Frederick C Draper

Flavia R C Costa

Gabriel Arellano

Oliver L Phillips

Alvaro Duque

Manuel J Macía

Hans Ter Steege

Gregory P Asner

Erika Berenguer

Juliana Schietti

Jacob B Socolar

Fernanda Coelho De Souza

Kyle G Dexter

Peter M Jørgensen

J Sebastian Tello

William E Magnusson

Timothy R Baker

Carolina V Castilho

Abel Monteagudo-Mendoza

Paul V A Fine

Kalle Ruokolainen

Euridice N Honorio Coronado

Gerardo Aymard

Nállarett Dávila

Mauricio Sánchez Sáenz

Marcos A Rios Paredes

Julien Engel

Claire Fortunel

C E Timothy Paine

Jean- Yves Goret

Aurelie Dourdain

Pascal Petronelli

Elodie Allie

Juan E Guevara Andino

Roel J W Brienen

Leslie Cayola Pérez

G Ângelo

Manzatto

Narel Y Paniagua Zambrana

Jean-François Molino

Daniel Sabatier

Jerôme Chave

Sophie Fauset

Roosevelt Garcia Villacorta

Maxime Réjou-Méchain

Paul E Berry

Karina Melgaço

Ted R Feldpausch

Elvis Valderamma Sandoval

Rodolfo Vasquez Martinez

Italo Mesones

André B Junqueira

Katherine H Roucoux

José J De Toledo

Ana C Andrade

José Luís Camargo

Jhon Del

Aguila Pasquel

Flávia D Santana

William F Laurance

Susan G Laurance

Thomas E Lovejoy

James A Comiskey

David R Galbraith

Michelle Kalamandeen

Gilberto E Navarro Aguilar

Jim Vega Arenas

Carlos A Amasifuen Guerra

Manuel Flores

Gerardo Flores Llampazo

Luis A Torres Montenegro

Ricardo Zarate Gomez

Marcelo P Pansonato

Victor Chama Moscoso

Jason Vleminckx

Oscar J Valverde Barrantes

Joost F Duivenvoorden

Sidney Araújo De Sousa

Luzmila Arroyo

Ricardo O Perdiz

Jessica Soares Cravo

Beatriz S Marimon

Ben Hur

Marimon Junior

Fernanda Antunes Carvalho

Gabriel Damasco

Mathias Disney

Marcos Salgado Vital

Pablo R Stevenson Diaz

Alberto Vicentini

Henrique Nascimento

Niro Higuchi

Tinde Van Andel

Yadvinder Malhi

Sabina Cerruto

Ribeiro 56

John W Terborgh

Raquel S Thomas

Francisco Dallmeier

Adriana Prieto

Renato R Hilário

Rafael P Salomão

Richarlly Da Costa Silva

Luisa F Casas

Ima C Guimarães Vieira

Alejandro Araujo-Murakami

Fredy Ramirez Arevalo

Hirma Ramírez-Angulo

Emilio Vilanova Torre

Maria C Peñuela

Timothy J Killeen

Guido Pardo

Eliana Jimenez-Rojas

Wenderson Castro

Darcy Galiano Cabrera

John Pipoly

Thaiane Rodrigues De Sousa

Marcos Silvera

Vincent Vos

David Neill

Percy Núñez Vargas

Dilys M Vela

Luiz E O C Aragão

Ricardo Keichi Umetsu

Rodrigo Sierra

Ophelia Wang

Kenneth R Young

Nayane C C S Prestes

Klécia G Massi

José Reyna Huaymacari

Germaine A Parada Gutierrez

Ana M Aldana

Miguel N Alexiades

Fabrício Baccaro

Carlos Céron

Adriane Esquivel Muelbert

Julio M Grandez Rios

Antonio S Lima

Nigel C A Pitman

Luis Valenzuela Gamarra

Cesar J Cordova Oroche

Alfredo F Fuentes

Walter Palacios

Sandra Patiño

Armando Torres-Lezama

Christopher Baraloto

Amazon tree dominance across forest strata

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

Abstract

The forests of Amazonia are among the most biodiverse plant communities on Earth. Given the immediate threats posed by climate and land-use change, an improved understanding of how this extraordinary biodiversity is spatially organized is urgently required to develop effective conservation strategies. Most Amazonian tree species are extremely rare, but a small number are common across the region. Indeed, just 227 "hyperdominant" species account for more than 50% of all individuals > 10 cm dbh. Yet, the degree to which the phenomenon of hyperdominance is sensitive to tree size, the extent to which the composition of dominant species changes with size-class, and how evolutionary history constrains tree hyperdominance, all remain unknown. Here, we use a unique floristic dataset to show that, while hyperdominance is a universal phenomenon across forest strata, different species dominate the forest understory, midstory and canopy. We further find that although species belonging to a range of phylogenetically dispersed lineages have become hyperdominant in small size-classes, hyperdominants in large size-classes are restricted to a few lineages. These results suggest that achieving hyperdominance over large geographic regions has been much more challenging for canopy and emergent tree species than for understorey species. Our results demonstrate that it is essential to consider all forest strata in order to understand regional patterns of dominance and composition in Amazonia. More generally, through the lens of 654 hyperdominant species, we outline a tractable pathway for understanding the functioning of half of Amazonian forests across vertical strata and geographical locations.

Main text

The immense diversity of Amazonian forests is one of Earth's great natural wonders, and underpins the functioning and resilience of ecosystems [START_REF] Sakschewski | Resilience of Amazon forests emerges from plant trait diversity[END_REF][START_REF] Poorter | Diversity enhances carbon storage in tropical forests[END_REF] that play a crucial role in the global carbon and water cycles [START_REF] Brienen | Long-term decline of the Amazon carbon sink[END_REF][START_REF] Spracklen | The Effects of Tropical Vegetation on Rainfall[END_REF][START_REF] Staal | Forest-rainfall cascades buffer against drought across the Amazon[END_REF] . Despite three centuries of investigation, however, our collective understanding of how this diversity is organized at regional scales remains limited [START_REF] Ter Steege | The discovery of the Amazonian tree flora with an updated checklist of all known tree taxa[END_REF] .

Confronted with such overwhelming diversity, the challenge of monitoring a few hundred hyperdominant species (i.e. those species that together account for 50% of individuals across Amazonia [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF] ) becomes more tractable than monitoring the many thousands of rare species, particularly given the pace of action required for contemporary management decisions [START_REF] Wilson | Biodiversity research requires more boots on the ground[END_REF][START_REF] Gomes | Amazonian tree species threatened by deforestation and climate change[END_REF] .

Understanding the ecology and distribution of hyperdominant species is essential because these species dominate key ecosystem processes (e.g. carbon storage and cycling [START_REF] Fauset | Hyperdominance in Amazonian forest carbon cycling[END_REF] ) and may serve as an effective proxy for general biodiversity patterns [START_REF] Draper | Dominant tree species drive beta diversity patterns in western Amazonia[END_REF] .

Existing studies of Amazonian hyperdominance and regional-scale dominance have been limited by excluding small-stemmed individuals (<10 cm diameter) and by considering all individuals as equivalent regardless of diameter size-class [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF][START_REF] Draper | Dominant tree species drive beta diversity patterns in western Amazonia[END_REF][START_REF] Pitman | Dominance and distribution of tree species in upper Amazonian terra firme forests[END_REF][START_REF] Pitman | Oligarchies in Amazonian tree communities: a ten-year review[END_REF][START_REF] Honorio Coronado | Multi-scale comparisons of tree composition in Amazonian terra firme forests[END_REF][START_REF] Pitman | Distribution and abundance of tree species in swamp forests of Amazonian ecuador[END_REF] . Excluding small-stemmed species represents an important oversight because several thousand Amazonian tree species rarely or never reach 10 cm in diameter [START_REF] Gentry | Geographical Variation in Fertility, Phenology, and Composition of the Understory of Neotropical Forests[END_REF][START_REF] Gentry | Patterns of Neotropical Plant Species Diversity[END_REF][START_REF] Gentry | Contribution of Nontrees to Species Richness of a Tropical Rain Forest[END_REF][START_REF] Duque | Insights into regional patterns of Amazonian forest structure, diversity, and dominance from three large terra-firme forest dynamics plots[END_REF][START_REF] Duque | Different floristic patterns of woody understorey and canopy plants in Colombian Amazonia[END_REF] . While local-scale and taxa-specific dominance has been documented in small size classes [21][22][START_REF] Vormisto | Diversity and dominance in palm (Arecaceae) communities in terra firme forests in the western Amazon basin[END_REF][START_REF] Burnham | Dominance, diversity and distribution of lianas in Yasuní, Ecuador: who is on top?[END_REF] , basin-wide hyperdominance in small size classes has not been confirmed. Consequently, species dominating the understory of Amazonian forests at a whole-Amazon scale are not yet identified. Treating all stems > 10 cm in diameter as equivalent is also likely to over emphasize the dominance of mid-statured tree species (e.g. 10-20 cm diameter). The power-law relationship between stem density and diameter means that small-stemmed individuals (e.g. < 20 cm) are at least an order of magnitude more abundant than larger individuals (e.g. > 50 cm) [START_REF] Farrior | Dominance of the suppressed: Power-law size structure in tropical forests[END_REF] . This skewed understanding of dominance is highlighted by the difference between lists of hyperdominant species assembled using biomass rather than abundance [START_REF] Fauset | Hyperdominance in Amazonian forest carbon cycling[END_REF] . This disparity suggests that a more nuanced approach that measures dominance separately across forest strata is required.

Based on existing local-scale studies and field observations, we expect the composition of hyperdominant species to vary substantially across forest strata due to different environmental filters (e.g. variable light profiles) and different species pools. Existing studies also suggest that compositional similarity between understory and canopy hyperdominant species may vary regionally [START_REF] Baker | Consistent, small effects of treefall disturbances on the composition and diversity of four Amazonian forests[END_REF][START_REF] Ter Steege | Continental-scale patterns of canopy tree composition and function across Amazonia[END_REF] , perhaps due to regional variation in forest structure and rates of turnover [START_REF] Quesada | Basin-wide variations in Amazon forest structure and function are mediated by both soils and climate[END_REF] . For example, western Amazonia is known to have a floristically distinct understory, whereas understory communities in central and eastern Amazonia are thought to be comprised primarily by juveniles of larger tree species [START_REF] Baker | Consistent, small effects of treefall disturbances on the composition and diversity of four Amazonian forests[END_REF] .

We know that some taxonomic tree clades contain many hyperdominant species [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF] , and that genus-level abundance has a significant phylogenetic signal [START_REF] Dexter | Evolutionary patterns of range size, abundance and species richness in Amazonian angiosperm trees[END_REF] , yet no formal analysis of the phylogenetic structure of hyperdominance has been undertaken. Moreover, we may expect that hyperdominant species in different strata will display different phylogenetic patterns. Specifically, we hypothesize that hyperdominant species in large size classes from across Amazonia will be phylogenetically clustered for several reasons. First, maximum potential tree size has a significant phylogenetic signal in Amazonia [START_REF] Coelho De Souza | Evolutionary heritage influences Amazon tree ecology[END_REF] , and those genera able to occupy canopy and emergent strata are concentrated in specific lineages (e.g. families or orders) that are primarily located within a few deep clades (e.g. Fabids and Ericales) [START_REF] Coelho De Souza | Evolutionary heritage influences Amazon tree ecology[END_REF] . Second, while there is climatic variation across Amazonia, the above-canopy environment consists of high solar radiation, high temperatures, low humidity, and high diurnal variability irrespective of location 31 . These harsh but spatially consistent environmental conditions provide limited niche space, and are likely to filter for a distinct suite of functional characteristics that may only have arisen in species belonging to a few specific lineages. Third, tall trees tend to disperse better than smaller trees and shrubs [START_REF] Thomson | Seed dispersal distance is more strongly correlated with plant height than with seed mass[END_REF][START_REF] Thomson | Can dispersal investment explain why tall plant species achieve longer dispersal distances than short plant species?[END_REF] , and at least some common large tree lineages have been well dispersed throughout Amazonia across evolutionary timescales [START_REF] Dexter | Dispersal assembly of rain forest tree communities across the Amazon basin[END_REF] . This greater dispersal ability may mean that the strongest competitors for the canopy strata have been able to disperse consistently throughout Amazonia for millennia, thereby becoming hyperdominant across regions.

On the other hand, we may expect hyperdominant species in small understory strata may be more phylogenetically dispersed. First, because small trees and shrubs typically are more likely to be dispersal limited [START_REF] Thomson | Seed dispersal distance is more strongly correlated with plant height than with seed mass[END_REF][START_REF] Thomson | Can dispersal investment explain why tall plant species achieve longer dispersal distances than short plant species?[END_REF] , the strongest understory competitors may be less likely to disperse across regions and outcompete functionally equivalent species in other locations.

Secondly, understory species are often locally abundant and frequently have fast generation times [START_REF] Rüger | Beyond the fast-slow continuum: demographic dimensions structuring a tropical tree community[END_REF] . Over evolutionary timescales these high abundances and fast generation times may be likely to increase diversification among locally-restricted understory species [START_REF] Baker | Fast demographic traits promote high diversification rates of Amazonian trees[END_REF] . Third, the below canopy environment is more spatially heterogeneous, due to variation in forest structure, and the frequency and size of forest gaps, potentially leading to increased niche partitioning in smaller size classes. Moreover, because forest structure varies across Amazonia (e.g. taller denser canopy in Guiana shield vs shorter more dynamic canopy in western Amazonia) [START_REF] Quesada | Basin-wide variations in Amazon forest structure and function are mediated by both soils and climate[END_REF] , smaller-statured species may be exposed to different abiotic and biotic filters across large spatial scales, and develop greater local specialization associated with distinct functional characteristics. Therefore, we further predict that understory hyperdominants from different regions should be more distantly related than hyperdominants in larger size classes.

Here we assemble a unique dataset of 1240 floristic inventory plots distributed across lowland Amazonia, which include stems as small as 2.5 cm (Figure 1). Based on individual diameter measurements and species level identifications, we implement a spatially-stratified resampling approach to estimate basin-wide relative abundances for all tree species across six diameter size classes from the understory (2.5 -5 cm) to the forest canopy (> 50 cm). Using this dataset, we identify those species dominating different strata of Amazonian forests and ask: Q1. Is hyperdominance a constant phenomenon across Amazonian tree strata? and Q2. Does the identity of hyperdominant species differ across Amazonian tree strata, and how does this vary regionally? We also used a recently developed genus-level molecular phylogeny [START_REF] Coelho De Souza | Evolutionary diversity is associated with wood productivity in Amazonian forests[END_REF][START_REF] Neves | Evolutionary diversity in tropical tree communities peaks at intermediate precipitation[END_REF] to ask Q.3 Do patterns of phylogenetic clustering in hyperdominant species vary across forest strata? And does this correspond with our expectations of increased clustering in large-stemmed canopy strata and increased phylogenetic dispersal in smallstemmed understory strata?

Results and Discussion

Consistent hyperdominance across strata

We find that hyperdominance occurs throughout the Amazonian flora across forest strata, but the proportion of species that qualify as hyperdominant (i.e. together account for 50% of individuals) varies across size classes and regions from 3 -12 % (Figure 2A). At the basinwide Amazonian scale, the proportion of species that qualify as hyperdominant in our dataset (~ 4%) is broadly consistent with empirically derived species counts from previous studies [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF][START_REF] Fauset | Hyperdominance in Amazonian forest carbon cycling[END_REF] . This consistency of hyperdominance across size classes suggests that regional dominance of tree communities is a feature shared across Amazonian forest strata.

A larger species pool will necessarily result in stronger patterns of dominance because it contains more rare species, which will decrease the proportion of species that qualify as hyperdominant, even if the abundance of the most dominant species remains constant. However, the relationship between species richness and the level of dominance we observed in a given size class or region is weak and primarily driven by the basin-wide data (Figure 2B). Therefore, our results suggest that variation in dominance among size classes and regions is not an artefact reflecting the variable sampling intensity among regions and size classes. Some size classes are consistently more 'dominated' than others (i.e. a lower proportion of species are required to account for 50% of individuals). In particular, the 10 -20 cm size class consistently displays the strongest dominance patterns (Figure 2). The two smallest size classes have weaker dominance patterns, perhaps because smaller-stemmed species are more dispersal limited than larger individuals [START_REF] Thomson | Seed dispersal distance is more strongly correlated with plant height than with seed mass[END_REF] , and therefore less likely to be dominant over large areas and more likely to locally diversify. A clear exception to this occurs in forests on the Guiana Shield, where patterns of dominance are stronger in larger size classes than smaller ones. This may be partially explained by the relatively low diversity in the understory of these forests, due perhaps to greater resource limitation imposed by extreme shade from the more structured canopy in addition to low fertility associated with oligotrophic soils in this region [START_REF] Gentry | Geographical Variation in Fertility, Phenology, and Composition of the Understory of Neotropical Forests[END_REF] .

The proportion of species that qualified as hyperdominant at the regional scale was generally higher than in basin-scale analyses, i.e. dominance patterns are weaker at the regional scale (Figure 2). This pattern is primarily driven by those exceptionally common and widespread species that achieve hyperdominance in two or more regions. However, several of these widespread hyperdominant species may be species complexes, as recently shown for Protium heptaphyllum and Astrocaryum murumuru [START_REF] Damasco | Reestablishment of Protium cordatum (Burseraceae) based on integrative taxonomy[END_REF][START_REF] Roncal | Palm diversification in two geologically contrasting regions of western[END_REF] . Solving these issues will require more integrative taxonomic studies (e.g. incorporating DNA analyses alongside spectroscopy [START_REF] Damasco | Reestablishment of Protium cordatum (Burseraceae) based on integrative taxonomy[END_REF][START_REF] Draper | Quantifying Tropical Plant Diversity Requires an Integrated Technological Approach[END_REF] of other widespread hyperdominant species, which would help to further assess the validity of hyperdominant species identifications.

Southwest Amazonia exhibits stronger patterns of dominance than all other Amazonian regions in all but the largest size class (Figure 2). It is not immediately clear why this region has such strong patterns of dominance. However, it may be due in part to less environmental heterogeneity in this region, which contains relatively few areas of white-sand forest, swamp forests or seasonally-inundated forests [START_REF] Junk | A classification of major naturally-occurring Amazonian lowland wetlands[END_REF][START_REF] Adeney | White-sand Ecosystems in Amazonia[END_REF] . Although we do not explicitly consider habitat type in this study, many hyperdominant species are known to be dominant only in a single habitat type [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF] . Therefore, less environmental heterogeneity should lead to fewer hyperdominant species. The strong dominance patterns in southwestern Amazonia matter because several landmark studies have focussed on patterns of dominance in this region [START_REF] Pitman | Dominance and distribution of tree species in upper Amazonian terra firme forests[END_REF][START_REF] Pitman | Oligarchies in Amazonian tree communities: a ten-year review[END_REF]21 , and these patterns may not be representative of Amazonia more generally [START_REF] Tuomisto | Dispersal, environment, and floristic variation of Western Amazonian forests[END_REF] .

Identity of hyperdominant species across strata and region

The identity of hyperdominant species varies substantially across forest strata and region.

Over a third (38 %) of hyperdominant species are only dominant in a single size class within a single region, and nearly two thirds (62 %) are dominant in two or fewer size classes and two or fewer regions (Figure 3). Only one species, Eschweilera coriacea, achieves hyperdominance across all six size classes and all five regions. These results provide clear evidence that hyperdominant tree species composition is vertically stratified throughout Amazonia. Therefore, considering all individuals greater than 10 cm in diameter as equivalent completely overlooks the nuanced vertical stratification of tropical forests.

Moreover, even though 10 cm diameter cutoff protocols are well-suited to monitor carbon fluxes [START_REF] Brienen | Long-term decline of the Amazon carbon sink[END_REF] , alternative plot designs or data treatments may be better suited to monitor spatial variation in floristic diversity and composition [START_REF] Baraloto | Rapid Simultaneous Estimation of Aboveground Biomass and Tree Diversity Across Neotropical Forests: A Comparison of Field Inventory Methods[END_REF][START_REF] Phillips | Efficient plot-based floristic assessment of tropical forests[END_REF][START_REF] Magnusson | RAPELD: a modification of the Gentry method for biodiversity surveys in long-term ecological research sites[END_REF] .

Our multivariate analysis illustrates two strong axes of compositional variation among hyperdominant tree species (Figure 4 panel A). The first axis differentiates the five regions, while the second represents a gradient across six tree size classes. This compositional variation across strata is important because our best current methods of observing forests at large scales are through either: 1. Remote sensing approaches, which detect only those trees that reach sky-facing canopy positions; or 2. Plot networks, which are heavily influenced by species dominant in smaller or intermediate size-classes. Our results demonstrate that species dominating the view from above the canopy are different from those that dominate the view from below, thereby emphasising the mismatch between remotely sensed and plot-based studies. Addressing this mismatch will be essential to successfully integrating field and remote sensing data at large scales in Amazonia.

Despite this compositional mismatch, our data also suggest that while canopy hyperdominants comprise different species from those that dominate the understory, there is an important association in hyperdominant species composition between size classes within regions, i.e. hyperdominant species clearly form distinct regional groups across the first NMDS axis. Therefore, remotely sensed data from forest canopies may serve as an effective proxy for compositional patterns in smaller size classes, as has been found recently in understory tree, fern and lycophyte communities [START_REF] Draper | Imaging spectroscopy predicts variable distance decay across contrasting Amazonian tree communities[END_REF][START_REF] Tuomisto | Discovering floristic and geoecological gradients across Amazonia[END_REF] .

Our results contrast with previous observations, which suggest that the understories of Eastern Amazonia are primarily composed of juvenile individuals of large-stemmed species whereas western Amazonia has a more specialist and compositionally distinct understory [START_REF] Baker | Consistent, small effects of treefall disturbances on the composition and diversity of four Amazonian forests[END_REF] .

Instead, we find that across all regions, tree species that dominate forest understory tree communities are compositionally distinct from those that dominate the canopy, i.e. in all regions hyperdominant species form a distinct compositional gradient across strata, as reflected by the second NMDS axis (Figure 4). Indeed, there is no difference in potential maximum size among understory dominant species from different regions (Figure S.2).

At the family level, there is a clear positive relationship between the number of hyperdominant species and total species richness per family (Figure S3). However, our statistical null modelling approach shows that at a basin-wide scale several plant families have significantly more or fewer hyperdominant species than would be expected based on their species richness. Moreover, some families have more hyperdominant species than expected across several size classes; for example, Arecaceae, Burseraceae and Myristicaceae have more hyperdominant species than expected across all but the largest size class. Other families are overrepresented in terms of hyperdominant species in only smaller (e.g. Violaceae and Siparunaceae) or larger size classes (e.g. Moraceae). Alternatively, commonly occurring tree families including Rubiaceae and Lauraceae have consistently fewer hyperdominant species than we would expect based on their species richness.

Our results further reinforce the importance of the legume family Fabaceae in dominating Amazonian forests. At a basin-wide scale Fabaceae is the family with the greatest number of hyperdominant species across all size-classes, largely because Fabaceae is by far the most species rich family. While Fabaceae species are less common than would be expected by chance given their high species richness in small size classes, in the largest size-class Fabaceae are significantly overrepresented, and account for more than 30% of hyperdominant species.

Phylogenetic structure of hyperdominance across Amazonian tree strata

Our phylogenetic analyses demonstrate that while many lineages contain hyperdominant species (Figure 5), those species that are hyperdominant in the canopy of Amazonian forests show contrasting phylogenetic patterns to those that are hyperdominant in small understory strata (Figure 6). We find overall support for our prediction that hyperdominant in larger size classes tend to be concentrated in a few closely related lineages, for example in Fabaceae and Moraceae as well as Lecythidaceae and Sapotaceae. This phylogenetic clustering of canopy hyperdominant species is highlighted by our mean pairwise phylogenetic distance (MPD) null modelling analysis (Figures 6 andS4), which shows that hyperdominant species in the largest size classes are consistently more closely related than would be expected by chance. Our phylogenetic composition results reveal that canopy strata across the basin are dominated by species belonging to closely related lineages (Figure 4 panel B). The close phylogenetic relationship among large-stemmed regionally dominant tree species across the basin suggests that these species have been well dispersed across the basin through evolutionary time, supporting previous studies that found evidence for widespread dispersal in several common Amazonian tree lineages [START_REF] Dexter | Dispersal assembly of rain forest tree communities across the Amazon basin[END_REF] . These findings have important implications: If we accept the premise that phylogenetic diversity is an effective proxy for ecological or indeed functional diversity [START_REF] Swenson | The assembly of tropical tree communitiesthe advances and shortcomings of phylogenetic and functional trait analyses[END_REF][START_REF] Webb | Exploring the Phylogenetic Structure of Ecological Communities: An Example for Rain Forest Trees[END_REF][START_REF] Baraloto | Using functional traits and phylogenetic trees to examine the assembly of tropical tree communities[END_REF] , the high phylogenetic similarity among canopy species suggests there is lower functional diversity within a large proportion of the canopy strata. This low functional diversity may in turn reduce the resilience of these canopy communities to climate change. Previous studies have shown that large canopy trees in Amazonia have distinct trait profiles (e.g. hydraulic traits) [START_REF] Poorter | The importance of wood traits and hydraulic conductance for the performance and life history strategies of 42 rainforest tree species[END_REF] , appear to be particularly affected by drought [START_REF] Rowland | Death from drought in tropical forests is triggered by hydraulics not carbon starvation[END_REF] , and play a crucial role in Amazonian forest carbon storage and cycling [START_REF] Fauset | Hyperdominance in Amazonian forest carbon cycling[END_REF] . We propose that future research should continue to uncover the functional diversity and potential vulnerability both within and among lineages of these canopy hyperdominant species.

We find contrasting phylogenetic patterns in smaller, understory size-classes, which are widespread across the phylogeny as we predicted. Indeed, at the basin-wide scale, hyperdominant species in understory size classes are less closely related than expected by chance (Figure 6). This dispersed phylogenetic pattern is largely due to understory hyperdominants occurring across the major angiosperm clades (Figure S.5), but may also be because our list of understory hyperdominant species is composed of both understory specialist taxa as well as larger-statured species that achieve dominance as juveniles. Hence, this mixture of life stages and functional strategies across distinct clades is more likely to lead to a more phylogenetically dispersed assemblage. Nevertheless, our findings highlight that several characteristic understory genera such as Piper (Piperaceae), Rinorea (Violaceae), and Miconia (Melastomataceae) contain numerous hyperdominant species, which have not been recognised by previous studies of large stem (>10 cm) dominance [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF][START_REF] Pitman | Dominance and distribution of tree species in upper Amazonian terra firme forests[END_REF][START_REF] Pitman | Oligarchies in Amazonian tree communities: a ten-year review[END_REF] .

Our phylogenetic compositional analysis also supports our hypothesis that within understory strata, hyperdominant species from different regions are distantly related (Figure 4 panel B).

These results are consistent with limited dispersal and diversification of understory hyperdominant species at a basin-wide scale over evolutionary timescales, as has been suggested by others [START_REF] Gonzalez-Caro | Scale-dependent drivers of the phylogenetic structure and similarity of tree communities in northwestern Amazonia[END_REF] . Furthermore, the ability to become dominant in the understory of Amazonian forests is found across a diverse range of lineages, and therefore is relatively common. Because many of these lineages are distantly related, this suggests that a range of functional strategies has evolved to achieve hyperdominance in Amazonian understories.

Furthermore, the high phylogenetic distance among understory hyperdominant species is consistent with the hypothesis that greater environmental niche space in the forest understory has contributed to higher phylogenetic diversity.

Future Directions

The mechanisms that allow certain species to become hyperdominant remain elusive. However, our results provide a basis for testing hypotheses related to specific ecological mechanisms. Future analyses should capitalise on increasingly available functional trait data to tackle these issues. We expect species that dominate the canopy to be functionally distinct from those that dominate the understory; therefore, a size-class constrained framework may help to illuminate the mechanisms that underpin hyperdominance. In particular, a large-scale assessment of quantitative dispersal traits across a range of species may help to unravel why hyperdominant species in understory size classes display such different phylogenetic patterns to those in the canopy.

Previous studies have presented a compelling case for pre-or post-Columbian peoples increasing the abundance of many hyperdominant species in order to extract products such as fruits, nuts or building materials [START_REF] Levis | Persistent effects of pre-Columbian plant domestication on Amazonian forest composition[END_REF][START_REF] Mcmichael | Ancient human disturbances may be skewing our understanding of Amazonian forests[END_REF] . Here, we show that many of these "domesticated" hyperdominants (e.g. Euterpe precatoria and Theobroma cacao) are in fact only dominant in smaller size classes. One possible explanation is that it is easier to harvest and manage small understory trees and shrubs than large canopy trees; therefore, species that dominate larger size classes may have been less influenced by human activity than species that dominate smaller size classes. It is important to note that this is not the case in all instances, and there is substantial evidence that some large-statured species (e.g. Bertholletia excelsa) were also managed during pre-Columbian times [START_REF] Levis | How people domesticated Amazonian forests[END_REF] . Further investigation into the role of humans in shaping the composition of Amazon understories may help explain why such distantly related species have become dominant in different Amazonia regions. For example, paleoecological records may reveal if different groups of indigenous peoples have propagated different tree species in different regions.

Conclusions

There is a pressing demand to quantify and monitor the biodiversity of Amazonia in the coming decades, however, we currently lack the resources necessary to undertake the 'Linnaean renaissance' required to fully document the biota of arguably Earth's most diverse forests. By identifying those species that are hyperdominant across forest strata, we outline a size-class based framework for understanding Amazonian forests, irrespective of strata or location. This framework has revealed that species dominating either the canopy or understories of Amazonian forests not only are taxonomically distinct but also represent different phylogenetic patterns. Species belonging to a range of phylogenetically dispersed lineages have become hyperdominant in small size classes, whereas species that are hyperdominant in large size-classes belong to a few specific lineages. Point colour refers to the plot size and diameter cut-off: 1. Red points: small plots < 1 ha and stems ≥ 2.5 cm; 2. Blue points: large plots > 1ha and stems ≥ 10 cm; 3. Gold points: large plots >1 ha and stems ≥ 10 cm with nested subplot for small stems ≥ 2.5 cm. Solid white lines indicate the border of the five sampling regions defined for analyses, dashed white lines show the further subdivision of sampling regions into 10 sampling zones. Sampling regions have been labelled as follows: Northwest Amazonia (NW); Southwest Amazonia (SW); Southern Amazonia (SA); Central Amazonia (CA); Guiana Shield (GS). The shaded area shows the area defined as Amazonia based on: 1. Annual precipitation > 1300 mm, 2. Elevation < 1000 516 m (above sea level), and 3. Forest cover > 70%. across the five Amazonian regions and the basin-wide 'Amazonia' dataset (Panel B). Dashed lines show linear regressions based on the five regional data sets (black line; R 2 = 0.01, P =0.26), and the five regional datasets plus the basin-wide 'Amazonia' dataset (grey line; R 2 = 0.08, P =0.05). In both panels, a lower proportion of hyperdominant species indicates stronger patterns of dominance. Sampling regions as indicated in Figure 1 are: Northwest Amazonia (NW); Southwest Amazonia (SW); Southern Amazonia (SA); Central Amazonia (CA); Guiana Shield (GS).

Methods

Floristic data

Floristic data came from 1240 forest inventory plots, distributed across the Amazon basin (Fig. 1). The plot data fall into two broad categories: Firstly, the RedGentry network consisted of 1027 small forest plots (typically 0.1ha but ranging from 0.04-0.25 ha) within which all stems with a diameter at 1.3 m in height (dbh) greater than 2.5 cm were measured and identified. Secondly, 520 larger forest plots (typically 1 ha but ranging from 0.5 to 9 ha) from the RAINFOR and ATDN networks were used. Within these larger plots all stems with a dbh greater than 10 cm were measured and identified. Many of these plots are curated and stored within ForestPlots.net, a cyber-infrastructure initiative that unites plot records and their contributing scientists from the world's tropical forests.

The RedGentry plot data came from a range of sources and therefore included a range of plot sizes and sampling protocols. Most plots were 0.1 ha in size and consisted of 10 transects of 2 X 50 m arranged systematically around a single transect baseline following the 'Gentry protocol' [START_REF] Phillips | Efficient plot-based floristic assessment of tropical forests[END_REF] . However, 307 plots were subplots nested in within larger 1 ha inventory plots (Fig. 1). The majority of these nested 0.1 ha plots were part of the PPBio network.

Taxonomic standardization

It was not possible to standardise morphospecies across datasets as plots were installed by many different botanical teams at different times, often without accompanying herbarium vouchers. Therefore, all individuals that were not identified to species level were excluded from all subsequent analysis. These exclusions lead to a substantial loss of individuals (mean 21 % of individuals per plot, Figure S.7) and were phylogenetically biased, i.e. some families had a higher degree of taxonomic uncertainty than others. Nevertheless, this approach renders our analysis comparable to other landscape analyses conducted on larger stems in this and other regions.

Species exclusions

Species names were checked for synonymy and spelling mistakes using the taxonomic names resolution service (TNRS) using the R package taxize [START_REF] Chamberlain | taxize: Taxonomic information from around the web[END_REF] . Any species that were not recognised in the automated process were checked manually for spelling mistakes. Identifications that could not be easily assigned to a species were considered unidentified morphospecies and were removed from further analysis. Finally, our list of legitimate species names was crosschecked against the most current published checklists [START_REF] Cardoso | Amazon plant diversity revealed by a taxonomically verified species list[END_REF][START_REF] Ter Steege | Towards a dynamic list of Amazonian tree species[END_REF] . Species that did not occur on this Amazon checklist (887 species) were checked manually against collection records in the Tropicos database 62 . 39 of these were confirmed to be illegitimate Amazonian species because they have ranges either outside of our region (i.e. on another continent). A further 579 species that were described as either epiphytes, lianas, herbs, or ferns were also excluded from our analysis. These lifeforms were included in some plot datasets and excluded from others. As individual datasets are normally geographically clustered, including them would likely lead to spatially biased species abundance estimates. A further 47 species were excluded because there was no recorded collection since their descriptions, we considered these individuals to be wrongly identified.

Species inclusions

We included 180 species in our analysis that had been excluded from previous analyses or checklists. The majority of these inclusions were small stemmed species that had previously been excluded for being shrubs or treelets. We considered these definitions to be subjective. 37 of these included species have previously been considered illegitimate because they occur primarily in Savanna or seasonally dry habitats. However, because several of our plots were located close to boundaries between ecosystem types, and many species are shared across these boundaries (not necessarily in their optimum habitat), we included these non-core rainforest species in our dataset.

Defining Amazonia

To ensure that our analysis included only plots located in lowland Amazonia and did not incorporate plots in marginal seasonally dry or montane environments, it was necessary to define our study area. We defined lowland Amazonia using four remotely sensed criteria: 1.

Watersheds were estimated using the hydrosheds data layer [START_REF] Lehner | New Global Hydrography Derived From Spaceborne Elevation Data[END_REF] , in addition to the Amazon basin; we also included eastern branches of the Orinoco and all watersheds to the east of that mark in the Guiana Shield. 2. Elevation was measured using the global SRTM digital elevation model [START_REF] Jarvis | Hole-filled SRTM for the globe Version 4[END_REF] , lowlands were considered to be land area below 1000m elevation following Cardoso et al. (2017) [START_REF] Cardoso | Amazon plant diversity revealed by a taxonomically verified species list[END_REF] . 3. Precipitation was estimated using the CHIRPS annual mean rainfall data [START_REF] Funk | The climate hazards infrared precipitation with stations -A new environmental record for monitoring extremes[END_REF] , and a minimum mean annual precipitation value of 1300 mm year -1 was used to define moist forests following Cardoso et al. (2017) [START_REF] Cardoso | Amazon plant diversity revealed by a taxonomically verified species list[END_REF] . 4. Tree cover was estimated using the 2010 global forest cover map [START_REF] Hansen | High-resolution global maps of 21st-century forest cover change[END_REF] , and all pixels with > 70 % forest cover were included. The four layers were overlaid, and the intersecting area was used to define Amazonia. This final layer was then sieved and filtered to reduce speckle, which was primarily driven by the complex patterns of deforestation along the southeastern border. All geospatial analysis was conducted using QGIS software [START_REF]QGIS Geographic Information System[END_REF] .

Spatial standardisation

Many species (27 %) occurred fewer than five times across the plot network. Therefore, we did not attempt to generate basin-wide population estimates for most species as other studies have done [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF] . Instead, we used only the empirical data from plots to estimate those species likely to be hyperdominant at basin-wide scales, under the assumption that this plot network is reasonably representative of abundances of the most common species.

Plots are not distributed evenly across Amazonia, but instead are clustered in space, for example, there are many more plots in western Amazonia than in Southern or Eastern Amazonia. Furthermore, plots varied in size and therefore so did the number of individuals per plot. To account for these biases, and to attempt to ensure the Amazonian flora was sampled as evenly as possible, we used a spatially-stratified bootstrap resampling approach.

All sampling procedures were performed in the statistical language R using the tidyverse packages dplyr, tidyr, and purrr [START_REF] Wickham | A Grammar of Data Manipulation[END_REF][START_REF] Wickham | tidyr: Easily Tidy Data with 'spread()' and 'gather()[END_REF][START_REF] Henry | purrr: Functional Programming Tools[END_REF] . This approach consisted of the following steps:

1.)

Greater Amazonia (as defined above) was divided into 5 regions roughly following previously defined boundaries [START_REF] Ter Steege | Hyperdominance in the Amazonian tree flora[END_REF][START_REF] Tuomisto | Discovering floristic and geoecological gradients across Amazonia[END_REF] . Each region was then split roughly in half to generate 10 total sampling zones that were broadly similar in area (Area varied from 210,000 to 1081,000 km 2 ). Each sampling zone contained at least 40 individual plots (at least 20 small plots and at least 10 large plots).

2.)

The entire dataset was then divided into six strata-specific datasets. This was done by stratifying the data by dbh into six size classes (2.5 -5 cm, 5 -10 cm, 10 -20 cm, 20 -30 cm, 30 -50 cm, >50 cm). Diameter was used as a proxy for tree height because tree height was not measured in most plots, and because of the strong allometric relationship between diameter and height.

3.) 20 small plots or large plots with nested subplots and five large plots were sampled from each sampling zone at random without replacement. This step ensured spatially even sampling across the basin, and the five additional large plots ensured a reasonable number of large individuals were sampled.

4.)

From each of these 25 plots a standard number of individuals (50% of the median individuals per plot per size class) were sampled with replacement, ensuring an even number of individuals was sampled for each plot.

5.)

These standardised samples from each plot were then assembled into a single species by plot matrix.

6.)

Steps 2 and 3 were repeated 10 6 times, generating 10 6 estimates of abundance for each species across the basin.

7.)

The mean and standard deviation of abundance for each species was calculated across the 10 6 estimates of abundance.

8.)

Hyperdominant species were then defined as those species that together account for 50 % of the mean total abundance of all individuals within each size class across all iterations.

To identify regionally dominant species, steps two-seven were repeated for each of the five predefined regions individually.

Phylogenetic analyses

To understand where hyperdominant species are situated across the Amazonian phylogeny, we used a published genus-level molecular phylogeny for Amazonian tree species [START_REF] Coelho De Souza | Evolutionary diversity is associated with wood productivity in Amazonian forests[END_REF][START_REF] Neves | Evolutionary diversity in tropical tree communities peaks at intermediate precipitation[END_REF] . A genus-level phylogeny was used because a species-level molecular phylogeny for the full Amazonian flora does not yet exist. Genera occurring in our lists of Amazon-wide hyperdominant species were mapped onto the phylogeny, which was then pruned to remove taxa not occurring in our dataset. The final phylogenetic tree contained 646 genus tips. We then plotted the phylogeny for all genera occurring in our dataset using the R package ggtree [START_REF] Yu | ggtree : an r package for visualization and annotation of phylogenetic trees with their covariates and other associated data[END_REF] .

The tips of genera that contained hyperdominant species were coloured to highlight their location. Tip colours corresponded to a continuous variable that was the mean size class for hyperdominant species that were in the given genus. Genus labels were given to all genera that contained three or more hyperdominant species.

We used the mean pairwise phylogenetic distance (MPD) metric and a null modelling approach to test if hyperdominant species are more closely related to one another than would be expected if hyperdominance were distributed across the phylogeny at random 51.72 .

Because a species-level molecular phylogeny has not been developed across Amazonian plant taxa, we first added species tips with a uniform branch length (0.1) to all tree genera. This approach allowed us to make species-level comparisons using a genus level phylogeny, while minimizing the assumptions made about within-genus phylogenetic structure. We then calculated MPD among hyperdominant species for each hyperdominant community, and compared this observed MPD to a null distribution of expected MPD if we sampled an equivalent number of species at random across a phylogeny with an equivalent species pool [START_REF] Tsirogiannis | PhyloMeasures: a package for computing phylogenetic biodiversity measures and their statistical moments[END_REF] .

Where the observed MPD fell outside two standard deviations of the null distribution, that hyperdominant community was considered to be significantly more clustered (lower MPD) or significantly more dispersed (higher MPD) than we expect by chance. All phylogenetic analysis was conducted in R, using packages phylomeasures, phytools, and caper [START_REF] Tsirogiannis | PhyloMeasures: a package for computing phylogenetic biodiversity measures and their statistical moments[END_REF][START_REF] Revell | phytools: an R package for phylogenetic comparative biology (and other things)[END_REF][START_REF] Orme | caper: Comparative Analyses of Phylogenetics and Evolution in R. R package version 1[END_REF] .

MPD is known to be influenced by the extent to which species are divided among the three major angiosperm clades (Magnoliids, Monocots and Eudicots) [START_REF] Coronado | Phylogenetic diversity of Amazonian tree communities[END_REF] . Large stemmed Amazonian tree species are predominantly found within the Eudicots, while small stemmed species are found across the three clades. These deep-clade distributions are therefore likely to increase phylogenetic clustering within the large-stemmed species and increase phylogenetic overdispersion within small stemmed species. In part we account for this in measurement of MPD as we remove genera from the tree that do not occur in the size class/region for which we are measuring MPD. However, to explore the effect of this deep-clade diversity further, we repeated our MPD analysis within Eudicots only (Figure S.5). This analysis demonstrates that the overall patterns of increased clustering in larger size classes is maintained within eudicots. The analysis also shows that the phylogenetic dispersion found within smallstemmed hyperdominant species is due to these understory hyperdominant species occurring across these deep phylogenetic nodes.

Compositional analyses

To understand how the composition of hyperdominant species varied across size classes and regions we used a multivariate statistical approach. Specifically, we used the Jaccard index as a metric of how similar or different the composition of hyperdominant species was among the 36 communities of a given size classes within a given region, e.g. northwest amazon -2.5-5 cm size class. For clarity, these regional and size class specific groups are hereafter referred to as hyperdominant communities. The Jaccard distances were generated using with the R package vegan [START_REF] Oksanen | Community Ecology Package. R package version 2[END_REF] .

We expanded these compositional analyses not only to consider how taxonomic composition varied among hyperdominant communities, but also, to quantify how phylogenetically similar hyperdominant communities were among size classes and regions. To do this we again used a multivariate statistical approach, however, this time using two metrics of phylogenetic beta diversity. Both phylo-beta diversity metrics were calculated at genus rather than species level as we used the genus-level phylogeny. To account for the fact that some genera contain several hyperdominant species, we used the number of hyperdominant species per genus for each hyperdominant community per size class as our input community matrix.

The first metric that we used was the abundance weighted MPD among hyperdominant communities, which provides a deep/basal node weighted assessment of phylogenetic beta diversity [START_REF] Swenson | The assembly of tropical tree communitiesthe advances and shortcomings of phylogenetic and functional trait analyses[END_REF] . The second metric of phylogenetic beta diversity that we used was the generalized version the unifrac method [START_REF] Chen | Associating microbiome composition with environmental covariates using generalized UniFrac distances[END_REF] , calculated with the R package GUniFrac 78 . We used an α value of 0.5, meaning that we moderately weighted genera by the number of hyperdominant species that they contained in that site/size class. The unifrac metric provides a stable tip-weighted assessment of phylogenetic beta-diversity.

Because the phylogenetic analysis was conducted using a genus-level phylogeny, we do not account for any within genus phylogenetic structure that could affect these metrics. However, any within-genus structure will have little effect on patterns of MPD, as this metric is heavily weighted towards deep-node differences among communities [START_REF] Swenson | The assembly of tropical tree communitiesthe advances and shortcomings of phylogenetic and functional trait analyses[END_REF] . The tip-weighted unifrac method is likely to be more heavily influenced by the missing within-genus structure, therefore these results are only presented in the supplementary information.

To reduce the dimensionally of this multivariate data and visualize the taxonomic and phylogenetic similarities among hyperdominant communities we used Non-metric multidimensional scaling (NMDS). NMDS analyses were run for at least 50 iterations and until a stable solution was reached (stress < 0.2). Each NMDS was optimized over three dimensions and displayed in an ordination plots. All NMDS ordinations were performed in the R package vegan [START_REF] Oksanen | Community Ecology Package. R package version 2[END_REF] .

Figure 1 .

 1 Figure 1. Map of study area and 1240 floristic inventory plots, represented by coloured

  517

Figure 2 .

 2 Figure 2. The proportion of species that are hyperdominant (i.e. together account for 50% of

Figure 3 .

 3 Figure 3. Two-dimensional histogram showing the number of species that are

Figure 4 .

 4 Figure 4. NMDS ordinations showing similarity in composition of hyperdominant species in

Figure 5 .

 5 Figure 5. Hyperdominant species mapped onto a genus level Amazonian tree phylogeny. All

Figure 6 .

 6 Figure 6. Divergence of mean pairwise phylogenetic distance (MPD) from null models for

Acknowledgements

We dedicate this study to the late Alwyn Gentry, who not only established 41 of the plots that form the foundation of our analyses, but also pioneered the synthetic approach that underpins our study. This paper is a product of the RedGentry, RAINFOR, PPBio and ATDN networks,

Forest Fragments (BDFFP-INPA). Finally, we would like to thank Frans Bongers, Julissa Roncal and one anonymous reviewer for their valuable comments on the manuscript.

Data availability

The datasets generated and/or analysed within this study are available from the corresponding author on reasonable request and with permission of relevant data owners.

Code availability

All custom written analytical code used in this study are available online (https://github.com/FreddieDraper/RedGentry)

Data from many of these networks are curated by ForestPlots.net, a cyber-infrastructure initiative that unites plot records and their contributing scientists from the world's tropical forests. These initiatives have been supported by numerous people and grants, but we are indebted to hundreds of institutions, field assistants, botanists, and local communities for help in establishing plots and identifying their >4600 species. We would especially like to thank the following for their important role: Eduardo Hase, Raimunda Nazaré Oliveira de Araújo, Samuel Almeida, Julio Serrano, João Batista de Silva, Katia Cangani, Ocírio Souza Pereira, Julio do Vale, Maria Carmozina, Everaldo da Costa Pereira, Sebastiao Salvino de Souza, Cecilia Ballón Falcón, Massiel Corrales Medina, Antônio Magalhães da Silva, Jose Farreras, Franklin Molina and Jon Lloyd.

Author Contribution

FCD and CB conceived the study. FCD, GPA and CB designed the study with input from FC, GA, OLP, and HtS. FCD and JBS performed the analysis with input from CB, GPA, GA, OLP, AD, FCdS and KD. FCD wrote the manuscript with input from CB, FC, GA, OLP, AD, MJM, GPA and HtS. Initials refer to the first 14 authors and the last author. All other coauthors contributed data and had the opportunity to comment on the manuscript.


Matthieu Nesme

Yohan Payan

François Faure

rd Workshop in Virtual Reality

Keywords: (according to ACM CCS): I.3.5 [Computer Graphics]: Physically based modeling I.3.7 [Computer Graphics]: Animation physically based animation, finite elements, deformable modeling, real time simulation

We present a new method for physically animating deformable shapes using finite element models (FEM). Contrary to commonly used methods based on tetrahedra, our finite elements are the bounding voxels of a given shape at arbitrary resolution. This alleviates the complexities and limitations of tetrahedral volume meshing and results in regular, well-conditionned meshes. We show how to build the voxels and how to set the masses and stiffnesses in order to model the physical properties as accurately as possible at any given resolution. Additionally, we extend a fast and robust tetrahedron-FEM approach to the case of hexahedral elements. This permits simulation of arbitrarily complex shapes at interactive rates in a manner that takes into account the distribution of material within the elements.

Introduction

Physically-based animation has generated a growing interest over the last twenty years, because it allows a virtual object to realistically react to external input rather than replay precomputed animations. However, it is not yet widely used in real-time applications such as video games, because physical models are difficult to tune and their animation is computationally expensive.

In this paper, we propose a novel approach to tackle these difficulties in the case of viscoelastic deformable bodies. Viscoelastic deformation can be used to model a wide range of real-world objects, including biological tissues, cloth, and a variety of deformable manufactured objects. They are modeled as a continuous material subject to physical laws relating local strain to local stress, and discretized over a finite number of degrees of freedom to allow numerical simulation. This has been deeply studied in the domain of mechanical engineering [START_REF] Bathe | Finite Element Procedures in Engineering Analysis[END_REF], and a wide variety of material laws and discretization methods has been proposed. In the computer graphics community, the seminal paper of Terzopoulos [START_REF] Terzopoulos | Elastically deformable models[END_REF] showed how to produce complex and visually realistic animations, at a high computational price. Since then, a lot of work has been done to reduce the complexity using alternative material laws, spatial discretization, and time integration. Our approach can be summarized as follows. We first build a high-resolution voxelization of the geometrical object (if not already available).We then recursively merge the voxels up to an arbitrarily coarser mechanical resolution. The merged voxels are then used as hexahedral finite elements embedding the detailed geometrical shape and animated using fast implicit time integration.At each level, the mass and stiffness of a merged voxel are deduced from its eight children, automatically taking into account the non-uniform distribution of material. The objects can simultaneously include volumetric parts, surface parts such as wings or ears, and one-dimensional parts such as tails. The animation is robust against degenerate configurations such as element inversion. We believe that these features put together make our apc The Eurographics Association 2006.

proach suitable for interactive virtual environments such as video games.

Our specific contributions are: the automatic voxelization of a surface model, the automatic tuning of the FEM parameters based on the distribution of material in each cell, and the robust animation of hexahedral elements.

The remainder of this paper is organized as follows. Related work is briefly discussed in section 2. In Section 3, we show how to construct the voxel mesh and embed the surface in the deformable voxels. In Section 4, we present a new method to apply fast dynamic FEM to the voxels and detail how to interact with the embedded surface. In Section 5, we explain how to tune the mass and stiffness of the voxels. We discuss results in Section 6 and conclude in Section 7.

Related Work

The complexity of the material laws have a high impact on computation time. T he most simple viscoelastic law is the spring model [CEO * 93, BHW94]. However, it has been shown that it can not accurately model 3D elasticity, and recent work focus on 3D finite elements [START_REF] Hauth | Visual Simulation of Deformable Models[END_REF]. Important gains in computation time have been obtained by replacing the complex Green-Lagrange rotationally invariant strain tensor by the product of a rotation with the linearized Cauchy strain tensor [MDM * 02, EK03, MG04]. This method is commonly referred as stiffness warping or co-rotational elements. Robustness in degenerate configurations such as flat or inverted elements has then been improved [START_REF] Irving | Invertible finite elements for robust simulation of large deformation[END_REF][START_REF] Nesme | Efficient, physically plausible finite elements[END_REF]. Spatial discretization directly impacts the number of degrees of freedom, and thus, the complexity of the system. Traditional FEM analysis requires an accurate mesh of the entire volume of the object. This is hardly applicable to visually pleasing detailed shapes. Moreover, most available computer graphics models are surface meshes, and it is hard to mesh the interior of such models with a controllable number of tetrahedra without creating nearly singular elements which result in unstable simulations, especially for small parts like the ears of the bunny or the tail of the mouse presented on the teaser. Moreover, it is difficult to animate surfaces (like clothes or dragon's wings) using a volume mesh. Some approaches try to separate rendering detail from the (possibly hierarchical) mechanical model, using an external [START_REF] Debunne | Dynamic real-time deformations using space and time adaptive sampling[END_REF] or embedded [CGC * 02b, CGC * 02a, MG04] rendering layer. Nevertheless, the tetrahedrization stage remains far from trivial. To avoid this volume meshing stage and control the number and shape of elements, some methods build automatically an optimized mechanical mesh using a 3D grid [START_REF] Müller | Physically based simulation of objects represented by surface meshes[END_REF][START_REF] James | Squashing cubes: Automating deformable model construction for graphics[END_REF] or an octree [START_REF] Dequidt | Time critical animation of deformable solids[END_REF][START_REF] Nesme | Hierarchical multiresolution finite element model for soft body simulation[END_REF]. However, the resulting mechanical properties are simplified and the meshes have to be very fine to model the objects accurately. Alternatively, it is possible to reduce the number of degrees of freedom of detailed shapes using modal anal-ysis [START_REF] James | Real-time subspace integration for st.venant-kirchhoff deformable models[END_REF] or global shape matching [START_REF] Müller | Meshless deformations based on shape matching[END_REF], however these methods fail to capture local deformation. Recently, meshless methods have been proposed [DC95, MKN * 04, PKA * 05] in order to more easily model fracture and tearing, however they are quite slow and not the best adequate for real-time animation of non-pasty soft bodies. Time integration is an important issue when dealing with stiff objects, which need very small time steps to avoid instabilities when explicit schemes are used [START_REF] Press | FLANNERY: Numerical Recipes in C[END_REF]. Baraff and Witkin [BW98] showed how to efficiently apply implicit time integration which allows large time steps. Collision detection and response is also a difficult topic related to time integration. A lot of detection methods have been proposed [TKH * 05] and response strategies have been discussed [START_REF] Bridson R | Simulation of clothing with folds and wrinkles[END_REF].

Deformable Bounding Voxels

Contrary to the traditional finite elements, in our approach, the volume mesh does not fit exactly the object, and all the nodes are not exactly under the surface, like in [CGC * 02b, CGC * 02a, MG04]. Some elements include the surface. Surface points are linearly interpolated within the cells. In the following, this cells are called bounding elements. The first step consists in building the mesh and computing the mechanical properties. Then the mechanical simulation can be performed. The global algorithm is presented in Algorithm 1.

Algorithm 1 PREPROCESSING

Building the Voxels

The first step consists of a voxelization of the object. Starting from the bounding box of the object, an octree decomposition is employed up to a given fine maximal resolution (see Figure 1). Only non-empty cells are considered. To detect the outside/inside/boundary cells, the bounding box is inflated a little in order to ensure that its vertices and edges are outside the object (in our implementation, an arbitrary inflation of 1% is used). All the boundary cells contain surface points or intersect surface polygons. All the non-boundary cells on the borders are outside. This outside state is propagated to all their neighbours until a boundary cell is met. All others cells are inside. This algorithm (illustrated in Figure 2) works well for volumes without holes inside, but the animation can be nevertheless interesting for them. It could be easily extended to more complex geometries, and generalized to non-hexahedral cells, like in [CGC * 02b]. Note that

First step consists of detecting boundary cell (grey) i.e., consurface points and/or intersecting surface triangles (red). Then, outside border cells (white) are propagated to their neighborhood (blue arrows). Remaining cells are inside (black).

it also automatically handles open surfaces, seen as object boundaries and animated like non-full cells.

Embedding the Shape

order to link the surface with bounding elements, surfaces vertices u(p) are trilinearly interpolated using the eight values u(q) defined at the nodes of the bounding element as illustrated in 2D in Figure 3:

u(p) = Hu(q) (1)
where the (3 × 24) interpolation matrix H of the considering element is the concatenation of the influences of the element vertices on a given vertex:

H =   h i 0 0 0 h i 0 0 0 h i   , h i (r, s,t) = 1 8 (1 ± r)(1 ± s)(1 ± t)
To simplify the computations, the interpolations are done in natural coordinates in a local frame (r, s,t) ∈ [-1, 1] 3 . The Jacobian operator J relating the world coordinates to the local coordinate is needed for consistency:

∂ h ∂ r = J ∂ h ∂ x .
In regular elements,

J =    2 length 0 0 0 2 width 0 0 0 2 height   
Large triangles remain flat even if they belong to several deformable voxels, so they have to be splitted where they cross voxel faces.

y x q 0 q 1 -1 -1 r 1 s q 2 q 3 1 p

Mechanical Animation

When the collection of cells is built, mechanical laws can be applied to bounding elements in order to animate cells and the interpolated shape.

Force Computation

The linear Finite Element Method on hexahedral elements is used to compute internal forces [START_REF] Bathe | Finite Element Procedures in Engineering Analysis[END_REF]. In this paper, we consider only isotropic linear elastic materials. According to Hooke's law, the material properties are only defined by the Young's Modulus and the Poisson's ratio. As explained in [MDM * 02], a great advantage of the linear formulation is that all stiffness matrices can be precomputed, because they do not evolve too much during the animation.

Since the standard linear approach is inaccurate for large rotation of the elements, stiffness warping has to be employed c The Eurographics Association 2006.

to avoid artificial inflating. The main idea is to compute the forces in an element's local rotated frame by decomposing the displacement into a rigid rotation combined with a deformation. The force applied by a deformed element to its sampling points is given then by

f = R T K(Rx -x 0 )
where K is the stiffness matrix, x and x 0 are the current and the initial positions, and matrix R, which encodes the rotation of a local frame with respect to its initial orientation, is updated at each frame.

Since the first approach proposed [MDM * 02], several methods have been proposed to compute R. Methods using eigenvectors and eigenvalues [EKS03, [START_REF] Müller | Interactive virtual materials[END_REF][START_REF] Irving | Invertible finite elements for robust simulation of large deformation[END_REF] give the smallest deformations for most accurate results. A significantly faster method has been proposed in [START_REF] Nesme | Efficient, physically plausible finite elements[END_REF]. However, the latter introduces some vertex ordering-dependent anisotropy and the evaluated strain is a slightly higher. However, in our case, a perfect application of the laws of physics is not necessary that is why we prefer the speed of this approach than the accuracy. An another important point of this approach is its robustness in degenerate configurations such as flat or inverted elements. An elegant treatment has been presented in [START_REF] Irving | Invertible finite elements for robust simulation of large deformation[END_REF] but it is not aimed at real-time applications. Using [START_REF] Nesme | Efficient, physically plausible finite elements[END_REF], the inversion of an element is detected for free, and modeled as a high compression. Stability is maintained and the elements can recover their initial shape without visible artifact. This is the essential and the desired effect in the case of non-physical situations, which can occur in real applications.

We extend to hexahedra the method [NPF05] initially designed for tetrahedra. For each hexahedron, three arbitrary edges could be selected in order to extract the rotation the same way as done for a tetrahedra. However, it is preferable to involve all the vertices in the computation to obtain a rotation that results in smaller measured deformations. To do this, we compute the average of four edges in the three directions, as illustrated Figure 4. Extreme element twisting can result unrealistic stable configurations, however this is unlikely to occur when several elements are used, and it does impact stability.

Time integration

The stable Euler implicit solver is employed to solve the ODE. The filtered conjugate gradient presented in [START_REF] Baraff | Large steps in cloth simulation[END_REF] gives the advantage to not necessary need an assembling stiffness matrix, since it only requires a product matrix/vector. It is well adapted to the stiffness warping formulation for which the assembling matrix should vary at each time step because of rotations. It is possible to process the elements one after another. Implicit integration is very wellsuited to our approach, where all elements are regular, with the same size. The corresponding equation system is welland can be solved iteratively. As usual, a large number of iterations can be necessary to accurately model materials with high stiffness. However, a reduced number of iterations (near 5 to 10) generally acceptable. This allows trade-offs between accuracy and computation speed.

Boundary Conditions

Applying classic boundary conditions to the cell nodes is trivial but is not always sufficient. To directly manipulate the surface, or to accurately handle contact constraints, the boundary conditions must be applied to the surface, and dispatched over the lattice vertices.

Penalty forces

In case of penalty forces, forces f(p) are applied to points of the surface. We dispatch these forces over the lattice vertices.

principle of virtual work implies that

f(q) = H T f(p)
where H is the interpolation matrix at point p.

Hard constraints

When a displacement is imposed at the surface, the corresponding lattice displacements have to be computed. We compute the smallest displacements of the control points in least-squares sense, as in direct manipulation of FFD [START_REF] Hsu | Direct manipulation of free-form deformations[END_REF]. A displacement constraint is written u(p) = c ⇔ Hu(q) = c where c the constraint value. When several constraints are applied, we build a system containing all constrained points p and all influenced control points q. The resulting system can be solved using a pseudo-inverse of the matrix H:

u(q) = H + c with H + = (H T H) -1 H T , dim(p) ≤ dim(q) H + = H T (HH T ) -1 , dim(p) > dim(q) .
When there are more constraints than control points, the best compromise (in the least-square sense) is computed. Note that in the specific case of at most one constraint applied to each lattice vertex, H T i H i is a scalar, and the solution of the system is efficient. c The Eurographics Association 2006.

Accurate Physical Parameters

To adapt the stiffness of a cell according to its content, it is necessary to refine the octree mesh more precisely than desired for the animation. The main idea is to pop up the information from fine cells to coarser cells.

Weighted Stiffness

In order to improve an approximation of the stiffness of an element, the amount of matter contained in a cell is taken into account, as a material stiffness factor:

D c = α c D
α c is the filling ratio of the mechanical cell c and D is the stress-strain matrix relating the material properties. The finest depth is considered close enough to the surface to chose a filling rate equal to 50% for boundary cells. Inside cells have a filling ratio equal to 100% and cells to 0%. Using the octree hierarchy, these ratios are averaged until the desired mechanical level, such as α parent = 1 8 ∑ 7 i=0 α child i . Thus, almost cells are softer than full cells. Figure 5 shows the poping up of filling ratios, from the finest depth to the mechanical level.

Weighted Masses

The given total mass of the object m total is distributed to each cell. do this, the masses of the finest leaves m lea f are computed by subdividing the total mass by the number of leaves, taking into account the filling ratio.

m lea f = m total 2 × #insideLeaves + #boundaryLeaves
The factor 2 comes from the filling ratio, i.e., an internal cell is twice heavier, so m internal = 2 × m lea f and m boundary = m lea f . The masses are then summed up from the finest level to the mechanical level. Let us consider the child cells as interpolations of their parent cell. In this context, it is possible to deduce the stiffness influence of a child cell K child on its parent K parent , and to the stiffness of each parent cell based on its eight children. In this section, we take into account the distribution of the material over the child cells. Differences with the uniform stiffness are illustrated in Figure 6. only the large cells are considered, that returns to remove degrees of freedom to child nodes. To some extent child nodes are dependent from their parents and can be deduced as an interpolation (child nodes are constrained to stay "in the middle"). In this case, we can define eight matrices L child which represent the interpolated child nodes u child based on their parent cell nodes u parent : u child = L child u parent . Reciprocally, forces applied to child nodes can be popped up to the parent nodes using the transpose of the interpolation:

f parent = L T child f child . However f child = K child u child , so f parent = L T child K child L child u parent .
Summing the influence of the eight children of a parent cell, we obtain:

K parent = 7 ∑ i=0 L T i K i L i
We do the same for the masses:

M parent = ∑ 7 i=0 L T i M i L i ,
where M is the mass matrix of a cell. At the mechanical level, we finally lump the mass matrix to obtain a diagonal matrix, which allows faster matrix products and easy weight computation.

Matrices L child only depend on the shape of the elements, so they can be defined once for all. Matrices for hexahedral subdivision are given in Appendix A.

Results

Non-Uniform Stiffness

Taking into account the contents of bounding elements improves animation quality without adding complexity. This allows us to perform plausible animations using a reduced number of elements. An extreme case is presented in Figure 7 where an object in form of 'c' is animated using one single boundary element. As expected, using a classical uniform stiffness, both parts of the object have the same properties, and the empty part is as stiff as the full part. Another example is Figure 8 for a more complex object. In contrast, using our precomputed non-uniform law, stiffness takes into account where the matter is, resulting in more realistic behaviors.

Performance

Table 1 gives results for several simulations that were run on a laptop Pentium M 2 GHz with 2 GB of RAM and a nVidia Quadro FX Go1400. Performances depend linearly on the numbers of surface points and the number of bounding elements since number of iterations is fixed. It is a great advantage, since it is possible to keep a fast frame rate even when a high detailed mesh is animated, by reducing the number of cells.

In our implementation, a large time is used to update surface vertices. It is possible to accelerate this by performing the interpolations on the GPU using a vertex shader. Preliminary results show that a very simple shader that only compute final interpolations for rendering, without reading back the results for collision management, accelerates the dragon animation by a factor of two.

Robustness

Thanks to the used Finite Element Method, our simulator is robust face to degenerated configuration (Figure 9).

Surfaces and Lines

An important contribution of our approach is to handle surfaces, like the dragon's wings and T-shirt (Figure 10), as well as lines, the same way as volume. A nice feature of our method applied to this kind of objects is that their rest shape can be straightforwardly chosen as the most familiar form. For example, the T-shirt tends to recover the shape of a worn cloth.

Conclusion

Our approach disconnects mechanical complexity from geometrical detail. The mechanical resolution is independent from the rendering model, allowing to adjust the mechanical precision in order to interactively animate very detailed objects. Additional data computed in a preprocessing stage (grid points, interpolation weights and stiffness matrices) allows to apply fast and robust finite element dynamics. It does not add complexity compared with traditional FEM. Moreover, we have presented novel hexahedral elements with stiffness warping and robust to inversion. A nice feature of our method is its ability to animate objects including both volumes (closed surfaces) and surfaces, since voxels can be built automatically inside the object and around the surfaces. Our approach is adaptable to more classic finite element meshes, like tetrahedra. It only needs to build meshes at different scales and the interpolation schemes to convert from one level to the other, which is available in all subdivision schemes. Currently, a linear interpolation of the surface inside cell is performed, which can introduce discontinuities of normals in case of large deformations. More continuous interpolation would be an improvement, for example by using Berstein polynomials. Moreover, in order to improve computational speed, interpolation could be performed by the GPU using a vertex shader.

All our examples are limited to a single mechanical level, but the precomputed non-uniform stiffness idea could be extended to meshes with non-uniform resolution, such as deformable octrees.

L 0 = 1 8              8 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 4 0 4 0 0 0 0 0 2 2 2 2 0 0 0 0 4 0 0 0 4 0 0 0 2 2 0 0 2 2 0 0 2 0 2 0 2 0 2 0 1 1 1 1 1 1 1 1              L 1 = 1 8             

  in: (surface, Young modulus, Poisson ratio, max depth, mechanical depth) out: (elements,masses,stiffnesses,interpolation weights) Build the octree at maximal resolution Detect boundary/outside/inside cells Compute the mass and stiffness of the leaves Pop up the mass and stiffness at desired mechanical level

Figure 1 :

 1 Figure 1: The original shape. The entire maximal depth octree. All finest non-empty cells used to precompute mechanical properties. The animated mechanical depth.

Figure 2 :

 2 Figure 2: Algorithm to detect outside/inside/boundary cells. First step consists of detecting boundary cell (grey) i.e., consurface points and/or intersecting surface triangles (red). Then, outside border cells (white) are propagated to their neighborhood (blue arrows). Remaining cells are inside (black).

Figure 3 :

 3 Figure 3: Interpolation of a surface vertex p inside a boundary element.

Figure 4 :

 4 Figure 4: Deformed local frame of an hexahedron.

Figure 5 :

 5 Figure 5: Red cells are boundary and green inside. Filling rates are shown in red transparency. From the maximal depth to the mechanical resolution

Figure 6 :

 6 Figure 6: 2D stiffness in level of gray. From left to right: a) four original cells, three empty, one full. b) the corresponding cell with uniform weighted behavior presented in section 5.1. c) the corresponding non-uniform cell presented in section 5.3, stiffness varies along the axes

Figure 7 :

 7 Figure 7: Two similar objects are simulated by one single cell and subject to gravity. The left one is simulated using a basic uniform law and the right one, using the precomputed non-uniform law.

Figure 8 :

 8 Figure 8: Two similar bunnies with the same mechanical mesh (left) are subject to gravity. The middle one is animated using a uniform stiffness, while the right one uses a nonuniform stiffness. Note the difference of ears behavior.

Figure 9 :

 9 Figure 9: Robustness face to degenerated configurations (top: element inversions, bottom: flat elements recover their rest form by increasing their stiffness).

Figure 10 :

 10 Figure 10: Animate surfaces like volumes.

  Figure 11: Indices.

c

  The Eurographics Association 2006.

Table 1 :

 1 Results.

	c The Eurographics Association 2006.

c The Eurographics Association 2006.

Appendix A: Matrices for Hexahedral Subdivision

The following presents the eight interpolation matrices L c that give values of child cells c nodes from their parent cell nodes, according with indices presented in figure 11. They are very easy to compute. If a i→c j is the influence of the th parent point on the j th point of its c th child, then