rank is chosen. The overall rank is obtained by summation of the individual ranking. The results are presented in table 10. As shown table 10, the lognormal distribution gives the best approximation for the critical defect size. This conclusion had been made also by YI et al. [START_REF] Yi | Scatter in fatigue life due to effects of porosity in cast A356-T6 aluminum-silicon alloys[END_REF].

Statistics analyses of the defects

Figure 21 shows the experimental data fitted using the lognormal distribution. The parameters of the distribution are obtained using the Maximum Likelihood Estimation method and are summarized in Table 12. 12]. These 𝜇 ≈ 5.8 µm results explain the different sensitivity of alloy A and B fatigue strength to the FAV. In terms of the defect spatial distribution, these results can be explained by the fact that, the largest defects in the alloy A are distributed in a way that when the volume is increased there is a higher probability of finding a large defect. However, for alloy B, the defects are so densely distributed that when increasing the volume of the specimen the probability of finding a large defect does not increase.

Statistics analyses of the fatigue strength

The same procedure was applied to the data obtained for the fatigue strength. Table 11 summarizes the results of the statistical tests. From theses analyses, the lognormal distribution was also chosen as the best distribution for the experimental data in terms of fatigue strength distribution for the different batches. The corresponding parameters are summarized in Table 12.

Cumulated probability function

In this section, the experimental data are plotted on a lognormal probability scale for the defect size and the fatigue strength distributions :the data should follow a straight line if it conforms to the lognormal distribution [START_REF] Koehler | Goodness-of-Fit Tests Based on P-P Probability Plots AU -Gan, F. F[END_REF]. In general, this approach is based on a linearization of the cumulative probability function of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:

(7) 𝑌 = 𝐴 + 𝐵𝑇 Where and are the variables. 𝑌 = ɸ -1 (𝐹) 𝑇 = ln (𝜎) Figure 22 shows lognormal probability paper resulting from this linearized cumulative probability function. Figure 22 shows the distributions of critical defect sizes in plotted on lognormal probability papers. 𝐴𝑟𝑒𝑎 Firstly, it can be seen that the critical defect sizes show good linearity on the lognormal probability paper. This confirms that the statistical analyses above are correct. Similar results were obtained in [START_REF] Yi | Scatter in fatigue life due to effects of porosity in cast A356-T6 aluminum-silicon alloys[END_REF]. Fig. 19, also shows that, for the three different specimen volumes, the critical defect distribution for alloy B are almost superimposed. This confirms the conclusions from section 3.3. For the case of alloy A, increasing the size of critical volume leads to an increase in the critical defect size, until the A-V2 volume at which the defect size stabilizes.

J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF
To sum up, for alloy A, the large defects are rare [see Figure 2]. Therefore, due to the small number of these defects, large defects are unlikely to be found in small specimens (i.e. small critical defects) but are more likely to be found in large specimens. In contrast, alloy B contains a high number of large defects that leads to almost the same probability of finding a large defect in both volumes (B-V1 and B-V2). Figure 23 shows that the fatigue strength follows a lognormal distribution. Especially the A-V1 and B-V1 data. This is due to the continuous variation in the section of the V1 specimen shape. In addition to the conclusions drawn in the previous section.

A-V1 A-V2 A-V3 B-V1 B-V2 A-VN-1 B-VN-1
J O U R N A L P R E -P R O O F JOURNAL PRE-PROOF 22
It is very clear thus that the size effect on HCF strength for alloy A is caused by differences in critical defect sizes between different specimens and their spatial distribution. For alloy B, there is an insignificant difference in terms of critical defect sizes between B-V1 and B-V2.

To sum up, the parameters of the distributions of defects in the material (size and spatial distribution) will control the magnitude of the statistical size effect.

 Higher number of large defects and low inter-pores distance leads to non-significant volume effect (case of alloy B): the volume effect is present but has little effect on the observed variables.

 Low number of large defects and relatively large inter-pores distance leads to strong presence of the volume effect which is facilitated by experimental means (case of alloy A).

Kitagawa-Takahashi diagram: defect size effect

In order to link the fatigue strength of the different specimens to the critical defect size, Kitagawa-Takahashi diagrams have been established. The estimation of fatigue strength for each specimen was carried out using the local stress. Results are shown in Figure 24.

In this work, the "hot spot" stress at the notch tip is used as a parameter to compare the different configurations.

Using classical logic, this implies that Kf is considered equal to Kt, where Kt is the elastic stress concentration factor and Kf is the fatigue notch factor, defined as the ratio of the smooth specimen fatigue strength (reference) to the notched specimen fatigue strength. However, the concept of Kf cannot easily be applied in this work because it has been experimentally shown that the plane specimen fatigue strength is a function of the volume (or the size of the smooth specimen). It is well known that Kf is generally inferior to Kt and that this is often contributed to a "notch effect" which is essentially a combination of the size effect the stress gradient effect and the presence of local plasticity. If the plasticity is considered to be negligible than understanding the notch effect corresponds perfectly to the objectives of this work. Moreover, given that 1<kf<kt, and Kt=1.78 is not very large, it is expected that the difference between Kt and Kf will not be large.

Figure 24 shows the Kitagawa-Takahashi diagram in terms of local stress amplitude as a function of the 𝐾 𝑡 𝜎 𝐷 square root of the critical defect size. The points on the graph are the critical defects observed on the fatigue failure surfaces of the tested specimens of the two alloys. Note that, for the notched specimens of alloy B (with multi-site initiation), only the largest defect is shown in Figure 24.

J O U R N A L P R E -P R O O F
From Figure 24, the following conclusions can be made:

i) The fatigue strength of alloy A tends to decrease when the size of the critical defects increases. Same conclusions were made by [START_REF] Hidalgo | Fatigue life estimation of cast aluminium alloys considering the effect of porosity on initiation and propagation phases[END_REF][START_REF] Tenkamp | Defect-correlated fatigue assessment of A356-T6 aluminum cast alloy using computed tomography based Kitagawa-Takahashi diagrams[END_REF][START_REF] Pessard | Microstructural heterogeneities and fatigue anisotropy of forged steels[END_REF][START_REF] Houria | Influence of casting defect and SDAS on the multiaxial fatigue behaviour of A356-T6 alloy including mean stress effect[END_REF]. Note also that the size of the critical defects increases when the critical volume increases. Indeed, the notched specimens (A-VN) with the lowest critical volume (5 mm 3 ) have the smallest defect sizes. By changing to a FAV of 110 mm 3 for the A-V1 specimens then to 375 mm 3 for the A-V2 specimens, the size of the critical defects has a tendency to increase and the fatigue strength drops.

ii) Between specimens of size A-V2 and A-V3, there is almost no difference in the average value of the fatigue strength as shown in Figure 12 and Figure 24. It can be concluded that the fatigue strength has stabilised, most probably due to the saturation in the maximum critical defect size.

iii) The size of the critical defects in alloy B is relatively large for all types of tested specimens. Their 𝐴𝑟𝑒𝑎 are greater than 220 μm. For this range of defect size, the fatigue strength of the specimens does not change significantly compared to the variation in alloy A.

Using the Kitagawa-Takahashi diagram represented with linear axes Figure 25 gives a better understanding of the small size effect in alloy B. The critical defects are generally at the level where the fatigue strength begins to stabilize. Therefore, and since the size of the critical defects remain relatively large ( greater than 220 𝐴𝑟𝑒𝑎 μm), the difference in the fatigue strength and the scatter remain low. Indeed, when the size goes from 220 μm to 620 μm the local stress amplitude goes from approximately 67 MPa to 60 MPa [see Figure 25]. So, even for the different specimen sizes, the fatigue strength does not significantly change leading to a slight size effect and a low scatter. Figure 26 is a schematic representation of the fatigue strength amplitude as a function of the average critical defect size for volumes A-V1, A-V2, B-V1 and B-V2. It shows that due to the large average critical defect size of the alloy B, The size effect and the scatter are low. In contrast, the average critical defect size of the A-V1 and A-V2 specimens are in the interval in which the curve has a strong slope. A large difference between the fatigue strength of these two volumes and high scatter can hence be expected. A large size effect in terms of fatigue strength is a result of large size effect in the corresponding distributions of critical defect size (which is the case for alloy A). However, a large size effect if terms of critical defects does not lead systematically to a large size effect in terms of fatigue strength, depending on the range of the critical defect size.

Study of the notch effect

As previously discussed, four of the five specimens tested from batch B-VN, have two crack initiation sites (see Figure 15 and Table 4). In this section, the notch effect is investigated for both alloys. Computer Tomography analyses are used to gain insight into the problem.

Post-mortem analyses

As shown in Figure 27, the defect density and the size of the defects in the notched section of the specimen taken from alloy B are much higher in comparison with alloy A. The high defect density in alloy B means that even by decreasing the FAV, there is still a high probability of finding large defects near to the surface of the notch. This explains why the averages of the distributions for alloy B in Figure 21 are close to each other for different FAVs. The opposite is observed for alloy A. By decreasing the FAV, the average size of the critical defect drops (see Figure 19). This leads to the activation of other failure mechanisms such as oxide, inclusion or PSB.

Furthermore, the higher probability of finding many large defects near to the surface of the notch in alloy B can lead to the multiple crack initiation sites. Two crack initiation sites were observed in four out of five notched specimens for batch B-VN. This phenomenon is also linked to the gradient stress effect introduced by the notch.

Crack initiation at the notch

In the following, the fractographic analyses have been matched to the tomographic analyses, the results are shown in Figure 30 and Figure 31. For alloy A, the results show that due to the lack of large defects near the surface, the specimen has failed from an inclusion located at the surface. For alloy B, due to the high density of large pores in this material, the specimen has failed from a pore at the surface. It can be seen that the critical pore is not the largest one contained in the fatigue active volume, as indicated in Figure 29. It is a different pore at the surface of the notch (see Figure 30 (b) and Figure 31).

Conclusions

In this study, an experimental investigation has been conducted to evaluate the statistical size effect (or volume eff ect) on the fatigue strength of cast Al-Si alloys. Statistical analyses of fatigue data were carried out by means of statistical tests (Log-likelihood, AD test and its corresponding P-value). The main results can be summarized as follows:

-The fatigue strength is strongly dependent on microstructure defects in cast aluminium alloys. The most harmful defects are the pores located on specimen surface or subsurface. The pore size is the key factor controlling fatigue strength, and the random distribution of pores can cause great scatter in fatigue strength. -A variation in the highly stressed volume is accompanied by a change in the fatigue strength of the material.

This result confirms the existence of a size effect in cast Al-Si alloys. Furthermore, alloy A has a more

J O U R N A L P R E -P R O O F
pronounced volume effect than alloy B. This is due to the characteristics of the defect population of each alloy (pore size and distance between large pores). -Good correlation between the local fatigue strength amplitude and the critical volume was found. Hence, 𝐾 𝑡 𝜎 𝐷 it can be concluded that the highly stressed volume approach is an effective way to describe the statistical size effect in cast Al-Si alloys.

-Both critical defects size and the fatigue strength follow a lognormal distribution. For the fatigue strength, this has consequences for fatigue design. Some works are based on the idea that the fatigue resistance follows a normal distribution [START_REF] Thomas | Fatigue design and reliability in the automotive industry[END_REF]. Which is not the case for Al-Si cast alloys. -The fatigue strength for alloy A stabilizes from a critical volume of 375 mm 3 , which can be considered as the Fatigue Representative Elementary Volume. -The fatigue strength of alloy B stabilizes at a smaller critical volume of 110 mm 3 which can be considered as the fatigue REV of this alloy. -The alloys do not show the same fatigue damage mechanisms. For the alloy B, all the specimens have shown fracture initiated from a pore. However, the alloy A shows several mechanisms of crack initiation (pores, oxide, PSB). Nevertheless, the initiation from the pore is the predominant mechanisms and the percentage of the other mechanisms increases by decreasing the FAV size. -Size effect and scatter of fatigue are strongly linked to the average size of the critical defects. If the average size of the critical defect is lower than 220 µm, the alloy will show high size effect and high scatter, which is the case of alloy A. If it is higher than 220 µm the alloy will show low size effect and low scatter, that is the case of alloy B. -Density of pores plays a role also in the fatigue size effect and scatter; more presence of large defects and low inter-pores distance (high-density) leads to non-significant volume effect (case of alloy B). On the other hand, less presence of large defects and relatively large inter-pores distance (low-density) leads to strong presence of the volume effect (case of alloy A).

A modelling work coupling the defect size and position distributions and the loaded volume is in progress and will be presented in a next paper.  High density and large size defects lead to minor size effect and less scatter.

 Low density and small size defects induce large size effect and high scatter.

Figure 1 :

 1 Figure 1: The microstructure of the investigated alloys (a) Alloy A (AlSi7Cu05Mg03 -T7) (b) Alloy B (AlSi7Mg03 -T7) and (c) Grain map showing grains for alloy A with 337 ± 143 µm. 𝑨𝒓𝒆𝒂 (𝒈𝒓𝒂𝒊𝒏) =

Figure 2 :

 2 Figure 2: Micro-tomography scans of the alloys (a) A alloy and (b) B alloy

Figure 3 :

 3 Figure 3: Distributions of the defect size of the pores from the scanned specimens in Fig. 2

Figure 4 :

 4 Figure 4: Defect size as a function of the sphericity of the defects

Figure 5 :

 5 Figure 5: Specimens for tensile fatigue tests

Figure 6 :Figure 7 :

 67 Figure 6: The definition of the depth of a critical defect

Figure 8 :

 8 Figure 8: Fatigue crack localization in the hour-glass shaped V1 specimens

Figure 10 :

 10 Figure 10: Highly Stressed Volume for (a) smooth specimens V1, V2 and V3 and (b) notched specimens

Figure 11 :

 11 Figure 11: SN-Curves plotted in semi-Linear diagram for the batches A-V1, A-V2, B-V1 and B-V2

Figure 12 :

 12 Figure 12: Local stress amplitude at fatigue limit as function of critical volume for alloys A and B

Figure 13 :

 13 Figure 13: Nominal stress amplitude as function of critical volume for alloys A and B

13 3. 3 Figure 14 :Figure 15 :

 1331415 Figure 14: Fatigue failure, specimen B-V1-05, Sa= 55 MPa, Nf=919402 cycles, (a) Macroscopic view, detailed view of (b) surface pore and (c) internal poreFigure14highlights the importance in fatigue of the free surface[START_REF] Serrano-Munoz | Location, location & size: defects close to surfaces dominate fatigue crack initiation[END_REF]. For this specimen, a very large defect can be seen in the middle of the specimen ( =575 µm) and a smaller one is visible near the surface ( 𝐴𝑟𝑒𝑎 =367 µm). The SEM observations clearly show that the smaller pore close to the surface is responsible 𝐴𝑟𝑒𝑎 for the fatigue failure.

•

  Mécanisme : Oxide at the surface + PSB •

Figure 16 :Figure 17 :Figure 18 :Table 5 :Figure 19 :

 161718519 Figure 16: Crack initiation site with the combination of two mechanisms, specimen A-V1-29, Sa=75 MPa, Nf=328135 cycles, crack initiation from an oxide at the surface and persistent slip band (PSB)

Figure 20 :

 20 Figure 20: Illustration of two distributions

Figure 21 :

 21 Figure 21: Cumulative probabilities functions of the critical defects for the different tested volumes (a) for alloy A and (b) for alloy B From Figure 21 (a) it can be seen that (i) for alloy A, the size of critical defects is increased drastically by increasing the size of the FAV. However, for alloy B in Figure 21 (b), the mean values of the distributions for the three different volumes (B-VN, B-V1 and B-V2) are almost the same ( ) [see Table12]. These 𝜇 ≈ 5.8 µm results explain the different sensitivity of alloy A and B fatigue strength to the FAV. In terms of the defect spatial distribution, these results can be explained by the fact that, the largest defects in the alloy A are distributed in a way that when the volume is increased there is a higher probability of finding a large defect. However, for alloy B, the defects are so densely distributed that when increasing the volume of the specimen the probability of finding a large defect does not increase.

21

 21

Figure 22 :

 22 Figure 22: Critical defect sizes in √area plotted on lognormal probability papers

Figure 23 :

 23 Figure 23: Fatigue strength in MPa plotted on lognormal probability papers

1 Figure 24 :

 124 Figure 24: Kitagawa-Takahashi diagram (R=0.1)

Figure 25 :

 25 Figure 25: Local fatigue resistance as a function of critical defect size (R=0.1)

Figure 26 :

 26 Figure 26: Schematic representation of fatigue resistance as a function of the means of critical defect size for each batch

Figure 27 :Figure 28 :Figure 29 :

 272829 Figure 27: CT scans of the notched section of (a) specimen A-VN-05, and (b) specimen B-VN-05 Figure 28 shows the location of defects in the notched specimens A-VN-05 and B-VN-05 as a function of their sizes in terms of . 𝐴𝑟𝑒𝑎 𝑒𝑞

Figure 30 :

 30 Figure 30: Fatigue failure, (a) specimen A-VN-05, Sa=70 MPa, Nf=1584002 cycles, crack initiation from an inclusion with of 27 µm (b) specimen B-VN-05, Sa=45 MPa, Nf=777402 cycles and crack initiation from a pore at the 𝑨𝒓𝒆𝒂 surface with of 313 µm 𝑨𝒓𝒆𝒂

Figure 31 :

 31 Figure 31: specimen B-VN-05 (a) tomographic image and (b) its corresponding fractographic image

Groupe

  PSA and the French National Agency for Research and Technology (ANRT) financially supported this work.

32 Highlights

 32 Effects of porosity on size effect and scatter of fatigue strength are investigated.  Two alloys with different degrees of porosity are used.  Magnitude of size effect and scatter are linked to the population defects.

Table 1 :

 1 Properties of the investigated cast Al-Si alloys

		Grade			Alloy A	Alloy B
	Designation					AlSi7Cu05Mg03 -T7	AlSi7Mg03 -T7
	Chemical composition (wt %)	7% Si, 0.3% Mg, 0.5% Cu, Al Rem.	7% Si, 0.3% Mg, Al Rem.
	Casting Process				Gravity Die	Lost Foam
	Heat treatment				T7	T7
	SDAS (µm)					42±10	77±19
	Young Modulus E (GPa)			77±6 [16]	68±5 [16]
	Yield stress	𝜎 𝑌0.2%	(MPa)		260±2	240±5
	Ultimate tensile strength	𝜎 𝑢	(MPa)	304±4	251±6
	Elongation A (%)				4.7±1.2	0.8±0.1
	Porosity (%)				0.03	0.28
	Micro-hardness (Hv25Gr)		115	100

Table 2 :

 2 Characteristics of the defect distributions from scanned volumes of 363 mm3

		Alloy A [Nb of pores = 454]	Alloy B [Nb of pores = 1569]
		𝐴𝑟𝑒𝑎 𝑒𝑞 [µm]	Sphericity	Nearest Neighbor [µm]	𝐴𝑟𝑒𝑎 𝑒𝑞 [µm]	Sphericity	Nearest Neighbor [µm]
	Min	20	0.48	60	20	0.17	40
	Max	170	0.99	1200	300	1.00	1200
	Mean	40	0.85	350	60	0.71	160
	Median	35	0.87	180	40	0.76	120
	Standard deviation	20	0.09	260	50	0.20	130
	Coefficient of variation	0.43	0.09	0.83	0.83	0.24	0.86

Table 3 :

 3 Characteristics of the samples

	Specimen	Number of specimens Alloy A Alloy B	Kt	FAV A in mm 3 for the sublayer of 500 µm	FAV B in mm 3 for the sublayer of 650 µm
	VN	5	5	1.78	5	6
	V1	47	11	1	90	110
	V2	46	11	1	375	450
	V3	12	0	1	1020	-

Table 4 :

 4 Critical defects for notched specimens of the alloy B

	JOURNAL PRE-PROOF
	Specimen	√Area : Site 1 [µm]	√Area : Site 2 [µm]
	B-VN 01	176	248
	B-VN 02	243	333
	B-VN 03	204	320
	B-VN 04	176	330
	B-VN 05	313	-
			F
			O O
		E -P R
	P R
	A L
	J O U R N
		14

Table 6 :

 6 descriptors for magnitudes of d = 0.01 to 2.0, as initially suggested by Cohen and expanded by Sawilowsky

	Magnitude of the effect	d	Reference
	Very small	0.01	Sawilowsky, 2009
	Small	0.20	Cohen, 1988
	Medium	0.50	Cohen, 1988
	Large	0.80	Cohen, 1988
	Very large	1.20	Sawilowsky, 2009
	Huge	2.0	Sawilowsky, 2009

Table 7 :

 7 Summarizes the results in terms of the defect distributions of the standard volumes

			Defect [µm]		Fatigue strength [MPa]
		A-V1 A-V2 B-V1 B-V2 A-V1 A-V2 B-V1 B-V2
	Mean	126	221	352	348	73	65	51	53
	Standard Deviation	53	138	100	94	9	8	5	3
	Pooled Standard Deviation	101		97		9		4

Table 8 :

 8

	Cohen "d" calculation
			Cohen "d"
		Defect	Fatigue strength
	Alloy A	0.93	0.97
	Alloy B	0.04	0.45

Table 9 :

 9 evaluated probability functions

	Distributions	𝒇(𝝈)	𝑭(𝝈)

Table 10 :

 10 Summary of the Goodness of Fit (GOF) statistics for different distribution models for the critical defects

	Batch	Number of specimens	Tests	Lognormal	Gumbel	Weibull
			P-value	0.58	0.01	0.17
	A-V1	40	AD Log-likelihood	0.31 -222	1.63 -232	0.54 -224
			Rank	1	3	2
			P-value	0.40	0.01	0.41
	A-V2	37	AD Log-likelihood	0.38 -207	1.12 -212	0.38 -207
			Rank	1	3	2
			P-value	0.74	0.68	0.90
	A-V3	12	AD Log-likelihood	0.24 -64	0.28 -65	0.20 -64
			Rank	2	3	1
			P-value	0.78	0.04	0.32
	B-V1	11	AD	0.23	0.78	0.43
			Log-likelihood	-59	-62	-61

Table 11 :

 11 comparative analyses of the fitting procedure for statistical distribution models of fatigue strength

	Batch	Sample size	Creteria	Lognormal	Gumbel	Weibull
			P-value	0.22	0.08	0.20
	A-V1	47	AD Log-likelihood	0.49 -147	0.66 -150	0.51 -148
			Rank	1	3	2
			P-value	0.01	0.14	0.09
	A-V2	46	AD Log-likelihood	1.05 -125	0.57 -124	0.65 -124
			Rank	3	1	2
			P-value	0.40	0.01	0.04
	A-V3	11	AD	0.36	0.93	0.76
			Log-likelihood	-35	-38	-37

Table 12 :

 12 summary of the parameters of the identified distributions for 95% confidence level

	Batch	Lognormal		Defect Weibull		Gumbel	Lognormal	Stress Weibull		Gumbel
	A-V1	µ s	4.75 0.43	α γ	143 2.60	υ β	54 56.86	µ s	4.29 0.11	α γ	77 9.90	υ β	77 7.75
	A-V2	µ s	5.21 0.48	α γ	230 2.43	υ β	250 93.85	µ s	4.16 0.12	α γ	68 10.13	υ β	68 6.48
	A-V3	µ s	5.14 0.50	α γ	214 2.58	υ β	230 75.75	µ s	4.12 0.10	α γ	65 7.22	υ β	65 7.22
	B-V1	µ s	5.83 0.28	α γ	389 3.58	υ β	406 117.53	µ s	3.91 0.10	α γ	52 11.28	υ β	53 4.63
	B-V2	µ s	5.82 0.27	α γ	382 4.22	υ β	393 90.15	µ s	3.93 0.06	α γ	52 2.08	υ β	52 2.08


Mahshad Mahdavi

Richard Zanibbi

Harold Mouchère
email: harold.mouchere@ls2n.frchristian.viard-gaudin@ls2n.fr

Christian Viard-Gaudin

Utpal Garain

ICDAR 2019 CROHME + TFD: Competition on Recognition of Handwritten Mathematical Expressions and Typeset Formula Detection

Keywords: mathematical expression recognition, handwriting recognition, formula detection, performance evaluation

) teams submitting results. The strongest formula recognition results were produced by the USTC-iFLYTEK research team, for both stroke-based (81%) and image-based (77%) input. For the new typeset formula detection task, the Samsung R&D Institute Ukraine (Team 2) obtained a very strong F-score (93%). System performance has improved since the last CROHME -still, the competition results suggest that recognition of handwritten formulae remains a difficult structural pattern recognition task.

I. INTRODUCTION

During its history, the CROHME competition has advanced the state-of-the-art for handwritten math recognition systems, and produced a standard benchmark for online handwritten math recognition research. An IJDAR paper summarizing the outcomes and innovations in evaluating handwritten mathematical recognition during the first four years of the competition (2011)(2012)(2013)(2014)2016) is available [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF], [START_REF] Mouchère | Icfhr2016 competition on recognition of handwritten mathematical expressions (crohme2016)[END_REF]. CROHME data has been used by research groups from around the world.

This sixth edition of CROHME was organized to continue encouraging activities in handwritten math recognition research, and to improve the available data, tools and benchmarks for research in this area. In CROHME 2019, there are three tasks: 1) online handwritten formula recognition (from strokes), 2) offline handwritten formula recognition (from images generated using strokes), and 3) typeset formula detection in document images.

To accommodate the growing number of encoder-decoder formula recognition systems producing L A T E X as output, in this edition we consider only symbolic formula structure, rather than in previous CROHMEs where systems were evaluated at the stroke level. For this purpose, we developed a new Symbolic Label Graph (symLG) representation that can be used with the existing evaluation tools for the competition (LgEval and CROHMELib).

Thirty-three (33) groups and individuals registered for the competition, and nineteen (19) groups submitted results. For the main task (online handwritten formula recognition), the highest recognition rate is 13% higher than in CROHME 2016 (67.65% vs. 80.73%). While encouraging, these rates suggest that recognizing handwritten math remains a difficult problem, likely due to the high number of symbol classses (101 for CROHME), and the complex two dimensional structure of math, which can include recursive and hierarchical structures.

In the following Sections we describe the competition tasks, dataset collection and encodings, evaluation metrics and tools, system descriptions, results, and then provide a brief conclusion.

II. TASKS Task 1. Online Handwritten Formula Recognition. For the main task in CROHME, participants convert a list of handwritten strokes captured from a tablet or similar device to a Symbol Layout Tree (SLT) [START_REF] Zanibbi | Recognition and retrieval of mathematical expressions[END_REF]. Participating systems are ranked based on the number of correctly recognized formulas (expression rate). to render images. Participants then convert these handwritten formula images to a Symbol Layout Tree. For evaluation, the same evaluation tools are used as for Task 1. Again, participants are ranked by the expression rate of their system.

• Task 2a (symbols): subtask where participants recognize isolated symbols, including 'junk' (invalid symbols).

Ranked by symbol recognition rate. • Task 2b (parsing from provided symbols): subtask where participants parse formulas from provided symbols (bounding boxes + labels). Ranked by expression rate. Task 3. Detection of Formulas in Document Pages. Given a document page along with the bounding boxes of characters on that page (as are available for born-digital PDF files), participating systems identify formulas using bounding boxes. Evaluation is performed using Intersection-over-Union (IoU, or equivalently the Jaccard similarity coefficient), and systems are ranked based on their F-measure after matching output formula boxes to ground truth formula regions.

III. DATASETS AND FORMULA ENCODINGS

In this Section we describe data used in the competition, how it was collected, and the encodings used. Table I summarizes the datasets used for the competition.

Handwritten Formulas: Input Data. For Task 1, we use online data in the same InkML and Label Graph (LG) file formats from previous CROHMEs. Strokes are defined by lists of (x,y) coordinates, representing sampled points as a stroke is written. Groupings of strokes into symbols, symbol labels, and formula structure are provided in both the InkML and LG formats. In InkML structure is represented using Presentation MathML (an XML-based representation), while in LG a simpler CSV-based representation is used. In both cases, formula structure is represented by a Symbol Layout Tree, as seen in Figure 1(b). Roughly speaking, this format represents the appearance of a formula by the placement of symbols on the different writing lines of the expression. Spatial relationships between symbols (e.g, 'R' for adjacent-at-right) are indicated using edge labels.

For Task 2, the offline formula data is provided as greyscale images. These were rendered automatically from the online data with 1000 × 1000 pixels with 5 pixels of padding. This format is used for the main task (Task 2) and Task 2b. For the isolated symbol sub-task, isolated symbols are rendered at 28 × 28 pixels with the same amount of padding (5 pixels). The resolution of the inputs files is fixed for Task 2, but participants were welcome to resize the original images using pre-processing methods of their choice.

Symbol Layout Graph (symLG) Formula Representation. The stroke-based LG files used in previous CROHMEs allow all segmentation, classification, and structural errors to be identified unambiguously, even when segmentations disagree [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF]. However, with the success of encoder-decoderbased systems that generate L A T E X output, a new representation is needed -these systems do not output information about stroke segmentation or the location of symbols in the input, instead producing Symbol Layout Trees directly.

Figure 1 shows two graph representations for the same expression '2+3 c .' In the Stroke Label Graph (LG), there are 5 nodes (one per stroke), and edges represent segmentation and spatial relationships between pairs of strokes (including 'no relationship'). For the Stroke Label Graph, node identifiers are for individual strokes. In the Symbol Label Graph (symLG), there are 4 nodes (one per symbol) and edges represent relationships between symbols (no segmentation information is provided). For symLG, node identifiers are constructed from the sequence of relation labels on the path from root to the symbol. For example, 'c' in Figure 1 has the identifier 'oRRSup' (origin/root, Right, Right, Superscript).

To compute the similarity of two Symbol Layout Trees in our symLG representation, we use an adjacency matrix. Labels on the diagonal define symbol labels, while off-diagonal elements represent spatial relationships between parent and child symbols. Using this representation, we can determine how formulas in an SLT representation differ in structure and symbol labels, but not the correspondence between symbols and relationships in a symLG file and the input data (i.e., strokes/images). Still, the symLG representation allows existing metrics and tools designed for evaluation of strokelevel LG files at the symbol level to be used directly. We note that symLG is closely related to the tree-based symbolic representation from earlier CROHME competitions [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF], but permits more detailed error analysis.

Formula Data Collection. We used the labeled handwritten formulae from previous CROHMEs that are publicly available

(s1) + (s2) + (s3) 3 (s4) c (s5) R R + + R R Sup 2 (o) + (oR) 3 (oRR) c (oRRSup) R R Sup (a) Stroke Label Graph (LG) (b) Symbol Label Graph (symLG)
Fig. 1: Different graph representations for formula '2 + 3 c ' written using 5 strokes. Node identifiers are shown in brackets.

[2] to generate training and validation sets for the formula recognition tasks. The new formula test set was produced by the organizers at the University of Nantes and the Indian Statistical Institute (ISI). Three input devices were used, with different sampling and scaling: TabletPC with pen-based touch screen, a whiteboard, and a graphic tablet. We used arXiv papers from 2002 and 2003 made available through the KDD 2003 Cup [START_REF] Gehrke | Overview of the 2003 KDD cup[END_REF]. A corpus of 1207 expressions was selected, according to the same expression grammar and symbol frequency constraints used to create the CROHME 2016 Test set using the algorithm described in [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF]. 1200 expressions were produced by 80 writers (15 expressions each), with only one expression being discarded. The data was annotated using a semi-automatic process [START_REF] Mouchère | Icfhr2016 competition on recognition of handwritten mathematical expressions (crohme2016)[END_REF]. Some expressions were cleaned, removing noisy strokes or splitting connected symbols, and sometimes the ground-truth L A T E X string was updated to fit what was actually written.

We Formula Detection Data. For typeset formula detection, we modified the GTDB-dataset 1 created by the infty group (Masakazu Suzuki et al. [START_REF] Suzuki | Infty: an integrated ocr system for mathematical documents[END_REF]). Documents in the collection are PDFs taken from scientific journals and textbooks with different font faces and notation styles. GTDB provides groundtruth at the character and formula level (including SLTs) in CSV format, with carefully annotated displayed (offset) formulas and formulas within text lines (inline formulas). Nearly 25% of the annotated formulas are single symbols (e.g., λ). The organizers at RIT modified the ground-truth data to compensate for differences in scale and translation found in the publicly available versions of PDFs in the collection. Details may be found in Table I. A script was provided to compile and render the competition PDFs at 600 dpi. In addition to the document images, character bounding boxes were provided; in the training set, character labels, and indication of whether a character belongs to a formula region were also provided. 1 https://github.com/uchidalab/GTDB-Dataset IV. EVALUATION METRICS AND TOOLS Submission. Beginning with CROHME 2016, we have used an online submission and evaluation tool2 implemented in Django. Multiple submissions were permitted for each team, with the highest recognition rates used to rank systems. After each submission, the performance metrics were computed, and a leaderboard visible to all participants was updated in real time. For the Test sets, only recognition rates/F-scores were visible to participants.

Tools. Updated versions of the CROHMELib and LgEval libraries used for the competition were made available to participants offline. These tools support converting between LG and InkML, and computing performance metrics [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF]. For formula detection, we developed a tool that ranked based on Intersection-over-Union (IoU). We also provided tools to let participants convert their outputs from MathML, L A T E X,or LG into SymLG.

Formula Recognition Metrics. LgEval metrics include formula and symbol recognition rates, along with recall and precision metrics for detection (segmentation) and detection + classification for both symbols and relationships [START_REF] Mouchère | Advancing the state of the art for handwritten math recognition: the CROHME competitions, 2011-2014[END_REF]. Recognition systems that produce L A T E X strings or stroke-level Label Graphs (LGs) both have their outputs converted to symLG (see Figure 1). This allows systems producing stroke-level and symbol-level results to be compared directly, albeit with a loss in the stroke segmentation information provided in the strokelevel representation. The symLG representation allows us to identify errors as symbol classification errors, relationship classification errors, or structure errors.

Note that this change in representation has several impacts on the results; in particular, segmentation information cannot be calculated -it is possible for a formula to be recognized with the correct Symbol Layout Tree, but without correctly segmenting symbols. This makes the expression rate less strict than at the stroke-level. Also, because labels on relationship paths identify symbols, when symbols do not appear at expected locations in the output they are treated as missing ('ABSENT' in LgEval), which leads to an underestimate of symbol recall.

Recognition rates were used to rank systems for Tasks 1, 2, and their subtasks.

Formula Detection Metric. We use an F-score based on matches with an IoU ≥ 0.75 to rank systems. For each ground truth box in a document page, if there is a detection box that has an IoU above the given threshold, the GT box is considered to be detected correctly. We enforce a one-to-one mapping, where each output bounding box is assigned to the GT bounding box with the highest IoU score, with each GT box matched to at most one output bounding box. We used an implementation by Padilla3 to calculate intersections for each pair of boxes. We then calculate precision, recall and Fscores. In the offline tool, all boxes missed in the output are automatically listed with the corresponding IoU score for error analysis. For each system, we calculate both a coarse detection score (IoU ≥ 0.5) and a fine detection score (IoU ≥ 0.75).

V. PARTICIPATING METHODS

The system descriptions in this Section are provided by the CROHME 2019 participants (note: some are missing). Systems are organized according by their order of appearance in Tables II and IV. Nineteen teams submitted results for CROHME 2019, from which 9 teams participated in Task 1 and its sub tasks, 8 teams participants in Task 2 and its sub tasks, and finally 4 teams participated in Task 3. Tasks participated in are indicated in square brackets before each team name.

[1, 2] USTC-NELSLIP and iFLYTEK Research. We utilize attention based encoder-decoder models: an RNN-based encoder for online formula recognition [START_REF] Zhang | Track, attend, and parse (tap): An end-to-end framework for online handwritten mathematical expression recognition[END_REF], and a CNN-based encoder for offline formula recognition [START_REF] Zhang | Watch, attend and parse: An end-to-end neural network based approach to handwritten mathematical expression recognition[END_REF]. We strengthen the online encoder by using a CNN-RNN architecture, and strengthen the offline encoder using an enhanced DenseNet architecture. For Task 1 we combine the online and offline recognition models. For Task 2, we preprocess the images by first extracting the image foreground, resize it using the estimated average height of each symbol based on information in original images. We train using only the official training dataset, and employ a data augmentation method [START_REF] Le | Pattern generation strategies for improving recognition of handwritten mathematical expressions[END_REF] to alleviate the problem of limited training data. Finally, we use the text dataset provided in the NTCIR-12 MathIR dataset [START_REF] Zanibbi | NTCIR-12 MathIR task overview[END_REF] to train a single layer RNN-based language model, and combine it with our encoder-decoder models [START_REF] Zhang | Track, attend, and parse (tap): An end-to-end framework for online handwritten mathematical expression recognition[END_REF].

[1, 1b] Samsung R&D Institute Ukraine -Team 1. This system is a deep-learning solution that employs multi-task recurrent neural networks (RNNs) for character recognition in combination with preprocessing and re-ordering, and a Probablistic Context-Free Grammar (PCFG) in combination with spatial relation analysis and language models. For Task 1b, we used a PCFG in combination with spatial relation analysis and language models. In order to cover the complex 2D structure of Handwritten Math equations, we have introduced two different bigram language models: language sequence model and language relation model. For parsing, we introduced heuristics into bottom-up parsing and the dynamic programming-based CYK parsing algorithm. The first heuristic exploits the concepts of dominance and mutual visibility for character candidates. The second heuristic applies dynamic pruning to the search area depending upon detected character candidates. We used an additional dataset to train our system for both tasks [START_REF] Zhelezniakov | Acceleration of online recognition of 2d sequences using deep bidirectional lstm and dynamic programming[END_REF].

[1] MyScript. The MyScript Math recognizer system is built on the principle that segmentation, recognition, and interpretation have to be handled concurrently and at the same level in order to produce the best candidates. The recognition engine analyzes the spatial relationships between all parts of the equation, in conformity with rules laid down in its grammar, to determine the segmentation of all of its parts. The grammar is defined by a set of rules describing how to parse an equation, each rule being associated with a specific spatial relationship. The symbol recognizer combines Neural Nets evaluating hypotheses with static and dynamic characteristics, and a BLSTM trained with the CTC algorithm, which processes the ink signal directly. The recognition also combines statistical and LSTM-based language models, to evaluate the contextual probabilities of symbols in the equation. This recognizer was trained specifically on the data provided for the online formulas recognition task (CROHME Train 2014, Test 2012, Test 2013). We also built a second system, by combining this specific recognizer with our public Math recognition engine, trained on more generic data. The combination improves the accuracy by solving some ambiguous cases.

[1, 2] Sun Yat-Sen University. This system recognizes offline formulas by extracting strokes from the input image and then performing online recognition. First, input images are re-scaled to 1500x1500 and binarized using Sauvola's method. Second, a thinned skeletonization of the image is decomposed into junctions and segments, with segments that are too short being discarded. Third, segments pairs sharing a junction with a minimal difference in direction are linked repeatedly, after which double-traced strokes are fixed using heuristic rules. Fourth, strokes are sorted using a recursive X-Y cut and topological sort. Finally, after extracted strokes are recognized by MyScript Interactive Ink (version 1.3) using a customized grammar and DPI of 576, exported MathMLs are converted to CROHME's conventions and then converted to symLGs [START_REF] Chan | Stroke extraction for offline handwritten mathematical expression recognition[END_REF].

[1, 3] Samsung R&D Institute Ukraine -Team 2. Our formula detector is based on graph-theoretic methods for determining the position of multi-character formulae, in combination with statistical and context-recognition-based approaches for detecting single-character mathematical symbols inside text [START_REF] Degtyarenko | Text/shape classifier for mobile applications with handwriting input[END_REF].

[1, 2] CASIA/NLPR PAL Group We entered two systems: PAL, and PAL-v2 which extends our previous work [START_REF] Wu | Imageto-markup generation via paired adversarial learning[END_REF]. The attention-based encoder-decoder model in PAL-v2 is trained using official data only. Training data is rendered as 1000x1000 offline images using the official tool, after which we extract the image foreground and normalize the height to 64 pixels, using dilation/erosion to thicken the strokes. We augment the training data set using rotations, perspective shift, distortion, and bevel, as well as the decomposition operation introduced by Le et al. [START_REF] Le | Pattern generation strategies for improving recognition of handwritten mathematical expressions[END_REF]. This expanded the training data to 330k images, which are then used for Paired Adversarial Learning [START_REF] Wu | Imageto-markup generation via paired adversarial learning[END_REF]. An ensemble of 6 models with different initializations produced the PAL-V2 results. We did not use an additional language model to assist with recognition. The PAL architecture for Task 2 is the same as PAL-v2, but is trained on both the augmented training set and extra data without Paired Adversarial Learning. The extra data was obtained using an additional 70k formulas from the Wikipedia formula corpus, from which we generated 120k formula images. These 120k Wiki formulas were combined with the original training data, and then augmented to produce 3 times more data using the procedure described above.

[1] MathType. The system is composed of a convolutional encoder and an attention-based LSTM decoder. The online input coordinates are encoded as an image, where additional channels account for trajectory information. It is a model designed to be fast and use low memory for deployment in realtime scenarios. We trained the system using the competition dataset and also with additional private data.

[1, 2] Tokyo University of Agriculture and Technology. Task 1: The system includes three main tasks: symbol segmentation, symbol recognition, and structure recognition. The symbol recognizer is a combination of LSTM models extractfeatures for both online and offline patterns (converted from online patterns). We improved the structure recognition system and added grammar rules from the version that participated in CROHME 2016. Additional detail of our system are presented elsewhere [START_REF] Le | A system for recognizing online handwritten mathematical expressions by using improved structural analysis[END_REF]. Task2: For offline handwritten mathematical expression recognition, we employ an attentionbased residual sequence-to-sequence (ARseq2seq) model [START_REF] Ly | An attention-based end-toend model for multiple text lines recognition in japanese historical documents[END_REF]. Our ARseq2seq model consists of 3 parts: 1) feature extraction by CNN, 2) encoding by BLSTM, and 3) decoding by LSTM with an attention mechanism. We also introduce using the residual connection in the decoder. Details of the Arseq2seq model are presented in Ly et al. [START_REF] Ly | An attention-based end-toend model for multiple text lines recognition in japanese historical documents[END_REF]. The input images are cropped and resized into a 192 × 192 image.

[2, 2b] University of Linz. This system is composed of two subsystems: the first is the symbol detector, and the second is the symbol parser. The symbol detector is a Faster RCNN object detection model using an Inception Resnet (v2) convolutional neural network as the backbone. The symbol parser is also implemented using a deep neural network, based on the Transformer model developed by Vaswani et al.,where instead of positional embeddings, we use bounding box embeddings.

[3] Rochester Institute of Technology -Team 1. We treat formula detection as an object recognition problem. A modified YOLOv3 model [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] containing a 53-layer CNN (Darknet-53) is used to define a 416 × 416 detection window, which we then pass over the large document image using a 10% stride at train and test time. A pixel-level confidence map for the document is obtained by averaging detection box confidences across window locations (i.e., average pooling). Detected formula bounding boxes are extracted from the document confidence map by thresholding followed by connected component extraction. A post-processing step is used to refine detections: math regions are clipped/grown to fit all connectedcomponents lying within a detected formula region. The model performs well for displayed formulas in particular, with some under-segmentation for formulas spanning multiple text-lines (e.g., in derivations).

[3] Rochester Institute of Technology -Team 2. We detect formulas using a single shot multi-box detector (SSD512) with two major modifications. First, we use wider default boxes in addition to default boxes from the original SSD architecture. Wider default boxes help us detect wide formulas. Second, we use focal loss instead of cross-entropy loss for training. Adding wide default boxes increased the number of default boxes to over 45k, which in turn increases the imbalance between positive and negative examples. Therefore, we use the focal loss to give increased weight to ambiguous examples near decision boundaries. We use a sliding window approach, splitting the original 600dpi image into images of size 512×512 using a 10% stride. Finally, detections obtained from SSD are used in a voting-based pooling strategy to get math bounding boxes at the scale of the original image.

VI. RESULTS

In this Section we summarize the results of the competition. No teams participated in the symbol recognition tasks (Tasks 1a and 2a), so we do not discuss them further.

Formula Recognition (Tasks 1 and 2). Table II shows the expression recognition rate for each system, with systems sorted in decreasing order of their correct expression rate. Also shown are the number of expressions correctly recognized when 1 to 2 errors in symbol class labels are permitted, along with the number of formulas with the correct recognized structure, ignoring symbol labels (i.e., treating all symbols as correct). Allowing just one labeling error, all rates increase substantially. USTC-iFLYTEK obtained the highest recognition rate for both strokes in task1 and images in task2. We should mention that MyScript (accidentally) withdrew a submission that obtained an expression rate of 80.82%, but this was determined after the competition was over. To be fair to other participants, we have removed this from the official results table.

A surprisingly small drop-off in recognition accuracy occurs when the stroke data is removed and raw pixels are provided in Task 2, decreasing the recognition rate from 80.73% for online recognition to 77.15% for offline recognition. A summary of confusion histograms that tabulate errors in symbol and relationship recognition is presented in Table III (for space, the largest subgraphs shown are pairs of symbols). The detailed results show that the most common error is missing symbols, which is due to symbols being identified by their absolute path: errors in structure lead to errors in symbol location and classification. Looking at the detailed confusion histogram results produced using the confHist tool from LgEval, in most cases, there is a shift in the structure causing missing nodes.

Typeset Formula Detection. Table IV presents the Recall and Precision rates, ranking participants by F-Score. There is a real gap between the winning system Samsung R&D-2 and the other three participants. One factor may be that the winning system made effective use of the provided character locations which the RIT teams ignored, for example. Using a more permissive metric (IoU ≥ 0.5) substantially increases the F-score for all but the winning system.

VII. CONCLUSION

For the formula recognition tasks, end-to-end neural networks produced very strong results, winning both tasks. However, hybrid systems combining grammatical and statistical approaches still obtain results close to the state-of-the-art (e.g., MyScript). Concerning the formulae detection in printed documents, the graph-based approaches outperformed participants' deep-learning approaches. The results of the competition lead us to conclude that math recognition remains a challenging task for state-of-the-art systems.

TABLE I :

 I CROHME 2019 DATA SETS

	Tasks	Training	Validation	Test
	Formulae (1, 2)	Train 2014 + Test 2013 + Test 2012	Test 2014	Test 2019
		9993 expr	986 expr	1199 expr
	Symbols (1a, 2a)	Train 2014 + Test 2013 + Test 2012	Test 2014	Test 2016
		180440 symbols + junks	18435 symbols + junks 15483 symbols + junks
	Structure (1b, 2b)	Train 2014 + Test 2013 + Test 2012	Test 2014	Test 2016
		9993 expr	986 expr	1147 expr
	Formula Detection (3) 36 rendered PDFs (600dpi): 569 pages n/a	10 rendered PDFs (600dpi): 236 pages
		26395 formula regions		11885 formula regions
		Character BBs and labels		Character BBs without labels

  expanded the CROHME 2019 Training set by adding previous test sets (from 2012 and 2013, see Table I). The Validation set for 2019 is the CROHME 2014 Test set. Test sets for Tasks 1a/b and Tasks 2a/b were the CROHME 2016 Test set.

TABLE II :

 II FORMULA RECOGNITION RESULTS

		Structure + Symbol Labels	Structure
	2019 Test Data	Correct ≤ 1 s.err ≤ 2 s.err	Correct
	Task 1: Strokes
	USTC-iFLYTEK	80.73	88.99	90.74	91.49
	Samsung R&D 1	79.82	87.82	89.15	89.32
	MyScript	79.15	86.82	89.82	90.66
	Sun Yat-Sen U.	77.40	85.82	87.99	88.82
	Samsung R&D 2	65.97	77.81	81.73	82.82
	PAL-v2	62.55	74.98	78.40	79.15
	MathType	60.13	74.40	78.57	79.15
	TUAT	39.95	52.21	56.54	58.22
	Task 2: Images
	USTC-iFLYTEK	77.15	86.82	88.99	89.49
	PAL	71.23	80.31	82.65	83.82
	Sun Yat-Sen U.	65.22	78.48	83.07	84.90
	PAL-v2	62.89	74.98	78.40	79.32
		62.14	75.06	78.23	78.32
	Univ. Linz	41.49	54.13	58.88	60.02
	TUAT	24.10	35.53	43.12	43.70
	2016 Test Data
	Task 1B: Strokes with Provided Symbols
	Samsung R&D 1	92.94	93.11	93.20	93.20
	fvg	85.88	85.88	85.96	85.88
	Task 2B: Images with Provided Symbols
	Univ. Linz	81.34	81.87	82.30	82.21
	s.err: symbol class label errors

TABLE III :

 III FORMULA REC. ERRORS (2019 TEST DATA)

			Symbols			Symbol Pairs (w. Relation)
		1 st	2 nd	3 rd		1 st	2 nd	3 rd
	Task 1: Strokes	E #	E #	E #	E	#	E	#	E	#
	USTC-iFLYTEK Samsung R&D 1	1 70 70	+ 54 -69	53 2 62	1 +1 15 16	2	15 14	+-13 -1 10
	MyScriptA	56	+ 51	48		14	-1 8	+2	8
	Sun Yat-Sen U.	71	2 67	-65		16		14	(x 10
	Samsung R&D 2	120	-98	+ 89	-1 20	+1 18	1	18
	Task 2: Images
	USTC-iFLYTEK PAL	1 66 127	+ 57 86	2 57 80	1 → ∞ 21 14	13 +1 19 2	C = 9 = 1 19
	Sun Yat-Sen U.	107	105	102	+1 25	-1 25	x+ 20
	PAL-v2	-819 944	2 724 828	1 706 818	1	164 205	2 130 153	(x 110 -1 120
	E: symbol/subexpresion error
	Note: repeated symbols and subexpressions omitted for readability.

TABLE IV :

 IV TYPESET FORMULA DETECTION RESULTS

			IoU ≥ 0.75			IoU ≥ 0.5
		F1	Recall	Prec.	F1	Recall	Prec.
	Samsung R&D-2 93.45	92.73	94.17	93.63	92.91	94.36
	RIT 2	68.29	62.49	75.29	75.41	67.00	83.14
	RIT 1	60.58	58.17	63.20	71.32	68.47	74.40
	michiking	16.14	13.99	19.10	31.18	27.00	36.87

http://crohme2019.cs.rit.edu

https://github.com/rafaelpadilla/Object-Detection-Metrics

Acknowledgments. Our sincerest thanks to all of the participants. The authors would also like to thank Puneeth Kukkadpu and Parag Mali for preparing data for the formula detection task, Soumen Kumar Koley for helping update the online submission system, and Rafael Padilla for the initial implementation of object detection metrics. This work was supported in part by the Alfred P. Sloan Foundation and the Centre for Artificial Intelligence and Machine Learning (CAIML) of ISI, Kolkata.