Lucie Mahaut
email: lucie.mahaut1@gmail.com

Florian Fort

Cyrille Violle

Grégoire T Freschet

Multiple facets of diversity effects on plant productivity: species richness, functional diversity, species identity and intraspecific competition

Keywords: biodiversity-ecosystem functioning, complementarity effect, functional trait, functional distinctiveness, niche difference, roots, selection effect, species coexistence

1. Deciphering the mechanisms that drive variation in biomass production across plant communities of contrasting species composition and diversity is a main challenge of biodiversity-ecosystem functioning research. Niche complementarity and selection effect have been widely investigated to address biodiversity-productivity relationships. However, the overlooking of the specific role played by key species have limited so far our capacity to comprehensively assess the relative importance of other potential drivers of biodiversity effects.

2. Here, we conducted a grassland diversity-productivity experiment to test how four potential facets of biodiversity effects, namely species richness, functional diversity, species identity and the relaxation of intraspecific competition, account for variations in above and root biomass production.

3. We grew six plant species in monoculture, as well as in every combinations of two, three and six species. Plant density was kept constant across the richness gradient but we additionally grew each species in half-density monoculture to estimate the strength of intraspecific competition for each studied species. We characterized eight functional traits, including root traits, related to nutrient and light acquisition and computed both the functional dissimilarity and the community weighted mean (CWM) of each trait. We further partitioned aboveground biodiversity effect into complementarity and selection effects. 4. We observed strong positive biodiversity effects on both aboveground and root biomass as well as strong positive complementarity effect. These arose largely from the presence of a particular species (Plantago lanceolata) and from CWM trait values more than from a higher functional dissimilarity in plant mixtures. P. lanceolata displayed the highest intraspecific competition, which was strongly relaxed in species mixtures. By contrast, the presence of Sanguisorba minor negatively affected the productivity of plant mixtures, this species suffering more from interspecific than intraspecific competition.

5. This study provides strong evidences that the search for key species is critical to understand the role of species diversity on ecosystem functioning and demonstrates the major role that the balance between intraspecific and interspecific competition plays in biodiversity-ecosystem functioning relationships. Developing more integrative approaches in community and ecosystem ecology can offer opportunities to better understand the role that species diversity plays on ecosystem functioning.

Introduction

Although numerous plant diversity-ecosystem functioning experiments have reported positive effects of plant species richness on ecosystem productivity (e.g., [START_REF] Tilman | Diversity and Productivity in a Long-Term Grassland Experiment[END_REF][START_REF] Cardinale | Biodiversity loss and its impact on humanity[END_REF]Lefchek et al., 2015), the nature of the mechanisms that cause this pattern remains highly controversial. It is widely accepted that niche complementarity can lead to higher productivity in plant mixtures compared to monocultures (biodiversity effects; [START_REF] Huston | Hidden treatments in ecological experiments: re-evaluating the ecosystem function of biodiversity[END_REF][START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF]. Such a complementarity effect may be due to species differences in the way they capture and use resources (so-called resource partitioning), due to species ability to alter their surrounding environment and to subsequently favour the fitness of other species (e.g., abiotic facilitation), or due to plant interactions with other trophic levels (Barry et al., 2018). Positive biodiversity effects can also result from an increased probability of selecting a species with a specific property as the size of the community increases, for example a highly productive species [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF]. The selection effect describes whether the species that dominate plant mixtures are the most productive species in monoculture (i.e. positive selection effect) or the least productive species in monoculture (i.e. negative selection effect; [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF]. However, the presence of some other species can also exert a disproportionate effect on ecosystem functioning irrespective of their biomass in monoculture [START_REF] Jaillard | A combinatorial analysis using observational data identifies species that govern ecosystem functioning[END_REF]. Such a species-specific effect underpins the wellknown concept of keystone species, i.e. species having 'disproportionately large effects relative to its abundance' [START_REF] Paine | A note on trophic complexity and community stability[END_REF]; see also Violle et al. 2017 for a revisiting concept in the light of functional ecology theory). Recently, [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF] extended this concept by defining "key species" as those species that are 'consistently and significantly associated to a certain level of ecosystem functioning or services' [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF]. Although the search for key species can reveal unsuspected mechanisms for ecosystem functioning [START_REF] Huston | Hidden treatments in ecological experiments: re-evaluating the ecosystem function of biodiversity[END_REF][START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF], their role in biodiversity-ecosystem functioning experiments have been largely neglected.

Because ecological niches are theoretically linked to a suite of functional traits (Violle & Jiang 2009), functional traits appear to be a promising tool for understanding diversityproductivity relationships. On the one hand, differences in functional traits (functional dissimilarity) can reflect differences in the use of resources that allow species to partition the local pool of resources and avoid interspecific competition [START_REF] Violle | The return of the variance: intraspecific variability in community ecology[END_REF]. For instance, differences in the vertical distribution of roots among species allow the capture of water and nutrients at different soil depths. On the other hand, the functional traits of dominant species in plant mixtures can be approximated using the community weighted mean (CWM) of functional trait values [START_REF] Garnier | Plant functional markers capture ecosystem properties during secondary succession[END_REF][START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF]. It has been argued that functional dissimilarity can mediate the complementarity effect while CWMs can mediate the selection effect by highlighting the role of the dominant species on ecosystem functioning [START_REF] Cadotte | Functional traits explain ecosystem function through opposing mechanisms[END_REF]. Disentangling the respective influence of both processes through a trait-based approach thus requires using a set of traits that are directly linked to species' resource use and competitive ability. For instance belowground, this requires studying root traits that reflect how species develop specialized strategies to explore the soil volume (e.g., deep root fraction, specific root length, root inter-branch distance) and to extract water and nutrients (e.g., root hair length, specific nutrient absorption rate; [START_REF] Freschet | Allocation, morphology, physiology, architecture: the multiple facets of plant above-and below-ground responses to resource stress[END_REF].

Finally, species may exert highly specific effects on the functioning of ecosystems that cannot be captured by metrics of functional diversity computed at the community scale [START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF]. Among others, the fact that every species has a specific density-productivity relationship in monoculture can be an important mechanism for ecosystem functioning. Indeed in most biodiversity-productivity experiments, species relative density -which directly drives the strength of intraspecific competition [START_REF] Chesson | Mechanisms of Maintenance of Species Diversity[END_REF] -decreases along the gradient of species richness (i.e. substitutive experimental design; Hector, 1998;Joliffe, 2000). In parallel, interspecific competition -which by essence is null in monoculture -increases along this gradient, so that weaker competitors can perform better in monoculture where they do not encounter other species [START_REF] Turnbull | Coexistence, niches and biodiversity effects on ecosystem functioning[END_REF]. The relative importance of intraspecific and interspecific competition on the productivity of each species could therefore be an important driver of biodiversity-ecosystem functioning relationships [START_REF] Benedetti-Cecchi | Increasing accuracy of causal inference in experimental analyses of biodiversity[END_REF][START_REF] Turnbull | Coexistence, niches and biodiversity effects on ecosystem functioning[END_REF][START_REF] Turnbull | Understanding the value of plant diversity for ecosystem functioning through niche theory[END_REF]. Despite these evidences, previous experimental biodiversityecosystem functioning studies have largely neglected the role of species intraspecific competition (but see [START_REF] Polley | Do species evenness and plant density influence the magnitude of selection and complementarity effects in annual plant species mixtures?[END_REF], leaving the question unanswered.

In this study, we tested the strength and significance of four drivers of plant biomass production in biodiversity-ecosystem functioning experiment, namely species richness, functional diversity, species identity and intraspecific competition. To do so, we experimentally designed grassland plant communities representing a gradient of species richness (from one to six) and functional diversity. We manipulated three different functional groups (namely 'grasses', 'forbs' and 'legumes'), and characterized eight functional traits (three shoot and five root traits) that were directly related to the acquisition of light and nitrogen [START_REF] Freschet | Allocation, morphology, physiology, architecture: the multiple facets of plant above-and below-ground responses to resource stress[END_REF]. We tested each above-mentioned effect separately by combining the conceptual framework of [START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF] and the statistical framework of [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF]. Briefly, to understand how species diversity affects ecosystem functioning, [START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF] suggests to first test for the role of functional diversity and, in a second step, to look for potentially remaining species-specific effects. In parallel, [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF] developed a statistical framework that aims at identifying key species that drive ecosystem functioning [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF]. In this framework, we separately tested the effect of the presence of a candidate species or the effect of a candidate functional trait by adding species presence or functional diversity as an explanatory variable to a baseline model that previously accounted for the effects of species richness. Finally, we estimated the strength of the effect of intraspecific competition by quantifying for every species the gain of individual biomass when decreasing plant density in monocultures.

Materials and Methods

Experimental design

The experiment was conducted at the Center for Functional and Evolutionary Ecology, Montpellier, France. We grew six plant species in monoculture (6 combinations), as well as in every combinations of two (15 combinations), three (20 combinations) and six species (1 combination) in a greenhouse with three replicate pots for the monoculture, two and three species combinations and six replicates for the six species combination. Plant species were common European herbaceous species (see Table 1): two grasses (Bromus erectus Huds., Dactylis glomerata L.), two forbs (Plantago lanceolata L., Sanguisorba minor Scop.) and two legumes (Lotus corniculatus L., Trifolium repens L). We chose this set of species to avoid functional redundancy, these species displaying contrasted functional traits. Seeds were collected from permanent grasslands located in southern France. Plant density was kept constant across the richness gradients (i.e. six individuals per pot with equal species relative abundance) but we additionally grew each species in three replicates of half-density monoculture (i.e. three individuals per pot) to estimate the strength of intraspecific competition for each studied species. Climate conditions in the greenhouse were semi-controlled.

Temperature was allowed to fluctuate between 15 and 19°C at night and between 21 and 25°C during the day. Natural light conditions were complemented for the duration of the experiment (with 400W Na-ion lamps) to provide a typical change in photoperiod during the plant growing season from 12 h initially to 14.5 h at the end of the experiment.

We used deep pots (depth 60 cm, diameter 15 cm) containing c. 17 kg (DW) of soil. Soil density was increased by compaction every 20 cm in depth (from 1.51 to 1.63 and 1.74 g.cm -3 ) to ensure realistic growth conditions for plants. The soil was a calcareous sandy loam (pH = 8.5) with rather low organic matter content (9.7 g.kg -1 ), cation-exchange capacity (0.5 g.kg -1 ) and total N content (0.5 g.kg -1 ). At the start of the experiment, in November 2015, a soil leachate solution was added to ensure the presence of symbiotic N2-fixing bacteria in the pot. Pots were watered three times a week to provide moisture conditions close to field capacity in the soil profiles; this corresponded to 0.1L of water at the start of the experiment and 0.6L at the end in order to account for increasing plant demand. In addition, all pots received three soil enrichments (after 1, 4 and 9 weeks) in phosphorus (P) and potassium (K) in increasing amount over time for a total of 10 g P m -2 and 24 g K m -1 . Note that nitrogen was not supplied so that it remained the main limiting resource for plant growth. We randomly placed pots on wheeled carriages and avoided side effects by rearranging carriages every two weeks.

Biomass measurements

Plants harvest took place 13 weeks after seedling transplantation after all species had shown first signs of flowering. We cut aboveground parts of plants at the base and separated the six plant individuals to measure aboveground biomass of each individual plant. We evaluated root biomass of each pot after splitting the column of soil in three equal cylinders, each 20 cm long, and careful retrieving and washing roots from each cylinder. Plant material was ovendried at 60°C for 48 h and weighed.

Trait measurements

We measured three aboveground traits and five belowground traits related to both nitrogen and light acquisition. For each trait, the detailed protocol is presented in [START_REF] Freschet | Allocation, morphology, physiology, architecture: the multiple facets of plant above-and below-ground responses to resource stress[END_REF]. Briefly, three weeks before harvest, we measured light-saturated leaf photosynthetic rate per area (Aarea, µmolCO2 m -2 s -1 ) -that provides the leaf maximal photosynthesis capacityon one individual per monoculture pot by quantifying the amount of C accumulated in a leaf exposed to a high light intensity for several minutes (C influx vs efflux). One week before the experiment harvest, we recorded the maximum height (cm) achieved by all plant individuals in all pots. Plant height is a good proxy for light depletion through the canopy (Violle et al. 2009). Specific leaf area (SLA, m².kg -1 ) was assessed at harvest based on two to four leaves (depending on the species) from each plant individual that we immediately scanned for leaf area measurement. Specific leaf area corresponds to the area of light capture per biomass invested in leaves and is related to the fundamental trade-off existing between species acquisition and conservation of resources [START_REF] Garnier | Leaf anatomy, specific mass and water content in congeneric annual and perennial grass species[END_REF].

Root functional traits were measured from monoculture pots only because of the difficulty and labour associated to separating roots among species in mixture pots. A subsample of roots was used to determine root length, mean root diameter and the proportion of very fine roots (<0.2 mm) using a digital image analysis system (WinRhizo, version 2009, Regent Instrument, Québec, Canada). Deep root fraction (DRF), which reflects the relative investment of species to take up nutrient from the deeper soil horizons, was estimated as the ratio of root biomass deeper than 20 cm to total root biomass. Root inter-branch distance (RID, cm) is a measure of root cost-efficiency to explore large soil volume (rather than exploit soil volume intensively) and was quantified as the average distance between first order roots. Specific root length (SRL, m.g -1 ) was estimated as the ratio of root length to root dry mass to represent the cost-efficiency of roots to explore and/or exploit soil volume. A second subsample of roots was stained with methyl violet in order to make root hairs visible and measure root hair length (RHL, mm) on 10 randomly selected first order roots, over stretches of 1 mm roots situated 2 mm away from the root tips, using ImageJ software. Root hair length is a proxy for the soil volume explored around the root. Finally, we used one replicate (pot) from the six species in monoculture to measure specific root nitrogen absorption rate (Nabs), which reflects the shortterm maximum nitrogen uptake capacity per unit root length. We calculated Nabs as the total amount of 15 N taken up by plants after injection of different forms of 15 N in the soil, per length of fine root and per h (µg 15 N m -1 h -1 ).

Biodiversity effects

We quantified the biodiversity effect (ΔY), which is the performance of plant mixture relative to that expected from monocultures, separately for aboveground and root biomass production. Then, following the equation proposed by [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF]:

𝛥𝑌 𝑗 = 𝑁 𝑗 . 𝛥𝑅𝑌 ̅̅̅̅̅̅ . 𝑀 ̅ + 𝑁 𝑗 . 𝐶𝑂𝑉(𝛥𝑅𝑌 𝑖,𝑗 , 𝑀 𝑖 ) Equation 1
we computed the two components of ΔY, the complementarity and selection effects, only for aboveground ΔY as we did not measure individual root biomass in plant mixtures. In this equation, 𝑁 𝑗 is the number of species in pot j. 𝛥𝑅𝑌 𝑖,𝑗 is the deviation from the expected relative yield of species i in pot j calculated as:

𝛥𝑅𝑌 𝑖,𝑗 = 𝑌 𝑖,𝑗 𝑀 𝑖 - 1 𝑁 𝑗 Equation 2
where 𝑌 𝑖,𝑗 is the biomass measured for species i in pot j and 𝑀 𝑖 is the average monoculture biomass for species i. As species were sown at constant density of individuals, the expected yield is simply the inverse of the number of species in pot j (1/𝑁 𝑗 ). The first component of the biodiversity effect equation (𝑁 𝑗 . 𝛥𝑅𝑌 ̅̅̅̅̅̅ . 𝑀 ̅ ) is the complementarity effect, which quantifies the performance of plant mixtures relative to the performance of the component monocultures. The covariance between species performance in monoculture and in plant mixture, 𝑁 𝑗 . 𝐶𝑂𝑉(𝛥𝑅𝑌 𝑖,𝑗 , 𝑀 𝑖 ), has been termed the selection effect.

Functional diversity

We assessed the effects of species traits on ecosystem functioning by computing two facets of functional diversity, namely functional dissimilarity and community weighted mean (CWM). We quantified functional dissimilarity using the functional dispersion (FDis) index computed separately for each trait using the fdisp function of the FD R package [START_REF] Laliberté | FD: Measuring functional diversity (FD) from multiple traits, and other tools for functional ecology (Version 1.0-12)[END_REF]. For each trait, we computed FDis based on mean species trait values measured in monoculture pots only to insure homogeneity between the different traits that were measured in all pots or in monoculture only (Table 1). Since plant height and SLA were quantified at the individual level, we further tested the role of intraspecific trait variability by computing FDisHeight and FDisSLA based on trait values measured on individuals from all pots. CWM were calculated for each trait by multiplying the mean species trait value measured in monoculture pots by the proportional abundance of each species in each community. Finally, we classified the six species into three functional groups (legumes, herbs and forbs).

Data analyses

We investigated the effect of species richness, functional dissimilarity, CWM and species identity on aboveground and root biomass production, aboveground and belowground biodiversity effects as well as aboveground complementarity and selection effects using linear models. Following [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF], we first tested the extent to which species richness affected biomass production, biodiversity effects, complementarity effect and selection effect in a baseline model (M0). Next, we built 'functional group', 'functional dissimilarity', 'CWM' and 'species identity' models to test the extent to which the data support the effect of a particular functional group, functional trait or species identity on these response variables. To do so, we added the presence of each functional group or species (coded as a binary variable) or FDis and CWM of each trait separately as an explanatory variable to M0. The resulting model (M1i), which is the importance of a candidate species i, functional group i or functional trait i to explain variation in productivity, was then evaluated according to its Akaike information Criterion (AIC). We considered a species, a functional group or a functional trait as important for productivity if ΔAIC (AICM0 -AICM1i) was greater than 4 [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF]. Although a commonly adopted rule of thumb states that a model with ΔAIC<2 is likely to be the best model [START_REF] Burnham | Model Selection and Multimodel Inference: A Practical Information Theoretic Approach[END_REF][START_REF] Richards | Testing ecological theory using the information-theoretic approach: Examples and cautionary results[END_REF], here we doubled this threshold to reduce the probability of having false positive. We standardized all variables before analysis to facilitate the interpretation of the regression coefficients [START_REF] Schielzeth | Simple means to improve the interpretability of regression coefficients[END_REF]. Because in the six species mixtures each species and functional group were systematically present, we ran the functional group and species identity models without the six-species mixture pots. The sample size were therefore 128 for the functional trait models and 110 for the functional group and species identity models. We further computed the posterior model probabilities to report the probability that each model M1i is better than M0 using the bicreg function of the BMA package [START_REF] Raftery | BMA: Bayesian model averaging (version 3.18[END_REF]. Posterior model probabilities were highly consistent with the ΔAIC analysis.

Consequently, we presented ΔAIC in the main manuscript and posterior model probabilities in supplementary Table S1.

In addition, we tested for the influence of intraspecific competition, i.e. species-specific density-productivity relationships, on biomass production by computing for each species intraspecific competition logarithmic response-ratio [START_REF] Hedges | The Meta-Analysis of Response Ratios in Experimental Ecology[END_REF]. To do so, for each species, we compared the biomass of the nine individuals grown in monocultures sown at half-density to the biomass of the 18 individuals grown in monocultures sown at full density. Positive logarithmic response-ratio corresponds to higher biomass for the individual of the monoculture sown at half-density while negative response-ratio corresponds to higher biomass for the individuals of the monoculture sown at full density. Finally, we computed the standardized difference (effect size) between individual plant biomass of species grown in monoculture with half density and monoculture and mixtures of two, three and six species grown at full density (six individuals). Since root biomass was not quantified at the level of individual plants in mixtures, such effect sizes were only calculated for aboveground biomass.

Confidence intervals (α=0.05) were computed to test whether individual plant biomass in full density pots significantly differed from individual plant biomass of the same species grown in monoculture at half-density. All analyses were conducted using R ver. 3.4.4. (www.rproject.org).

Results

Effects of species richness on plant aboveground and belowground biomass production

Among all mixtures, aboveground productivity was the main fraction of plant biomass production (Fig. 1a,b), accounting for c. 70% of the biomass produced per pot. Aboveground biomass significantly increased with species richness (P < 0.001, r² = 0.08) while root biomass did not (P = 0.17, r² = 0.01; Supplementary Table S2). Most importantly, both mean aboveground and belowground biodiversity effects (ΔY) were positive at all species richness levels (Fig. 1b). However, they did not increase with species richness (P = 0.09, r² = 0.02; P = 0.70, r² = 0.01 for aboveground and belowground net effects, respectively; Table S2).

The additive partitioning of aboveground biodiversity effect (ΔY) revealed that among all mixtures, complementarity effect was the main fraction of ΔY (Fig. 1c), accounting for ~ 83% of aboveground ΔY. Complementarity effect did not increase with species richness (P = 0.34, r² = 0.01; Table S2) while the selection effect did (P = 0.02, r² = 0.04; Table S2).

Removing the six species mixtures from the analyses did not change the effects of species richness on aboveground biomass production, aboveground and belowground biodiversity effects and complementarity effects (Table S2). However, root biomass significantly increased with species richness while the positive effect of species richness on selection effect disappeared when the six species mixtures were removed from the analyses (Table S2).

Stronger effects of CWMs compared to functional dissimilarity and functional group diversity

ΔAIC values revealed that the main facet of functional diversity affecting biomass production was the CWM of the studied traits (Table 2). After controlling for species richness effect, CWMSLA and CWMAmax exerted a strong influence on all the components of biomass production -except the selection effect (Table 2). Productivity, biodiversity effects and complementarity effect consistently decreased with CWMSLA and CWMAmax (Table 2; Figs. S1, S2 and S3). In addition, aboveground productivity, root productivity and belowground net effect increased with CWMHeight (Table 2; Figs. S1 andS2). Aboveground net effect, belowground net effect and complementarity effect also increased with CWMNabs and CWMDRF and decreased with CWMRHL (Table 2; Figs. S2 andS3). By contrast, plant functional group identity poorly affected plant productivity and biodiversity effects beyond the overall effect of species richness (Table 2). The only exception was the presence of leguminous species that exerted a negative effect on root productivity in plant mixtures (P < 0.001). The ΔAIC analysis also revealed that functional dissimilarity slightly affected plant productivity and biodiversity effects (Table 2).

After controlling for species richness effect, root productivity significantly increased with FDisHeight, FDisSRL and FDisNabs (Table 2; Fig. S4). Finally, aboveground net effect, belowground net effect and complementarity effect significantly decreased with FDiRID (Table 2; Figs. S5 andS6).

Key species associated to changes in productivity

The presence of P. lanceolata in the mixture was the principal driver of change in 2). All these components of biomass production were significantly higher when P. lanceolata was in the plant mixtures (Fig. 2). The presence of S.

minor and T. repens in plant mixtures also markedly affected biomass production. Aboveground and root productivity, net effects and complementarity effects were lower in presence of S.

minor while the selection effect was higher. Aboveground productivity was higher in the presence of T. repens while root productivity and selection effects were lower (Fig. 2).

Nevertheless their influence on net effects, complementarity and selection effects were substantially (from two to ten times) lower than to those of P. lanceolata (Table 2).

Species-specific effect and intraspecific competition

Species' logarithmic response-ratio calculated based on aboveground biomass production was positive for all species (Fig. 3), meaning that biomass production was in average higher in half-density plots for all species. However, we found significant differences between species (ANOVA, F=41.3, P < 0.001), mean logarithmic response-ratio being the highest for P.

lanceolata (1.36±0.45) and the lowest for S. minor (0.41±0.69; Fig. 3). In other words, the biomass of P. lanceolata individual plants was four times lower in the full-density monoculture than in the half-density monoculture. In addition, we found that the biomass of individual plants of P. lanceolata in the three and six species mixtures did not significantly differ from their biomass in the half-density monoculture (Fig. 4). Similarly, the biomass of individual plants of T. repens in the six species mixture was equivalent to their biomass in the half-density monoculture, whereas it was otherwise lower (Fig. 4). By contrast, the biomass of individual plants of S. minor in full-density monoculture, two, three and six species mixtures was lower than in half-density monoculture but the lowest difference was observed between full-density monoculture and half-density monoculture (Fig. 4).

Discussion

Our study revealed contrasting influences of the four studied facets of biodiversity effects on ecosystem biomass production. Species richness and functional dissimilarity showed only moderate influence, whereas community weighted trait means (CWM), species identity and the relaxation of intraspecific competition accounted for a large part of observed biodiversity effects.

The significantly higher aboveground and root biomass production observed here in plant mixtures compared to monoculture is a common pattern in biodiversity-ecosystem functioning studies (e.g., [START_REF] Tilman | Diversity and Productivity in a Long-Term Grassland Experiment[END_REF][START_REF] Cardinale | Biodiversity loss and its impact on humanity[END_REF]Lefcheck et al., 2015). We also reported a strong dominance of the complementarity effect over the selection effect, which has been frequently observed, at least in long-term experiments (Cardinale et al., 2007;Reich et al., 2012). Furthermore, the selection effect increased with species richness, confirming that the probability of selecting a highly productive species that over-produce in mixture increases with the size of the community [START_REF] Huston | Hidden treatments in ecological experiments: re-evaluating the ecosystem function of biodiversity[END_REF]. However, species richness per se did not affect aboveground and belowground biodiversity effects nor complementarity effect (Fig. 1). This reveals the limited value of species number per se to predict biodiversity-ecosystem functioning relationships, and emphasizes the importance of considering other biodiversity facets.

Since niche complementarity theoretically relies on functional trait differences (Violle & Jiang, 2009), we expected that biodiversity and complementarity effects would increase with functional dissimilarity [START_REF] Cadotte | Functional traits explain ecosystem function through opposing mechanisms[END_REF]. Surprisingly, we found weak effects of functional dissimilarity, except for a decrease of the biodiversity and complementarity effects with the dissimilarity in root inter-branch distance. The use of trait-by-trait dispersion indices to compute functional dissimilarity can explain the weakness of such relationships. However, while the combinations of traits may better describe species differentiation along the multiple ecological dimensions of species niche [START_REF] Kraft | Community assembly, coexistence and the environmental filtering metaphor[END_REF], using a multi-dimensional functional dispersion index did not improve the observed relationship between functional dissimilarity and biodiversity effects (See Supplementary Table S3). Another reason may be that we computed functional dissimilarity in plant mixtures based on mean species trait values in monoculture.

Doing so, we did not account for intraspecific trait variations, which can vary in conditions of competition and affect resource uptake (e.g., belowground, Mommer, van Ruijven, Jansen, van de Steeg & de [START_REF] Mommer | Interactive effects of nutrient heterogeneity and competition: implications for root foraging theory[END_REF][START_REF] Schenk | Root competition: beyond resource depletion[END_REF]. However, aboveground, accounting for intraspecific trait variations in plant height and specific leaf area, both traits being measured at the individual level, did not change the results (see Supplementary Table S3). Overall, given our targeted choice of traits capturing multiple facets of light and nutrient acquisition above and belowground [START_REF] Freschet | Allocation, morphology, physiology, architecture: the multiple facets of plant above-and below-ground responses to resource stress[END_REF], the weak relationship between functional dissimilarity and the complementarity effect confirms that the complementarity effect does not only reflect resources partitioning but a larger set of biotic interactions in species mixtures [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF][START_REF] Carroll | Niche and fitness differences relate the maintenance of diversity to ecosystem function[END_REF][START_REF] Niklaus | Can niche plasticity promote biodiversity-productivity relationships through increased complementarity[END_REF]Garry et al., 2018).

By contrast, we reported strong influence of five out of eight community weighed trait means on the complementarity effect -and by extent on biodiversity effects. Species ability to take up nitrogen resources was a main driver of biomass productivity, as suggested by the positive relationships between the community weighted mean of both nitrogen absorption rate and deep root fraction and biodiversity effects. Furthermore, biodiversity and complementary effects increased with decreasing the community weighted mean of specific leaf area and maximum photosynthetic rate, implying that biomass production was maximum when plant communities were dominated by slow-growing, resource conservative species (Wright et al., 2004). Although such a result might be surprising given the short-term nature of our experiment, it may reflect the fact that low nutrient availability in this experiment have favored plants with the more conservative resource strategies (Wright et al., 2004;[START_REF] Carmona | Trait hierachies and intraspecific variability drive competitive interactions in Mediterranean annual plants[END_REF]. More globally, the importance of functional trait values gives evidence that in this experiment, ecosystem functioning is strongly driven by the identity of the dominant species (mass-ratio hypothesis, [START_REF] Grime | Benefits of plant diversity to ecosystems: immediate, filter and founder effects[END_REF]. However, we did not find significant relationship between the CWMs and the selection effect while this is a main expectation under the mass-ratio hypothesis [START_REF] Cadotte | Functional traits explain ecosystem function through opposing mechanisms[END_REF]. This shows that the effect of the dominant species on ecosystem functioning differs from the selection effect sensu [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF] and calls for a more mechanistic approach to understand the relationships between biodiversity and ecosystem functioning.

We found a disproportionate contribution of one species, Plantago lanceolata, on the production of biomass and biodiversity effects in plant mixtures. Among the six species of this experiment, P. lanceolata displayed the lowest biomass in monoculture but the highest biomass in the 6x6-plant mixtures (Figure 4). Consequently, the specific effect of selecting P. lanceolata in mixture differs conceptually from the selection effect, which can be positive only when the most productive species in monoculture produce even more biomass in mixture [START_REF] Loreau | Partitioning selection and complementarity in biodiversity experiments[END_REF]. Despite this, the selection of P. lanceolata in mixture was the major determinant of positive biodiversity effect on productivity. Its low biomass in monoculture was due to high intraspecific competition, which was relieved in half-density monocultures and even more in species mixtures (Figure 3). Consequently, the much lower level of interspecific competition as compared to intraspecific competition for this particular species, appeared as the most critical driver of biodiversity effects in our experiment. Since the strength of intraspecific competition may be generally stronger than the strength of interspecific competition in plant communities [START_REF] Adler | Competition and coexistence in plant communities: intraspecific competition is stronger than interspecific competition[END_REF], we expect that such a mechanism may be widespread in biodiversitybiomass production relationships. The same pattern, although much less strong, was also observed here for T. repens, and the reverse pattern was recorded for S. minor, which suffered more from interspecific competition than from intraspecific competition (see also [START_REF] Turnbull | Coexistence, niches and biodiversity effects on ecosystem functioning[END_REF]. Interestingly, we observed that, across our six species, the effect of intraspecific competition on species production was inversely related to the effect of interspecific competition (Figure 4), suggesting that a priori knowledge of species-specific densitydependence production may be particularly useful to explain (and potentially predict) biomass gains in mixtures. An increasing number of biodiversity-productivity studies already accounts for the effect of negative density-dependence mechanisms such as density-dependence plant disease (e.g., [START_REF] Schnitzer | Soil microbes drive the classic plant diversity-productivity pattern[END_REF][START_REF] Mommer | Lost in diversity: the interactions between soil-borne fungi, biodiversity and plant productivity[END_REF]. By contrast, the role of intraspecific competition has received less attention (but see [START_REF] Polley | Do species evenness and plant density influence the magnitude of selection and complementarity effects in annual plant species mixtures?[END_REF]. Our study shows that accounting for the balance between interspecific and intraspecific competition in experimental studies is essential to better predict the effect of species diversity on ecosystem functioning.

Since the relative strength of intraspecific compared to interspecific competition should increase with niche differences [START_REF] Chesson | Mechanisms of Maintenance of Species Diversity[END_REF][START_REF] Kraft | Community assembly, coexistence and the environmental filtering metaphor[END_REF], we might have expected that P. lanceolata occupied a functional niche that is highly different from the one of the other species. However, computing species functional distinctiveness (sensu Violle et al., 2017) based on the traits used in this study (Supplementary Table S4) revealed that P. lanceolata is not particularly different from the other species. This suggests that other aspects of plant ecophysiology (e.g., three-dimensional architecture, [START_REF] Schenk | Root competition: beyond resource depletion[END_REF]) may drive the response of P.

lanceolata to intraspecific and interspecific competition.

Finally, we found that the presence of legume species in plant mixtures negatively affects belowground biomass production. By improving the availability of nitrogen in the mixture (relief of competition and transfer to neighboring plants; [START_REF] Temperton | Positive interactions between nitrogen-fixing legumes and four different neighbouring species in a biodiversity experiment[END_REF] legumes may lower the typical balance between root versus shoot biomass investments [START_REF] Freschet | Integrated plant phenotypic responses to contrasting above-and below-ground resources: key roles of specific leaf area and root mass fraction[END_REF]. Indeed, legumes generally exert a positive effect on aboveground biomass production [START_REF] Temperton | Positive interactions between nitrogen-fixing legumes and four different neighbouring species in a biodiversity experiment[END_REF][START_REF] Marquard | Plant species richness and functional composition drive overyielding in a six-year grassland experiment[END_REF]. Taking together, these results highlight the importance of considering both aboveground and belowground biomass production in biodiversity-ecosystem functioning analyses as both compartments can differentially respond to species diversity.

To conclude, our study brings novel evidence that species do not equally contribute to ecosystem functioning and that the search for key species (sensu [START_REF] Maire | Community-wide scan identifies fish species associated with coral reef services across the Indo-Pacific[END_REF] is a critical issue to understand the effects of species diversity on ecosystem functioning [START_REF] Diaz | Incorporating plant functional diversity effects in ecosystem service assessments[END_REF].

More precisely, by looking for a species-specific effect, we highlight the important role that intraspecific competition plays in shaping biodiversity-ecosystem functioning relationships, although the traits underlying species intraspecific competitive ability remain unknown. The fact that intraspecific competition also plays a dominant role for species coexistence [START_REF] Adler | Competition and coexistence in plant communities: intraspecific competition is stronger than interspecific competition[END_REF] claims for a more integrative approach in community and ecosystem ecology to better understand biodiversity-ecosystem functioning relationships [START_REF] Turnbull | Coexistence, niches and biodiversity effects on ecosystem functioning[END_REF][START_REF] Turnbull | Understanding the value of plant diversity for ecosystem functioning through niche theory[END_REF].

Accounting for non-linear species-specific density-productivity relationships will be critical to predict the effect of species diversity on ecosystem functioning (Baert, Jaspers, Janssen, De Laender & Aert, 2017). Tables Table 1: Species list and average trait values (± sd) as grown in monoculture conditions. g: grass; f: forb; l: legume. ***: p-value <0.001. For the functional group and species identity models, estimates represent the effect of the presence of each functional group and species in the mixture, respectively, while for the functional dissimilarity and CWM models, estimates represent the slope of the effect of increasing functional dissimilarity (CWM) in the mixture. better than the baseline model that only accounts for species richness (M0). Bold values correspond to models that have been selected according to the difference in AIC between M0 and M1i (i.e. ΔAIC, see Table 2).

Table S2: Model outputs of the baseline models (M0) that account for the effect of species richness on aboveground and root biomass production, aboveground and root biodiversity effects as well as aboveground complementarity and selection effects in every treatment (from monoculture to six-species mixtures for biomass; from two-species to six-species mixtures for biodiversity effects). In addition, we tested for the effect of species richness without the six-species plant mixtures. Table S4: Species functional distinctiveness. Functional distinctiveness is the average functional distance from a species to all other in a given community using the distinctiveness function of funrar R library [START_REF] Grenié | funrar: Functional Rarity Indices Computation[END_REF]. Functional distinctiveness varies from 0 when a species is, on average, functionally close to the other species to 1 when a species is highly distant from others.

  productivity (for aboveground and root productivity: ΔAIC = 20.307, r² = 0.225 and ΔAIC = 69.211, r² = 0.471, respectively), net effects (for aboveground and belowground net effects: ΔAIC = 26.224, r² = 0.229 and ΔAIC = 45.967, r² = 0.366, respectively) and complementarity effect (ΔAIC = 26.578, r² = 0.230; Table

Figure 1 :

 1 Figure 1: Effect of species richness on mean (A) aboveground and root productivity and (B)

Figure 2 :

 2 Figure 2: Species-specific effect on aboveground (red) and belowground (blue) productivity

Figure 3 :

 3 Figure 3: Variation of the intraspecific competition log-response ratio between species. We

Figure 4 :

 4 Figure 4: Standardized difference between individual plant biomass of species grown in

Figure 1 :

 1 Figure 1: Effect of species richness on mean (A) aboveground and root productivity and (B)

Figure 2 :

 2 Figure 2: Species-specific effect on aboveground (red) and belowground (blue) productivity

Figure 3 :

 3 Figure 3: Variation of the intraspecific competition log-response ratio between species. We

Figure 4 :

 4 Figure 4: Standardized difference between individual plant biomass of species grown in

Figure S1 :Figure S3 :Figure S6 :

 S1S3S6 Figure S1: Effects of community weighted mean trait value (CWM) on aboveground (red) and root (blue) productivity. ***: p<0.001; **: p<0.01; 18 *<0.05; ns: non-significant 19 20

  Violle, C., & Jiang, L. (2009). Towards a trait-based quantification of species niche. Journal of Plant Ecology, 2(2), 87-93. doi: 10.1093/jpe/rtp007 Violle, C., Thuiller, W.,Mouquet, N., Munoz, F., Kraft, N. J. B., Cadotte, M. W., … Mouillot,

	D. (2017). Functional Rarity: The Ecology of Outliers. Trends in Ecology & Evolution, 32(5),
	356-367. doi: 10.1016/j.tree.2017.02.002
	Williams, L. J., Paquette, A., Cavender-Bares, J., Messier, C., & Reich, P. B. (2017). Spatial
	complementarity in tree crowns explains overyielding in species mixtures. Nature Ecology &
	Evolution, 1(4), 0063. doi: 10.1038/s41559-016-0063
	Wright, I. J., Reich, P. B., Westoby, M., Ackerly, D. D., Baruch, Z., Bongers, F., … Villar, R.
	(2004). The worldwide leaf economics spectrum. Nature, 428(6985), 821-827. doi:
	10.1038/nature02403

Table 2 :

 2 Effects of functional group, functional dissimilarity, community weighted mean of trait (CWM) and species identity on biomass production (productivity), biodiversity effects (net effect), complementarity effect and selection effect.

		Specific leaf area (m 2 kg -1 )	Max photo-synthetic capacity (µmol-CO 2 m -2 s -1 )	Plant height (cm)	Specific root length (m g -1 )	Deep root (< 20 cm) fraction	Specific N absorption rate (µg m -1 h -1 )	Root inter-branch distance (cm)	Root hair length (mm)
	B. erectus (g) 26.63 ±2.5	14.64 ±1.0	24.50 ±4.2	150.12 ±53.7	0.38 ±0.02	0.06 ±0.01	0.35 ±0.04	0.24 ±0.03
	D. glomerata (g) 29.23 ±0.8	5.70 ±0.7	41.89 ±3.7	275.50 ±27.6	0.41 ±0.03	0.04 ±0.01	0.46 ±0.04	0.24 ±0.04
	P. lanceolata (f) 18.58 ±1.0	9.97 ±1.7	27.78 ±1.4	151.54 ±15.6	0.55 ±0.03	0.08 ±0.06	0.17 ±0.01	0.18 ±0.01
	S. minor (f)	24.83 ±0.3	24.85 ±0.9	15.44 ±0.9	130.26 ±45.8	0.61 ±0.03	0.06 ±0.02	0.14 ±0.01	0.13 ±0.01
	L. corniculatus (l) 42.75 ±5.1	14.53 ±2.1	15.33 ±1.1	87.52 ±19.1	0.38 ±0.11	0.09 ±0.03	0.36 ±0.09	0.20 ±0.01
	T. repens (l)	31.26 ±2.3	20.60 ±7.5	22.89 ±0.9	140.19 ±24.8	0.28 ±0.03	0.08 ±0.04	0.20 ±0.01	0.16 ±0.02

1.20 *** 9.53 0.11 -0.57 ** 6.72 0.06 -1.05 ** 7.38

  ΔAIC is the difference in AIC between M0 that accounts for the effect of species richness and models that further accounts for the effect of species functional group, functional dissimilarity, CWM of traits or species identity. Data significantly supported the model if ΔAIC (i.e. difference in AIC value between baseline model and a model accounting for either functional group, functional dissimilarity or species identity) >4 (bold values). *: p-value <0.05; **: p-value <0.01;

																	0.07	-0.15	-0.49	0.05
		DRF	0.92 *	3.52	0.11	0.03	-1.99	0.01 1.18 ** 9.12	0.10	0.50 *	4.48	0.04 1.03 ** 7.02	0.07	0.14	-0.51	0.05
		RID	-0.41	0.92	0.08	0.42	1.17	0.02	-0.66	1.45	0.04	-0.26	-0.31	0.00	-0.47	-0.17	0.01	-0.19	0.76	0.06
		B. erectus	-2.31 ** 5.64	0.13	-0.186 -1.86	0.05	-1.07	-0.02	0.01 -1.10 ** 5.26	0.06	-1.11	0.26	0.01	0.04	-1.97	0.02
		D. glomerata	0.46	-1.71	0.07 1.53 ** 7.44	0.12	-0.59	-1.41	0.01	0.27	-1.57	0.00	-0.21	-1.92	0.01	-0.38	0.33	0.00
	Species identity	P. lanceolata 3.89 *** 20.31 S. minor -3.74 *** 19.03	0.23 3.76 *** 69.21 0.22 -1.78 *** 11.04	0.47 3.79 *** 26.22 0.15 -1.95 * 4.77	0.23 2.57 *** 45.97 0.05 -1.05 * 4.61	0.37 3.71 *** 26.58 0.06 -2.60 *** 11.00	0.23 0.11 0.64 ** 4.94 0.08 -1.89	0.17 0.05
		L. corniculatus -2.031 * 3.77	0.11 -1.58 ** 7.98	0.12	-1.19	0.41	0.01	-0.35	-1.28	0.00	-1.49 *	2.02	0.03	0.30	-0.55	0.00
		T. repens	3.31 *** 19.93	0.22 -1.66 *** 9.27	0.13	0.98	-0.34	0.00	-0.35	-1.31	0.00	1.67 *	3.16	0.00 -0.69 ** 5.99	0.06

Table S3 :

 S3 Model outputs of the model accounting for the addition effect of species richness and functional dispersion indices (M1k) based on eight traits (multi-traits) or on single traits for Height and SLA measure at the individual plant levels thus accounting for intraspecific variability (Height ISV and SLA ISV). ΔAIC is the difference in AIC between M0 that accounts for the effect of species richness and M1k that further accounts for the effect of functional dispersion. Data significantly supported the model if delta AIC >4 (bold values).

		Above ground productivity Above ground productivity	Root productivity Belowground productivity	Aboveground ΔY Aboveground ΔY	Belowground ΔY Belowground ΔY Complementarity effect Complementarity effect	Selection effect Selection effect
		Estimate r² AIC Estimate r² AIC Estimate r² AIC Estimate r² AIC Estimate r² AIC Estimate r² AIC Estimate r² ΔAIC Estimate r² ΔAIC Estimate r² ΔAIC Estimate r² ΔAIC Estimate r² ΔAIC Estimate r² ΔAIC
		Baseline models (M0) Multitraits 0.75 With 6x6 Height -0.02 0.07 -2.00 0.93 *** 0.10 11.64 0.02 0.08 0.08 0.76 * 0.04 3.93 -0.64 0.03 0.73 1.29 *** 0.08 744.04 0.31 0.01 615.36 0.68 0.01 609.60 -0.09 0.01 476.59 0.37 0.01 602.84 0.32 * 0.04 363.83 0.08 0.01 -1.85 -0.32 0.01 -4.27 -0.32 0.90 4.38 0.01 -1.99 0.50 * 0.04 4.33 -0.12 0.01 -1.89 0.13 0.05 -0.78 (ISV) Without 6x6 1.88 ** 0.08 710.60 0.97 ** 0.06 584.84 0.68 0.01 579.10 0.51 0.01 452.22 0.59 0.01 573.49 0.09 0.01 345.50 SLA (ISV) 0.11 0.07 -1.94 0.2 0.01 -1.43 0.17 0.01 -1.79 0.27 0.01 -0.23 0.08 0.01 -1.94 0.08 0.04 -1.51
	745
	746	*: p-value <0.05; **: p-value <0.01; ***: p-value <0.001
						40

*: p-value <0.05; **: p-value <0.01; ***: p-value <0.001.

Acknowledgements

This study was supported by the EC2CO grant 'MULTIVERS' to G.T.F. and by the European Research Council (ERC) Starting Grant Project 'Ecophysiological and biophysical constraints on domestication of crop plants' (Grant-ERC-StG-2014-639706-CONSTRAINTS) to CV. We are grateful to Pascal Chapon, Annick Lucas, Malick Diao, Océane Cobelli, David Degueldre, Pauline Durbin, Jérémy Devaux, Pierrick Aury and Elodie Renaut for their proficient help with the experimental setup, harvest and plant measurements. This experiment was supported by Thierry Matthieu and the CEFE 'terrain d'expériences' (TE) team, Nicolas Barthès, Bruno Buatois and Raphaëlle Leclerc within the 'plateforme d'analyses chimiques' (PACE) and by the CEFE 'plateforme long-terme' (PLT).

Data Availability Statement

Data deposited in the Dryad Digital Repository: DOI https://doi.org/10.5061/dryad.1jwstqjqh [START_REF] Mahaut | Data from: Multiple facets of diversity effects on plant productivity: species richness, functional diversity, species identity and intraspecific competition[END_REF]

Author contributions

G.T.F., F.F. and C.V. planned and designed the research. G.T.F. and F.F. performed the experiments. L.M. analysed the data and wrote the manuscript. G.T.F., C.V., F.F. and L.M. contributed substantially to revisions. Posterior model probability report the probability that each functional group, functional dissimilarity, CWM and species identity models (M1i) is

Supplementary informations


Matteo Rucco
email: ruccomatteo@gmail.com

Katia Lupinetti
email: katia.lupinetti@ge.imati.cnr.it

Franca Giannini
email: giannini@ge.imati.cnr.it

Marina Monti
email: monti@ge.imati.cnr.it

Jean-Philippe Pernot
email: jean-philippe.pernot@ensam.eu

J.-P Pernot

CAD Assembly Retrieval and Browsing

Keywords: Assembly retrieval, shape matching, information visualization

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The large use of CAD (Computer Aided Design) and CAM (Computer Aided Manufacturing) systems in industries has generated a number of 3D databases making available a large amount of 3D digital models. The reuse of these models, either single parts or assemblies, and the exploitation of the knowledge associated with them are becoming an important way to facilitate new designs. To track and organize data related to a product and its lifecycle, modern CAD systems are integrated into PDM (Product Data Management) and PLM (Product Lifecycle Management) systems. Among others, the associated data usually involve the technical specifications of the product, provisions for its manufacturing and assembling, types of materials used for its production, costs and versioning. These systems efficiently manage a search based on textual metadata, which cannot be sufficient to effectively retrieving the searched data. Actually, standard parts, text-based annotation and naming convention are company-or operator-specific, thus difficult to generalize as search keys. To overcome these limitations, content-based algorithms for 3D model retrieval are being developed based on shape characteristics. A wide literature is available and some commer-cial systems provide shape-based model retrieval. [START_REF] Biasotti | Retrieval and clas-sification methods for textured 3D models: a comparative study[END_REF][START_REF] Cardone | A survey of shape similarity assessment algorithms for product design and manufacturing applications[END_REF][START_REF] Iyer | Three-dimensional shape searching: state-of-the-art review and future trends In[END_REF] provide an overview of the 3D shape descriptors most used in the CAD domain. However, these descriptors focus solely on the shape of a single component, which is not adapted for more complex products obtained as assemblies. An effective assembly search cannot be limited to simple shape comparison among components, but requires also information that is not always explicitly encoded in the CAD models, e.g. the relationships and the joint constraints between assembly components.

In this paper, we present methods for the retrieval of globally and/or partially similar assembly models according to different user-specified search criteria [START_REF] Lupinetti | CAD assembly descriptors for knowledge capitalization and model retrieval[END_REF] and for the inspection of the provided results. The proposed approach creates and exploits an assembly descriptor, called Enriched Assembly Model (EAM), organized in several layers that enable multi-level queries and described in section 4.1. The rest of the paper is organized as follows. Section 2 provides an overview of related works. Issues related to assembly retrieval are described in Section 3, while Section 4 presents the assembly descriptor and the comparison procedure. Section 5 reports some of the obtained results, focusing on the developed inspection capabilities. Section 6 concludes the paper discussing on current limits and future work.

Related works

Shape retrieval has been investigated far and wide in the recent years [START_REF] Biasotti | Retrieval and clas-sification methods for textured 3D models: a comparative study[END_REF][START_REF] Cardone | A survey of shape similarity assessment algorithms for product design and manufacturing applications[END_REF][START_REF] Iyer | Three-dimensional shape searching: state-of-the-art review and future trends In[END_REF][START_REF] Tangelder | A survey of content based 3D shape retrieval methods[END_REF]. However, most of the work present in literature deal with the shape of a single component and do not consider other relevant information of the assembly such as the relationships between the parts. One of the pioneer works dealing with assembly retrieval was presented by Deshmukh, et al. [START_REF] Deshmukh | Content-based assembly search: A step towards assembly reuse In[END_REF]. They investigate the possible usage scenarios for assembly retrieval and proposed a flexible retrieval system exploiting the explicit assembly data stored in a commercial CAD system. Hu et al. [START_REF] Hu | Relaxed lightweight assembly retrieval using vector space model In[END_REF] propose a tool to retrieve assemblies represented as vectors of watertight polygon meshes. Identical parts are merged and a weight based on the number of occurrences is attached to each part in the vector. Relative positions of parts and constraints are ignored, thus the method is weak in local matching. Miura and Kanai [START_REF] Miura | 3D Shape retrieval considering assembly structure In[END_REF] extend their assembly model by including structural information and other useful data, e.g. contact and interference stages and geometric constraints. However, it does not consider high-level information, such as kinematic pairs and some information must be made explicit by the user. A more complete system is proposed by Chen et al. [START_REF] Chen | A flexible assembly retrieval approach for model reuse In[END_REF]. It relays on the product structure and the relationships between the different parts of the assembly. The adopted assembly descriptor considers different information levels including the topological structure, the relationships between the components of the assembly, as well as the geometric information. Thus, the provided search is very flexible accepting rough and incomplete queries. Anyhow, most of the work require user support for the insertion of the required information and weakly support the analysis and browsing of the obtained results, which for large assemblies can be very critical. To overcome these limitations, in this paper, we present an assembly descriptor (i.e. the Enriched Assembly Model), which can support user requests based on different search criteria not restrained to the identification of assembly models with the same structure in terms of sub-assemblies, and tools for facilitating the inspection and browsing of the results of the retrieval process.

Assembly retrieval issues

Retrieving similar CAD assembly models can support various activities ranging from the re-use of the associated knowledge, such as production or assembly costs and operations, to part standardization and maintenance planning. For instance, knowing that a specific subassembly, which includes parts having a high consumption rate due to their part surrounding and usage, is present in various larger products may help in defining more appropriate maintenance actions and better planning of the warehouse stocks. Similarly, knowing that different products having problems share similar configurations can help in detecting critical configurations. Considering these scenarios, it is clear that simply looking for products (i.e. assemblies) that are completely similar to a given one is important but limited. It is therefore necessary to have the possibility to detect if an assembly is contained into another as well as local similarities among assemblies, i.e. assemblies that contain similar sub-assemblies. These relations can be described using the set theory. Being ≅ the symbol indicating the similarity according to given criteria, given two assemblies A and B, we say that:

A is globally similar to B iff for each component a Depending on the retrieval purpose, not only the criteria change but also the interest on the similarity among the parts or on their connections can have different priority. It is therefore important to provide flexible retrieval tools that can be adapted to the specific need and thus able to consider the various elements characterizing the assembly despite on how the assembly was described by the user (e.g. structural organization) or on the information available on the CAD model itself (e.g. explicit mating conditions).

i ∈ A, ∃ b h ∈ B s.t. a i ≅ b h , for each relation (a i, a j ) ∈ A, ∃ (b h, b k ) ∈ B s.t. (a i,
In addition, it might be difficult to assess the effective similarity when various elements contribute to it. It is crucial to provide tools for gathering results according to the various criteria and for their inspection. This is very important in the case of large assemblies, where detecting the parts considered similar to a given assembly might be particularly difficult.

4

The proposed approach

Based on the above considerations, we propose a method for the comparison of assembly models exploiting various levels of information of the assembly. Differently from most of the work presented in literature, our method can evaluate all the three types of similarity described above. It uses a multilayer information model, the socalled Enriched Assembly Model (see section 4.1), which stores the data describing the assembly according to three different layers, in turns specified at different level of details thus allowing a refinement of the similarity investigation. Depending on the type of requested similarity, an association graph is build putting in relation the elements of the EAM of two CAD models to be compared. The similar subset of these two models are then corresponding to the maximal clique of the association graph (see section 4.2). To analyze the retrieved results, a visualization tool has been developed; it highlights the correspondences of the parts and provides statistics on the matched elements (see section 4.3).

Enriched Assembly Model (EAM)

The EAM is an attributed graph, where nodes are the components and/or composing sub-assemblies while arcs represent their adjacency relations. It uses four information layers: structure, interface, shape and statistics [START_REF] Lupinetti | CAD assembly descriptors for knowledge capitalization and model retrieval[END_REF].

The structural layer encodes the hierarchical assembly structure as specified at the design stage. In this organization, the structure is represented as a tree where the root corresponds to the entire assembly model, the intermediate nodes are associated with the sub-assemblies and the leaves characterize the parts. Attributes to specify parts arrangement (regular patterns of repeated parts) are attached to the entire assembly and to its sub-assemblies [START_REF] Lupinetti | Use of regular patterns of repeated elements in CAD assembly models retrieval In[END_REF]. The organization in sub-assemblies is not always present and may vary according to the designer's objectives.

The interface layer specifies the relationships among the parts in the assembly. It is organized in two levels: contacts and joints. The first contains the faces involved in the contact between two parts and the degree of freedom between them. The joint level describes the potentially multiple motions resulting from several contacts between two parts [START_REF] Lupinetti | Automatic Extraction of Assembly Component Relationships for Assembly Model Retrieval[END_REF].

The shape layer describes the shape of the part assembly by several dedicated descriptors. Using several shape descriptors helps answering different assembly retrieval scenarios, which can consider different shape characteristics and at different level of details. They include information like shape volume, bounding surface area, bounding box and spherical harmonics [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3 d shape descriptors[END_REF].

The statistics layer contains values that roughly characterize and discern assembly models. Statistics are associated as attributes to the various elements of the EAM. For the entire assembly and for each sub-assembly, they include: the numbers of subassemblies, of principal parts, of fasteners, of patterns of a specific type, of a specific joint type. To each node corresponding to a component, the statistics considered are: percentage of a specific type of surface (i.e. planar, cylindrical, conical spherical, toroidal, free form), number of maximal faces of a specific type of surface. Finally, for each arc corresponding to a joint between parts, the stored statistics include the number of elements in contact for a specific contact type.

The E.A.M. is created using ad hoc developed modules [START_REF] Lupinetti | CAD assembly descriptors for knowledge capitalization and model retrieval[END_REF][START_REF] Lupinetti | Use of regular patterns of repeated elements in CAD assembly models retrieval In[END_REF][START_REF] Lupinetti | Automatic Extraction of Assembly Component Relationships for Assembly Model Retrieval[END_REF], which analyze the content of the STEP (ISO 10303-203 and ISO 10303-214) representation of the assembly and extract the required information.

EAM comparison

Adopting this representation, if two models are similar, then their attribute graphs must have a common sub-graph. The similarity assessment between two EAMs can then be performed by matching their attribute graphs and finding their maximum common subgraph (MCS). The identification of the MCS is a well-known NP-hard problem and among the various techniques proposed for its solution [START_REF] Bunke | A comparison of algorithms for maximum common subgraph on randomly connected graphs[END_REF] we chose the detection of the maximal clique of the association graph, since it allows identifying also locally similarities.

The association graph is a support graph that reflects the adopted high-level similarity criteria. Each node in the association graph corresponds to a pair of compatible nodes in the two attributed graphs according to the specified criteria. Associated arcs connect nodes if they have equivalent relations expressed as arcs connecting the corresponding nodes in the attribute graphs.

A clique is a sub-graph in which for each couple of nodes a connecting arc exists. For the clique detection we applied the Eppstein-Strash algorithm [START_REF] Eppstein | Listing all maximal cliques in large sparse real-world graphs[END_REF]. This algorithm represents an improved version of the algorithm by Tomita [START_REF] Tomita | The worst-case time complexity for generating all maximal cliques and computational experiments[END_REF], which is in turn based on the Bron-Kerbosch algorithm for the detection of all maximal cliques in graphs [START_REF] Bron | Algorithm 457: finding all cliques of an undirected graph[END_REF]. As far as we know, Eppstein-Strash algorithm is up to now the best algorithm for listing all maximal cliques in undirected graphs, even in dense graphs. The performances of the algorithm are in general guaranteed by the degeneracy ordering.

The algorithm of Eppstein-Strash improves Tomita's algorithm by using the concept of degeneracy. The degeneracy of a graph G is the smallest number d such that every subgraph of G(V, E) contains a node of degree at most d. Moreover, every graph with degeneracy d has a degeneracy ordering: a linear ordering of the vertices such that each node has at most d neighbors after it in the ordering. Eppstein-Strash algorithm first computes the degeneracy ordering; then for each node v in the order, starting from the first, the algorithm of Tomita is used to compute all cliques containing v and v's later neighbors. Other improvements depend on the use of adjacency lists for data representation. For more details we refer to [START_REF] Eppstein | Listing all maximal cliques in large sparse real-world graphs[END_REF].

Among all the maximal cliques present in the associated graph, we consider as interesting candidates of the similar sub-graphs only those having: 1) the majority of arcs corresponding to real joints between the corresponding components, 2) a number of nodes bigger than a specified value. In this way, priority is given to sub-graphs which contain a significant number of joined similar components, thus possibly corresponding to sub-assemblies. Then, for each selected clique, a measure vector is computed. The first element of the vector indicates the degree of the clique, while the others report the similarity of the various assembly characteristics taken into consideration for the similarity assessment. Depending on the search objectives the set of characteristics to consider may change. The default characteristics are the shape of the components and the type of joint between them. The examples and results discussed in the next section consider the default characteristic selection.

Result visualisation

The proposed retrieval system has been implemented in a multi-module prototype system. The creation of the EAM description is developed by using Microsoft Visual C# 2013 and exploiting the Application Programming Interface (API) of the commercial CAD system SolidWorks. The matching and the similarity assessment module is developed by using Java and is invoked during the retrieval as a jar file. In the end, to analyze the obtained results, a browser view has been implemented. It is obtained by multiple dynamic web pages that are based on HTML5, jQuery, Ajax and PHP. Moreover, Mysql is used as database system, while X3D library is used for the STEP model visualization.

The system has been tested on assembly models obtained from on-line repositories [17, 18, 19] and from university students' tests. 2 shows an example of the developed user interface, where the design can choose an assembly model as query and set the required criteria of similarity. In this example, it is required to retrieve models similar for shape and joint. Some results of this query are shown in Fig. 3. The first model in the picture (top-left) coincides with the query model. The retrieved models are gathered together in the other views of Fig. 3. Each retrieved and displayed assemblies has a clique that has been detected in the association graph and satisfies the required conditions. The assemblies are visualized in X3D view that allows rotating, zooming and selecting the various 3D components of the retrieved assembly. Components are visualized in the transparency mode to make possible to see also the internal ones. Under each model, three bars are shown to quickly get an idea of how similar to the query the retrieved assemblies are. The first two indicate the percentage of coverage (i.e. percentage of matched elements) with respect to both the query and target model. Thank to these bars, the user can see the type of similarity (i.e. global, partial or local). If the green is not complete, it means that just a subset of the query model is matched, thus the similarity is locally. The global similarity is shown by the purple bar, if this bar is not complete, then the similarity is partial. The last bar shows the average shape similarity among the components associated with the displayed clique. Simply looking at the reported model and checking the purple bar, the user can notice that (except the first model which represents the query model) no models are globally similar to the query one according to the criteria he/she has specified. The first model in the second row is partially similar with the query one, since the query is completely included in it (see the green bar). Other models are locally similar with the query model, thus just a subset of the query model is included in them. If the user wants to further analyze the levels of similarity of the chosen characteristics or to visualize all the subsets of matched parts, he/she can select one of the retrieved assemblies and a new browser page is prompted. Once selected, a new page as in Fig. 4 is available, where the user can get the list of all the interesting clique, using the sliders at the top of window. With these sliders, the user can choose some thresholds that the proposed results have to satisfy. In particular, they refer to the dimension of the matched portion, the shape similarity measure and the joint similarity measure. After the setting of those parameters, the button "Clique finding" can be pressed to get the results displayed in a table as visible on the left of Fig. 5. The rows of the table gather together all the matching portions that satisfy the required criteria. In this example, four information are accessible for each matching portion: an identification number, the number of matched parts, the shape similarity and the joint similarity. Selecting one of them, the corresponding clique is visualized within the assembly. It highlights the component correspondence with the query model using same colors for corresponding components in the two objects, as shown on the right part of Fig. 4. To easy the comparison according to several available criteria (here reported just the default ones), a radar chart is used. It illustrates the shape and joint similarities among the overall assembly in relation to the clique degree. This type of visualization is very useful to compare multiple data and to have a global evaluation in just a look. Moreover, the radar charts are convenient to compare two or more models on various features expressed through numerical values. The larger is the covered area, the more the two assemblies are similar. In the reported case, the user can observe immediately that the two models are not completely matched, even if they look very similar. This is because the gears in the two models have a significant different shape, which avoids including those parts among the matched one, thus decreasing the global level of similarity. On the other side, the retrieved portion completely satisfies the requests, thus reporting an assemment of 1.

Conclusions

In this paper, methods for the identification and evaluation of similarities between CAD assemblies are presented. While almost all of the products are made of assembled parts, most of the works present in literature are addressing the problem of similarity among single parts. For assemblies, the shape of the components is not the only characteristic to be considered. Increasing the number of elements to consider, augments from the one hand the possibility of adapting the search to specific user needs, and on the other hand the difficulty to evaluate the results. The method here described can consider all or a subset of the various aspects of the assembly, namely the shape of the components, their arrangements (i.e. patterns), their mating contacts and joints. The evaluation of the retrieved results is supported by exploiting colour variations in the 3D visualisation of the components in correspondence between the compared assemblies. Measures and statistics quantifying the similarity of the overall assemblies and of the matched subparts are reported according to the various considered characteristics.

In future work, we plan to introduce graph databases, such as Neo4j, for speedingup the search of local similarity among big assembly models. We also intend to improve the clique-finding algorithm by allowing it to select automatically the dimension of the biggest clique. Moreover, we intend to involve the definition of a single measure for the overall ranking of the retrieved assemblies similar to a query one. This information will be displayed in an ad-hoc infographics, which will be developed for improving the user-experience.

Fig. 1 .

 1 Fig. 1. Example of different type of similarities

Fig. 2 .

 2 Fig. 2. An assembly model and the similarity criteria used for the matching

Figure

  Figure2shows an example of the developed user interface, where the design can choose an assembly model as query and set the required criteria of similarity. In this example, it is required to retrieve models similar for shape and joint. Some results of this query are shown in Fig.3. The first model in the picture (top-left) coincides with the query model. The retrieved models are gathered together in the other views of Fig.3. Each retrieved and displayed assemblies has a clique that has been detected in the association graph and satisfies the required conditions. The assemblies are visualized in X3D view that allows rotating, zooming and selecting the various 3D components of the retrieved assembly. Components are visualized in the transparency mode to make possible to see also the internal ones. Under each model, three bars are shown to quickly get an idea of how similar to the query the retrieved assemblies are. The first two indicate the percentage of coverage (i.e. percentage of matched elements) with respect to both the query and target model. Thank to these bars, the user can see the type of similarity (i.e. global, partial or local). If the green is not complete, it means that just a subset of the query model is matched, thus the similarity is locally. The global similarity is shown by the purple bar, if this bar is not complete, then the similarity is partial. The last bar shows the average shape similarity among the components associated with the displayed clique. Simply looking at the reported model and checking the purple bar, the user can notice that (except the first model which represents the query model) no models are globally similar to the query one according to the criteria he/she has specified. The first model in the second row is partially similar with the query one, since the query is completely included in it (see the green bar). Other models are locally similar with the query model, thus just a subset of the query model is included in them.

Fig. 3 .

 3 Fig. 3. A sample of the retrieved models for the proposed speed reducer query (top left)

Fig. 4 .

 4 Fig. 4. Initial page for investigating the model similarity

Fig. 5 .

 5 Fig. 5. Example of matching browsing