Alice Lemoinne
email: alice.lemoinne@inrae.com

Guillaume Dirberg

Myriam Georges

Tony Robinet
email: tony.robinet@mnhn.fr

Fine-scale congruence in bacterial community structure from marine sediments sequenced by short-reads on Illumina and long-reads on Nanopore

Keywords: microbial metabarcoding, environmental DNA, methods, primers, diversity 5 10

Following the development of high-throughput sequencers, environmental prokaryotic communities are usually described by metabarcoding with genetic markers on the 16S domain. However, shortread sequencing encounters a limitation in phylogenetic coverage and taxonomic resolution, due to the primers choice and read length. On these critical points, nanopore sequencing, a rising technology, suitable for long-read metabarcoding, was much undervalued because of its relatively higher error rate per read. Here we compared the prokaryotic community structure in a mock community and 52 sediment samples from two contrasted mangrove sites, described by short-reads on 16SV4-V5 marker (ca. 0.4kpb) analyzed by Illumina sequencing (MiSeq, V3), with those described by long-reads on bacterial nearly complete 16S (ca. 1.5 kpb) analyzed by Oxford Nanopore (MinION, R9.2). Short-and long-reads retrieved all the bacterial genera from the mock, although both showing similar deviations from the awaited proportions. From the sediment samples, with a coverage-based rarefaction of reads and after singletons filtering, co-inertia and Procrustean tests showed that bacterial community structures inferred from short-and long-reads were significantly similar, showing both a comparable contrast between sites and a coherent sea-land orientation within sites. In our dataset, 84.7 and 98.8% of the short-reads were assigned strictly to the same species and genus, respectively, than those detected by long-reads. Primer specificities of long-16S allowed it to detect 92.2% of the 309 families and 87.7% of the 448 genera that were detected by the short 16SV4-V5. Long-reads recorded 973 additional taxa not detected by shortreads, among which 91.7% were identified to the genus rank, some belonging to 11 exclusive phyla, albeit accounting for only 0.2% of total long-reads.

Introduction

The composition and structure of microbial communities are nowadays studied in environmental samples through culture-independent methods, based on nucleic acid sequencing, either in bulk DNA extracted from environmental samples (metagenomic) or only for DNA markers of interest (gene fragments), amplified from environmental samples before they are sequenced (metabarcoding). The metagenomic approach is exempt from amplification bias inherent to metabarcoding (marker specificities, PCR-induced stochasticity) and can produce the useful MAGs (metagenome assembled genomes), but it still faces technical and cost challenges [START_REF] Taş | Metagenomic Tools in Microbial Ecology Research[END_REF]. The metabarcoding approach remains more widely used, much cheaper, but amplification bias are recurrent : (i) primers choice is crucial and constrained by the maximum size of inserts for second-generation sequencers (400bp for Ion Torrent PGM, 550bp for Illumina MiSeq, up to 800 bp for Roche 454, [START_REF] Luo | Direct Comparisons of Illumina vs. Roche 454 Sequencing Technologies on the Same Microbial Community DNA Sample[END_REF]) ; (ii) taxa diversity can be overestimated, because of the nontargeted DNA present in the sample (i.e. DNA from the eukaryotic digestive tracts, or extracellular "relic" DNA, [START_REF] Carini | Relic DNA Is Abundant in Soil and Obscures Estimates of Soil Microbial Diversity[END_REF]) and also because of the ribosomal DNA polymorphism, hidden in individual genomes, an intragenomic variability in the number of duplicates of ribosomal operon, bearing differences in allelic variants between copies [START_REF] Pereira | The Impact of Intragenomic RRNA Variation on Metabarcoding-derived Diversity Estimates: A Case Study from Marine Nematodes[END_REF] ; and (iii) relative abundances of reads per taxa are somehow inaccurate, compared to awaited abundances in the mock samples, a probable consequence of PCR stochasticity and primers specificity.

In high-throughput sequencing (HTS) metabarcoding, primer choice is known to be crucial for taxa resolution, phylogenetic coverage and sensitivity to fine community structure. For prokaryotes, none of all the primer pairs that amplifies markers at a convenient size for short-reads (SR) strategies (> 550bp for Illumina) can give a complete phylogenetic coverage. Primers spanning over more than one 16S V-region are often preferred, because they improve taxonomic resolution. However, any of these combinations (V1-V2, V3-V4, V4-V5, V6-V8, V7-V9, etc.) showed bias in phylogenetic coverage [START_REF] Abellan-Schneyder | Primer, Pipelines, Parameters: Issues in 16S RRNA Gene Sequencing[END_REF]. The 412 bp V4-V5 marker (515F-926R, [START_REF] Parada | Every Base Matters: Assessing Small Subunit RRNA Primers for Marine Microbiomes with Mock Communities, Time Series and Global Field Samples: Primers for Marine Microbiome Studies[END_REF]) covers more broadly the prokaryotic domains (bacteria and archaea), whereas the 438 bp V6-V8 (B969F-BA1406R, [START_REF] Willis | Influence of 16S RRNA Variable Region on Perceived Diversity of Marine Microbial Communities of the Northern North Atlantic[END_REF]) amplifies additional bacterial clades, leading some authors to consider as a best method to combine several short regions along the prokaryotic 16S to minimize these bias [START_REF] Fuks | Combining 16S RRNA Gene Variable Regions Enables High-Resolution Microbial Community Profiling[END_REF]. However, the multiplication of marker standards for bacteria and archaea also plays against intercomparability.

Third-generation DNA sequencers marked a significant progress for metabarcoding studies, in the fact that the marker size was no longer a technical limitation (tens of kpb for PacBio Sequel II, and no theoretical limit for Nanopore devices), and one can target much more binding sites for primers, improving considerably taxonomic resolution and phylogenetic coverage [START_REF] Furneaux | Long-and Short-Read Metabarcoding Technologies Reveal Similar Spatiotemporal Structures in Fungal Communities[END_REF][START_REF] Tedersoo | Perspectives and Benefits of High-Throughput Long-Read Sequencing in Microbial Ecology[END_REF][START_REF] Sahraei | Effects of Operational Taxonomic Unit Inference Methods on Soil Microeukaryote Community Analysis Using Long-Read Metabarcoding[END_REF]).

These long-read (LR) high-throughput sequencers have been first implemented for sequencing markers from cultivated organisms [START_REF] Schlaeppi | High-Resolution Community Profiling of Arbuscular Mycorrhizal Fungi[END_REF][START_REF] Loit | Relative Performance of MinION (Oxford Nanopore Technologies) versus Sequel (Pacific Biosciences) Third-Generation Sequencing Instruments in Identification of Agricultural and Forest Fungal Pathogens'. Edited by Irina S. Druzhinina[END_REF][START_REF] Maestri | A Rapid and Accurate MinION-Based Workflow for Tracking Species Biodiversity in the Field[END_REF]. LR environmental metabarcoding has been usually performed on PacBio sequencers, because the Single-Molecule Real-Time (SMRT) technology offers a read quality similar to those of SR platforms. LR metabarcoding is mostly used for taxonomic groups in which SR are too short for a descent assignment, like micro-eukaryotes and specially fungi [START_REF] Tedersoo | PacBio Metabarcoding of Fungi and Other Eukaryotes: Errors, Biases and Perspectives[END_REF][START_REF] Furneaux | Long-and Short-Read Metabarcoding Technologies Reveal Similar Spatiotemporal Structures in Fungal Communities[END_REF][START_REF] Kolaříková | PacBio Sequencing of Glomeromycota RDNA: A Novel Amplicon Covering All Widely Used Ribosomal Barcoding Regions and Its Applicability in Taxonomy and Ecology of Arbuscular Mycorrhizal Fungi[END_REF][START_REF] Sahraei | Effects of Operational Taxonomic Unit Inference Methods on Soil Microeukaryote Community Analysis Using Long-Read Metabarcoding[END_REF][START_REF] Gueidan | A Long-Read Amplicon Approach to Scaling up the Metabarcoding of Lichen Herbarium Specimens[END_REF], but also a few bacterial phyla [START_REF] Katiraei | Evaluation of Full-Length Versus V4-Region 16S RRNA Sequencing for Phylogenetic Analysis of Mouse Intestinal Microbiota After a Dietary Intervention[END_REF]. Despite several published works showed the possibility to use Nanopore sequencing for environmental or food metabarcoding, by sequencing mock communities of known composition [START_REF] Benítez-Páez | Species-Level Resolution of16S RRNA Gene Amplicons Sequenced through the MinIONTM Portable Nanopore Sequencer[END_REF][START_REF] Davidov | Identification of Plastic-Associated Species in the Mediterranean Sea Using DNA Metabarcoding with Nanopore MinION[END_REF][START_REF] Urban | María Mercedes Zambrano, and Alejandro Sanchez-Flores[END_REF][START_REF] Rodríguez | Evaluation of DNA Metabarcoding Using Oxford Nanopore Sequencing for Authentication of Mixed Seafood Products[END_REF] or by comparing it with an Illumina library sequenced concurrently (J. [START_REF] Shin | Analysis of the Mouse Gut Microbiome Using Full-Length 16S RRNA Amplicon Sequencing[END_REF]H. Shin et al. 2018), the great majority of works that we found in literature did not use the Nanopore platform for environmental LR metabarcoding.

Despite raw reads accuracy are similar for PacBio (88-90%) and Nanopore (95-98% on the R9 flowcells, above 99% for R10.4), the fact that PacBio circular consensus sequence technology (CCS) can align several reads of the same amplicon brings it to an accuracy of >99.9% at 10-fold consensus [START_REF] Tedersoo | Perspectives and Benefits of High-Throughput Long-Read Sequencing in Microbial Ecology[END_REF]. The first LR third-generation sequencer acknowledged to be suitable for metabarcoding was PacBio Sequel II on fungal complete rRNA operon (ca. 3000 bp, Tedersoo, Tooming-Klunderud and Anslan 2018). Despite its error rate being slightly higher than Illumina, the PacBio LR sequencing allowed a much better taxonomic resolution, due to the joint powers of ITS1-ITS2 and SSU-LSU flanking regions on the same amplicon.

Promising attempts were made to reach a satisfactory accuracy with Nanopore, by mimicking PacBio with a rolling circle amplification (RCA, [START_REF] Baloğlu | A Workflow for Accurate Metabarcoding Using Nanopore MinION Sequencing'. Edited by Erica Leder[END_REF] or by flanking, at the two first steps of PCR, each single amplicon with a unique molecular identifier (UMI, [START_REF] Karst | High-Accuracy Long-Read Amplicon Sequences Using Unique Molecular Identifiers with Nanopore or PacBio Sequencing[END_REF]. RCA and UMI methods produce a consensus error rate of 0.7% (coverage > 45x) and 0.01% (> 25x) respectively, offering a quality similar to PacBio or Illumina standards [START_REF] Baloğlu | A Workflow for Accurate Metabarcoding Using Nanopore MinION Sequencing'. Edited by Erica Leder[END_REF][START_REF] Karst | High-Accuracy Long-Read Amplicon Sequences Using Unique Molecular Identifiers with Nanopore or PacBio Sequencing[END_REF]. The consensus, compared with BLAST [START_REF] Camacho | BLAST+: Architecture and Applications[END_REF] to reference sequences of a curated database, could be assigned more accurately to a taxa than standard short markers do (reviewed by Kerkhof 2021). However, lab and downstream bioinformatic workflows are quite complex to implement for ecology scientists, requiring a higher technicity in library preparations and in downstream bioinformatics than for directly sequencing amplicons from environmental samples, as we tested it without success. To date, no environmental metabarcoding based on RCA or UMI protocols have been published.

In community ecology, Nanopore was initially used for simply barcoding individuals with LR [START_REF] Maestri | A Rapid and Accurate MinION-Based Workflow for Tracking Species Biodiversity in the Field[END_REF]), but quickly metabarcoding appeared with Nanopore sequencing alone, to detect pathogen bacterial strains, mostly by a metagenomic approach [START_REF] Brown | MinIONTM Nanopore Sequencing of Environmental Metagenomes: A Synthetic Approach[END_REF][START_REF] Charalampous | Nanopore Metagenomics Enables Rapid Clinical Diagnosis of Bacterial Lower Respiratory Infection[END_REF][START_REF] Cuscó | Shallow Metagenomics with Nanopore Sequencing in Canine Fecal Microbiota Improved Bacterial Taxonomy and Identified an Uncultured CrAssphage[END_REF], or eukaryotic communities on the more or less complete rRNA operon (H. [START_REF] Lu | Oxford Nanopore MinION Sequencing and Genome Assembly[END_REF][START_REF] Rodríguez | Evaluation of DNA Metabarcoding Using Oxford Nanopore Sequencing for Authentication of Mixed Seafood Products[END_REF]). For bacterial communities, studies with a metabarcoding workflow on environmental samples and relying only on Nanopore MinION, aimed at characterizing mouse gut or human respiratory bacteriomes (J. [START_REF] Shin | Analysis of the Mouse Gut Microbiome Using Full-Length 16S RRNA Amplicon Sequencing[END_REF][START_REF] Ibironke | Species-Level Evaluation of the Human Respiratory Microbiome[END_REF], bacteria associated with algae or plastic debris at sea (H. [START_REF] Shin | Elucidation of the Bacterial Communities Associated with the Harmful Microalgae Alexandrium Tamarense and Cochlodinium Polykrikoides Using Nanopore Sequencing[END_REF][START_REF] Davidov | Identification of Plastic-Associated Species in the Mediterranean Sea Using DNA Metabarcoding with Nanopore MinION[END_REF][START_REF] Van Der Loos | Characterizing Algal Microbiomes Using Long-Read Nanopore Sequencing[END_REF], pathogenic bacteria in food [START_REF] Planý | Evaluation of Bacterial Consortia Associated with Dairy Fermentation by Ribosomal RNA (Rrn) Operon Metabarcoding Strategy Using MinION Device[END_REF], fungal communities and biotic interactions [START_REF] Vass | Co-occurrences enhance our understanding of aquatic fungal metacommunity assembly and reveal potential host-parasite interactions[END_REF] or pelagic bacteriomes in freshwaters [START_REF] Urban | María Mercedes Zambrano, and Alejandro Sanchez-Flores[END_REF].

In all studies we found, Nanopore was used alone, except for two. [START_REF] Loit | Relative Performance of MinION (Oxford Nanopore Technologies) versus Sequel (Pacific Biosciences) Third-Generation Sequencing Instruments in Identification of Agricultural and Forest Fungal Pathogens'. Edited by Irina S. Druzhinina[END_REF] compared it with PacBio CCS for detecting fungal pathogens in plants, concluding that "MinION could be used for rapid and accurate identification of dominant pathogenic organisms and other associated organisms from plant tissues following both amplicon-based and PCR-free metagenomics approaches". J. [START_REF] Lu | Nanopore Sequencing of Full RRNA Operon Improves Resolution in Mycobiome Analysis and Reveals High Diversity in Both Human Gut and Environments[END_REF] characterized mycobiomes of fungal isolates and environmental samples by sequencing in parallel the full rRNA operon on MinION and the shorter ITS2 on Illumina HiSeq. They concluded that "ITS2 sequencing [was] more biased than full operon sequencing".

At the time of this study, there was no sequencing platform in the research center nearby the place where sediment samples were collected and stored (Guadeloupe Island). The estimated cost of 1 Gb PacBio sequencing (17€) was lower than Illumina NovaSeq (44€) and MiSeq (56€), but the accessibility to a PacBio sequencer was difficult for this remote place, because of the instrument cost (650 k€ for a PacBio Sequel II) and technicity. Today, the MinION device of Oxford Nanopore Technologies is accessible for 900€, the estimated cost for 1Gb is about 12€, and its smartphone size allows scientists to use it as a field lab device. The portability of the MinIon device is advantageous for molecular ecology scientists located far away from a research center, opening possibilities for studying microbial communities from a field lab, i.e. equipped with usual devices for DNA extraction (mortar, mini-centrifuge, spectrophotometer for DNA drops), PCR (freezer, thermocycler, electrophoresis tank, UV table, ultra-pure water), and libraries making (DNA fluorometer, DNA dryer). Such a field lab is affordable and quite simple to set up for molecular ecologists in remote places or for proposing environmental metabarcoding in the frame of engineering consultancy.

To date, no published work has compared Nanopore LR to Illumina SR bacterial metabarcoding on the same environmental samples, perhaps because of the impossibility to disentangle the effects of primers from those of sequencing technologies. Here we propose to overcome this issue by evaluating the similarity of bacterial communities described from marine sediments sequenced by 16S LR on Nanopore MinION device, and by sequencing in parallel the 16S-V4V5 SR amplified from same DNA extracts on Illumina MiSeq, and by addressing a simple question: will these sequencing strategies conserve (i) the structures of bacterial communities between two neighboring mangrove sites, and (ii) the sea-land orientation of bacterial communities within sites?

Materials and methods

Sampling sites and sample collection

In June 2019, two sites were selected in the mangrove of Guadeloupe Island, at 6 km of distance each other, for their a priori difference in the level of direct and indirect human pressures (Fig. 1a-b) : the impacted "Rivière salée" site was located on the foreshore of a salty river, close to the city of Pointe-à-Pitre, to its dump and its airport (latitude -61,5469; longitude 16,2594) ; the less-impacted "Babin" site was located in a Ramsar-protected area close to coral reefs (latitude -61,5294 ; longitude 16,3388).

A total of 54 samples of surface sediment were collected on intertidal zone, on 3 lines of 3 points each respectively in each site, each line separated by 3 m to the neighboring line. Points were separated by 12.5 m within a line. Each point was composed of 3 biological replicates (a, b and c), analyzed in the workflow separately (Fig. 1c). The line closest to the sea was the "seaward line", those closest to the inland mangrove was the "landward line", the "middle line" was in between. Therefore, each line showed a different time of marine immersion per day. Each replicate was sampled with a sterile syringe and appropriated microbiological precautions, stored in a 50ml Falcon tube, freezed a couple of hours after sampling and preserved at -20° C.

DNA extraction

Samples were sent in France, freeze-dried and crushed to powder in a mortar, carefully cleaned with an alcoholic tissue between each sample processing. Total genomic DNA from 50mg of dried samples and a standard microbial community (zymoBIOMICS Microbial Community Standard D6300, by ZYMO RESEARCH), here named "Ze", were extracted using the NucleoSpin Soil kit (Macherey-Nagel) with a final elution volume of 50 µl following the manufacturer instructions. After this DNA extraction of samples and Ze, nucleic acid yield and purity were checked using a Nanodrop spectrophotometer (Thermo Fisher Scientific) and the concentration of each sample was equalized to final concentration of 10ng.µl -1 on a PCR plate of 96 wells.

Illumina library

In order to limit PCR biases, the first round of PCR consisted in 3 PCR replicates per sample, targeting the DNA coding for the V4-V5 hypervariable region of 16S RNA ribosomal with degenerate primers [START_REF] Parada | Every Base Matters: Assessing Small Subunit RRNA Primers for Marine Microbiomes with Mock Communities, Time Series and Global Field Samples: Primers for Marine Microbiome Studies[END_REF]) : 515F (GTGYCAGCMGCCGCGGTAA) and 926R (CCGYCAATTYMTTTRAGTTT). Two other primer pairs (18SV9 and ITS2) were amplified, added to libraries and sequenced together with 16SV4-V5. They were in raw reads but not presented here.

Each primer was flanked in its 5'-end by a nucleotide sequence used for indexing at a later step, according to a protocol proposed by Nag et al. (2017). At this stage, 2 additional PCR blanks were done with water instead of extracted DNA. Each 12,5 µl reaction mix contained 1 µl of DNA (~10ng.µl -1 ), 0,25 µl of forward primer, 0,25 µl of reverse primer (10nM), 6,25µl of 2✕ Promega Green Master mix G2, 4,25µl of milliQ water. The PCR cycles consisted of of initial denaturing for 2 min at 94°C, followed by 30 cycles (denaturation 30 s at 94°C, hybridization 30 s at 51°C, elongation 45 s at 72 °C) and a final elongation during 5 min at 72°C. First PCR products were verified by electrophoresis on 1% agarose gel, re-amplified if negative until they were positive. Each PCR triplicate was pooled into one before the indexing PCR. Indexation PCR was realized in a 27.5 µl reaction mix containing 2 µl of first PCR products, 5 µl of reverse and forward index, 12,5µl of NEB Q5 2X mix and 8µl of milliQ water. This second PCR consisted of a initial denaturing for 30s at 98°C, followed by 30 cycles (denaturation 20s at 98°C, hybridization 20s at 60 °C, elongation 10s at 72°C) and final elongation 10s at 72°C. At this stage, one PCR blank was added with water instead of first PCR products. All indexed samples were pooled into a single low-bind tube and purified with magnetic beads (Nucleomag, Macherey Nagel, 1:1 ratio). Size range of final PCR products was verified by electrophoresis (Agilent BioAnalyzer, High-sensitivity), with an waited size peak around 420bp, then pooled in a final library, and sequenced on an Illumina MiSeq (one Miseq Reagent v3 kit 600 cycles and one nano MiSeq Reagent kit v2 kit 500 cycles for resequencing) in the Concarneau marine station (MNHN) to output demultiplexed fastq files.

Nanopore library

The same DNA extracts were processed in parallel for Nanopore LR sequencing, with the following 16S markers : V1-V9 regions (nearly complete 16S for bacteria, ~1.45 kpb; [START_REF] Weisburg | '16S Ribosomal DNA Amplification for Phylogenetic Study[END_REF]; 27F:AGAGTTTGATCMTGGCTCAG ; 1492R: TACGGYTACCTTGTTACGACTT). PCRs were performed in 3 small-volume replicates of 12,5 µl each, containing 6,25µl of LongAmp Taq 2✕ 165Master Mix (NEB), 4,25µl of milliQ water, 1 µl of DNA (~10ng.µl -1 ), 0,25 µl of forward primer, 0,25 µl of reverse primer (10nM each). PCR cycles consisted of initial denaturing for 3 min at 94°C, followed by 30 cycles composed of denaturation for 30s at 94°C, hybridization for 30s at 51°C, and elongation for 45s at 65°C and final elongation for 10 min at 65°C. All first PCR products were verified by agarose gel electrophoresis, re-amplified if negative until they were positive, and positive triplicates were pooled into one before the indexation PCR. Concentrations were measured by the Qubit fluorometer (dsDNA BR kit) and brought back to a concentration of 1ng.µl -1 . Indexation PCR was realized according to the Nanopore « PCR barcoding ( 96) amplicons (SQK-LSK109) » protocol. Indexed amplicons were pooled into one tube per primer/marker and purified with magnetic beads (Nucleomag Macherey Nagel, 1:0.8 ratio). Indexed and purified products were verified on agarose gel electrophoresis.

DNA concentration was measured by phospho-luminescence (Qubit), then diluted in order to have 1µg of DNA into 47µl of water. Final ligation of Nanopore sequencing adapters was done following the "SQK-LSK109 with EXP-PBC096" protocol. 16S V1-V9 library was sequenced on two R9.4.1

MinION flow cells (half of the samples + Ze for each). Flow cells were loaded on MinION Mk-1C and sequenced for approximately 48h, until no further sequencing reads could be collected above Q10 quality score. Fast5 files were basecalled and demultiplexed using Guppy 6.4.2 high-accuracy model on a local GPU (Nvidia Quadro K4000) and DNA sequence reads were output with >Q10 flag, as fastq files. For Illumina SR and Nanopore LR, indexes with less than 1500 reads were resequenced.

Sequence data are available in NCBI with BioProject accession number PRJNA985243.

Processing of raw reads

Fastq files from Illumina SR were filtered with R package DADA2 v 1.16.0 [START_REF] Callahan | DADA2: High-Resolution Sample Inference from Illumina Amplicon Data[END_REF].

Reads R1 and R2 were filtered using the filterAndTrim function (minLen=200, matchIDs=TRUE, maxN=0, maxEE=c(3,3)), then merged to unique sequences (ASVs) with at least 12 overlapping nucleotides between R1 and R2. Chimeric sequences were removed using the removeBimeraDenovo function. A matrix of 16SV4-V5 ASVs per sample was obtained and processed by Qiime2 tools, after 16SV4-V5 ASVs were extracted from fasta files containing sequences from other primers (18SV9 and ITS2, not presented here). Nanopore LR fastq sequences (>Q10) were filtered with Nanofilt : all reads shorter than 1.4 kpb and longer than 1.6 kpb for 16S V1-V9 were removed. Then, ASVs with 97% of similarity were clustered into OTUs using the Vsearch (version 2023.7.0) tool. OTUs were taxonomically assigned with a trained Qiime2 classifier, inferring to the SILVA NR 99 reference database v138.1 [START_REF] Quast | The SILVA Ribosomal RNA Gene Database Project: Improved Data Processing and Web-Based Tools[END_REF], formatted for this specific marker.

Community structures analysis

Chloroplastic, mitochondrial and eukaryotic assignments, contaminants detected from blanks and singletons (OTUs with only one read in all samples) were removed from OTU tables. Tables of filtered OTU read abundances, OTU taxonomy and sample data were imported to make phyloseq objects in R, one for each marker (R package phyloseq, McMurdie and Holmes 2013). Bacterial taxa known to be present in Ze samples were all above a relative abundance threshold of 1.8% for Illumina 16SV4-V5 and of 1.0% for ONT (Fig. 2), so relative abundances in all phyloseq objects were filtered above these thresholds.

The prokaryotic community structure of environmental samples depends tightly on the read number in each sample. The conventional rarefaction consists in randomly depleting reads in each sample, until all samples reach the number of reads of the poorest one [START_REF] Simberloff | Properties of the Rarefaction Diversity Measurement[END_REF]. This method is known to have major bias: non-reproducibility since reads are removed randomly, and alteration of community structures due to the random sorting of rare species [START_REF] Coddington | Undersampling Bias: The Null Hypothesis for Singleton Species in Tropical Arthropod Surveys[END_REF]. In soil or sediment microbiotas, sample OTU richness depends strongly on sample size, therefore we opted for the rarefaction method developed by [START_REF] Chao | Coverage-Based Rarefaction and Extrapolation: Standardizing Samples by Completeness Rather than Size[END_REF], consisting in comparing samples of equal completeness (equal coverage), not of equal size. "When samples are standardized by their coverage (a measure of sample completeness [...]) instead of by their size, the estimated richnesses approximately satisfy a replication principle, which is an essential property for characterizing diversity" [START_REF] Chao | Coverage-Based Rarefaction and Extrapolation: Standardizing Samples by Completeness Rather than Size[END_REF]. This coverage-based rarefaction was used by the function phyloseq_coverage_raref (R package metagMisc, Mikryukov 2019).

Since the SR primers (16SV4-V5) amplified both bacteria and archaea, but LR primers (16SV1-V9) amplified bacteria only, archaeal taxa obtained by SR were removed for the present analysis (deleted after rarefaction). Further analyses with archaeal taxa are however proposed in Supplementary Materials.

Analyses were carried out on filtered OTU tables after coverage-based rarefaction, except for Fig. 5, in which both rarefaction methods are shown. In sediment samples, core members were identified by their prevalence among all samples (≥50%, i.e. they were present in 50% or more of the samples, Fig. 5). For exploring dissimilarities between datasets, a Principal Coordinate Analysis (PCoA, from phyloseq ordinate function, equivalent to MDS -Metric Multidimensional Scaling) was performed on matrices of Bray-Curtis distances between communities. To identify the most contributing OTUs to the different parts of the communities, a Principal Component Analysis (PCA, from R package ade4, dudi.pca function) was performed on relative abundances. In order to assess the similarity of community structures described by both sequencing methods, a Procrustes analysis was carried out on their respective PCoA scores, with procrustes and protest functions (R package vegan). In parallel, a co-inertia analysis on PCA two first components was done, with coinertia and RV.rtest (999 permutations) from ade4. A Mantel permutation test was performed on two matrices of Bray-Curtis distances, for SR and LR bacterial communities (Pearson method, 999 permutations), with vegan. Classification trees were used to characterize the genus and species contributing the most to the [site x (sea-land orientation] effect in each dataset by the R package randomForest [START_REF] Liaw | Classification and regression by randomForest[END_REF].

Results

Mock Community

OTUs were filtered on a minimum of 50 reads/OTU for Ze. The 8 prokaryotic taxa of the Ze community have all been found with each sequencing method (Fig. 2). However, the proportions of reads assigned at each taxa were not those supposed to have been initially introduced (12% for each prokaryote). The SR and LR methods both overestimated some genus (Lactobacillus, Limosilactobacillus, Salmonella) and underestimated some others (Escherichia, Listeria, Enterococcus). Both methods found undetermined reads (5.1% for SR, 11.0 and 10.4% for LR_1 and LR_2, respectively). The re-sequencing of Ze on two different LR flow-cells showed a stability in the results.

Samples read coverage

With conventional read rarefaction, for bacteria only, all samples were standardized at 1582 reads for both sequencers, resulting in a total of 570 (16SV4-V5) and 967 (full-length 16S) bacterial species, so in proportion full-length 16S counted 170% of the species detected by 16SV4-V5. With coverage-based rarefaction, samples analyzed by SR presented 749 species for 2609 reads in average (min 1338, max 4991), samples analyzed by LR showed 1495 species for 5108 reads in average (min 4451, max 6019, Table 1). In proportion with this rarefaction method, LR detected twice more species than SR. For the rest of this section, only results obtained by the coveragebased rarefaction method are presented.

Variations in community structures

Community composition and multivariate analyses showed that both technologies detected a marked difference between bacterial communities from Babin and Rivière salée sites, but also their fine-scale orientation, from sea-to land-oriented samples. Communities sequenced by SR and LR described the same global patterns, i.e. a preponderance of Pirellulales (Planctomycetota) in Rivière salée, of Pseudomonadales (Gammaproteobacteria) and Bacteroidales (Bacteroidota) in Babin, separating clearly the two sites in ordination (Fig. 3a-b). Babin showed the most structured community along the tidal gradient, with the presence of Pseudomonadales in seaward samples and of Bacteroidales (Bacteroidota) in landward samples. Biological replicates were relatively close to each other in the PCoAs (Fig. 4a-b), but Bray-Curtis dissimilarity indexes of communities within replicates were always higher for SR than for LR, either for Babin or Rivière salée (Fig. 4c, anova p<0.001).

The Procrustes analysis of the two first axes of multivariates showed a significantly strong similarity between structures drawn by SR and LR (Fig. 4d-e, p<0.001), confirmed by a co-inertia analysis on PCA's two first axes (p<0.001). The Mantel test indicated a significant correlation coefficient of 0.7248 (p<0.001) between the Bray-Curtis dissimilarity matrices obtained from LR and SR communities using OTUs at species taxonomic rank. In order to point out the similarity of taxa 270 280 contributing to the [site (sea-land orientation] effect, ✕ classification trees were made by a random forest approach on the 393 genus and 285 families shared between SR and LR. Models found 48% of similarity among the top-100 contributing genus and 63% among the top-100 contributing families between sequencers. However, taxa contributing in the same way to the [site (sea-land ✕ orientation] effect were scarce (Fig. S2, Table S1). Archaean communities described with LR specific primers followed roughly the structure obtained with bacteria (Fig. S3).

Phylogenetic diversity

Over the 56 bacterial phyla detected in total, 54 were detected by LR and 45 by SR (Table 1). At high taxonomic levels, 16SV4-V5 and full-16S were approximately ⅘ alike for phyla, LR detected 11 exclusive phyla over a total of 54 for this primer (20% of exclusive among those detected by LR), when SR only had 2 (4.4%). However, LR-exclusive subcommunity represented only 0.2% of reads in the full community (Fig. S1). The 11 phyla only detected by LR were Acetothermia, WS2, LCP-89, WOR-1, Armatimonadota, Margulisbacteria, Nitrospinota, Fermentibacterota, Methylomirabilota, Caldatribacteriota, WPS-2, whereas the only 2 detected by SR were Cloacimonadota and CK-2C2-2, with a coverage-based rarefaction (Fig. 5). At lower taxonomic levels, 92.2% and 87.7% of respectively the family and bacterial genus detected by 16SV4-V5 were detected by full-16S. LR detected twice more species than SR, with only 34.9% of the species and 50.1% of the genus detected shared with SR. The trend that LR detected almost all SR taxa plus a certain number of LR original taxa decreased with lowering taxonomic ranks (Fig. 6).

All the 54 LR-detected phyla were more diversified based on full-16S, but four : NB1-j, SAR324, Dadabacteria and Hydrogendentes. The most diversified phylum, the Proteobacteria, presented more than 4 times more species with full-16S than with 16SV4-V5. Overall, communities described by the two primer-sets were phylogenetically very similar when considering shared taxa at the family and genus level (92.2% and 87.7% of taxa similarity for SR vs. LR, respectively). 11.7% of the LR (bacterial 16S) were unassigned at the phylum level, versus 0.36% for SR. 53.1% of the LR unassigned at the species level (35.4% of total LR bacterial OTUs), versus 46.0% for SR (34.7% of total SR bacterial OTUs, Table 1). For shared genera, the unassigned reads were much lower for SR (5.8%) than LR (35.5%). All core-phyla detected by SR were also parts of core-phyla detected by LR, whatever the rarefaction method used (Fig. 5).

Discussion

In this study, bacteria were amplified on their rRNA gene by 16SV4-V5 and full-16S primers from the same DNA extractions of environmental samples, sequenced on Illumina and Nanopore respectively, and assigned on the same database of reference sequences (Silva 138.1 SSU LR99). Filtered and standardized with a coverage-based rarefaction, the bacterial communities described by both primer pairs and sequencing tools were similar in their coarse structure (site effect) and fine structure (sea-land orientation), with nonetheless a couple of constant differences, already noticed with SR vs. LR sequencing on PacBio [START_REF] Katiraei | Evaluation of Full-Length Versus V4-Region 16S RRNA Sequencing for Phylogenetic Analysis of Mouse Intestinal Microbiota After a Dietary Intervention[END_REF]): (i) communities described by full-16S were twice more species-diversified than those described by 16SV4-V5, reflecting differences in primer pairs ; (ii) abundances of OTUs based on LR were slightly less variable within biological replicates than those based on SR. The present work suggests that LR can be used for characterizing bacteria communities by metabarcoding environmental samples on a Nanopore sequencing device.

Similar site-and sea-orientation patterns, based on short-or long-reads

Coarse and fine spatial structures were overall significantly similar, since the site effect and the sealand orientation were conserved in ordinations.

Differences in abundances for the same taxa were obvious in the structure of mock communities, i.e. coming from the same DNA extraction but followed by separate amplification on different primers, different library preparation and sequencing. This discrepancy is typical and outlines the semi-quantitative trait of any microbial HTS sequencing. However, all qualitative elements (betadiversity) of mocks were preserved, allowing us to extend this observation to communities described from environmental samples processed with the same workflow as for the mock. This assumption of a correct taxa detection in spite of abundance discrepancies may explain differences observed in top-20 bacterial taxa influencing structures (Fig. 3a), and is reinforced by the relative orientation of samples, preserved between the two sequencing workflows on the same ordination (Fig. 3b).

On the other hand, it is noteworthy that LR communities contained twice more species than SR ones and this did not change the overall structure of ordinations, providing evidence that corecommunities in both sequencing strategies were congruent and that additional taxa detected by LR did not significantly changed this ordination. In another perspective, 16SV4-V5 Illumina's communities, albeit reduced, were sufficient and contained the smallest share of taxa needed to correctly describe the assemblages at play.

Long-reads outperformed short-reads for taxonomic diversity [START_REF] Katiraei | Evaluation of Full-Length Versus V4-Region 16S RRNA Sequencing for Phylogenetic Analysis of Mouse Intestinal Microbiota After a Dietary Intervention[END_REF] sequenced full-16S amplicons on a PacBio system, and extracted afterward in silico the 16SV4 fragments. In-silico-extracted V4 dataset had approximately half of the read count per sample, compared to those of the full-16S PacBio dataset, indicating that a significant proportion of the taxa that were identified by full-16S were not detected by extracting the V4-region from the same initial sequences. In this way, the length of the 16S fragment can modify the taxonomic assignment, a longer fragment increasing the diversity of taxa assigned, albeit not figuring if they were true taxa or not. Our study confirmed that there were much more taxa detected by full-16S than by 16SV4-V5, but also that a certain proportion of taxa sequenced with 16SV4-V5 were not detected with full-16S dataset (30.3% of the species and 13.3% of the genus). However, this observation is much tempered when the proportion of reads involved in these non-detected taxa is considered, concerning 15.3% of the reads for species and only 1.2% for genus.

350

When considering non-shared taxa, the present study illustrated the assignation power of a longer bacterial 16S rRNA, compared to a restricted 16S V-region, incidentally acknowledged to have the most appropriate cover for bacteria among SR primers [START_REF] Parada | Every Base Matters: Assessing Small Subunit RRNA Primers for Marine Microbiomes with Mock Communities, Time Series and Global Field Samples: Primers for Marine Microbiome Studies[END_REF][START_REF] Walters | Improved Bacterial 16S RRNA Gene (V4 and V4-5) and Fungal Internal Transcribed Spacer Marker Gene Primers for Microbial Community Surveys[END_REF][START_REF] Willis | Influence of 16S RRNA Variable Region on Perceived Diversity of Marine Microbial Communities of the Northern North Atlantic[END_REF]). Taxa assignment rates were lower at species level whatever the read length, probably due to the incomplete databases that are constantly being updated, or pseudogenes and intra-genome 16S polymorphism [START_REF] Pei | Diversity of 16S RRNA Genes within Individual Prokaryotic Genomes[END_REF][START_REF] Větrovský | The Variability of the 16S RRNA Gene in Bacterial Genomes and Its Consequences for Bacterial Community Analyses[END_REF], impossible to evaluate with our approach.

Our study on marine sediment samples could not provide evidence that full-16S improved the taxonomic assignment, as it was done with human gut microbial communities [START_REF] Jeong | The Effect of Taxonomic Classification by Full-Length 16S RRNA Sequencing with a Synthetic Long-Read Technology[END_REF][START_REF] Matsuo | Full-Length 16S RRNA Gene Amplicon Analysis of Human Gut Microbiota Using 430 445 455 465 MinIONTM Nanopore Sequencing Confers Species-Level Resolution[END_REF]). However, if genus level is considered as the maximum resolution of 16S sequencing for a correct taxonomic assignment, a proper primers comparison should start from genus toward higher levels. The fact that read assignment was always lower for LR-exclusive taxa probably reflects more the fact that mangrove sediments contain a high diversity of uncultivated microbes with presently unavailable full-16S in reference databases, than a lower sequencing accuracy of Nanopore (and therefore a plausible sequencing-platform effect). showing the degree of matching between the two ordinations ; empty dots show the position of the samples in the LR ordination and arrows point to their positions in the SR ordination ; the plot also shows the rotations between the axis (solid vs. dashed), necessary to make ordinations match as closely as possible ; (e) residuals for each sample between the ordinations (this time, on the 20 first axis); the horizontal lines, from bottom to top, are the 25% (dashed), 50% (solid), and 75% (dashed) quantiles of the residuals. (a) Venn diagrams showing the proportions of bacterial taxa shared and unshared between both primer pairs, at each phylogenetic rank (numbers in the discs refer to the numbers of taxa of the portion of the disc it is written on) ; (b) On left axis, proportion of bacterial taxa shared between both primer pairs (lines with dots) and proportion of reads assigned to taxa shared between both primer pairs (lines with stars), number of shared and unshared taxa (bars, right axis), at each phylogenetic rank ; more details in Table S1. 1. Statistics on bacterial taxa detected by SR and LR primer pairs, shared or unshared taxa between primer pairs, by taxonomic rank, percentage of taxa or reads assigned. Here only the bacterial dataset was coverage-based rarefied and singletons filtered. *among taxa assigned at this rank (non-assigned are not counted). In the bacterial communities analyzed, 84.7 and 98.8% of SR were assigned to the same species and genus, respectively, than those detected by LR (in orange). Conversely, 68.5 and 89.6% of LR were assigned to the same species and genus, respectively, than those detected by LR (in blue).

640

Figure 1 .

 1 Figure 1. (a) Location of sampling sites on Guadeloupe Island (red squares) ; (b) zoom on the two sampling sites, with site names ; (c) sampling protocol in each site : 3 lines of 3 points, each composed of 3 biological replicates (a, b and c), at 12.5m of distance between each point on each line.

Figure 2 .

 2 Figure 2. Relative abundances found in a mock community sequenced by SR (1 Illumina flow-cell, 16SV4), LR1 and LR2 (2 Nanopore flow-cells, full-16S), after singletons filtering ; the theoretical abundances are shown at left. *Eukaryotic taxa, which are present in the mock but not supposed to be amplified with 16S bacterial markers. Mock datasets were filtered at a minimum depth of 50 reads per OTU, at species level (97%), then aggregated to genus level.

Figure 3 .

 3 Figure 3. (a) Top-20 bacterial orders in samples for both sequencing devices, ranked by their overall relative abundances in samples ; (b) biplot of sample scores from a nMDS on abundances of bacterial OTUs agglomerated at genus level, for both sequencing devices (stress=14.1%) ; for this common ordination, shared OTUs were named differently between SR and LR on purpose, in order to separate the two datasets for a better visualization. Number of reads per sample was rarefied with the coverage-based method (Chao and Jost 2012).

Figure 4 .

 4 Figure 4. (a-b) PCoA on coverage-based rarefied abundances of bacterial communities at species level, (a) sequenced by SR, showing biological replicates (polygons) ; (b) sequenced by LR ; (c) Dispersion of Bray-Curtis dissimilarity index within biological replicates, salmon boxplots for SR, cyan for LR ; thick horizontal lines : mean ; box plots : 75% range ; whiskers : 95% range ; dots: outliers ; (d) Procrustes analysis of the 2 first components of both PCoAs (presented in a-b),showing the degree of matching between the two ordinations ; empty dots show the position of the samples in the LR ordination and arrows point to their positions in the SR ordination ; the plot also shows the rotations between the axis (solid vs. dashed), necessary to make ordinations match as closely as possible ; (e) residuals for each sample between the ordinations (this time, on the 20 first axis); the horizontal lines, from bottom to top, are the 25% (dashed), 50% (solid), and 75% (dashed) quantiles of the residuals.

  615

Figure 5 .

 5 Figure 5. Number of OTUs (97% similarity, singleton-filtered) for each prokaryotic phylum in environmental samples analyzed here (bacteria only), depending on the sequencing strategy : (a) with conventional equal-rarefaction (1582 reads for all samples of both strategies, see Methods section for comments on inner bias) ; (b) with coverage-based read rarefaction (see Results section for details). Phylum names in red or blue were detected only by SR or only by LR, respectively. Red or blue dots indicate core-phyla, i.e. phyla with a minimum prevalence of 50% in the respective datasets. Red or blue arrows indicate phyla that were not detected with read equal-rarefaction, for SR or LR respectively.

Figure 6 (

 6 Figure 6 (next page). (a)Venn diagrams showing the proportions of bacterial taxa shared and unshared between both primer pairs, at each phylogenetic rank (numbers in the discs refer to the numbers of taxa of the portion of the disc it is written on) ; (b) On left axis, proportion of bacterial taxa shared between both primer pairs (lines with dots) and proportion of reads assigned to taxa shared between both primer pairs (lines with stars), number of shared and unshared taxa (bars, right axis), at each phylogenetic rank ; more details in TableS1.

  Figure 6.

Table

	Primers / Sequencer	Taxa / Reads	Phylum	Class	Order	Family	Genus	Species
		Taxa detected	45	106	209	309	448	749
		% taxa assigned	97.8%	92.5%	91.9%	88.0%	82.4%	65.3%
	Short-reads	% reads assigned	99.6%	99.6%	98.3%	98.0%	94.0%	54.0%
	135 692 reads Illumina	Taxa exclusive % taxa assigned % reads assigned	2 100% 100%	7 100% 100%	15 93.3% 97.1%	24 87.5% 83.9%	55 81.8% 84.5%	227 80.6% 85.8%
	(515F + 926R)	% taxa shared with long-reads	95.6%	93.4%	92.8%	92.2%	87.7%	69.7%
		% reads with same assignment	99.99%	99.97%	99.7%	99.7%	98.8%	84.7%
		than LR*
		Taxa detected	54	140	316	483	785	1495
		% taxa assigned	98.1%	89.3%	89.2%	85.9%	80.9%	64.6%
	Long-reads	% reads assigned	88.3%	86.0%	74.9%	73.3%	67.8%	46.9%
	265 650 reads Nanopore	Taxa exclusive % taxa assigned % reads assigned	11 100% 100%	41 82.9% 70.5%	122 85.2% 87.2%	198 82.8% 83.7%	392 79.3% 81.8%	973 67.8% 67.4%
	(27F + 1492R)	% taxa shared with short-reads	79.6%	70.7%	61.4%	59.0%	50.1%	34.9%
		% reads with same assignment	99.7%	99.3%	95.7%	94.7%	89.6%	68.5%
		than SR*
	Shared taxa	Shared taxa % taxa shared % taxa assigned among shared	43 76.8% 97.6%	99 67.3% 91.9%	194 58.6% 91.8%	285 56.2% 88.1%	393 46.8% 82.4%	522 30.3% 58.6%

Acknowledgements

This work was lead by the GT DCE Mangroves (Groupe de Travail Directive Cadre sur l'Eau Mangroves), founded by the OFB (Office Français de la Biodiversité, Olivier Monnier), lead by the MNHN (Muséum National d'Histoire Naturelle, GD). Lab work was achieved by AL during her Master practice, with BOREA financial support. Thanks to our colleague Cédric Hubas (BOREA), for discussion and statistical advice. Thanks to associate-editor and reviewers of PCI microbiology for their pertinent suggestions.

Funding

This work has been funded by the OFB (Office Français de la Biodiversité, Groupe de Travail Mangroves) for field work and G.D. appointment, and by our research unit BOREA for A.L. Master grant and genetics consumables 550 565

Data accessibility

All data presented and scripts in this manuscript are available from the GitHub repository : https://github.com/tonyrobinet/nanopore_metabarcoding. Raw sequences (fastq format) are available in NCBI with BioProject accession number PRJNA985243.

Competing interests

The authors declare no competing interests Supplementary material Available at https://github.com/tonyrobinet/nanopore_metabarcoding S2). Mean Decrease Gini is a measure of how each variable contributes to the homogeneity of the nodes and leaves in the resulting random forest (see Methods for details) ; the higher the value of MDG score, the higher the importance of the variable in the model. Table S1. Archaea detected by SR were mentioned but the read coverage by sample was much lower than those for archaeal LR.

Methods and results for archaea

Table S2. Bacterial genus contributing the most importantly to the site effect, after a random forest analysis on Illumina and Nanopore datasets. In green : OTUs common to both datasets. MDG : mean decrease in Gini coefficient, a measure of how each variable contributes to the homogeneity of the nodes and leaves in the resulting random forest ; the higher the value of MDG score, the higher the importance of the variable in the model.


D Zeka

P.-A Guidault

D Néron

M Guiton

G Enchery

Preliminary study for the simulation of wire ropes using a model reduction approach suitable for multiple contacts

Keywords: Proper Generalized Decomposition, LATIN method, frictional contact

Le contexte de cet article concerne la simulation de lŠévolution complexe des conditions de contact et de frottement entre les Ąls des câbles des liignes dŠancrage dŠéoliennes Ćottantes offshore pour prédire avec précision leur durée de vie en fatigue. AĄn de faire face aux fortes non-linéarités du problème et aux temps de calcul élevés nécessaires, le solveur LATIN non linéaire et la réduction de modèle Proper Generalized Decomposition (PGD) sont proposés pour résoudre le problème. Dans ce contexte, un cas test simple unidimensionnel, représentatif du problème posé, est analysé dans le cadre LATIN-PGD : différents indicateurs de convergence pour le solveur LATIN sont étudiés aĄn de vériĄer et assurer la convergence des quantités dŠinterface de contact, cruciales pour la détermination de la durée de vie en fatigue. Ensuite, la réductibilité du problème est étudiée, en se concentrant sur le tri efficace et le contrôle de la taille et de la qualité de la base réduite tout au long des itérations du solveur non linéaire.

Introduction

Floating offshore wind turbines (FOWTs) are basically constituted by a wind turbine installed on a Ćoater, which is linked to the seabed by means of mooring lines, and, despite the offshore mooring technology has been used for many years, mooring line break events are still relevant concerns today. Mooring lines may be constituted by spiral strand wire ropes, which consist in a complex assembly of steel wires usually wrapped together in a twisted helical assembly of different wire layers around a single core wire. Due to their peculiar architecture, tension and bending loadings induce complex frictional phenomena between the wires that may induce fretting fatigue damage. A direct Ąnite element analysis of the wire mechanics in order to predict fatigue life is computationally costly, since such kind of problem involves simulating the complex evolution of contact and friction conditions between the wires of the wire rope [START_REF] Bussolati | Modèle multi-échelle de la fatigue des lignes dŠancrage câblées pour lŠéolien offshore Ćottant[END_REF][START_REF] Bussolati | Robust contact and friction model for the fatigue estimate of a wire rope in the mooring line of a Floating Offshore Wind Turbine[END_REF]. In order to overcome these difficulties, reduced-order model (ROM) techniques can be used. Over the years, reduced-order models have proven themselves as reliable tools in reducing computational complexity in the context of linear and non-linear problems [START_REF] Hesthaven | CertiĄed reduced basis methods for parametrized partial differential equations[END_REF][START_REF] Chinesta | Separated representations and pgd-based model reduction. Fundamentals and Applications[END_REF]. Nevertheless, the use of reduced-order modeling can be problematic for contact problems, as a reduced-order basis may not easily and efficiently capture non-regular and propagating multiscale phenomena that occur at contact interfaces, sliding, sticking and separation zones being difficult to follow. For contact problems, pertinent model reduction techniques rest on a posteriori methods. They cover POD projection-based methods for displacements and contact forces [START_REF] Balajewicz | Projection-based model reduction for contact problems[END_REF], adopting a non-negative matrix factorization scheme to the construction of a positive reduced basis for the contact forces. Reduced Basis methods have also been used with a greedy algorithm for the construction of a non-negative basis for the contact forces [START_REF] Benaceur | A reduced basis method for parametrized variational inequalities applied to contact mechanics[END_REF]. Enrichment techniques with POD modes for parametric problems have been succesfully used for the simulation of fretting fatigue in [START_REF] Cardoso | An enrichment-based approach for the simulation of fretting problems[END_REF]. In [START_REF] Giacoma | A multiscale large time increment/fas algorithm with time-space model reduction for frictional contact problems[END_REF], the LATIN method [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -new approaches and nonincremental methods of calculation[END_REF] combined with a multigrid solver is applied to frictional contact problems by making use of a pre-computed reduced basis. A priori model reduction techniques instead rely on the Proper Generalized Decomposition (PGD) [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving timedependent partial differential equations[END_REF] and its application to contact problems with the LATIN method can be found for instance in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF][START_REF] Ladevèze | A multiscale computational approach for contact problems[END_REF][START_REF] Passieux | Approximation radiale et méthode LATIN multiéchelle en temps et espace[END_REF]. The LATIN (Large Time INcrement) is a non-incremental solver for non-linear problems which iterates on the whole time-space domain [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -new approaches and nonincremental methods of calculation[END_REF], a peculiarity that makes it natively suitable for model-order reduction. Moreover, the resulting iterative scheme with two search directions features a robust treatment of contact non-linearities due to the fact that it shares similarities with augmented Lagrangian formulations for contact problems and Uzawa-like algorithms [START_REF] Simo | An augmented lagrangian treatment of contact problems involving friction[END_REF]. For the problem at hand, model reduction based on the LATIN-PGD method is exploited in this paper in order to efficiently deal with the non-linear content of the problem. Within the target application on FOWTs, several scientiĄc challenges arise. First of all convergence criteria for the non-linear solution method must assure a good convergence for local contact quantities: a global convergence criterion does not ensure local convergence of the interface quantities, which are crucial for fretting fatigue life prediction. Moreover, the LATIN convergence rate for contact problems strongly depends on search directions, and updating search directions is a challenging issue [START_REF] Zavarise | A superlinear convergent augmented lagrangian procedure for contact problems[END_REF]. Another issue concerns the efficient treatment and accurate representation of the multiscale content for contact interface quantities. The use of a multiscale strategy in this case is appealing as discussed in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF] and proposed in [START_REF] Giacoma | A multiscale large time increment/fas algorithm with time-space model reduction for frictional contact problems[END_REF]. A multiscale mixed domain decomposition method that builds a reduced basis per subdomain has been successfully proposed in [START_REF] Ladevèze | On a mixed and multiscale domain decomposition method[END_REF][START_REF] Ladevèze | A multiscale computational approach for contact problems[END_REF][START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF][START_REF] Nouy | Multiscale computational strategy with time and space homogenization: A radial-type approximation technique for solving microproblems[END_REF][START_REF] Oumaziz | Analysis, improvement and limits of the multiscale latin method[END_REF]. In these approaches, the deĄnition and updating of a suitable coarse problem is crucial. In this paper, the application of a domain decomposition method is not addressed but it will be revisited and studied in future works. This paper is organized as follows: in Section 2 the LATIN method is brieĆy introduced along with a summary of the equations to solve at the global and the local stages in the case of a frictional contact problem. In Section 3, the LATIN method is applied on a simple one-dimensional benchmark representative of the target application, where only one frictional contact interface is considered. Various convergence and error criteria are investigated with respect to the accuracy of contact interface quantities. A study of the sensitivity of the convergence rate to the choice for the search direction is also proposed. Section 4 reports the application of PGD in order to check the reducibility of the benchmark problem following [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF]. It is shown in particular how a dedicated sorting strategy for the reduced basis can improve the representation of the contact interface quantities and consequently the efficiency of the LATIN-PGD method. Conclusions and perspectives are provided in Section 5.

LATIN method applied to frictional contact problems

In this section the reference frictional contact problem is presented, as well as the LATIN method adopted to solve it.

Reference contact problem

Let consider, under the assumptions of small displacements and an isothermal quasi-static state, the equilibrium of a solid body occupying domain Ω over a time interval ∥0, T ∥ and whose bounday ∂Ω can be split into three complementary parts: ∂ 1 Ω where displacements u p are prescribed, ∂ 2 Ω with prescribed time-dependent external loadings f ext and ∂ 3 Ω where contact with a rigid wall may occur (Fig. 1). Let also consider a linear elastic regime for the solid body and frictional contact conditions at the potential contact interface ∂ 3 Ω. At the contact interface ∂ 3 Ω, the trace of the displacement Ąeld u(x, t) is v(x, t) ≙ u| ∂ 3 Ω×[0,T ] , and one can distinguish between normal an tangential components of displacements and contact forces λ(x, t) with respect to ∂ 3 Ω, as follows:

𝜕 3 Ω Ω 𝜕 1 Ω 𝜕 2 Ω 𝑔
∀(x, t) ∈ ∂ 3 Ω × ∥0, T ∥ ∶ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ v(x, t) ≙ v N n + v T λ(x, t) ≙ λ N n + λ T , (1)
with n being the outward normal vector to ∂Ω.

The problem consists in Ąnding the displacement Ąeld u(x, t) and the CauchyŠs stress Ąeld satisfying kinematic admissibility, static admissibility and constitutive laws for the solid body and the contact interface. By using the Ąnite element approximation in space for the displacement Ąeld, and by discretizing the time interval into a regular time stepping t 0≤i≤Nt such that t i+1 ≙ t i + ∆t, the discretized reference frictional contact problem is to look for the vector of nodal displacements u(t) and contact forces λ(t)

verifying ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ku ≙ f ext + B T λ R(v, λ) ≙ 0 (2)
for every time step, with K being the stiffness matrix. The boolean matrix B maps global nodal quantities to their value on the contact interface nodes, v ≙ Bu is the trace of the displacement Ąeld over the contact interface and R represents the Signorini-Coulomb non-smooth constitutive law on the contact interface. Signorini conditions [START_REF] Signorini | Questioni di elasticità non linearizzata e semilinearizzata[END_REF] are applied to normal contact quantities, while Coulomb frictional law [START_REF] Coulomb | Théorie des machines simples en ayant égard au frottement de leurs parties et à la roideur des cordages[END_REF] is used for the tangential contact behaviour.

The LATIN method

The LATIN method is a non-incremental solver for non-linear problems [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -new approaches and nonincremental methods of calculation[END_REF]. The main idea of the LATIN method is to separate a given problem in order to avoid the simultaneity of the global character and the non-linear (local) character of the problem. Thus, the mechanical properties of the equations are considered in order to deĄne two manifolds: the local, and possibly non-linear, equations which belong to some non-linear manifold Γ, and the linear, and possibly global, equations which belong to some linear manifold A. The search of the solution is based on a two-search direction algorithm: starting from an initial guess of the solution s 0 ∈ A, at each iteration a solution is alternately built in each of the manifolds as sketched in Fig. 2. The Ąrst step, called local stage, is to look for the solution ŝn+ 1 2 ∈ Γ ⋂ E + knowing the global solution s n ∈ A at the previous iteration by making use of an ascent search direction E + . The second step, called global stage, consists in seeking the global solution s n+1 ∈ A ⋂ E -by using a descent search direction E -. The exact solution s exact belongs to the intersection of the two manifolds. Two types of formulations can be adopted, based on the used unknows: the formulation in displacement and the formulation in velocity. The velocity formulation is usually adopted in the context of material non-linearities, where constitutive relations are expressed in strain-rate formulation (see [START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF][START_REF] Nouy | Multiscale computational strategy with time and space homogenization: A radial-type approximation technique for solving microproblems[END_REF][START_REF] Passieux | Approximation radiale et méthode LATIN multiéchelle en temps et espace[END_REF]), while the displacement formulation is usually more adequate in the context of linear elastic behaviour (see [START_REF] Cardoso | An enrichment-based approach for the simulation of fretting problems[END_REF][START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF]).

Here the equations to be solved for the global and the local stage for the displacement formulation of the LATIN method are summarized, where one makes use of the unknowns s ≙ (u, λ). The velocity formulation, based on the unknowns s ≙ ( u, λ), differs from the displacement formulation mainly from the fact that displacements have to be reconstructed from the velocities by making use of a pertinent time integration scheme.

Local stage

Given s n ≙ (u n , λ n ) ∈ A by the global stage at iteration n, the local stage is a corrective stage occurring at the contact interface and consists in Ąnding

ŝn+ 1 2 ≙ (û n+ 1 2 , λn+ 1 2
) ∈ Γ by following the ascent search

direction E + : (ŝ n+ 1 2 -s n ) ∈ E + ⇐⇒ λn+ 1 2 -λ n ≙ k(v n+ 1 2 -v n ), (3)
with k being the search direction parameter, homogeneous to a stiffness. The search direction parameter may be different from node to node and from time step to time step, it may also be updated according to the contact status; however, in the present study, a constant search direction in space and time along the iterations is adopted. The contact status on the interface can be detected by means of contact indicators C N and C T , deĄned on every contact node for the normal and tangential directions, (see [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF][START_REF] Cardoso | An enrichment-based approach for the simulation of fretting problems[END_REF][START_REF] Champaney | Une nouvelle approche modulaire pour lŠanalyse dŠassemblages de structures tridimensionnelles[END_REF] for more details). In the present work only the tangential contact is analyzed, and the deĄnition of the tangential contact indicator is the following:

C i T ≙ λ i T -k(v i T -vi-1 T ), (4)
where, in order to alleviate the notation, subscripts of the iterations have been removed and superscript i refers to time step t i . At the current iteration and at the current time step, all the previous quantities are known, and, based on the value of the contact indicator, two conditions can occur, which are summarized in Tab. 1, where µ represents the friction coefficient and λN is the current value of the normal contact force at the local stage.

Global stage

Given the solution ŝn+

1 2 ≙ (û n+ 1 2 , λn+ 1 2
) ∈ Γ known from the previous local stage, the global stage at the current iteration consists in Ąnding s n+1 ≙ (u n+1 , λ n+1 ) ∈ A by following the descente search direction

E -: (s n+1 -ŝn+ 1 2 ) ∈ E -⇐⇒ λ n+1 -λn+ 1 2 ≙ k(v n+ 1 2 -v n+1 ). (5)
Local stage: tangential components sliding:

∥C i T ∥ > µ| λi N | sticking: ∥C i T ∥ ≤ µ| λi N | ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ λi T ≙ µ| λi N | C i T ∥C i T ∥ vi T ≙ v i T -1 k (λ i T - λi T ) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ λi T ≙ C i T vi T ≙ v i T -1 k (λ i T - λi T ) ≙ vi-1
T Table 1: Solution of the local stage for the displacement formulation of the LATIN method.

Linear constitutive law, kinematic admissibility and static admissibility have to be veriĄed and, by taking into account the search direction (5), the following discretized problem has to be solved at the global stage for each time step t i :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∥K + B T kB∥u i ≙ f i ext + B T ( λi + kv i ) v i ≙ Bu i λ i ≙ λi + k(v i -v i ). (6)

Initialization and error and convergence indicators

The LATIN algorithm can be initialized with the linear contactless elastic solution s 0 ≙ (u 0 , λ 0 ) ∈ A and, by denoting with s ref ≙ (u ref , λ ref ) the reference solution of the problem, obtained with a high number of iterations of the LATIN method, one can deĄne:

• Reference solution error:

η ref ≙ ∥s -s ref ∥ 2 ∥s ref ∥ 2 (7)
• LATIN convergence indicator:

η ≙ ∥s -ŝ∥ 2 1 2 (∥s∥ 2 + ∥ŝ∥ 2 ) (8)
• Displacement and force convergence indicators:

η u ≙ ∥u -û∥ 2 u 1 2 (∥u∥ 2 u + ∥û∥ 2 u ) (9)
η λ ≙ ∥λ -λ∥ 2 λ 1 2 (∥λ∥ 2 λ + ∥ λ∥ 2 λ ) (10)
• Time error indicator with respect to reference solution:

η t ≙ ∫ ∂ 3 Ω ∥k(u(x, t) -u ref (x, t)) 2 + 1 k (λ(x, t) -λ ref (x, t)) 2 ∥dS ∫ ∂ 3 Ω (ku 2 ref (x, t) + 1 k λ 2 ref (x, t))dS (11)
with the following norms:

∥s∥ 2 ≙ ∫ ∂ 3 Ω ∫ [0,T ] (ku 2 (x, t) + λ 2 (x, t) k )dSdt (12)
∥u∥ 2 u ≙ ∫ ∂ 3 Ω ∫ [0,T ] ku 2 (x, t)dSdt ( 13
)
∥λ∥ 2 λ ≙ ∫ ∂ 3 Ω ∫ [0,T ] 1 k λ 2 (x, t)dSdt (14)

One-dimensional numerical application

In the following section a one-dimensional benchmark problem, represented in Fig. 3a, is analyzed, that is, a one dimensional clamped bar subjected to a time dependent traction loading F (t) (see Fig. 3b) and in contact with a frictional surface by means of a normal pressure p(t) acting on it. The used parameters are shown in Tab. 2. In this particular case, the whole domain coincides with the contact interface (i.e., the boolean matrix B reduces to the identity matrix), and the normal contact force λ N is prescribed and corresponds to p(t) so that only tangential behaviour has to be analyzed. The considered benchmark problem can be fairly seen as representative of the wire rope mechanics: in fact the rope is subjected to cyclic traction loadings from the sea state and, because of the helical geometry of the rope, traction and bending phenomena cause pressure loads between the different layers of wires which slide with respect to each other determining frictional contact phenomena. Some snapshots of the reference solution of the problem obtained with a very high number of iterations of the LATIN method (overkill solution) are shown in Fig. 4. One can notice that the sliding front propagates as the traction force increases, during the loading stage. Even during unloading, another sliding front propagates from the right top of the bar, until the traction force becomes zero, and, because of the presence of friction, the bar does not get back to its original position. x/L Fig. 5 and Fig. 6 represent some space modes of the singular value decomposition (SVD) of the quantity k(u ref -u 0 ) + (λ ref -λ 0 ), which will be used in the next section for the model-order reduction of the problem. We recall that the number of modes M obtained with the SVD is M ≙ min(N x , N t ) ≙ 50, according to Tab. 2. As found in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF], the Ąrst modes depict a global behaviour of the solution, the subsequent modes, on the other hand, bring localized corrections at speciĄc points, which illustrates the multiscale content of frictional contact problems.

Analysis of the convergence indicators

Here the behaviour of the different error and convergence indicators deĄned previously in section 2.2.3 for the LATIN method are analyzed with respect to the benchmark problem. A Ąrst guess value k 0 of the search direction parameter, related to the mechanical properties of the considered problem (see [START_REF] Boucard | A suitable computational strategy for the parametric analysis of problems with multiple contact[END_REF]), has been used for this analysis, i.e.,

k 0 ≙ ES l el , (15)
with l el being the length of the Ąnite elements. Fig. 7 shows the evolution of the different error and convergence indicators along the iterations of the LATIN method. From Fig. 7a it can be noticed that the LATIN convergence indicator underestimates the real error for this particular problem, and, in general, one may notice that displacements converge faster than forces . Fig. 7b shows how the reference solution error behaves at different time-steps: the error presents a time propagating behaviour, at lower time steps the error is lower than the one at higher time steps. Mastering convergence behaviour is crucial for the targeted application concerning the fatigue life prediction of spiral strands: an accurate computation of local contact conditions is crucial to achieve the desired goal and, for this reason, convergence criteria must assure a good convergence of local quantities both in space and time. As presented in this study this is a challenging issue, in fact the classical LATIN convergence indicator, which is a global indicator in space and time, may not be appropriate for our purpose since different convergence rates are observed for contact forces and displacements, as also for the different time steps. For this reason different convergence indicators can be deĄned, see for example [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF], where a convergence indicator is deĄned as the maximum norm of s over all the contact nodes and all the time steps. In [START_REF] Passieux | Approximation radiale et méthode LATIN multiéchelle en temps et espace[END_REF] a new convergence indicator is deĄned in the context of domain decomposition by taking into account the different behaviour of every interface of every subdomain (perfect interface, imposed displacement, imposed force, contact interface, etc...). These different convergence indicators will be tested in future works.

Determination of the optimal search direction

Since the LATIN method shares similarities with an augmented lagrangian formulation where the search direction parameter is analogous to the augmentation parameter, the convergence rate of the LATIN method depends on the value of the search parameter k. In the following, by considering a constant k in space and time, ones looks for the optimal value k opt of search direction k which minimizes the required number of iterations to reach an error with respect to the reference solution of η ref ≙ 10 -4 .

Starting with a Ąrst guess value k 0 for the search parameter, a parametric study on the LATIN method has been performed by varying the search direction with respect to k 0 deĄned in Eq. ( 15). Fig. 8a displays the evolution of the number of iterations with respect to the search direction, and one can notice that, at least for the problem here considered, an optimal value exists and that the required number of iterations to reach convergence with this value of the search direction are far less than the reference value k 0 , as it can be seen from Fig. 8b. In this study it was shown how the LATIN method convergence rate is dependent on the search parameter, as for augmented lagrangian techniques [24] [25]. Here a constant value of the search direction in space and time has been adopted, however, the search parameter is related to local contact conditions at the contact interface, i.e., sticking, sliding, contact detachment. Updating the search direction along the iterations according to the contact status on every node and every time step may be an efficient strategy to achieve a better convergence rate, and will be investigated in future works. The search parameter, however, not only affects the convergence rate of the solution but also the quality of the solution [START_REF] Passieux | Approximation radiale et méthode LATIN multiéchelle en temps et espace[END_REF], for the targeted applications it is crucial to ensure, along with a fast convergence rate, also a good quality of the converged solution.

LATIN-PGD for frictional contact problems

The LATIN-PGD approach applied to frictional contact mechanics has been discussed in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF][START_REF] Cardoso | An enrichment-based approach for the simulation of fretting problems[END_REF][START_REF] Passieux | Approximation radiale et méthode LATIN multiéchelle en temps et espace[END_REF].

Here, following [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF], a sorting algorithm is used to capture the best PGD modes according to the SVD. Let reformulate the global stage of the LATIN method by seeking for corrections (∆u, ∆λ) ≙ (u nu n-1 , λ n -λ n-1 ) between two consecutive global iterations. Given an initial linear elastic solution Ku 0 ≙ f ext and λ 0 ≙ 0, the reformulated global stage of the LATIN method at the current iteration n consists in solving the following problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ K∆u ≙ B T ∆λ ∆v ≙ B∆u ∆λ + k∆v -r sd ≙ 0, (16)
with the residual on the search direction r sd deĄned as r sd ≙ λ-λ n-1 +k(v-v n-1 ), known at this stage.

Once the global stage is solved, displacements and contact forces are corrected accordingly. One can then introduce a space-time separated representation of both forces and displacements in the global stage of the LATIN method, that is, one looks for a separated representation of corrections ∆u ≙ Vϕ(t) and ∆λ ≙ Lψ(t). By inserting this requirement in the equilibrium equation ( 16), one obtains the following condition of admissibility for space and time functions of forces and displacements:

∀t ∈ ∥0, T ∥ ∶ KVϕ(t) ≙ B T Lψ(t) ⇐⇒ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ KV ≙ B T L ϕ(t) ≙ ψ(t), (17)
which means that space and time functions of forces and displacements are not independent. Problem [START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF], including the separated representation of corrections respecting admissibility [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving timedependent partial differential equations[END_REF], is overconstrained, as a consequence, search direction equation is veriĄed at best in a weak sense by solving the following problem:

{W, ϕ(t)} ≙ arg min W,ϕ ∥W φ(t) -r sd ∥ F , (18)
with W ≙ L + kBV being an auxiliary space mode introduced to take account of the linear relationship between V and L (see [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving timedependent partial differential equations[END_REF]. With this separated representation the search equation is veriĄed in a weak sense, but the constraint between space and time modes of forces and displacements exactly satisĄes the admissibility conditions. Different approaches can be exploited, for example, by requiring the search direction to be exactly satisĄed and the admissibility conditions to be satisĄed in a weak sense; although, since the search direction is just a parameter of the problem, it is more reasonable to require it to be veriĄed in a weak sense. Alg. 1 shows the so-called quasi-optimal LATIN-PGD scheme for contact problems introduced in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF]. It consists of three different stages: an updating stage for the whole set of time modes everytime a new couple is added to the basis (see [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving timedependent partial differential equations[END_REF] on how this stage is cheap to compute and largely improves the quality of the solution), an orthonormalization stage applied to space modes every time a new couple {W, ϕ(t)} is added to the basis, and a downsizing stage applied to time modes in order to reduce their redundancy after orthonormalization. For further details, the interested reader can refer to [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF].

SVD modes [START_REF] Pastor | Modal assurance criterion[END_REF]. Shortly, given two sets of vectors of the same dimension (X i ) p 1 and (Y i ) q 1 , the MAC matrix M is deĄned as:

M ij ≙ |X T i Y j | 2 ∥X i ∥ 2 ∥Y j ∥ 2 ∈ ∥0, 1∥. (19)
M ij measures the correlation between mode X i and mode Y j . M ij ≙ 1 means that the modes are collinear, that is highly correlated, otherwise M ij ≙ 0 means that the modes are orthogonal, that is highly uncorrelated.

Conclusions

The reducibility of a one-dimensional frictional contact benchmark problem with the LATIN non-linear solver and the Proper Generalized Decomposition has been studied. The study on the convergence indicators shows that assuring a good convergence of local contact quantities in space and time is a challenging task: forces and displacements converge at a different rate as also convergence differs from one time step to another. Moreover the global LATIN convergence indicator can underestimate the real error with respect to the reference solution. The analysis for the search direction shows that the the convergence rate of the LATIN method is highly dependent on the search direction. A well suited search direction can improve convergence and computational time. Updating the search direction based on the contact interface status can be a valuable approach that is currently being investigated. The application of PGD shows some interesting results in agreement with [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF]. With the downsizing algorithm proposed in [START_REF] Giacoma | Toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver[END_REF], the reduced basis dimension and quality can be controlled and enables to select the best (quasi-optimal) modes with the respect to the Singular Value decomposition of the problem. For future works, the multiscale version of the LATIN-based domain decomposition method [START_REF] Ladevèze | A multiscale computational approach for contact problems[END_REF][START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF][START_REF] Ladevèze | On a mixed and multiscale domain decomposition method[END_REF][START_REF] Oumaziz | Analysis, improvement and limits of the multiscale latin method[END_REF] will be exploited in order to introduce a coarse scale problem and to generate reduced basis per subdomain, by considering problems with multiple contact interfaces. This strategy will be revisited in order to take beneĄt from the multiscale content of the problem, with global and local space and time modes, while paying attention especially on the following aspects: improve convergence rate of the solution and control the qualtity of interface quantities, which are crucial for the target application concerning the fatigue life prediction of spiral strand wire ropes for FOWTs, with numerous and complex loadings on long time intervals.

Figure 1 :

 1 Figure 1: Frictional contact problem setting.

Figure 2 :

 2 Figure 2: LATIN iteration scheme between the two manifolds.

  Loading and unloading stages for the traction force F (t).

Figure 3 :

 3 Figure 3: Benchmark problem set: (a) sketch of the problem, (b) time evolution of traction loading F (t).

  Snapshots of the solution during unloading.

Figure 4 :

 4 Figure 4: Reference solution of the benchmark problem: (a) during loading, (b) during unloading.

3 Figure 5 :Figure 6 :

 356 Figure 5: First three SVD space modes of k(u ref -u 0 ) + (λ ref -λ 0 ).

  Evolution of error indicators η, η ref , η λ , η v along the iterations of the LATIN method. Evolution of time error η t along the iterations of the LATIN method.

Figure 7 :

 7 Figure 7: Error indicators for the benchmark problem with k ≙ k 0 : (a) global space-time errors, (b) time error η t .

  Evolution of the number of iterations with respect to the search direction. Trend of reference error η ref with k ≙ k 0 and k ≙ k opt .

Figure 8 :

 8 Figure 8: Parametric study on the search direction: (a) evolution of the number of iterations with respect to the search direction to reach an error level of η ref ≙ 10 -4 , (b) reference error with k ≙ k 0 and k ≙ k opt .

  Evolution of PGD basis size with orthonormalization and time functions update.

Figure 9 :Figure 10 :

 910 Figure 9: Evolution of PGD basis size along the iterations of the LATIN method: (a) with orthonormalization, (b) with orthonormalization and time functions update.

Fig. 11 Figure 11 :

 1111 Fig. 11 displays the MAC diagrams for the different stages used in the LATIN-PGD in Alg. 1. By

Table 2 :

 2 Used parameters for the benchmark problem.

											Parameters
						Young modulus, E		210 GPa
						Bar cross section, S		3.14 mm 2
						Bar length, L				1 m
						Number of elements, N x	50
						Number of time steps, N t	100
						Time interval, T			1 s
						Friction coefficient, µ		0.3
						Traction force, F (t)		1000 sin( πt T ) N
						Pressure load, p(t)		5000 N/L
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

S32

Algorithm 1: Quasi-optimal LATIN-PGD algorithm for frictional contact Input: Tolerances: ε, ε 1 , ε The LATIN-PGD scheme illustrated in Alg. 1 is applied to the one-dimensional benchmark problem, and the effect of the different stages (updating, orthonormalization, downsizing) of the algorithm on the quality of reduced basis is analyzed. Fig. 9 and Fig. 10 show the evolution of the PGD basis along the iterations by making use of the different stages of Alg. 1. Without any additional stage, a new mode is generated at each iteration and the dimension of the PGD basis largely exceeds the dimension of the problem. This approach is not useful since modes are highly redundant and no computational saving is achieved. With orthonormalization of the space modes the dimension of the basis is controlled in size (see Fig. 9a). In fact, in the orthonormalization stage, every time a new couple is generated, the new space mode is projected onto the previously computed basis and, if the new space mode is not independent from the others, the new couple is not added to the basis according to a tolerance ε 1 . By using both orthonormalization and time function update, a full basis is generated at the Ąrst iterations (see Fig. 9b). With downsizing (see Fig. 10), one can sort and control the basis quality and size by capturing only the best PGD modes. The size of the generated basis depends on a tolerance ε 2 on the norm of the time modes. The behaviour of the PGD basis along the iterations can be easily visualized by making use of the Modal Assurance Criterion (MAC) diagrams between the PGD modes and the