Eric Garnier
email: eric.garnier@cefe.cnrs.fr

Denis Vile

Catherine Roumet

Sandra Lavorel

Karl Grigulis

Marie-Laure Navas

Francisco Lloret

Inter-and intra-specific trait shifts among sites differing in drought conditions at the north western edge of the Mediterranean Region

Keywords: leaf traits, life form, multivariate functional space, plant height, seed mass, water availability, within-species trait variability

Identifying consistent and predictable associations between traits and environment is one of the oldest quest of ecology. Yet, there are few formal and robust quantification of such associations, which seriously impedes our capacity to predict how ecological systems respond to global changes, including climate. This study was designed to assess how differences in environmental conditions affect plant form and function in a wide array of species.

Twelve traits were measured on 40 species in three Mediterranean sites differing in drought conditions. Some species being common among sites, 78 species belonging to four major Raunkiaer life form categories were studied. These traits correspond to: (i) plant size: vegetative and maximum plant height, (ii) seed mass, (iii) leaf morpho-anatomical traits: leaf area, specific leaf area, dry matter content and thickness, (iv) leaf chemical composition: mass based nitrogen, phosphorus and carbon contents, and carbon isotopic fraction.

On average, there was a shift in the phenotypic space towards more resource conservative and taller species in the drier sites. These changes were not always consistent for hemicryptophytes and chamaephytes on the one hand, and for phanerophytes on the other hand. This is interpreted as different species responding to different aspects of complex changes in environmental factors.

Intraspecific trait variation differed among species, and was lower than interspecific variation.

Changes in site-average trait values were therefore mostly driven by species turnover among sites.

The traits selected do not respond strongly to the differences in environmental conditions however, resulting in a moderate shift in the phenotypic space between sites. We argue that traits more directly related to plant water economy should be considered for an improved description of plant phenotypic response to the environmental factors at stake. The implications for the prediction of plant responses to climate changes likely to occur in the Mediterranean Region are discussed.

Highlights for

"Inter-and intra-specific trait shifts among sites differing in drought conditions at the north western edge of the Mediterranean Region" by Garnier et al. (Special Issue of Flora)  Species from different life forms react differently to complex ecological gradients  Average trait values among sites are mostly driven by interspecific variation  The combination of traits studied is moderately sensitive to changes in environmental factors  More conservative species are not found to be systematically less plastic  Plant responses to climate depend on interactions with other environmental factors Identifying consistent and predictable associations between traits and environment is one of the oldest quest of ecology (e.g. [START_REF] Schimper | Plant Geography upon a Physiological Basis[END_REF], which has been boosted by the pressing need to improve our understanding and modelling of global change impacts on vegetation (e.g. Smith et al., 1997;[START_REF] Verheijen | Impacts of trait variation through observed trait-climate relationships on performance of an Earth system model: a conceptual analysis[END_REF]. A central assumption of trait-based ecology is that environmental factors are major selective forces that shape the form and function of organisms [START_REF] Grime | Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory[END_REF][START_REF] Westoby | Plant ecological strategies: Some leading dimensions of variation between species[END_REF]: a given set of environmental conditions sorts individuals in a local community based on the value of their traits [START_REF] Keddy | Assembly and response rules: two goals for predictive community ecology[END_REF][START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: revisiting the Holy Grail[END_REF], thereby favoring particular trait values or combinations thereof ("strategies" sensu [START_REF] Grime | Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory[END_REF] in these specific conditions. However, and in spite of the wealth of studies that have identified relationships between traits and various environmental factors [START_REF] Garnier | Plant Functional Diversity -Organism Traits, Community Structure, and Ecosystem Properties[END_REF] for a review), there are surprisingly few formal and robust quantifications of such relationships [START_REF] Weiher | Advances, challenges and a developing synthesis of ecological community assembly theory[END_REF][START_REF] Shipley | Reinforcing loose foundation stones in trait-based plant ecology[END_REF].

Trait-environment associations may arise as a consequence of differences in trait values both among species and among populations of a same species under different environmental conditions (e.g. [START_REF] Ackerly | A trait-based approach to community assembly: partitioning of species trait values into within-and among-community components[END_REF][START_REF] Lepš | Community trait response to environment: disentangling species turnover vs intraspecific trait variability effects[END_REF]. Under the simple hypothesis that a low trait value confers a selective advantage in one environment while a high trait value confers a selective advantage in another environment, it can be predicted that the direction of trait-environment relationships should be the same (i) for all species and (ii) within and among species. Although this is often the case, different groups of species have also frequently been found to react differently to the same environmental cue (e.g. [START_REF] Wright | Modulation of leaf economic traits and trait relationships by climate[END_REF][START_REF] Read | Convergent effects of elevation on functional leaf traits within and among species[END_REF][START_REF] Šímová | Spatial patterns and climate relationships of major plant traits in the New World differ between woody and herbaceous species[END_REF]. Diverse combinations of responses within and among species have also been observed, including: (i) same directional change within and among species, in agreement with the hypothesis stated above, (ii) directional changes among species with no change within species, (iii) directional changes among species and idiosyncratic changes within species, and (iv) opposite directional changes among and within species (reviewed in [START_REF] Vellend | Drawing ecological inferences from coincident patterns of population-and community-level biodiversity[END_REF]. Both genetic and environmental determinants might be responsible for these varying patterns, such as gene flow within species opposing local selection or different species responding to different aspects of complex natural gradients [START_REF] Vellend | Drawing ecological inferences from coincident patterns of population-and community-level biodiversity[END_REF].

The chief objective of this study is to test whether the variations in trait values are consistent within and among species of different life forms found in sites with differing drought conditions within the Mediterranean Region. Beyond the general bimodal pattern of cool wet winters and warm dry summers, Mediterranean-type climates show a wide range of temperature and rainfall variations (e.g. [START_REF] Nahal | The mediterranean climate from a biological viewpoint[END_REF][START_REF] Daget | Mediterranean bioclimate and its variation in the palaeartic region[END_REF] which impact the spatial distribution of vegetation across the Mediterranean Region (cf. [START_REF] Quézel | Ecologie et Biogéographie des Forêts du Bassin Méditerranéen[END_REF]. Understanding associations between traits and environment induced by such variations would help predict the response of vegetation to climate change in this region, considered as one of the most vulnerable areas of the planet (Thiébault and Moatti, 2016). More arid climates are expected in the future in the Mediterranean Region, with increases in both temperature and length of the summer dry period [START_REF] Kovats | Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part B: Regional Aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF] leading to potentially stronger impacts of drought on vegetation [START_REF] Quintana-Seguί | Drought: observed trends, future projections[END_REF]. We thus selected sites differing in drought conditions, using a space-for-time substitution to assess the potential impacts of predicted climate changes on vegetation (discussed in [START_REF] Koch | Terrestrial transects for global change research[END_REF][START_REF] Fukami | Long-term ecological dynamics: reciprocal insights from natural and anthropogenic gradients[END_REF].

Numerous aspects of plant form and function differ between species from environments of low and high water availability, such as overall plant stature and allocation patterns, leaf and root structure, anatomy and physiology, hydraulic properties of the different organs, etc. [START_REF] Reich | The evolution of plant functional variation: traits, spectra and strategies[END_REF]Craine, 2009 for reviews). Studies that have considered simultaneously traits from these different aspects are rare however (but see [START_REF] Nunes | Which plant traits respond to aridity? A critical step to assess functional diversity in Mediterranean drylands[END_REF][START_REF] De La Riva | A multidimensional functional trait approach reveals the imprint of environmental stress in Mediterranean woody communities[END_REF]. Here, we test how traits capturing various dimensions of plant functioning differ between species and populations of species from sites with differing drought conditions. In particular, we assess how the phenotypic space defined with the traits of the recently identified "global spectrum of plant form and function"

("global spectrum" hereafter; [START_REF] Díaz | The global spectrum of plant form and function[END_REF] is modified in response to changes in these conditions. This spectrum distributes species along two major axes: the first one organizes species

A C C E P T E D M A N U S C R I P T
according to their stature, organ size (leaf and seed) and stem specific density, while the second one describes the trade-off between resource acquisition and conservation in leaves ("fast-slow continuum" hereafter: cf. Wright et al., 2004;[START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: a traits manifesto[END_REF]. From less to more arid sites, we expect a shift along the two axes of the global spectrum, from high to low stature on the one hand, and from species acquiring resources rapidly to species conserving resources efficiently on the other hand.

Traits specific to plant water economy are not explicitly taken into account in the fast-slow continuum however (but see Reich, 2014 for a discussion), while a high water use efficiency (indicative of efficient water conservation) is expected under more arid conditions (Prentice et al., 2014 and references therein). We thus evaluated whether this was indeed the case, and also assessed the position of leaf carbon isotopic fraction (Lδ 13 C, used as a surrogate of water-use efficiency: cf. [START_REF] Farquhar | Carbon isotope discrimination and photosynthesis[END_REF] in the phenotypic space of the global spectrum. The expected differences between sites of contrasting drought conditions were examined both within and among species, making the further and surprisingly rarely tested hypothesis that within species variations are less pronounced for species which conserve resources efficiently [START_REF] Grime | The role of plasticity in resource capture by plants[END_REF]discussed in Valladares et al., 2007).

To address these questions, we selected three sites with differing drought conditions at the north-western edge of the Mediterranean Region. These sites were chosen so as to limit as much as possible potential confounding factors which might influence trait variation. However, controlling for disturbances in a region where long-standing and widespread human activities and recurring fires are common selecting forces acting on vegetation (cf. [START_REF] Thompson | Plant Evolution in the Mediterranean[END_REF] is not straightforward. We thus also qualitatively account for potential effects of differences in disturbance regimes among sites in addition to those of drought. Using trait values measured for 40 species of different life forms in each site, we test the following hypotheses: (1) changes in average trait values between sites are due to changes in both species identity and life forms and to trait variation within species, (2) the major traits structuring the two axes of the global spectrum of plant form and function (plant size and the fast-slow continuum) capture adequately plant response to drought, (3) leaf carbon isotopic fraction

A C C E P T E D M A N U S C R I P T
is higher in more arid sites (indicating higher water use efficiency) and scales with the fast-slow axis of the global spectrum, and (4) the magnitude of intraspecific trait variation is smaller in more conservative species.

Material and methods

Study sites

The study was conducted at three sites located in the north western part of the Mediterranean Region: Garraf, in north eastern Spain, Cazarils and La Fage in southern France (Fig. 1, Table 1). In the three sites, the climate is Mediterranean with mild wet winters and warm dry summers. Mean annual temperature decreases from 16.1 °C in Garraf to 9.9 °C in La Fage, while mean annual precipitation is almost twice as high in the French sites as compared to the Spanish one (Table 1). This results in a gradual increase in the aridity index (the ratio between mean annual rainfall and mean annual evapotranspiration: United Nations Environment Programme, 1992) from Garraf (dry sub-humid climate) to Cazarils and La Fage (both classified as humid). The length of the dry period, as assessed by the period during which mean monthly rainfall < 2 × mean monthly temperature [START_REF] Gaussen | L'indice xérothermique[END_REF], also gradually increases from Garraf to La Fage (see climate diagrams in Fig. 1). While still classified as Mediterranean, the climate in La Fage is at the limit of the temperate zone. >>insert Fig. 1 here >>insert Table 1 here At the three sites, the bedrock is limestone and soils have comparable pH values (Table 1). In Garraf, the vegetation consists in a mosaic of shrublands and Pinus halepensis open forests growing on rocky soils and on terraces that were established over marls and cultivated in the past centuries.

Agricultural abandonment started at the end of the 19 th century and was complete around the mid-20 th century. Several wildfires had burnt since then, particularly in 1982 and 1994, affecting most of

A C C E P T E D M A N U S C R I P T
the area. Shrubs (macrophanerophytes), scrubs (nanophanerophytes) and herbaceous species are important components of the community (Lloret and Vilá, 2003 for details). In Cazarils, the site consists in a large clearing in an otherwise mostly wooded area, with Quercus pubescens and Q. ilex as dominant tree species. This clearing, which was cultivated until the mid-20 th century, is surrounded by degraded Mediterranean shrublands that were regularly cut for fuel. Over the past 70 years or so, recreational hunting, selective clearcutting and burning, and extensive grazing by sheep and goats have been the main disturbances at this site. Former and current land uses have resulted in a complex spatial organization of the vegetation with plant communities composed of species from different growth and life forms [START_REF] Floc'h | Biodiversity and ecosystem trajectories: first results from a new LTER in southern France[END_REF]Navas et al., 2010 for details). In La Fage, the area was probably cultivated until the 19 th century, at least partially. It was then extensively grazed until 1972, both by local and transhumant sheep herds, allowing woody species to recolonize a large part of the area. Since 1972, it has been subdivided into paddocks either grazed by sheep at a low stocking rate or fenced to prevent grazing. The vegetation consists primarily in rangelands dominated by perennial grasses and forbs, with a woody component covering approximately 20-25% of the area [START_REF] Molénat | Pour un élevage ovin allaitant performant et durable sur parcours: un système expérimental sur le Causse du Larzac[END_REF] for details).

Species selection

At all three sites, the vegetation consists in a mosaic of patches with different vegetation types, composed of various mixtures of trees, shrubs and herbaceous species (Fig. 1). Forty species per site, making a total of 120 "populations" (40 x 3), were selected for study on the basis of their frequent occurrence in these different types of vegetation (Table A.1 in Appendix). 21 species were shared between Garraf and Cazarils, 10 between Garraf and La Fage, 19 between Cazarils and La Fage, and eight species were common to the three sites. The 120 populations represented a total of 78 species across the three sites, belonging to 26 botanical families, mostly represented by Poaceae and Fabaceae (11 species each), Lamiaceae (10 species), Cistaceae (7 species), Asteraceae and Rosaceae (5 species each). All species were perennials, and could be assigned to five life form [START_REF] Raunkiaer | The Life Forms of Plants and Statistical Plant Geography, English Edition[END_REF]: 29 chamaephytes, 28 hemicryptophytes, 14 evergreen phanerophytes, 6 deciduous phanerophytes and one geophyte. Chamaephytes and hemicryptophytes were represented by at least 10 species in each site, while evergreen phanerophytes and hemicryptophytes were substantially more numerous in Garraf and La Fage respectively, reflecting the composition of the vegetation at these sites (Fig. A.1 in Appendix); apart from geophytes, the deciduous phanerophyte life form was the least represented overall: 3, 6 and 5 species in Garraf, Cazarils and La Fage, respectively (Fig. A.1 and Table A.1 in Appendix). In some analyses (see below), hemicryptophytes, chamaephytes and the geophyte were grouped into "low stature species", while deciduous and evergreen phanerophytes were grouped into "high stature" species.

A C C E P T E D M A N U S C R I P T categories sensu

Trait selection and measurements

Eleven traits were measured on healthy individuals of the 40 species selected in the three sites (Table 2), and one -maximum plant height -was taken from a reference flora of the Mediterranean region concerned [START_REF] Tison | Flore de la France Méditerranéenne Continentale[END_REF]. These traits were selected on the basis of their relevance for plant functioning (Table 2; [START_REF] Pérez-Harguindeguy | New handbook for standardised measurement of plant functional traits worldwide[END_REF]Garnier et al., 2016 for details). They were measured during the spring peak of growth (between May and July) over the 1999-2007 period (Table 1). Methods, which mostly follow [START_REF] Pérez-Harguindeguy | New handbook for standardised measurement of plant functional traits worldwide[END_REF], are briefly summarized below. Terminology, which follows [START_REF] Garnier | Towards a thesaurus of plant characteristics: an ecological contribution[END_REF], and number of replicates are given in Table 2. Traits were grouped into three categories:

Whole plant and seed size. Vegetative plant height (plant height hereafter) was recorded as the upper 20% of the foliage of adult plant, taking only vegetative structures into account. The ratio between these measured values and maximum plant height (PHmax) drawn from the flora (one single value per species) was taken as an index of development of the individuals selected for measurement (Hratio): the closer to 1, the more advanced the individuals toward the potential height the species can reach in a Mediterranean context. The dry mass of mature seeds (SM) was measured on samples of Dispersal structures, fruit flesh and sclerotesta were removed as carefully as possible, and only the seed coat (testa) was left on the cleaned seeds. Missing seed mass data (for 5, 11 and 10 species in Garraf, Cazarils and La Fage, respectively) were completed by values taken primarily from BROT, a trait data base for species of the Mediterranean Region [START_REF] Paula | Fire-related traits for plant species of the Mediterranean Basin[END_REF][START_REF] Paula | BROT: a plant trait database for Mediterranean Basin species[END_REF] and from the Seed Information Database (Royal Botanic Gardens Kew, 2018), for the two species for which there were no data in BROT (Ruta chalepensis and Satureja montana).

Leaf morpho-anatomy. Leaf area (LA), specific leaf area (SLA) and leaf dry matter content (LDMC) were assessed on intact, full-grown leaves taken from plants in full light situations (i.e. not under tree cover) or from the outer canopy for tall woody species, following complete leaf rehydration. SLA and LDMC were calculated as the ratio of leaf area to leaf dry mass, and of leaf dry mass to leaf water saturated fresh mass, respectively. Leaf thickness was estimated as 1/(SLA*LDMC) [START_REF] Vile | Specific leaf area and dry matter content estimate thickness in laminar leaves[END_REF].

We also measured leaf length and width, but do not show results for these two traits since all analyses were very similar to those found for leaf area.

Leaf chemical composition. Leaves sampled to determine the traits described above were pooled to obtain three to four bulked samples for nitrogen, carbon and phosphorus analyses. These samples were ground individually, and their total mass-based nitrogen and carbon content (LNCm and LCCm, respectively) determined with an elemental analyzer (Carlo Erba Instruments, model EA 1108, Milano, Italy). The carbon isotopic fractions (Lδ 13 C values relative to PDB standard) were determined on separate batches from the same samples, using a CHN elemental analyzer coupled online to an isotope mass spectrometer (ANCA-MS system, Europa Scientific Ltd, Crewe, UK). Mass-based leaf phosphorus content (LPCm) was determined on additional leaf samples similar to those previously described. Approximately 20 mg of leaf material were ground and digested using concentrated sulphuric acid and hydrogen peroxide at 100 °C for 35 min and 360 °C for 2 h. Mass based phosphorus content was then determined with a colorimetric autoanalyzer (Alliance Instrument, Evolution II, Frépillon, France), using the molybdenum blue method [START_REF] Grimshaw | Nutrient elements[END_REF]. 2 here

A C C E P T E D M A N U S C R I P T >>insert Table

Data analyses

Differences between sites in mean trait values per species were tested using one-way ANOVAs followed by post-hoc Tukey tests using different subsets for all species, for each life form separately (Ranunculus gramineus, the only geophyte in the data set, was merged with hemicryptophytes in all analyses), for low and high stature species, and then for shared and sitespecific (unshared) species. For each trait, we calculated standardized effect sizes (SES hereafter) for all three sites taken together (contrast weights set to the differences in aridity index between sites) and between pairs of sites using the Hedges's d metric:

𝑑 = 𝑌 1 -𝑌 2 √ (𝑛 1 -1)𝑠 1 2 -(𝑛 2 -1)𝑠 2 2 𝑛 1 + 𝑛 2 -2 𝐽 (𝑒𝑞. 1)
where Y1 and Y2 are the site average values for sites 1 and 2, respectively, n1 and n2 are the number of species sampled in sites 1 and 2 respectively, s1 and s2 are the standard deviations of the means, and J i a correction factor for small sample sizes (cf. [START_REF] Rosenberg | Effect sizes: conventional choices and calculations[END_REF]. By convention, site 1 was always taken as the less arid site in all calculations, i.e. Cazarils for the Garraf-Cazarils contrast, and La Fage for the Cazarils-La Fage and Garraf-La Fage contrasts (cf. Table 1). Bootstrap confidence intervals of Hedges's d were calculated using the function bootES (R/bootES package: [START_REF] Kirby | BootES: An R package for bootstrap confidence intervals on effect sizes[END_REF].

Principal component analyses (PCA) were performed to investigate the phenotypic space as shaped by the relationships between traits across sites. Between-site PCA was performed on all studied traits to test for differences between sites within the phenotypic space [START_REF] Chessel | The ade4 package -I : One-table methods[END_REF].

The null hypothesis that there is no difference between sites was tested with a randomization test (randtest.between in the R/ade4 package). A second PCA ("global spectrum" PCA hereafter) included

A C C E P T E D M A N U S C R I P T
only the traits used in [START_REF] Díaz | The global spectrum of plant form and function[END_REF] to describe the global spectrum of plant form and function (cf. Table 2).

The relative contributions of intra-and interspecific trait variability effects on site-average trait values were assessed following [START_REF] Lepš | Community trait response to environment: disentangling species turnover vs intraspecific trait variability effects[END_REF] for each site contrast. The method is based on the decomposition of the total sum of squares (SStotal) of the site-level trait variance into "fixed" (SSfixed), "intraspecific" (SSintraspecific) and "covariation" (SScov) effects, so that SStotal = SSfixed + SSintraspecific + SScov. Briefly, for each site and trait, we calculated site-average trait values using species trait values as measured in that site (which includes both inter-and intraspecific effects), and "fixed" site average trait values using species trait values averaged over the pair of sites considered (which does not account for the intraspecific variability effect). Then, we extracted the SS for each of the three site-average measures (SStotal, SSfixed, and SSintraspecific), and calculated the SScov component, which is the effect of covariation between inter-and intraspecific trait variability by subtracting SSfixed and SSintraspecific from SStotal. Positive (resp. negative) covariation indicates that intra-and interspecific variations are responding in the same (resp. opposite) direction between sites.

Next, we tested the hypothesis that intraspecific trait variation was lower in more conservative species. To do so, we assumed that the score of species on the global spectrum PCA axis describing the slow-fast continuum of leaf functioning was indicative of the position of this species on the acquisitive/conservation axis of functional variation (cf. Wright et al., 2004), and followed a three step procedure. First, this score was calculated for each site and averaged across sites (for species present in more than one site) after varimax rotation to maximize factor loadings on the dimension defined by the combination of SLA, LDMC and LNCm (see Fig. 3B). Second, to quantify intraspecific trait variation, we calculated the absolute value of SES for each species present in at least two sites, using individual replicate trait values (as opposed to species averages in the analyses described in the previous paragraph). Finally, relationships between the score of species on the slow-fast axis of the PCA and SES values were tested for each trait. These relationships gave qualitatively similar results to those established using the plasticity index proposed by Valladares et (2006). We thus retained SES as estimates of intraspecific variation to maintain a consistency of methods. Note that in all analyses pertaining to intraspecific variability (variance partitioning and variability among species), environmental heterogeneity within sites was not accounted for.

A C C E P T E D M A N U S C R I P T al.
All analyzes were performed in the programming environment R (R Development Core Team, 2017).

Results

Site differences in trait values among and within life forms

The direction of changes in trait values averaged for the 40 species per site were generally similar between Garraf and Cazarils and between Cazarils and La Fage ( 3), and (ii)

variations in trait values within life-forms among sites (Fig. 2), especially for hemicryptophytes and chamaephytes; in phanerophytes, differences were significant only between Garraf and Cazarils for few traits (VPH and LNCm in evergreens: Fig. 2A and 2I, LPCm and LCCm in deciduous: Fig. 2J and2K).

>>insert Fig. 2 here >>insert Table 3 here A striking result is that the direction of change in VPH and Hratio between Garraf and the two other sites was negative for hemicryptophytes and chamaephytes (low stature species), while the trend was opposite for evergreen phanerophytes (Fig. 2 1). Differences among sites for the other traits were generally more consistent in the two groups of species (Fig. A.2B and C;Table A.3 in Appendix). One exception concerns Lδ 13 C, which was higher (less negative values) in Garraf than in Cazarils in low stature species, suggesting higher water use efficiency in the more arid site for these species (Fig. 2L andA.2B). This tendency was not confirmed between Cazarils and La Fage however, and was not found in the case of high stature species.

Changes in the multivariate trait space among sites

Fifty-seven percent of the variation in functional space defined by the 12 traits was accounted for by the plane defined by the first two principal components of the PCA (Fig. 3A, Table 4). The traits contributing most to the first axis of this plane were VPH, PHmax, LDMC, SLA, Hratio, SM and Lδ 13 C, while those contributing most to the second axis were LPCm, LNCm, LTest, SM, SLA and LA to a lower extent (Table 4). Lδ 13 C was weakly correlated with SLA (r = -0.23, P < 0.05) and LNCm (r = -0.30, P < 0.01), and more strongly with LDMC (r = -0.41, P < 0.001). When only the six traits used to define the phenotypic space of the "global spectrum of plant form and function" were selected (cf.

Table 2), the first two principal components explained 72% of the variation (Fig. 3B, Table 4). The >>insert Fig. 3

here >>insert Table 4 here

There was substantial overlap among the multivariate trait spaces of the three sites, but also a significant shift of the barycenters of the spaces toward species of lower stature showing lower LDMC, higher SLA, LNCm and LPCm in less arid sites (Fig. 3C&D). The displacements of the barycenters were somewhat different for low and high stature species. In the former, there were comparable shifts along the first two components of the PCA (white symbols on Fig. 3C&D), while in the latter the shift was almost parallel to axis 2 (dark grey symbols on Fig. 3C&D). This further illustrates that in

Intra-and interspecific differences in trait values

There were striking differences among species shared between sites both in the direction and magnitude of intraspecific differences in trait values (Fig ). As a consequence, differences in site-average trait values were mostly explained by differences in the trait values of species unshared between sites: for the three site contrasts, species turnover explained the largest percentage of variance for the majority of traits (Fig. 4). The only exceptions were for LCCm in the case of the Garraf-Cazarils contrast (Fig. 4A) and for Lδ 13 C in the case of the Cazarils-La Fage contrast (Fig. 4B). For the two extreme sites (Garraf and La Fage), the percentage of variance explained by species turnover was always higher than 60%, and intraspecific variation did not explain more than 10% of the variance for any trait (Fig. 4C).

>>insert Fig. 4 here Fig. 5 shows the relationships between the score of the species on the "slow-fast" dimension of the PCA conducted on the six traits of the global spectrum (Fig. 3B; cf. Methods section for means of calculation) and the SES calculated for each trait and for each species common to at least two sites, used as a means to quantify intraspecific variation. Various patterns were found among traits, and for a given trait, between site contrasts. Out of the 30 relationships tested (10 traits × 3 site contrasts), only four were significant (Fig. 5 andTable A.4 in Appendix): three were positive (higher

A C C E P T E D M A N U S C R I P T
intraspecific variation for species from the fast end of the continuum), for SLA, LNCm and LPCm for the Cazarils-La Fage contrast; one was negative, for Lδ 13 C in the case of the Garraf-La Fage contrast.

>>insert Fig. 5 here

Discussion

Variations in traits values among sites differing in environmental conditions were found to differ for species from the different of life forms. Size related traits varied in opposite directions in low and high stature plants, while variations for leaf morpho-anatomical and chemical traits were more consistent. This translated into different shifts in the first plane of multivariate phenotypic space for low and high stature species. Trait values did not always vary in the same direction within and among species. This, combined with the different proportion of life forms with contrasting trait values across the three sites, resulted in changes in site-average trait values being mostly driven by species turnover. We further discuss these different points below.

Differences in trait values among sites

Our first result is that trait values do not always change in a consistent way for species of the different life forms. Plant height, leaf dimensions and seed mass of low stature species (hemicryptophytes and chamaephytes) were all higher under more arid conditions, while for high stature species, the trend was either opposite (evergreen phanerophytes) or non-significant (deciduous phanerophytes). This suggests either differences in response to the same environmental factor among groups of species, or differences in environmental conditions sensed by the two groups of species among sites. The tendency observed for plant height in low stature species is unexpected, as plant stature generally decreases with decreasing rainfall (e.g. [START_REF] Mooney | Photosynthetic capacity and carbon allocation patterns in diverse growth forms of Eucalyptus[END_REF]Gross et height) in less arid sites for these species is suggestive of higher disturbance levels in the low stratum of vegetation in Cazarils and La Fage as compared to Garraf. The last wildfire, the primary disturbance factor in Garraf, burnt in 1994 (7 years before measurements at this site), and likely affected all vegetation strata. By contrast, the most recurrent disturbance in the two French sites is grazing, likely to affect mostly low stature species. Our results thus suggest that over a common background of differences in drought conditions, the nature and level of disturbances differ for low and high stature species among sites (see [START_REF] Navas | Suites of plant traits in species from different stages of a Mediterranean secondary succession[END_REF] for a specific analysis at the Cazarils site), with a stronger impact on height in the former (see [START_REF] Díaz | Plant trait responses to grazing -A global synthesis[END_REF][START_REF] Garnier | Plant Functional Diversity -Organism Traits, Community Structure, and Ecosystem Properties[END_REF][START_REF] Herben | Effects of disturbance frequency and severity on plant traits: An assessment across a temperate flora[END_REF], for syntheses of the impact of disturbance on plant height). Further differences in trait values should thus be interpreted in the light of this complex combination of differences in environmental factors among the three sites (cf. [START_REF] Vellend | Drawing ecological inferences from coincident patterns of population-and community-level biodiversity[END_REF].

Trends in leaf area variations among sites were qualitatively comparable to those for plant height, both for low (smaller leaf area in less arid sites) and high (larger leaf area in less arid sites) stature plants. The recent worldwide analysis conducted by [START_REF] Wright | Global climatic drivers of leaf size[END_REF] shows that climate controls on leaf area involve a complex interaction between mean annual rainfall and mean temperature of the warmest month. Under the combination of these climate variables experienced by plants in the three sites studied here, we would expect smaller leaf area in Garraf (588 mm/28.5 °C) than in Cazarils (1075 mm/30 °C) and no difference between Cazarils and La Fage (1039 mm/24.4 °C). This is precisely what was found for high stature species. The reverse trend observed for low stature species might be related to the disturbance issues discussed above, as higher disturbancesincluding grazing -select for species with small leaves (reviewed by [START_REF] Garnier | Plant Functional Diversity -Organism Traits, Community Structure, and Ecosystem Properties[END_REF].

Higher seed mass is assumed to offer an advantage in drier conditions because the seedlings produced are more capable of resisting environmental hazards and reserves are needed for droughtresistance mechanisms (e.g. [START_REF] Leishman | The role of large seed size in shaded conditions -Experimental evidence[END_REF][START_REF] Lloret | Seedling survival of Mediterranean shrubland species in relation to root:shoot ratio, seed size and water and nitrogen use[END_REF]. Higher seed mass in drier conditions have sometimes been observed [START_REF] Wright | Differences in seedling growth behaviour among species: trait correlations across species, and trait shifts along nutrient compared to rainfall gradients[END_REF][START_REF] Pakeman | Impact of abundance weighing on the response of seed traits to climate and land use change[END_REF],

A C C E P T E D M A N U S C R I P T
but as found in the present study, this is not always the case [START_REF] Mazer | Ecological, taxonomic, and life history correlates of seed mass among Indiana dune Angiosperms[END_REF][START_REF] Leishman | The role of large seed size in shaded conditions -Experimental evidence[END_REF].

Differences in leaf morpho-anatomical traits (SLA, LDMC and LTest) among sites were similar in hemicryptophytes, chamaephytes and evergreen phanerophytes. For these species, thicker leaves with high LDMC values (indicative of high leaf density; cf. [START_REF] Garnier | Leaf anatomy, specific mass and water content in congeneric annual and perennial grass species[END_REF] resulting in lower SLA values were found in more arid sites, which were also the less disturbed. Comparable trends have been found in response to water availability [START_REF] Mooney | Photosynthetic capacity and carbon allocation patterns in diverse growth forms of Eucalyptus[END_REF][START_REF] Cunningham | Evolutionary divergences in leaf structure and chemistry, comparing rainfall and soil nutrient gradients[END_REF][START_REF] Niinemets | Global-scale climatic controls of leaf dry mass per area, density, and thickness in trees and shrubs[END_REF] and disturbance [START_REF] Garnier | Assessing the effects of land use change on plant traits, communities and ecosystem functioning in grasslands: a standardized methodology and lessons from an application to 11 European sites[END_REF][START_REF] Herben | Effects of disturbance frequency and severity on plant traits: An assessment across a temperate flora[END_REF]. For water, this is interpreted as follows: cells of leaves developed under low water availability are smaller and more tightly packed, resulting in a lower fraction of air space in the leaf (Poorter et al., 2009 and references therein). Such leaves are stiffer, and are less prone to wilting under dry conditions (discussed in [START_REF] Niinemets | Global-scale climatic controls of leaf dry mass per area, density, and thickness in trees and shrubs[END_REF][START_REF] Poorter | Causes and consequences of variation in leaf mass per area (LMA): a meta-analysis[END_REF]. Lower SLA leaves under drier conditions have not systematically been found however [START_REF] Wright | Modulation of leaf economic traits and trait relationships by climate[END_REF][START_REF] Ordoñez | Plant strategies in relation to resource supply in mesic to wet environments: Does theory mirror nature?[END_REF][START_REF] Nunes | Which plant traits respond to aridity? A critical step to assess functional diversity in Mediterranean drylands[END_REF], with non-linear trends between SLA and rainfall showing increasing SLA at the drier end of gradients sometimes observed [START_REF] Gross | Uncovering multiscale effects of aridity and biotic interactions on the functional structure of Mediterranean shrublands[END_REF][START_REF] De La Riva | A multidimensional functional trait approach reveals the imprint of environmental stress in Mediterranean woody communities[END_REF]. Slightly higher SLA in the more arid site was also found here in deciduous phanerophytes, in which the two components of SLA -leaf density (approximated by LDMC) and thickness (cf. [START_REF] Witkowski | Leaf specific mass confounds leaf density and thickness[END_REF][START_REF] Vile | Specific leaf area and dry matter content estimate thickness in laminar leaves[END_REF] vary in opposite directions: LDMC tends to be higher in the more arid site (as expected) but leaves also tend to be thinner there, resulting in no net change in SLA values among sites. While the response of LDMC and/or leaf density to water availability appears consistent among studies, the response of leaf thickness is more controversial (see [START_REF] Poorter | Causes and consequences of variation in leaf mass per area (LMA): a meta-analysis[END_REF], calling for more studies to better understand the effects of water availability on the underlying components of SLA.

LNCm and LPCm were consistently higher in less arid sites, irrespective of the level of disturbance experienced by species from the different life-forms. This does not appear to be systematic however [START_REF] Ordoñez | Plant strategies in relation to resource supply in mesic to wet environments: Does theory mirror nature?[END_REF][START_REF] Rumman | Variation in bulk-leaf 13 C discrimination, leaf traits and water-use efficiency-trait relationships along a continental-scale climate gradient in Australia[END_REF], with non-linear responses sometimes observed (de la [START_REF] De La Riva | A multidimensional functional trait approach reveals the imprint of environmental stress in Mediterranean woody communities[END_REF]. Theoretical considerations actually suggest that low-

A C C E P T E D M A N U S C R I P T
rainfall species should operate at higher leaf nitrogen per unit leaf area (LNCa), which has been verified in several studies [START_REF] Wright | Least-cost input mixtures of water and nitrogen for photosynthesis[END_REF]Schulze et al., 2006a;[START_REF] Rumman | Variation in bulk-leaf 13 C discrimination, leaf traits and water-use efficiency-trait relationships along a continental-scale climate gradient in Australia[END_REF]. By contrast we did not find any significant difference in LNCa values among the three sites studied here

(1.66 ± 0.08, 1.64 ± 0.08 and 1.70 ± 0.09 g m -2 in Garraf, Cazarils and La Fage, respectively). This is due to changes in both SLA and LNCm going in the same direction and with a comparable amplitude across sites (LNCa = LNCm/SLA). The same conclusion holds for LPC expressed per unit leaf area (data not shown). This might be due to nitrogen and phosphorus availability in soils being higher in less arid sites, but whether this is a direct effect of the higher water availability is unknown.

Variations in LCCm among sites were small, which might be why we did not observe higher LCCm in less arid sites (see [START_REF] Ma | Variations and determinants of carbon content in plants: a global synthesis[END_REF]. Contrary to expectations (e.g. [START_REF] Prentice | Balancing the costs of carbon gain and water transport: testing a new theoretical framework for plant functional ecology[END_REF], Lδ 13 C values were not systematically higher (less negative values indicating higher water use efficiency) in more arid sites: this was only the case for low stature species in the case of the Garraf-Cazarils contrast. The trend was even reversed for species of all four life-forms between Cazarils and La Fage.

Although relationships between Lδ 13 C and several indicators of water availability have been found in several studies (e.g. [START_REF] Turner | Annual rainfall does not directly determine the carbon isotope ratio of leaves of Eucalyptus species[END_REF][START_REF] Prentice | Balancing the costs of carbon gain and water transport: testing a new theoretical framework for plant functional ecology[END_REF][START_REF] Rumman | Variation in bulk-leaf 13 C discrimination, leaf traits and water-use efficiency-trait relationships along a continental-scale climate gradient in Australia[END_REF], it has been argued that these might be mediated through indirect effects of water availability on SLA and LNCm (Schulze et al., 2006b). Since the pattern of correlations between Lδ 13 C and these two traits observed in our study is comparable to that found by these authors, we may hypothesize that differences in Lδ 13 C among sites are driven by differences in leaf traits induced by complex environmental variations rather than by water availability per se.

The shape of the phenotypic space and its changes with environmental conditions

The shapes of the phenotypic spaces assessed using either the 12 traits or the six traits of the global spectrum were very similar: Axis 1 of the full PCA was mostly structured by plant size and seed mass on the one hand, and leaf tissue density and carbon composition on the other hand; Axis 2 mostly described leaf characteristics (SLA, LTest, LNCm, LPCm and LA). The reduction to six traits did not [START_REF] Ackerly | Functional strategies of chaparral shrubs in relation to seasonal water deficit and disturbance[END_REF][START_REF] Díaz | The global spectrum of plant form and function[END_REF]. The strongest bivariate relationship was found with LDMC, one of the components of SLA (cf. [START_REF] Witkowski | Leaf specific mass confounds leaf density and thickness[END_REF][START_REF] Garnier | Leaf anatomy, specific mass and water content in congeneric annual and perennial grass species[END_REF]. Higher Lδ 13 C values in high LDMC leaves have also been observed by [START_REF] Prieto | Leaf carbon and oxygen isotopes are coordinated with the leaf economics spectrum in Mediterranean rangeland species[END_REF] in species from Mediterranean rangelands and by [START_REF] Rumman | Variation in bulk-leaf 13 C discrimination, leaf traits and water-use efficiency-trait relationships along a continental-scale climate gradient in Australia[END_REF] in woody species across a large climate gradient in Australia.

Relationships between Lδ 13 C and other leaf traits (SLA, LNCm and LPCm in particular) were not consistent in the different studies (Schulze et al., 2006b;[START_REF] Prieto | Leaf carbon and oxygen isotopes are coordinated with the leaf economics spectrum in Mediterranean rangeland species[END_REF][START_REF] Rumman | Variation in bulk-leaf 13 C discrimination, leaf traits and water-use efficiency-trait relationships along a continental-scale climate gradient in Australia[END_REF].

We can thus conclude that water use efficiency better relates to traits describing tissue density than to traits of the slow-fast continuum of leaf functioning.

Whether patterns of co-variations among traits identified at a global scale (e.g. the leaf economics spectrum: Wright et al., 2004; the global spectrum of plant form and function: [START_REF] Díaz | The global spectrum of plant form and function[END_REF] bear some relevance to interpret variation at the scale of local communities has recently been questioned [START_REF] Funk | Leaf traits within communities: Context may affect the mapping of traits to function[END_REF][START_REF] Messier | Trait variation and integration across scales: is the leaf economic spectrum present at local scales?[END_REF]. Here, we show a close agreement between the phenotypic space of the global spectrum as detected in the worldwide analysis conducted by [START_REF] Díaz | The global spectrum of plant form and function[END_REF] and the organization of variables in the PCAs conducted in each of the three sites. This might be due to species belonging to contrasting life forms in our study, which allowed us to cover a relatively wide range of values for both plant size (from 2 to 738 cm for plant height, a major component of Axis 1 of the global spectrum PCA) and leaf traits (e.g. from 4.3 to 30 m 2 kg -1 for SLA, a major component of Axis 2 of the global spectrum PCA). Funk and Corwell (2013) further postulated that in communities dominated either by herbaceous or woody species, relationships among the traits of the leaf economics spectrum might be weak because within community variation in leaf life span is low. In one of the sites studied here (Cazarils), leaf life span was found to vary between 59 and 802 days for a subset of the species screened [START_REF] Navas | Leaf life span, dynamics and construction cost of species from Mediterranean old-fields differing in successional status[END_REF].

The relative standard deviation of leaf life span for this data set is about 0.6, a value for which correlations among the traits of the leaf economics spectrum are high (Funk and Corwell, 2013). We Although differences in environmental conditions among the three sites induced an overall shift toward smaller and less conservative plants from Garraf to La Fage, the overlap in the phenotypic spaces of the three sites was substantial. This indicates that the selected traits do not respond very strongly to the differences in environmental conditions between sites, although these correspond to a doubling of annual rainfall and aridity index from the more to the less humid site.

The six core traits of the global spectrum, which allow us to qualify functionally a broad range of species using size and resource acquisition and use at the leaf level, therefore appear only marginally sensitive to these environmental factors. This calls for the inclusion of traits more specific to the ecophysiological processes underlying the response, which would probably improve the detection of the environmental signal on plant functioning (discussed in [START_REF] Field | Questions of scale[END_REF][START_REF] Rosado | Going back to basics: Importance of ecophysiology when choosing functional traits for studying communities and ecosystems[END_REF]. For example, functional strategies of shrubs in relation to water deficit and disturbance have been shown to be better described using the minimum water potential reached during the growing season and the life span of leaves than several of the traits used in the present study [START_REF] Ackerly | Functional strategies of chaparral shrubs in relation to seasonal water deficit and disturbance[END_REF].

More generally, traits pertaining to the recently proposed hydraulic framework based on the efficiency of water transport throughout the plant [START_REF] Mcdowell | Mechanisms of plant survival and mortality during drought: why do some plants survive while others succumb to drought?[END_REF][START_REF] Sack | Plant hydraulics as a central hub integrating plant and ecosystem function: meeting report for 'Emerging Frontiers in Plant Hydraulics[END_REF][START_REF] Martínez-Vilalta | Water potential regulation, stomatal behaviour and hydraulic transport under drought: deconstructing the iso/anisohydric concept[END_REF] might be better candidates than the traits of the global spectrum to assess the response of plants to changes in water availability (see also [START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: a traits manifesto[END_REF].

Intraspecific variations among sites: patterns and consistency among species

Our results show that differences in site-average trait values were mostly driven by species turnover for almost all traits. Although changes in trait values within species have been found along gradients of water availability in a number of studies (e.g. [START_REF] Abrams | Genotypic and phenotypic variation as stress adaptations in temperate tree species: a review of several case studies[END_REF][START_REF] Rubio De Casas | Field patterns of leaf plasticity in adults of the long-lived evergreen Quercus coccifera[END_REF][START_REF] Bagousse-Pinguet | Traits of neighbouring plants and space limitation determine intraspecific trait variability in semi-arid shrublands[END_REF], the results found here suggest that such intraspecific variation may potentially have a low impact on changes in community-average trait values. This was also found by [START_REF] Gross | Uncovering multiscale effects of aridity and biotic interactions on the functional structure of Mediterranean shrublands[END_REF] in shrublands distributed along a rainfall gradient in Spain, and is in line with results reported in [START_REF] Siefert | A global meta-analysis of the relative extent of intraspecific trait variation in plant communities[END_REF] meta-analysis showing that on average, the contribution of intra-specific trait variation was substantially lower than that of interspecific variation to explain total trait variation among communities.

A C C E P T E D M A N U S C R I P T
In several cases, trait values changed in opposite directions among species, which implies that for certain species, values changed in a direction opposite to site-average values (see [START_REF] Vellend | Drawing ecological inferences from coincident patterns of population-and community-level biodiversity[END_REF] for other examples). For some traits (plant height, leaf area), opposite patterns were found for species from the high compared to the low stratum, suggesting that species respond to different aspects of the complex gradient studied [START_REF] Vellend | Drawing ecological inferences from coincident patterns of population-and community-level biodiversity[END_REF], corresponding here to various combinations of drought and disturbance (see section 4.1. above). For other traits (e.g. SLA, Lδ 13 C), trait variation was found to differ for species from the same stratum. This echoes results showing idiosyncratic trait variation within species along elevational gradients (e.g. [START_REF] Kichenin | Contrasting effects of plant inter-and intraspecific variation on community-level trait measures along an environmental gradient[END_REF][START_REF] Lajoie | Understanding context dependence in the contribution of intraspecific variation to community trait-environment matching[END_REF], even for co-occurring dominant species [START_REF] Lajoie | Understanding context dependence in the contribution of intraspecific variation to community trait-environment matching[END_REF]. One hypothesis that can be put forward to explain this counter-intuitive finding is that ecological sorting does not directly operate on such traits. In fact, and as shown by the multivariate analysis conducted in the present study, the response of plants to environmental factors involves the concomitant response of multiple, sometimes non-independent, traits [START_REF] Chapin | Evolution of suites of traits in response to environmental stress[END_REF]Garnier et al., 2016 for syntheses). As recognized for long [START_REF] Grime | Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory[END_REF][START_REF] Chapin | Evolution of suites of traits in response to environmental stress[END_REF], ecological sorting is an inherently multidimensional process which calls for an approach based on the whole-organism phenotype [START_REF] Kleyer | Why functional ecology should consider all plant organs: An allocationbased perspective[END_REF][START_REF] Laughlin | Fitness of multidimensional phenotypes in dynamic adaptive landscapes[END_REF]. Inconsistent changes in trait values with changes in environmental factors further reinforces the idea that different trait combinations might lead to success in a given environment [START_REF] Marks | Alternative designs and the evolution of functional diversity[END_REF][START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: a traits manifesto[END_REF].

We found evidence of smaller intraspecific variation in more conservative species for three traits of the fast-slow continuum of leaf functioning (SLA, LNCm and LPCm) and for one of the site contrasts. Such low responsiveness has been considered as part of a conservative resource-use

A C C E P T E D M A N U S C R I P T
strategy, assumed to be adaptive in resource-poor environments [START_REF] Grime | The role of plasticity in resource capture by plants[END_REF][START_REF] Valladares | Ecological limits to plant phenotypic plasticity[END_REF][START_REF] Grassein | Plant resource-use strategies: the importance of phenotypic plasticity in response to a productivity gradient for two subalpine species[END_REF]. Our results provide only weak support for this hypothesis however, since for traits other than the three mentioned above, we did not find any significant relationship between the position of the species on the fast-slow continuum and our assessment of intraspecific trait variation (which was even larger in more conservative species for Lδ 13 C). This can be discussed in the light of the "maladaptive plasticity" concept, which states that low or no plasticity might be adaptive under certain circumstances [START_REF] Valladares | Ecological limits to plant phenotypic plasticity[END_REF]. In particular, when changes in environmental conditions involve multiple factors, variations of trait values in one direction might prove beneficial for one factor and detrimental for another one.

Manipulative experiments separating the effects of drought and disturbance would be necessary to test whether this applies in the case of the main environmental factors identified as driving trait responses in the present study.

Implications for the prediction of plant responses to climatic changes

Under a scenario of median emissions of greenhouse gases, climate change predictions for the northern part of the Mediterranean Region by the end of the 21 st century include an increase in temperature by 2.5 to 3.5 °C in winter and 4 to 5 °C in summer, and a limited change of precipitation in winter, with a 20% decrease in summer [START_REF] Planton | The climate of the Mediterranean regions in the future climate projections[END_REF]. This will lead to increased impacts of drought on vegetation at least in summer, both via direct effects of reduced rainfall and indirect effects of increased temperature on potential evaporation. The spatial variations of climatic variables from the French to the Spanish sites studied here are qualitatively comparable to the temporal changes predicted for the end of the century, albeit with a larger spatial variation for precipitation.

What can be inferred from the present study to forecast the impacts of predicted climatic change on vegetation in this part of the world?

A first important issue raised by our study is that the effects of climate will be modulated by local environmental factors. That local conditions strongly interact with mesoclimate to control plant

A C C E P T E D M A N U S C R I P T
traits and vegetation structure has long been recognized (e.g. [START_REF] Cain | Life-forms and phytoclimate[END_REF], and has been explicitly

shown for several of the traits measured here [START_REF] Pakeman | Relative climatic, edaphic and management controls of plant functional trait signatures[END_REF]Gardarin et al., 2014 for disturbance;[START_REF] Simpson | Soil-climate interactions explain variation in foliar, stem, root and reproductive traits across temperate forests[END_REF][START_REF] Borgy | Plant community structure and nitrogen inputs modulate the climate signal on leaf traits[END_REF] for soil nutrient availability). In areas such as the Mediterranean Region, where fire and/or human activities have shaped ecosystem structure and function for long and will continue to do so in the future, potential modifications of the disturbance regime should thus be taken into account in addition to, or in interaction with climatic factors for a proper assessment of vegetation changes. This is all the more important that different species (e.g. low vs. high stature species) are likely to experience changes corresponding to various combinations of environmental factors, comparable to those observed across the complex spatial gradient studied here.

Assuming that the combinations of factors likely to change are properly identified, the next step is to assess which species will be selected under the new environmental conditions. Our study suggests that more pronounced and longer droughts should select for species with a more conservative leaf syndrome (low SLA, LNCm and LPCm, high LDMC). Predictions concerning plant stature are less straightforward, since these will depend on the changes in the disturbance regime that will go with the changes in climate. Whatever the case, we argue that the phenotypic space described using traits pertaining to the global spectrum of plant form and function is not particularly sensitive to variations in the availability of water. Improved predictions should thus be based on traits that better account for the underlying ecophysiological mechanisms allowing plants to cope with water limitations. In particular, several hydraulic traits such as the hydraulic safety margin-the difference between typical minimum xylem water potential and that causing xylem dysfunction-and xylem vulnerability to embolism, have been shown to predict drought-induced tree mortality at large spatial scales [START_REF] Choat | Global convergence in the vulnerability of forests to drought[END_REF][START_REF] Anderegg | Metaanalysis reveals that hydraulic traits explain cross-species patterns of drought-induced tree mortality across the globe[END_REF]. A better integration of such traits into an improved multidimensional description of the phenotype is clearly one of the key challenges for functional ecology in the near future (see discussion in [START_REF] Laughlin | The intrinsic dimensionality of plant traits and its relevance to community assembly[END_REF][START_REF] Kleyer | Why functional ecology should consider all plant organs: An allocationbased perspective[END_REF]. Table 1). Relationships between the score of the species on the slow-fast axis of the PCA shown in Fig. 3B ("global spectrum" PCA; see methods section for means of calculation) (x-axis) and their intraspecific trait variation (y-axis), assessed by calculating SES within species for each trait listed in Table 2, except PHmax (one value per species) and Hratio, for which results are the same as for VPH.

A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C
Colours correspond to the different site contrasts (G-C: Garraf-Cazarils; C-L: Cazarils-La Fage; G-L: Garraf-La Fage). Solid and dashed lines show significant (P < 0.05) and non-significant relationships, respectively (Table A.4 for statistics). 1. Main characteristics of the three study sites: Garraf, Cazarils and La Fage (see Fig. 1 for pictures). Temperature and precipitation data over the 1981-2010 period were taken from the meteorological stations of the Barcelona airport for Garraf (20 km east of the field site) and of Saint-Martin-de-Londres for Cazarils (3.5 km north-east of the field site); on-site data were available for La Fage. PET was calculated using the Hargreaves equation [START_REF] Hargreaves | Reference crop evapotranspiration from temperature[END_REF]. Hratio (VPH/PHmax: see Table 2) of low and high strata are respectively average values for hemicryptophytes and chamaephytes, and for phanerophytes (deciduous and evergreen). The aridity index and climate classes follow UNEP (1992). [START_REF] Pérez-Harguindeguy | New handbook for standardised measurement of plant functional traits worldwide[END_REF][START_REF] Garnier | Plant Functional Diversity -Organism Traits, Community Structure, and Ecosystem Properties[END_REF] for details), together with their abbreviations and units. Traits used in the analysis of the global spectrum of plant form and function [START_REF] Díaz | The global spectrum of plant form and function[END_REF] are in bold italics. All traits were measured from samples taken on-site, except maximum plant height, which was taken from [START_REF] Tison | Flore de la France Méditerranéenne Continentale[END_REF].

  seeds depending on species reproductive outputs and collected on at least ten individuals.

  Fig. 2 all species, and Fig. A.2A in Appendix). For most traits, the standardized effect sizes were larger between the two extreme sites (Garraf and La Fage), intermediate between Garraf and Cazarils and lower between Cazarils and La Fage (Fig. A.2A). Differences in site average values result from the combination of: (i) differences in the proportions of life-forms across the three sites (Fig. A.1), as a consequence of significant differences in average trait values among the four life forms for many traits (Table

  organization of variables in this space was strikingly similar between the global data set used by Dìaz et al. (2016) and the data set of the present study (Fig. A.3 in Appendix).

  , the values of size-related traits (e.g. height and seed size) decreased toward the less arid sites, while in phanerophytes, there was only little differences for those traits among sites (see alsoFig. 2 and Fig. A.2C & B). Shifts in leaf morpho-anatomical and chemical traits were more comparable for the two groups of species.

  . A.4 in Appendix): while for some traits, changes in values were relatively consistent (e.g. most species showed higher LNCm and LPCm in Cazarils than in Garraf on the one hand and in La Fage compared to Cazarils on the other hand: Fig.A.4I and J), for many others, different species showed different directions of changes between sites (e.g. SLA between Cazarils and La Fage: Fig. A.4F

  la[START_REF] De La Riva | A multidimensional functional trait approach reveals the imprint of environmental stress in Mediterranean woody communities[END_REF]. The lower Hratio (individuals further away from their maximum potential

  organization. Contrary to our hypothesis, Lδ 13 C scaled only weakly with the traits commonly used to capture the fast-slow continuum of leaf functioning (SLA and LNCm: Wright et al.

  the large range of trait values spanned in our data set explains the close agreement between the global and local phenotypic spaces found here.

Fig. 1 .Fig. 2 .

 12 Figure captions

Fig. 3 .

 3 Fig. 3. The first two components of a Principal Component Analysis run on the fourteen traits listed in Table2(panels A and C), and on the six traits used to define the phenotypic space of the global spectrum of plant form and function (cf.[START_REF] Díaz | The global spectrum of plant form and function[END_REF] (panels B and D). On panels A and B, black arrows indicate direction and weighing of vectors representing the traits considered; panels C and D show the position of species in the plane defined by the two first components of the PCA (dots), color coded in red for Garraf, green for Cazarils and blue for La Fage. Inertia ellipses are drawn around the barycenter of each group using the "s.class function" of the ade4 package. The barycenters of the three groups calculated for the 40 species per site are shown as a red circle (Garraf), green triangle (Cazarils) and blue square (La Fage, respectively. The corresponding barycenters for low stature species (hemicryptophytes and chamaephytes) are drawn in white, while those for high stature species (phanerophytes) are drawn in dark grey; arrows connect barycenters in the direction of increasing values of the aridity index (from more to less severe summer drought: cf. Table1). Table2for abbreviations.

Fig. 4 .

 4 Fig. 4.Variance partitioning of site-average trait differences for the three site contrasts: A, Garraf-Cazarils; B, Cazarils-La Fage, and C, Garraf-La Fage. The total between-site variance in trait values was partitioned into species turnover (dark grey bars), intraspecific trait variation (medium gray bars) and covariation components (light grey bars). Variance partitioning was conducted for 10 of the traits studied (PHmax and Hratio [= VPH/PHmax] were not included in the analyses because the former is constant for a given species, and the partitioning of the latter gives the same results as for VPH). Table2for abbreviations.

  Fig.5. Relationships between the score of the species on the slow-fast axis of the PCA shown in Fig.3B("global spectrum" PCA; see methods section for means of calculation) (x-axis) and their intraspecific trait variation (y-axis), assessed by calculating SES within species for each trait listed in Table2, except PHmax (one value per species) and Hratio, for which results are the same as for VPH. Colours correspond to the different site contrasts (G-C: Garraf-Cazarils; C-L: Cazarils-La Fage; G-L: Garraf-La Fage). Solid and dashed lines show significant (P < 0.05) and non-significant relationships, respectively (TableA.4 for statistics).

  Percentage of variance explained, eigenvalues and trait loadings of principal components (PC1, PC2 and PC3) for the two PCAs shown on Fig.4. The first analysis corresponds to the PCA performed with the 12 traits shown on Fig.1, while the second one is for the PCA run on the six traits of the global spectrum of plant form and function (cf.

  and compare Fig. A.2B and A.2C; see Table A.3 in Appendix for significance levels of ANOVAs run on low and high stature species). This means

	U S C R I P T
	N
	A
	M
	C C E P T E D
	A

that individuals of low stature species were taller and closer to their potential height in Garraf (cf.

"Low stratum Hratio" in Table

1

), while the reverse was true for evergreen phanerophytes (cf. "High stratum Hratio" in Table

Table 1 )

 1 : Garraf (dark grey); Cazarils (medium gray); La Fage (light grey). Hem: hemicryptophytes; Cham: chamaephytes; DPh: deciduous phanerophytes; EPh: evergreen phanerophytes. Within each group of three boxplots, different letters indicate significant differences among sites (see alsoTable A.2 in Appendix). Table2for abbreviations.

  Table 2 for abbreviations.

  Table 2 for abbreviations.

Table Table

 Table

Reference for detailed description of site

	Name Site location Latitude Longitude Altitude range (m a.s.l.) Average soil pH (water) Mean annual temperature (°C) Mean annual precipitation (P, mm) Mean annual evapotranspiration (PET, mm) Aridity Index (P/PET) Mean temperature of the coldest month (°C) Mean temperature of the warmest month (°C) Mean precipitation of the driest month (mm) Climate class	Garraf Spain, Catalonia 41°18'N 1°54'E 250-350 7.2-8.2 16.1 588 987 0.60 4.7 28.5 21 Dry sub-humid M A N U Cazarils France, Bas Languedoc 43°46'N 3°42'E 270-310 7.1 13.5 1075 1102 0.98 0.2 30.0 26 Humid S C R I P T La Fage France, Grands Causses 43°56'N 3°06'E 765-830 7.3 1039 1.23 24.4 44 Humid -0.6 843 9.9
	Site-average Hratio Low stratum Hratio High stratum Hratio Year and period of trait measurements C C E P T E D	0.45 0.51 0.35 2001 (June) Lloret and Vilá (2003)	0.31 0.23 0.51 1999 and 2001 (May-July) Le Floc'h et al. (1998)	0.18 0.15 0.41 2006 and 2007 (May-July) Molénat et al. (2005)
	A

Table 2 .

 2 Traits measured and their functional significance (Pérez

  Stem specific density rather than leaf dry matter content is used inDìaz et al. (2016)

	Type of trait	Trait	Functional	Abbreviation	Unit	Number of
			significance			replicates
	Whole plant	Vegetative plant height	Light capture, above-	VPH	cm	8-30
			ground competition,
			dispersal distance
		Maximum plant height	Potential height	PHmax	cm	-
			adult plants can
			reach under
			Mediterranean
			climate conditions
		Vegetative to maximum	Developmental index	Hratio	Unitless	8-30
		plant height ratio	of the individuals	= VPH / PHmax
			selected for
			measurement
	Leaf morpho-anatomy	Area	Light interception,	LA	cm 2	5-14
			energy balance
		Specific leaf area	Photosynthetic rate,	SLA	m 2 kg -1	5-14
			leaf longevity,
			relative growth rate
		Dry matter content*	Tissue density,	LDMC	mg g -1	5-14
			physical resistance,
			stress tolerance
		Thickness (estimated)	Physical resistance,	LTest	µm	5-14
			gas diffusion inside
			the leaf
	Leaf chemical	Nitrogen content/mass	Protein content	LNCm	mg g -1	3-8
	composition		(RubisCO),
			photosynthetic rate
		Phosphorus content/mass Content in nucleic	LPCm	mg g -1	3-8
			acids, bioenergetic
			molecules (e.g. ATP),
			membranes
		Carbon content/mass	Conversion factor	LCCm	mg g -1	3-8
			from gas exchange to
			mass based growth
			rate, lignin content
		Carbon isotopic fraction	Gas exchange, water-	Lδ 13 C	‰	3-8
			use efficiency
	Reproductive	Seed mass	Dispersal and	SM	mg	3-15
			regeneration
			strategy, seedling
			competition

*

Table 2 ;

 2 [START_REF] Díaz | The global spectrum of plant form and function[END_REF]. Loadings for these six traits are shown in italics.

		PCA on 12 traits	PCA on traits of the
				global spectrum
	Variation explained Eigenvalue Variable loadings Vegetative plant height Maximum plant height Hratio Seed mass Specific leaf area Leaf thickness (est.) Leaf phosphorus content Leaf carbon content Leaf carbon isotopic fraction Leaf nitrogen content Leaf dry matter content Leaf area	PC1 38.2 4.59 -0.89 -0.76 -0.63 -0.62 0.75 -0.29 0.34 -0.55 -0.62 0.55 -0.76 -0.31	PC2 19.1 2.29 -0.30 -0.47 0.07 -0.55 -0.52 0.58 -0.68 -0.06 -0.07 M -0.59 A PC3 10.6 1.27 0.10 0.12 0.02 0.14 -0.27 0.73 0.21 -0.17 -0.33 0.11 0.05 -0.52 N PC1 47.1 2.83 -0.84 -0.64 0.76 0.60 -0.80 -0.49 0.40 -0.36 U S C R I P T PC2 PC3 24.9 12.0 1.49 0.72 -0.31 0.10 -0.59 0.36 -0.41 0.07 -0.65 0.30 0.21 0.20 -0.63 -0.67
	C C E P T E D
	A

Acknowledgements

We thank Gérard Laurent, Anabelle Dos Santos, Jean Richarte, Lorraine Bottin, Elodie Chapuis, Magdy El-Bana, Pablo Cruz, Adeline Fayolle and the numerous students who contributed to data collection during the successive field campaigns. This work was partially supported by the Laboratoire Européen Associé "Mediterranean Ecosystems in a Changing World" and the INRA EcoGer project "DivHerbe". We thank the staff of the La Fage INRA experimental station for maintaining the site and for support to carry out part of the fieldwork, as well as James Aronson and Edouard Le Floc'h for creating favorable conditions for research at the Cazarils site. This study contributes to the "Biogéographie fonctionnelle des plantes du bassin méditerranéen" project funded by the BioDivMex initiative and was conducted with the support of LabEx CeMEB (convention ANR-10-LABX-0004) through the funding of the "Plant functional biogeography in the Mediterranean" project.


Benoît Delinchant
email: benoit.delinchant@g2elab.inpg.fr

Harijaona Lalao Rakotoarison

Vincent Ardon

O Chadebec

O Cugat

Gradient based optimization of semi-numerical models with symbolic sensitivity: Application to a simple ferromagnetic MEMS switch device

Keywords: Optimization, Gradient, Sensitivity, Multi-level, MEMS. I

come

Gradient based optimization of semi-numerical models

with symbolic sensitivity: Application to a simple ferromagnetic MEMS switch device

Introduction

Designing an electromagnetic device requires optimization algorithms and adapted models. In this paper, the virtual design of a ferromagnetic MEMS switch -and especially its preliminary sizing-is addressed, using MacMMems software [START_REF] Rakotoarison | Methodology and tool for generating semi-analytical models used to pre-design electromagnetic MEMS[END_REF] for the modelling, and CADES framework [START_REF] Delinchant | An optimizer using the software component paradigm for the optimization of engineering systems[END_REF] for the optimization. The modelling software can produce parameterized semi-analytical modelling of the device with symbolic sensitivity. Thanks to accuracy parameters, a multi-level strategy can be deployed in the optimization framework.

The ferromagnetic switch (also called "magnetic latch") is used as an optical switch in our case ; it is based on the rocking actuation of a ferromagnetic plate under the influence of a magnetic field variation. In our device, the static magnetic field is produced by a permanent magnet and altered by planar coils, as sketched in Fig. 1.

I. A software for semi-numerical modelling with formal sensitivity A. Semi-numerical modelling of magnetic MEMS

Modelling magnetic fields from permanent magnets and coils

Assumptions are constant magnetization of magnets and constant current density in conductors.

For magnets, a Coulombian equivalent charge approach is used with the following surface and volume charges :

n M S .   (1)   M div V    (2)
The resulting magnetic excitation is the following.

 

          S s V V ds r r r r dv r r r r r H 3 3 ) .( . ) .( ) (   (3)
As div(M)=0, volume charges are equal to zero leading to quick surface integration instead of classical volume integration. For conductors, Biot and Savart law is used.

                 dv r r r r j r H V 3 ) ( 4 1 ) (   (4)
Magnetic excitations were symbolically computed, as far as possible, for some basic shapes and magnetization direction [3][4].

Computing forces and torques on magnet, coil and ferromagnetic bodies

For magnet, an equivalent charge method is used.

dv H ds H F v ext v s ext s     . .   (5)     dv H AP ds H AP v ext v v s ext s s        . .   (6) For conductors, Laplace law is used. dv H j F v ext    . 0  (7)   dv H j AP v ext      . 0  (8)
For specifics configurations like parallelepiped magnets interaction [START_REF] Akoun | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF], force can be fully analytical. In our software, due to space transformation between interacting objects, and the possibility of there displacement during an optimization, force and torque computation will use numerical integration method.

For ferromagnetic material, a steady state Method of Moments (MoM) is applied. It consists in the meshed of ferromagnetic body along the X, Y and Z axes, with uniform induced magnetized elementary blocks [START_REF] Rakotoarison | Using Coulombian approach for Modeling Scalar Potential and Magnetic Field of a Permanent Magnet with Radial Polarization[END_REF]. This method doesn't require to mesh air and is particularly performing for radiating system. The issue of such an integral method is it's full matrices and then it's computation memory limitation.

H ext

M Q H H ext    (9)
Blocks interaction is defined by the interaction matrix (Q in equation 9). Q is a square matrix of size (3m)x(3m) (m is the number of blocks), composed of 3x3 square matrices Q ab. This matrix represented the magnetic excitation created by block (a) to the barycentre of block (b) and correpond to a well-known point matching approach.

           mm m ab m Q Q Q Q Q Q     1 1 11 ,            zz
Q Q Q Q Q Q Q Q Q Q (10) with ]² ; 1 [ ) , ( m b a 
In the case of parallelepiped block [START_REF] Euleaume | Computing 3D magnetic fields from insertion devices[END_REF], Q jk are defined by the following equations: (w x , w y , w z ) is emitting block dimensions.

    2 1 2 2 2 1 2 1 , , 1 1 4 1 tan ) 1 (             k j i k j i k j i k j i xx z y x z y x Q  (11)                     2 1 , , ) 1 ( 2 2 2 4 1 1 2 1 ln k j i k j i k xy k j i z y x z Q  (12)
(x 0 , y 0 , z 0 ) is target block barycentre. Q yy , Q zz are deduced from Q xx and Q xz , Q yz are deduced from Q xy .
The ferromagnetic material law is defined by H=f(M) curve, parameterized by the saturation induction and initial permeability as shown in equation 13.

) 2 ( ) 1 ( 2 ) ( sat r sat M M tg M M f        (13)
The unknowns, ferromagnetic magnetizations, can then be found by solving the non-linear equation 14 :

M H M f ext       Q ) ( ( 14
)
To do it, a Newton-Raphson solver has been used with a relaxation method to ensure convergence.

When magnetizations are known, each block can be seen as parallelepiped magnets, to compute magnetic field or force and torque as explained in figure 3. a generic one based on a numeric solver where complex shapes can be introduced thanks to a meshing tool. The second, which is detailed in the following, is MacMMems [START_REF] Rakotoarison | Efficient and accurate modelling of Mag-MEMS for the optimization of system design[END_REF][5] (Macro Modeller for Magnetic MEMS) dedicated to provide analytical models which aim to be derivate for optimization process. It is limited to simple shapes in order to produce symbolic derivates.

In MacMMems, geometry and physics are described easily thanks to an intuitive GUI (Graphical User Interface). Each body owns its equations depending on the material (magnet, coil, ferromagnetic, diamagnetic) and its shape (parallelepiped block, part or full cylinder). Space transformations can be added (translations and rotations). Magnetic field, force and torques can be computed. Depending on desired computations, MacMMems Dimensions, physics and spaces transformations can be parameterized leading to a very useful modeling.

Since a fully parameterized model, constituted of equations and call to functions, a symbolic treatment can be done to express parameter sensitivities. This task is realized in CADES Generator detailed in section C.

Comparison with FEM Modelling

The modeling of our device with a FEM software is very uneasy because a new heavy air mesh generation is needed at each switch position. For example, two hours are needed to plot the curve on figure 5 . In contrast, the semi-numerical model needs only 5 seconds for a ferromagnetic part decomposed in 6x3x1 blocks which sufficient accuracy for optimization needs.

CADES Framework with pure algebraic models

First of all, CADES framework is able to compute formal sensitivities from algebraic equations based on [START_REF] Atienza | A Methodology for the Sizing and the Optimization of an Electromagnetic Release[END_REF] work. To do so, model is analysed, treating equations and functions, finding parameters required for the computation (input parameters such as geometry or physics parameters) and the resulting output parameters (such as magnetic field forces and torques). Partial derivatives for each equation or function are symbolically determined thanks to derivation rules. Model partial derivatives are then determined by the composition of equations and functions. Model's jacobian matrix is then explicitly defined by equations and functions.

CADES Framework with semi-numerical models

For numerical functions such as integrals (needed to compute torques and forces or fields produced by complex shapes) or numerical algorithm as previous ferromagnetic magnetization solver, it is easy to compute numerical sensitivities, but it is known that they brought optimization difficulties. For this reason, in CADES Framework, each numerical function is able to return partial derivatives determined symbolically as far as possible.

In the case of integration functions, partial derivatives can be fully expressed. Indeed, considering function f(d,c,x) to integrate with 'x' variable, from 'a' to 'b' which can depend on geometric parameters like 'd'. But for 'd' parameter, from which depends integration limits, the expression is defined by more complex equation 17. It can be noticed that the derivative of an integral can be computed without integral (this occurs, if the function f does not depend on parameter 'd') which is far faster than finite differences techniques.

)

) ( , , ( ) ( )) ( , , ( ) ( . ) , , ( ) , ( ) ( )
( d a c d f d d a d b c d f d d b dx d x c d f d c d I d b d a               (17)
In the case of moment method algorithm, a symbolic derivation can be expressed for field, force or torque calculation depending on optimizatable parameters (most of time geometric). This symbolic derivation is possible despite numerical algorithm, such as non linear equation solving to solve ferromagnetic magnetization state. Indeed, torque, which can be constrained or optimized, is a function of the magnetization of the ferromagnetic material, the external field and system parameters. Its partial derivative is defined by equation 18.

 

ext ext H M p M p H ext p dp d dp d dp p H M d , , , , ,                ext ext H H M M   (18)
dM/dp is the harder term to compute since it is obtained by numerical algorithm such as Newton-Raphson (see equation [START_REF] Gitosusastro | Performance Derivative Calculations And Optimization Process[END_REF]. But implicit theorem demonstrates that dM/dp can be symbolically expressed.

M p p f dp d M f dp d dp d dp d           ) ( . ) ( M M M M Q M Q H ext (19)
dM/dp can then be found by solving the following linear system.

M Q H M M M Q ext                 dp d dp d p f dp d M f M p ) ( ) ( (20)
Due to computation improvement requirements, the method used to compute d/dp is more complex [START_REF] Akoun | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF].

dM/dp is not computed since it is not really necessary and it requires solving a linear system for each parameter 'p'. Instead of dM/dp, an adjoin state vector, as described in [START_REF] Gitosusastro | Performance Derivative Calculations And Optimization Process[END_REF], is computed once by solving a linear system, and d/dp is computed quickly for each parameter 'p'.

B. Multi-level optimization

To improve speed convergence and accuracy, optimization strategies often use several modelling level. A two level strategy consists on using a coarse model for its calculation speed, and a fine model for its calculation accuracy. Mixing both, in an optimization strategy leads to faster results.

Our modeling method allows fine and coarse model thanks to meshing size of ferromagnetic part, and adaptive numerical integration accuracy parameter. A minimal accuracy of the coarse model has been validated in figure 5. Its calculation speed is 20 times faster than fin model calculation speed.

The coarse model can be used to find a first solution which can be followed by the fine model optimization in order to adjust solution regarding accuracy approximation of the coarse model.

C. Global optimization strategies

Evolutionary Strategy optimization

Meta-heuristic algorithms, like genetic ones, try to find the global optimum but they lack of accuracy and need lot of model computations. To show this, an optimization has been done with an Evolutionary Strategy (ES) algorithm [START_REF] Bäck | Evolutionary Algorithms in Theory and Practice[END_REF] (15 parents, 100 children, 75 generations leading to 7516 iterations during 47 minutes). The optimization converges to a quite good solution meeting constraints requirements, but the simulation of this solution with the fine model gave unfeasible constraints. In our study, this solution (Fig. 9) is good enough, but this methodology is not safe regarding constraints. Programming) but its drawback is the initial configuration given to the algorithm. Considering a high speed convergence, several tries can be done with randomized initial configurations. 6 SQP were done independently and a good solution was found in few times with a so simple strategy. A good solution was founded in less than 8 minutes (6 times better). Theses results highlight potentiality of gradient based strategies in the scope of global algorithm.

Mixed Genetic/SQP strategy

Another solution regarding SQP initialization issue is a genetic/gradient mixed strategy 0. Instead of random starting configurations, a genetic algorithm solution can be given to SQP in order to improve it. For the genetic optimization, the generation number has been reduced as well as constraints satisfaction since precision will be reached with gradients (SQP algorithm). Such a methodology, detailed on a model using integrals and method of moment solver, can be extended to

Fig. 1 -

 1 Fig. 1-Picture of a Mag-Latch switch (©MicroLab / MagFusion Inc.) and schematics.

Fig. 2 -

 2 Fig. 2-Magnetization of ferromagnetic blocs with external field (eg. magnets or coils).

Figure 2

 2 Figure 2 shows magnetization of each block which depends on the external field (H ext produced by field sources like magnets or conductors) and the field produced by other blocks depending on their own magnetization.

  z 1,2 are similar. (x c , y c , z c ) is barycentre of emitting block.

Fig. 3 -

 3 Fig.3-Surface charge approach, used to compute force and torque with the ferromagnetic magnetizations.

(

  the global model by the assembly of equations given by bodies and space transformations.

Fig. 4 -

 4 Fig. 4 -Left: MacMMems (Macro Modeller for Magnetic MEMS), Right: CADES Generator

Fig. 5 -

 5 Fig. 5 -Left: 3D FEM solver (Flux3D). Right: comparison between FEM and semi-numeric solvers

Fig. 6 -

 6 Fig. 6 -Modelling tools sequence diagram

  depends on the position of the varying parameter. For instance, 'c' parameter is only on the integrand part, the expression is defined by equation 16.

Fig. 7 -

 7 Fig. 7geometrical design parameters Constant parameters are:  magnet magnetization : 1 T,  ferromagnetic plate dimensions : 1000x500x25 µm 3  ferromagnetic saturation induction : 1T  ferromagnetic initial permeability : 10000 T/(A.m)  commutation angle : 5°  resistive beam torque 0.227 µN.m

Fig. 8 -

 8 Fig. 8discretizating for coarse model is 6x3x1=18 blocks, for fine model 12x6x2=144 blocks

Fig. 9 -

 9 Fig. 9 -Evolutionary Strategy solution geometry

Fig. 12 -

 12 Fig. 12 -Multi-start strategy, 6 minimizations of Copper Losses (best found at iteration 47)

Fig. 12 -Fig. 13 -Fig. 13 -

 121313 Fig. 12 -Mixed Genetic-SQP strategy-Minimizations of Copper Losses

Table 1 :

 1 Design specifications

	Objective	Copper losses	To minimize
	Constraints	Holding torque	< -0.1 µN.m
		Actuation torque	> 0.025 µN.m
	Optimizable parameters	Magnet high	[50 ;100] µm
		Magnet width	[1000 ;2000] µm
		Magnet length	[1000 ;2000] µm
		Magnet gap	[1 ;100] µm

Table 2 :

 2 Evolutionary Strategy (ES) optimization

		ES on coarse	Computation of fine
	Copper losses (mW)	18,1	18,1
	Holding torque (µN.m)	0,10015	0,093
	Actuation torque (µN.m)	0,0254	0,0249
	Iterations number	7516	1
	Duration (sec)	2820	15

Table 4 :

 4 Multi-start SQP optimization strategy In the following figure, copper losses, holding torque and actuation torque are plotted during optimization iterations. 6 Solutions are marked with a circle. Each satisfies constraints but one has a better minimization than other at the iteration 47.

		Best of 6 SQP on coarse	SQP on fine
	Copper losses (mW)	13,1	14,7
	Holding torque (µN.m)	0,1	0,1
	Actuation torque (µN.m)	0,025	0,025
	Iterations number	89	4
	Duration (sec)	235	227

Table 3 :

 3 Mixed Genetic-SQP optimization strategyIn this last strategy, ES part remains the most time consuming (80%). Indeed, SQP on coarse model converge to a good solution very quickly as it can be seen in the following figures. But mixing both global and local approaches allows finding a global accurate solution.

		Genetic on coarse	SQP on coarse	SQP on fine
	Copper losses (mW)	14.1	13.1	14,7
	Holding torque (µN.m)	0,099	0.1	0,1
	Actuation torque (µN.m)	0,0253	0,025	0,025
	Iterations number	150	8	4
	Duration (sec)	400	42	227

much more numerical solver. Symbolic sensitivities calculation is a hard work to do but it leads to optimization improvements regarding direct optimizations like genetic algorithms. It can be interesting to compare our methodology with other derivation techniques like automatic differentiation [15].