A Tressaud

T Roisnel

L Fournes

J Grannec

H Touhara

H Imoto

H Dexpert

LOCAL ENVIRONMENT OF Sn IN TIN FLUORIDE-GIC

When carried out in anhydrous HF, the intercalation of tin tetrafluoride into graphite yields a stage-2 compound with a repeat distance I = 11.53 c The +IV oxidation state of the intercalated Sn species has been unambigu-119 ously confirmed by Sn Mbssbauer spectroscopy. The study of X-ray precession patterns of a crystal of composition C30SnF 5 has shown that the lattice of the intercalate is incommensurate with the a-parameter of graphite. The local environment of tin in both SnF 4 and C30SnF 5 has been determined at room temperature from X-ray absorption (EXAFS) spectra. The first coordination shell is formed of axially flattened octahedra of fluoride ions.

INTRODUCTION

A large number of fluoride-based graphite intercalation compounds (GIC) are currently obtained by heating graphite in the presence of volatile fluorides.

A convenient method permitting the intercalation of involatile fluorides has been recently developed by using a dissolution process of fluorides in anhydrous HF (AHF) saturated with F 2 [1][2][3]. Using this method, SnF4-based GICs have been previously prepared and characterized by powder X-ray diffraction, NMR and in-plane electrical conductivity measurements [4].

This work deals with the characterization of C SnF using single crystal x y precession analysis, EXAFS and M~ssbauer spectroscopy.

EXPERIMENTAL

SnF4-based GICs have been prepared in Kyoto from AHF saturated with fluorine gas using a procedure which has been already described [2][3][4]. Various pristine graphites have been used : HOPG chips (5 x 5 mm 2 , 0.5-1mm thick) and flaky natural graphite (NG) from Madagascar ( < i ~ m large).

The characterization of the materials has been carried out using X-ray diffraction, electron microprobe and elemental analysis of C, Sn and F. The crystallographic arrangement of the intercalated species has been determined from precession photographs using Mo-Ke radiation.

The local environment of tin has been obtained from EXAFS experiments performed at LURE-Orsay. The absorption spectra were recorded using the synchrotron radiation emitted by the DCI ring which permits the study of the Sn K-edge at 29200 eV. Various problems arose from this high energy value, which is close to the limit of the emission spectrum of DCI : intensity of the flux which was detected by the ionization chambers, stabilization of the incident beams.

In order to increase the diffraction angle and to insure a better energy resolution, the X-ray beam was monochromatized using a (511) Si single crystal. The other experimental requirements have been previously described [3,5].

The M~ssbauer parameters were obtained from a transmission spectrometer 119 with constant acceleration using a Sn radioactive source. The temperature dependence of the spectra has been studied down to 4.2 K in Bordeaux using a cryostat. All isomer shifts given below refer to CaSnO 3 at 293 K. Since the ratio of the parameter of this sublattice to the a-parameter of o graphite, aorthorh./agraph" (with agraph" = 2.46 A), is not an integer, the two cells are incommensurate in this direction. The surface ratio is :

STRUCTURAL CHARACTERIZATION OF C SnF

Sinterc./SperC~tOm = 67.2. This value can be taken into account by assuming that 6 Sn atoms are present in the large cell. This hypothesis is consistent with a stage-2 compound having a composition C22SnFy,with 4~y~6. This value is in good agreementwith that found from elemental analyses, i.e. CxSnF 5 with

x =25-30.

The excess of fluorine relative to the SnF 4 stoichiometry may be accounted for by at least three assumptions:

-partial filling of the intercalated layers, with the presence of islands, chains, and of variable size, -presence of HF and/or HF2 arising from the intercalation method.

EXAFS STUDY

Choice of the standard For a Sn(IV) standard, we have chosen Na2SnF 6 because of its stability in air. This compound which is isostructural with Na2NbF 6 crystallizes in the trirutile-type structure (P42/mnm space group) [6]. The absorption spectrum of HOPG-based C30SnF 5 (second stage compound) at Sn K-edge is shown in Fig. 1 for the 29000-30000 eV energy range. The normalized EXAFS modulations have been treated using standard procedures [3,5,8].

4.

2.

1.97 2.04 0 0

Using the same calculation procedure the fitting of the experimental absorption spectrum of SnF 4 leads to R and N parameters given also in Table i.

From the comparison of the above results, it can be seen that the coordination shells of the first nearest neighbors in both SnF 4 and C SnF are very similar x y and can be described as having axially flattened octahedra. The assumption of an octahedral environment of tin in the GIC is therefore confirmed as well as the axial flattening of SnF 6 octahedra in SnF 4.

Only the calculation of the first coordination shell has been fully treated.

However, the complete radial distribution functions around Sn atoms for the two samples and for the standard are grouped in Fig. 3 119

Sn MOSSBAUER STUDY

The ll9sn M~ssbauer spectrum measured at 293 K shows a relatively narrow -I line and leads to an isomer shift 6 = -0.33 mms (Fig. 4). The isomer shift of C30SnF 5 lies in between those of SnF 4 (6 = -0.36 mms -I) and tetravalent tin in ~-Sn3F 8 (~snIV = -0.315 mms-1), which unambiguously indicates that in the GIC the oxidation number of tin is (+IV).

The M~ssbauer spectrum obtained at 4.2 K is also reported in Fig. In other words, the interactions between SnF 6

octahedra may have a more pronounced "long range" character. On the other hand, those particles which have a smaller size than the critical value exhibit the same type of behavior all along the temperature range (singlet line).

  or from NG pristine materials. The X-ray powder pattern has o been previously characterized with a repeat distance I =11.53 A, which may c o indicate that the intercalant thickness d = I -3.35 n =4.83A is mainly go-1 c verned by octahedral SnF 6 units. Room-temperature precession photographs of a stage-2 single crystal show weak spots corresponding to the intercalated fluoride sublattice in addition to the graphite network. All these spots can be indexed with a rectangular cell of parameters a =11.93 A and b = 14.76 A. Three different orientations of this cell are observed which are related to each other by a 60 ° rotation.

  The network of Na2SnF 6 consists of a three-dimensional arrangement of distorted NaF 6 and SnF 6 octahedra which are linked together by edges and corners. Within the SnF 6 polyhedra, o the Sn-F distances are the following : 4 x 1.97 A ; 2 x 2.04 A [6]. Determination of the environment of tin in SnF 4 and in stage-2 C SnF x--y The structure of SnF 4 has been determined from powder data by Hoppe and Dahne [7]. The tetrafluoride crystallizes in the tetragonal system (I4/mmm space group ; a = 4.048 A and c = 7.930 A). The structure consists of layers of SnF 6 octahedra which are interconnected by four of their corners. The four equatorial distances are longer than the two axial distances: 4d :

Fig. i .Fig. 2 .

 i2 Fig. i. Absorption spectrum of HOPG-based C30SnF 5 at Sn K-edge (29000-30000 eV energy range).

Fig. 3 .

 3 Fig. 3. Radial distribution functions (Fourier transform) around tin for several compounds (a : SnF 4 ; b : SnF4-based GIC ; c : Na2SnF6).

Fig. 4 .

 4 Fig. 4. ll9sn M~ssbauer spectra of C30SnF 5 at 4.2 and 293 K.

TABLE 1

 1

	EXAFS results for pristine and intercalated SnF4, compared with crystallogra-
	phic data of Na2SnF 6 (first coordination shell)
		C30SnF 5	SnF 4	Na2SnF 6 [from 6"
	N I o	2.2	3.1	1.9	3.5
	R 1 (A)	1.95	2.07	1.92	2.06
		0.01	0.02	0.02	0.02
	bE Reliab.	0. i0 -4 0.36xi0	0.7 0.6x10	-5
	factor

ACKNOWLEDGEMENTS

We are indebted to Professor Hagenmuller and Dr S. Flandrois for their fruitful discussions and to Dr C. Hauw for his help on X-ray diffraction measurements.


Pierre Gosselet
email: gosselet@lmt.ens-cachan.fr

Maxime Blanchard
email: blanchard@ens-cachan.fr

Olivier Allix
email: allix@lmt.ens-cachan.fr

Guillaume Guguin
email: guguin@lmt.ens-cachan.fr

Non-invasive global-local coupling as a

Keywords: non-invasive coupling, Schwarz domain decomposition, non-conforming patch, Krylov acceleration

come

Non-invasive global-local coupling as a Schwarz domain decomposition method: acceleration and generalization 1 Introduction

The non-invasive local-global coupling technique proposed by Allix and first implemented in [START_REF] Gendre | Non-intrusive and exact global/local techniques for structural problems with local plasticity[END_REF] is an iterative technique which aims making accurate the well known submodeling technique [START_REF] Kelley | Mesh requirements for the analysis of a stress concentration by the specified boundary displacement method[END_REF][START_REF] Ransom | Computational methods for global/local analysis[END_REF][START_REF] Cormier | Aggressive submodelling of stress concentrations[END_REF]. It is strongly related to many reanalysis techniques [START_REF] Jara-Almonte | The specified boundary stiffness/force SBSF method for finite element subregion analysis[END_REF][START_REF] Whitcomb | Iterative global/local finite element analysis[END_REF][START_REF] Whitcomb | Application of iterative global/local finite-element analysis. part 1: linear analysis[END_REF] and domain decomposition methods [START_REF] Hecht | Numerical zoom and the Schwarz algorithm[END_REF].

The aim of this technique is to evaluate the effect of local modifications inside a computational model (geometry, material and load) without requiring heavy developments. More precisely the objective is to use an industrial model solved on a given commercial software and to simulate the presence of local alterations by iteratively spawning computations with only extra traction loads inside the model. Moreover, the alterations can be computed on any chosen software including dedicated research codes.

This philosophy was successfully applied in many different contexts like: the introduction of local plasticity and geometrical refinements [START_REF] Gendre | Non-intrusive and exact global/local techniques for structural problems with local plasticity[END_REF], the computation of the propagation of cracks in a sound model [START_REF] Duval | Non-intrusive coupling: recent advances and scalable nonlinear domain decomposition[END_REF], the evaluation of stochastic effects with deterministic computations [START_REF] Chevreuil | A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties[END_REF], the taking into account of the exact geometry of connectors in an assembly of plates [START_REF] Guguin | On the computation of plate assemblies using realistic 3d joint model: a non-intrusive approach[END_REF]. In [START_REF] Duval | Local/global non-intrusive parallel coupling for large scale mechanical analysis[END_REF] the method was used in order to implement a nonlinear domain decomposition method [START_REF] Keyes | Aerodynamic applications of Newton-Krylov-Schwarz solvers[END_REF][START_REF] Cresta | Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses[END_REF][START_REF] Hinojosa | Domain decomposition methods with nonlinear localization for the buckling and post-buckling analyses of large structures[END_REF][START_REF] Negrello | Substructured formulations of nonlinear structure problems -influence of the interface condition[END_REF] in a non-invasive manner in code aster. Extension of the approach to explicit dynamics was proposed in [START_REF] Bettinotti | A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics[END_REF], improved in [START_REF] Bettinotti | A fast weakly intrusive multiscale method in explicit dynamics[END_REF] and applied to the prediction of delamination under impact loading in [START_REF] Bettinotti | Simulation of delamination under impact using a global local method in explicit dynamics[END_REF]. Alternative non-invasive strategies can be derived from the extended finite element method [START_REF] Plews | An improved non-intrusive global-local approach for sharp thermal gradients in a standard FEA platform[END_REF][START_REF] Kim | A new generalized finite element method for two-scale simulations of propagating cohesive fractures in 3-d[END_REF].

After a description of the method (section 2), this paper provides several contributions. First the non-invasive coupling algorithm is proved to realize a simple implementation of the optimized non-overlapping Schwarz domain decomposition method (section 3). Then several accelerations techniques are proposed (section 4), some are classical but the linear and nonlinear conjugate gradient is new in this framework. The algorithms are described in a very programmer-friendly manner. Last an overlapping version of the method is proposed (section 5) which can be used to handle fully non-conforming meshes.

Derivation of the non-invasive algorithm

The algorithm we study is very general and applies to the study of many PDEs; in order to fix the ideas, we consider problems of nonlinear quasi-static structure mechanics under the small strain hypothesis. We note u the displacement field, ε the symmetric part of the gradient, σ Cauchy's stress tensor. For a domain Ω submitted to given body force f , Dirichlet conditions u d is imposed on the part ∂ d Ω of the boundary and Neumann condition g is imposed on the complement part ∂ n Ω. In order to manage viscous materials, the study is conducted over a time interval T = [0, T ], and the following equations are meant to be satisfied at any time t ∈ T , which we omit to write except when necessary.

Let V (Ω) = v ∈ H 1 (Ω), v = u d on ∂ d Ω be the affine space of admissible displacement and V 0 (Ω) the associated vector space. The conservation of momentum can be written as:

Ω σ(u) : ε(v) dx = Ω f • v dx + ∂nΩ g • v dS, ∀v ∈ V 0 (Ω) (1)
The notation σ(u) stands for local or non linear constitutive laws defined under the following functional expression: σ(x, t) = B(ε(u(x, τ )), τ < t), x ∈ Ω, t ∈ T

This modeling of the mechanical behavior is typically suited for elastoviscoplastic materials. For most models an alternative description by internal variables summarizing the effect of the past history can be found.

The mechanical problem above takes the following classical form:

Find u ∈ V (Ω) / a(u, v) = l(v), ∀v ∈ V 0 (Ω) ( 3
)
where l is a continuous linear form, and a is a continuous coercive form, linear in the second variable, note that a may be nonlinear in the first variable.

In the following we handle several space subdomains and models, when any quantity is specifically attached to one model, a superscript mentions it.

The Reference problem (superscript R ) is set on the domain Ω R which is the assembly of two non-overlapping subdomains: the zone of interest where a Fine model is required for a reliable simulation (superscript F ), and a Complement zone (superscript C ) where a simpler model is sufficient (and which in general covers most of the structure). The interface is Γ = ∂Ω C ∩ ∂Ω F , it is thus immersed in Ω R . Note that using several zones of interest presents no difficulty as long as they do not overlap; note also that the Reference problem is never formed in practice.

Reference problem

Find

u ∈ V (Ω R ), such that ∀v ∈ V 0 (Ω R ), a R (u, v) := a C (u, v) + a F (u, v) = l C (v) + l F (v) =: l R (v) (4)
We assume that we have another representation of the zone of interest, named Auxiliary representation (superscript A ) which shares the same characteristics as the Complement zone, and which is thus coarser than the Fine representation. Typically if Ω F was a zone where material coefficients have strong variations, the Fine representation would follow the exact distribution whereas the Auxiliary representation could use a homogeneized behavior; the load could also be simplified. An application is the case where the Fine model is stochastic whereas the Auxiliary model is deterministic [START_REF] Chevreuil | A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties[END_REF]. We insert the Auxiliary representation of the zone of interest in the Reference problem:

Find u ∈ V (Ω R ), such that ∀v ∈ V 0 (Ω R ), a C (u, v) + a F (u, v) = l C (v) + l F (v) a C (u, v) + a A (u, v) a G (u,v) = l C (v) + l A (v) l G (v) + a A (u, v) -l A (v) -a F (u, v) -l F (v) (5)
The Global problem, (supersript G ), is the assembly of the Complement zone with the Auxiliary (coarse) representation of the zone of interest, this problem is in practice assembled and dealt with by commercial software.

From the previous equation, we could derive the following stationary iteration:

a G (u n+1 , v) = l G (v) + a A (u n , v) -l A (v) -a F (u n , v) -l F (v) (6)
which would correspond to a fixed point of the Reference problem preconditioned by the coarse Global system. Not only convergence would be slow but also the right-hand-side terms would not be easy to compute in practice. Moreover, this iteration needs the Auxiliary domain Ω A to be coincident with the Fine domain Ω F which is a limitation we want to get rid of. In the following, we only assume that the interface is on the boundary of the Auxiliary domain Γ ⊂ ∂Ω A , so that Γ is on the boundary of all subdomains. We note V Γ the trace space of displacements on Γ and V * Γ its dual space of interface tractions, λ, v Γ is the associated duality bracket with λ ∈ V * Γ and v ∈ V Γ . We thus choose to associate the right hand side of (5) with the evaluation of local problems. Starting from p 0 = 0 (the mathematical space for p n is discussed later), the basic global/local iteration is then the following:

Global problem Find u G n ∈ V (Ω G ), such that ∀v ∈ V 0 (Ω G ), a G (u G n , v) = l G (v) + p n , v (7a)
Fine problem

     Find (u F n , λ F n ) ∈ V (Ω F ) × V * Γ , s.t. ∀(v, µ) ∈ V 0 (Ω F ) × V * Γ , a F (u F n , v) = l F (v) + λ F n , v Γ µ, u F n -u G n Γ = 0 (7b) Auxiliary problem      Find (u A n , λ A n ) ∈ V (Ω A ) × V * Γ , s.t. ∀(v, µ) ∈ V 0 (Ω A ) × V * Γ , a A (u A n , v) = l A (v) + λ A n , v Γ µ, u A n -u G n Γ = 0 (7c) Update p n+1 , v = a A (u A n , v A ) -l A (v A ) -a F (u F n , v F ) -l F (v F ) with v A |Γ = v F |Γ = v (7d)
In words, the Global problem is the coarse problem with extra load p, the Fine and Auxiliary systems are resolutions on the domain of interest with imposed Dirichlet conditions on Γ. We chose a Lagrangian formulation for these problems in order to make appear the reaction forces λ F and λ A . The update is simply the equivalent of ( 6) with fields issuing from the local solves instead of the global one.

Remark 1. Of course, the Lagrange multipliers are equal to the normal stress:

λ X = σ X • n X (8)
where X ∈ {A, F } and n X is the outer normal vector.

We have the following properties:

Assuming the fine and auxiliary problems were solved exactly, we have:

p n+1 = λ A n -λ F n ∈ V * Γ (9)
the corrective load p is then an immersed surface traction. In the following, we always assume the exactness of the resolution; note that using inexact solvers was investigated in [START_REF] Passieux | Local/global nonintrusive crack propagation simulation using a multigrid X-FEM solver[END_REF] where the method is identified with a localized multigrid iteration.

Because the Auxiliary problem corresponds to the restriction of the Global problem on the zone of interest with global displacement imposed, we directly have:

u A n = u G n|Ω A (10)
The introduction of the Auxiliary problem is thus not mandatory, it is just a workaround in case of software unable to compute the reaction in an immersed surface. Of course, the Auxiliary problem can be solved in parallel with the Fine problem.

We can also define the reaction from the Complement zone:

a C (u G n , v) = l C (v) + λ C n , v Γ , ∀v ∈ V (Ω C ) (11)
Then we see that:

λ C n + λ A n = p n (12)
The surface traction p n generates a discontinuity in the normal stress of the Global problem.

If we replace the auxiliary reaction by the complement one, we have:

p n+1 = p n + r n with r n+1 = -λ F n + λ C n ( 13
)
in words, the correction brought to p n+1 corresponds to the lack of balance between the Complement zone and the Fine representation of the zone of interest. This lack of balance is the residual r of the algorithm. The algorithm converges when the two representations are in equilibrium (r = 0, in which case the extra load p shall not evolve anymore).

The algorithm makes no use of domain integrals to communicate between subdomains; only interface data (on Γ) are exchanged, namely the displacement u G and the reactions λ F and λ A (or λ C ). As long as the interface Γ is well represented in all models, it is not necessary to use the exact Fine domain Ω F in the Auxiliary problem, any coarser representation is possible (Ω A ). Typically micro-perforations or micro cracks need not be represented in the Auxiliary problem. Of course modifying the representation of the zone of interest may have consequences on the convergence of the algorithm (but not on its limit which is the reference solution).

3 Connexion with alternate non-overlapping Schwarz method

The question of linking the non-invasive global-local coupling method to the many variants of domain decomposition and associated algorithms, like chimera, was studied in other publications like [START_REF] Hecht | Numerical zoom and the Schwarz algorithm[END_REF]. Here we propose to connect the method with the iterations of a non-overlapping optimized Schwarz method. The theoretical framework of Schwarz method will allow us natural extensions to the method, in particular the use of overlaps to treat mesh incompatibilities.

We consider two non-overlapping subdomains, Ω C and Ω F , connected by Interface Γ. The decomposed problem to solve can be written as:

Fine equilibrium (u F , λ F ) ∈ V (Ω F ) × V * Γ , s.t. ∀v ∈ V 0 (Ω F ), a F (u F , v) = l F (v) + λ F , v Γ (14a) Complement equilibrium (u C , λ C ) ∈ V (Ω C ) × V * Γ , s.t. ∀v ∈ V 0 (Ω C ), a C (u C , v) = l C (v) + λ C , v Γ (14b) Interface conditions λ F + λ C , v Γ = 0, ∀v ∈ V Γ µ, u F -u C Γ = 0, ∀µ ∈ V * Γ (14c)
In words subdomains must be in mechanical equilibrium while displacements shall be equal on the interface and force fluxes shall be balanced. The optimized Schwarz method consists in using Robin conditions at the interface. The Robin conditions are materialized by operators called interface impedances (or interface stiffnesses):

Q C and Q F from V Γ to V * Γ .
The interface conditions are rewritten as:

(λ F + λ C ) -Q C (u F ) + Q C (u C ) = 0 (λ F + λ C ) + Q F (u F ) -Q F (u C ) = 0 ( 15
)
where we need in particular (Q C + Q F ) to be injective for the equivalence with initial conditions to hold. In general Q C and Q F are chosen to be such that each associated form

V 2 Γ (u, v) → Q X (u), v Γ is bilinear symmetric continuous coercive.
The new conditions can be combined with the equilibrium:

a F (u F , v) + Q F (u F ), v Γ = l F (v) + Q F (u C ) -λ C , v Γ , ∀v ∈ V (Ω F ) a C (u C , v) + Q C (u C ), v Γ = l C (v) + Q C (u F ) -λ F , v Γ , ∀v ∈ V (Ω C ) (16)
Hence the alternate optimized Schwarz stationary iterations (λ F 0 = 0, u F 0 = 0):

Find u C n+ 1 2 ∈ V (Ω C ) s.t. ∀v ∈ V (Ω C ), a C (u C n+ 1 2 , v) + Q C (u C n+ 1 2 ), v Γ = l C (v) + Q C (u F n ) -λ F n , v Γ Compute λ C n+ 1 2 ∈ V * Γ s.t. ∀v ∈ V Γ , a C (u C n+ 1 2 , v) = l C (v) + λ C n+ 1 2 , v Γ Find u F n+1 ∈ V (Ω F ) s.t. ∀v ∈ V (Ω F ), a F (u F n+1 , v) + Q F (u F n+1 ), v Γ = l C (v) + Q F (u C n+ 1 2 ) -λ C n+ 1 2 , v Γ Compute λ F n+1 ∈ V * Γ s.t. ∀v ∈ V Γ , a F (u F n+1 , v) = l F (v) + λ F n+1 , v Γ (17)
It is well known that the optimal value for one subdomain's impedance is the Dirichlet-to-Neumann operator of the other subdomain which we note S X . Typically for Ω C , we have:

Q C opt = S F : V Γ u Γ → S F (u Γ ) = λ F ∈ V * Γ , where (u F , λ F ) ∈ V (Ω F ) × V * Γ are such that ∀(v, µ) ∈ V 0 (Ω F ) × V * Γ , a F (u F , v) = l F (v) + λ F , v Γ and µ, u F -u Γ Γ = 0 (18)
Note that in for linear problems, S X is an affine operator (and not just a linear operator) since it also takes into account the effect of the load. The existence of this operator is conditioned to the well-posedness of the Dirichlet problem over subdomains. There are many contexts where this well-posedness can be proved, at least locally, see for instance [START_REF] Ciarlet | Linear and nonlinear functional analysis with applications[END_REF] for details. The global existence of the operator can be proved in the case of coercive continuous monotone operators; see for instance [START_REF] Showalter | Hilbert space methods for partial differential equations[END_REF][START_REF] Showalter | Monotone operators in Banach space and nonlinear partial differential equations[END_REF] for an analysis at the level of the variational formulation and [START_REF] Feistauer | Finite element solution of nonlinear elliptic problems[END_REF] for the analysis of the finite element approximation. Mechanically this case is associated with positive hardening behaviors and certain contact laws, in small strains [START_REF] Ladevèze | Sur une famille d'algorithmes en mécanique des structures[END_REF][START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. Moreover, the Dirichlet-to-Neumann operator inherits properties from the initial problem (typically monotonicity, coercivity and continuity; see for instance [START_REF] Hauer | The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems[END_REF] and associated bibliography).

The global-local algorithm corresponds to the choice Q C = S A and formally Q F = ∞ (the Dirichlet condition being seem as the limit case of an infinite interface impedance). The choice Q C = S A is extremely strong because the can expect S A to be a good approximation of S F ; not only in term of stiffness (a A vs a F ) but also in term of load (l A vs l F ) which corresponds to providing a good initialization to the algorithm.

The framework of Schwarz method enables us to recover the following features: Krylov acceleration: replacing stationary iterations by Krylov solvers is classical in Schwarz methods [START_REF] Gander | Mathématiques pour l'ingénieur, chapter Méthodes de décomposition de domaines[END_REF]. The Dirichlet condition Q F = ∞ preserves some symmetry so that we can derive a conjugate gradient algorithm, see section 4.4.

Mixed approach: the condition Q F = ∞ is a poor approximation of the optimal choice. Many work exist on how to improve this approximation [START_REF] Gander | Optimized Schwarz methods without overlap for the Helmoltz equation[END_REF]. In [START_REF] Gendre | A two-scale approximation of the Schur complement and its use for non-intrusive coupling[END_REF] a two scale approximation of S C was proposed.

Parallel processing: the global-local method corresponds to the multiplicative version of the optimized Schwarz method. The additive (parallel) version could be tried in the non-invasive context. Note that this would only make sense in the presence of multiple Fine zones with finite Fine impedance Q F < ∞.

Nonlinearity: stationary iterations can directly be transferred to nonlinear problems, in particular the ones with monotone operators (positive hardening) [START_REF] Badea | On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems[END_REF][START_REF] Ladevèze | Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation[END_REF]. The local-global method was successfully applied in many nonlinear problems like plasticity or cracking [START_REF] Gendre | Non-intrusive and exact global/local techniques for structural problems with local plasticity[END_REF][START_REF] Passieux | Local/global nonintrusive crack propagation simulation using a multigrid X-FEM solver[END_REF] Overlapping version: optimized Schwarz methods also exist with overlaps. In [START_REF] Guguin | Nonintrusive coupling of 3d and 2d laminated composite models based on finite element 3d recovery[END_REF], the overlap was used as a buffer zone to dampen edge effects in plate/3D coupling. In section 5, we present another application, the handling of non-matching meshes.

Analysis and acceleration of the global/local algorithm 4.1 Notations

In order to further analyze the algorithm and be more practical, we now consider the finite element discretization of the problem. We use the following notations: f for the generalized forces, u for the nodal displacement and λ for the nodal reactions and p for the nodal component on the immersed surface effort. When indexing degrees of freedom, F, A, C stand for the internal degrees of freedom whereas Γ stands for nodes on the interface (whose description is identical in all models). We tried to use minimal notations, but sometimes a quantity defined on the interface is issued from one side specifically, in which case we make it clear by an extra superscript. In the linear(ized) case notation K is used for the stiffness matrices. Remark 2. We recall that the nodal reaction is not the discretization of the Lagrange multiplier. Indeed for a boundary degree of freedom i associated with shape function φ i , we have:

λ X i = Ω X (σ h : ε(φ i ) -f • φ i ) dx - ∂nΩ X g • φ i dS = Γ (σ X h • n X ) • φ i dS - Ω X (div(σ h ) + f ) • φ i dx + ∂nΩ σ h • n X -g • φ i dS (19)
where X ∈ {C, A, F }, n X is the outer normal vector and σ h is the stress tensor obtained from the finite element computation. Thus the nodal reactions λ X can be computed either by using a Lagrangian formulation for the Dirichlet condition (which is fairly common in commercial software) or by using the formula above to postprocess it from the finite element stress (which may be complex to implement in legacy software); hence the use of the Auxiliary model to compute reactions on the immersed interface.

If we assume that the Reference and the Global problems are well-posed, then Dirichlet problems are well posed on all subdomains, at least locally near the solution. We can then define the following nonlinear discrete Dirichlet-to-Neumann operators S X (we use the same notation as in the continuous case) which compute the reactions λ X from a given interface displacement u Γ : Fine problem (7b) :

λ F = S F (u Γ ; f F ) Auxiliary problem (7c) : λ A = S A (u Γ ; f A ) Complement problem : λ C = S C (u Γ ; f C ) (20)
Because of the nonlinearity, the effects of given loads appear as parameter of the method, they will be omitted in the absence of ambiguity.

Remark 3. In the case of linear problems, it is possible to give an explicit formula for the Dirichlet-to-Neumann operators. As an illustration, the equilibrium of the Fine problem writes:

K F ΓΓ K ΓF K F Γ K F F u Γ u F = f F Γ f F + λ F 0 ( 21
)
which can be condensed as:

λ F = S F (u, f F ) = S F u Γ -b F with S F = K F ΓΓ -K ΓF K -1 F F K F Γ b F = f F Γ -K ΓF K -1 F F f F (22)
In that case, S F is an affine operator: S F is the well known Schur complement of the Fine domain on the interface. Linearity allows to set apart the contrition of the given load, with b F the condensed right-hand side.

Note that the internal displacement in the Fine domain was implicitly computed as:

u F = K -1 F F f F -K F Γ u Γ (23)
Because of the additivity of integral with respect to the domain, the Global operator verifies the following decomposition S G = S C + S A and the Reference operator writes S R = S C + S F , and we can rephrase the Global and Reference problems in a condensed manner:

Reference problem (4), u Γ ? S R (u Γ ; f R ) = S C (u Γ ; f C ) + S F (u Γ ; f F ) = 0 Global problem (7a), u Γ ? S G (u Γ ; f G ) = S C (u Γ ; f C ) + S A (u Γ ; f A ) = p (24)
Note that each time one of the condensed operators is employed, the displacement inside the subdomains is implicitly computed: for instance, u G in a by-product of [START_REF] Jara-Almonte | The specified boundary stiffness/force SBSF method for finite element subregion analysis[END_REF] and u F in a by-product of [START_REF] Guguin | On the computation of plate assemblies using realistic 3d joint model: a non-intrusive approach[END_REF]. To make it clearer, we will use notations S X when analyzing the methods, whereas we will use the following functional notations when describing the algorithms:

[u G ] = SolveGlobal(p; f G ), u G is
defined on the whole Global model and in particular we have u

G Γ = S G -1 (p; f G ). [u F , λ F ] = SolveFine(u G ; f F ), u F is defined in the Fine model and we have λ F = S F (u G Γ ; f F ) [λ A ] = SolveAux(u G ; f A ), which in corresponds to λ A = S A (u G Γ ; f A ).
When authorized by the software, it can be replaced by the post-processing of the stress [START_REF] Guguin | Nonintrusive coupling of 3d and 2d laminated composite models based on finite element 3d recovery[END_REF].

The Fine and Auxiliary solves are in general gathered in one line because the computations can be run in parallel.

In order to keep notations simple, we assume the coarse and fine meshes are conforming at the interface. In most cases the zone of interest is deduced from an initial coarse computation and it is defined as a subset of coarse elements. Then the interface Γ lies on faces of coarse elements and it defines the boundary of the Fine domain. In that context, even if the Fine discretization is chosen to be finer than the Global one on Γ, a simple Global-master -Fine-slave strategy may give satisfying results: a transfer matrix T is computed (for instance using interpolation or Mortar techniques) such that the interface conditions can be written as:

Tu C Γ -u F Γ = 0 and λ C + T T λ F = 0 ( 25
)
so that the mechanical work is preserved. The algorithms presented below are unchanged, one just need to consider the Global interface as the master interface. Note that Section 5 presents a technique to handle non-coincident interfaces.

Stationary iterations

The global/local coupling iterations of equation ( 7) can formally be written as:

p n+1 = (S A -S F ) • S G -1 (p n ) = p n -S R • S G -1 (p n ) ( 26
)
One recognizes fixed point iterations. The convergence is controlled by the contraction property of the operator

I -S R • S G -1 = (S A -S F ) • (S A + S C ) -1 .
Remark 4. In the linear case, the operator that controls the convergence can be written as:

(S A -S F )(S A + S C ) -1 (27)
since all Schur complements are symmetric positive definite, a trivial sufficient condition for the operator to be a contraction is S A S F (for the quadratic norm ordering). Mechanically speaking this means that the Auxiliary model shall be stiffer that the Fine one; this is usually the case when the Fine model has a refined mesh or holes. Moreover, we can expect the Auxiliary model to be a good approximation of the Fine model leading to (S A S F ) and fast convergence.

A classical tweak for fixed point iterations is to use relaxation. This enables to grant contraction property or to improve the convergence rate. In that case, the iteration writes:

p n+1 = p n -ω n S R • S G -1 (p n ), |ω| > > 0 ( 28
)
For linear problems, it is well known that convergence is ensured for 0 < ω < 2/ρ S G -1 S R and the optimal value is ω = 2/(λ min + λ max ) where the λs' stand for the minimal and maximal eigenvalues of S G -1 S R . An alternative, when the area to be reanalyzed is known a priori, would be to choose the global model in order for the auxiliary model to be slightly stiffer than the local one.

The results of the existence of sufficient and of optimal relaxations can be extended to the case of a monotone problem. Indeed in that case, the method can be interpreted as an operator splitting technique [START_REF] Ryu | Primer on monotone operator methods[END_REF] on the condensed problem which inherits the useful properties of the original system (in particular monotonicity and coercivity). Reader may refer to [START_REF] Nouy | A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities[END_REF] for detailed proof with weak assumptions.

In practice, it is convenient to have ω adapted at each step. A good heuristic for the sequence (ω n ) is provided by Aitken's ∆ 2 . It was first tried in the global/local framework in [START_REF] Liu | A non-intrusive global/local algorithm with non-matching interface: Derivation and numerical validation[END_REF]. The strategy is summed-up in algorithm 1.

Quasi-Newton's approaches for linear Global model

The system to solve associated to the fixed point iterations (26) writes:

Find p / S R • S G -1 (p) = 0 ( 29
)
which mechanically means that we seek the surface traction to impose inside the Global coarse model (ie the stress discontinuity) such that the Fine model with Dirichlet conditions issued from the Global solve is in balance with the Complement zone.

Algorithm 1: Non-invasive stationary iterations with relaxation Arbitrary initialization p 0

for j ∈ [0, • • • , m] do [u G j ] = SolveGlobal(p j ; f G ) [u F j , λ F j ] = SolveFine(u G j ; f F ) // [λ A j ] = SolveAux(u G j ; f A ) Residual: r j = -(λ F j + p j -λ A j ) Update: p j+1 = p j + r j ; Aitken ∆ 2 : p j+1 ← ω j+1 p j+1 + (1 -ω j+1 )p j with ω j+1 = -ω j r T j-1 (rj -rj-1) rj -rj-1 2
end Applying a Newton iteration to system (29) leads to the sequence:

DS R DS G -1 (p n+1 -p n ) = -S R • S G -1 (p n ) ( 30
)
which was investigated in nonlinear relocalization techniques [START_REF] Negrello | Substructured formulations of nonlinear structure problems -influence of the interface condition[END_REF][START_REF] Cresta | Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses[END_REF][START_REF] Pebrel | Une approche par patchs pour les non linéarités localisées en calcul de structures[END_REF][START_REF] Hinojosa | Domain decomposition methods with nonlinear localization for the buckling and post-buckling analyses of large structures[END_REF]. but which is is not possible in general in a non-invasive framework. Anyhow, in the case of a linear Global model, it is possible to derive a quasi-Newton approach.

If the Global problem is linear then the differential of the Global problem is constant and it is equal to the Schur complement DS G = S G . Regarding the nonlinear part, we have:

DS R = S C + DS F = S C + S A S G + DS F -S A X (31)
of course X is not computable in a non-invasive manner, but a low rank approximation is possible using quasi-Newton formulas. In particular, SR1 formula was tried with success in [START_REF] Gendre | Non-intrusive and exact global/local techniques for structural problems with local plasticity[END_REF]. In practice, line search is not applied which makes the low rank update slightly lighter than usual, we note δ u j = (u ju j-1 ) |Γ and δ p j = (p jp j-1 ), the increment of the interface quantities:

DS R,0 = S G DS R,i = DS R,i-1 + r i-1 r T i-1 r T i-1 δ u i-1 = S G + R i ∆ -1 i R T i ( 32
)
with, for i > 0, R i = [r 1 . . . r i ] and ∆ i = diag(r T i δ u i ). Sherman-Morrison formula leads to:

DS -1 R,i = S G -1 -S G -1 R i ∆ i + R T i S G -1 R i -1 R T i S G -1 (33)
It makes sense to first evaluate S G -1 r i then apply corrections. For efficiency reasons, we also store the matrix

W i := S G -1 R i . The factorization of Matrix ∆ i + R T i S G -1 R
i is reused from one iteration to another, only one row and column must be computed. The method is recapitulated in the algorithm 2. Note that this algorithm is written in a way which makes no use of the linearity of the Global problem, so that it will be also tested in the full nonlinear case.

Conjugate gradient

Full linear case

This case occurs when all models are linear. Non-invasive global/local coupling can still be of interest in order to introduce complex local heterogeneities, stochastic behaviors or complex geometries in the Fine model.

For linear problems, it is rather classical to use Krylov accelerator on a stationary iteration. In our case, the problem to solve ( 29) is governed by the operator S R S G -1 which is symmetric in the S G -1 inner-product. We then can derive a right-preconditioned conjugate gradient. The algorithm being not so standard, it is given in algorithm 3.

Beside the improved convergence compared to stationary iterations, using conjugate gradient allows an unconditional convergence (without necessity for the Auxiliary model to be stiffer than the Fine model).

[u G 0 ] = SolveGlobal(p 0 ; f G ) [u F 0 , λ F 0 ] = SolveFine(u G 0 ; f F ) // [λ A 0 ] = SolveAux(u G 0 ; f A ) Residual: r 0 = -(λ F 0 + p 0 -λ A 0 ) Set W 0 = [], R 0 = [], x 1 = 0 for j ∈ [1, • • • , m] do p j = λ A j-1 -λ F j-1 [ũ G j ] = SolveGlobal(p j ; f G ) if j > 1 then Set: W j-1 = [W j-2 , ũG j -u G j ] Compute: x j = (∆ j-1 + W T j-1 R j-1 ) -1 W T j-1 r j-1 end Corrector u G j = ũG j -W j x j Set: δ u j = u j|Γ -u j-1|Γ , ∆ j = diag(δ u i ) 1 i j [u F j , λ F j ] = SolveFine(u G j ; f F ) // [λ A j ] = SolveAux(u G j ; f A ) Residual: r j = -(λ F j + p j -λ A j ), R j = [R j-1 , r j ] end Algorithm 3: Non-invasive right-preconditioned conjugate gradient Arbitrary initialization p 0 = 0 [u G 0 ] = SolveGlobal(p 0 ; f G ) [u F 0 , λ F 0 ] = SolveFine(u G 0 ; f F ) // [λ A 0 ] = SolveAux(u G 0 ; f A ) Post-process: λ C 0 = p 0 -λ A 0 r 0 = -(λ F 0 + λ C 0 ) Preconditioning: [u G 0 ] = SolveGlobal(p 0 + r 0 ; f G ) -[u G 0 ] y = u G 0 for j ∈ [0, • • • , m] do [u F j , λ F j ] = SolveFine(y + u G j ; f F ) -[u F j , λ F j ] // . . . [λ A j ] = SolveAux(y + u G j ; f A ) -[λ A j ] Post-process: λ C j = p j + p j -[λ A j + λ A j ] -λ C j
Direction of the variation of the residual:

r j = λ F j + λ C j
Optimal step size: α j = (r T j y)/(y T r j ) Updates:

p j+1 = p j + α j p j ; r j+1 = r j -α j r j u G j+1 = u G j + α j y ; u F j+1 = u F j + α j u F j ; λ C,A,F j+1 = λ C,A,F j + α j λ C,A,F j Preconditioning: [ũ G j+1 ] = SolveGlobal(p j+1 + r j+1 ; f G ) -[u G j+1
] Orthogonalization: p j+1 = r j+1β j p j with β j = -(r T j ũj+1 )/(r T j y) Correction: y = ũG j+1 + β j y end

Nonlinear case

Conjugate gradient can be extended to nonlinear cases using two ingredients:

A line search algorithm to optimize the length of the steps. For a given search direction p, one tries to find the optimal length α in term of the minimization of some norm of the residual. This can be done in a non-invasive manner by a sampling technique with several lengths (α i ) being tested in parallel. Classically these samples are used to interpolate the objective function and decide the final α. Because of the cost of the estimation of one configuration (one global solve followed by one local solve), we prefer to use directly the best sample already computed (except if the interpolated minimal let us expect a significantly better configuration).

A "conjugation" technique for the new search direction p j+1 = -r j+1 + β j p j given by a heuristic (using the notations of algorithm 4) like:

Fletcher-Reeves:

β j = r T j+1 r j+1 r T j r j
Polac-Ribière:

β j = r T j+1 (r j+1 -r j ) r T j r j
Dai-Yuan:

β j = r T j+1 r j+1 p T j (r j+1 -r j )
Hestenes-Stiefel:

β j = r T j+1 (r j+1 -r j ) p T j (r j+1 -r j ) (34)
Moreover it is often chosen to avoid negative steps by using β j ← max(0, β j ). The reader may refer to [START_REF] Glowinski | Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems[END_REF] and associated bibliography for more details. In our examples, the Polac-Ribière formula appeared to be more stable.

Algorithm 4: Non-invasive nonlinear conjugate gradient Arbitrary initialization p 0 [u G 0 ] = SolveGlobal(p 0 ; f G ) [u F 0 , λ F 0 ] = SolveFine(u G 0 ; f F ) // [λ A 0 ] = SolveAux(u G 0 ; f A ) Post-process: λ C 0 = p 0 -λ A 0 , r 0 = -(λ F 0 + λ C 0 ) Initialization: p 0 = r 0 for j ∈ [0, • • • , m] do for samples (α j,i ) i do Preconditioning: [u G j,i , λ C j,i ] = SolveGlobal(p j + α j,i p j ; f G ) [u F j,i , λ F j,i ] = SolveFine(u G j,i ; f F ) // [λ A j,i ] = SolveAux(u G j,i ; f A ) Post-process: λ C j,i = p j,i -λ A j,i , r j,i = -(λ F j,i + λ C j,i ) end
Optimal step size (Line search): choose I = arg min i r i,j Updates: p j+1 = p j + α j,I p j ; u F,G j+1 ← u F,G j,I ; r j+1 ← r j,I Pseudo-orthogonalization: p j+1 = r j+1β j p j with β j given by formula (34) end

Numerical illustration

The method is first illustrated on an academic 2D test case modeling a high pressure turbine blade of a plane engine (see figure 1). The Reference model possesses local perforations with adapted mesh which are not present in the Global model. Note that in that case, the Fine model is naturally more flexible than the Auxiliary model (because of the holes and the refined mesh). The Fine model is granted an elastoviscoplastic behavior of the form of [START_REF] Longuet | Incremental lifing method applied to high temperature aeronautical component[END_REF] modeling a realistic IN100 material at hot temperature ( 1 500 • C). We consider two configurations: in the first case the Complement model is linear (and so is the Global model), in the second case all models are elastoviscoplastic. We study the various acceleration strategies presented above in the case where the Complement zone is linear and the Auxiliary model is chosen to be so. Figure 2a presents the evolution of the Euclidean norm of the residual on the interface. As expected conjugate gradient is faster than other acceleration techniques. Figure 2b presents the evolution of the error measured by the Mises stress on the most loaded element with respect to the Reference model (which should not be available in production cases). We observe an important practical difficulty: Abaqus' truncation of Gauss point data makes it impossible to observe convergence beyond a relative precision of 10 -6 . This problem would appear much later on the residual which only involves nodal computations (which can be manipulated in double precision). Table 1 compares the duration of the computation. In that simple case all accelerations have close performance, but we observe that CG is faster than Aitken which is faster than SR1 which is 25% faster than Stationary iteration.

Method

Stat. SR1 Aitken CG CPU time 2'55 2 '11 1'57 1'47 Global/local iterations In that case, all models are granted the same nonlinear elastoviscoplastic behavior. As a consequence, plasticity may spread in the Complement zone. In that case the Fine and Auxiliary models only differ by their topology and their mesh. Before comparing the acceleration techniques, we specifically study the choice of the parameters of the nonlinear conjugate gradient. Prior experiments showed that, for problems close to the one considered here, the optimal line-search always often belongs to the interval [.8, 1.4], we thus use either 4 sampling points {.8, 1., 1.2, 1.4} or 9 sampling points {.8, 1., 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4} where the sampling will be performed.

Figure 3 presents the performance of conjugate gradient for various conjugation techniques and various samplings for the linesearch. We observe that, in that case, the Polac-Ribière conjugation gives best results, and testing 9 lengths seems to be significantly superior to only testing 4. We recall that the line search is conducted in parallel so that oversampling does not take more wallclock time, it only "wastes" machine time and software licenses. We compare, in Figure 4, the better configuration of conjugate gradient to other acceleration techniques. We observe that nonlinear conjugate gradient also behave better than the other techniques. Note that since the sampling of linesearch can be conducted in parallel, so that in practice one iteration of CG is faster than the one of the very cheap Aitken's method.

Overlapping version

In previous sections, we had assumed that the interface was described as the boundary of elements for all models. In practice this hypothesis is not so restrictive because most often the zone of interest is detected after an initial computation on the coarse global model, and it is constituted as a set of coarse elements satisfying a certain criterion. Even after remeshing, the boundary of the Fine description of the zone of interest matches a set of coarse faces (edges in 2D). Then a "simple" transfer matrix T can be sufficient to communicate between models on the interface. In particular, the easy choice of T being the interpolation matrix of the coarse kinematics in the fine kinematics can be implemented in most software. More evolved choices like mortar connections can also be employed in certain software.

We propose here an alternative strategy which makes use of the possibility to have the models overlap. In that case, there is no restriction on the definition of the meshes. This idea can directly be connected to overlapping optimized Schwarz methods, yet we propose a mechanical interpretation of it.

Note that the use of the overlap can be advantageous in the situations where edge effects can affect the fine model, even if meshes are conforming at the interfaces [START_REF] Guguin | Nonintrusive coupling of 3d and 2d laminated composite models based on finite element 3d recovery[END_REF].

Handling of incompatible patches

The starting point is the observation that the method can be formulated as the search for p which is the stress discontinuity on the Global model between the Complement zone and the Auxiliary description of the zone of interest. This discontinuity must be such that the Complement zone is in equilibrium with the Fine description of zone of interest loaded with Dirichlet conditions [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. Since p is a discontinuity, in order it to be well described in the coarse finite element model, it must be supported by the boundary of coarse elements. But there is no need for the support of p to match the boundary of the zone of interest. We thus propose to follow the figure 5. The Fine subdomain Ω F is positioned where needed in the zone of interest, its mesh is independent from the coarse mesh. We note Γ F = ∂Ω F the boundary of the Fine subdomain. The Auxiliary subdomain is the largest set of coarse elements fully contained in the zone of interest. We note Γ A the boundary of the Auxiliary zone. The two interfaces Γ F and Γ A thus do not coincide. Ω C is defined as the Complement to Ω A in the Global problem.

Algorithm 5 gives the basic stationary iteration in the presence of overlap, all accelerations can be considered. In order to distinguish between the interfaces, the Auxiliary and the Fine problems are written on separate lines even if they can be solved in parallel.

The main difficulty of this algorithm is the computation of the Fine reaction on Γ A with Ω F not exactly represented on the coarse grid. This computation mixes the Fine stress σ F h and the coarse shape functions φ G i .

Algorithm 5: Non-invasive stationary iterations with overlap Arbitrary initialization p 0

for j ∈ [0, • • • , m] do [u G j ] = SolveGlobal(p j on Γ A ; f G ) [λ A j ] = SolveAux(u G j on Γ A ; f A ) u F Γ F ,i = u G (x i ) for i a Fine dof of Γ F [u F j , σ F h,j ] = SolveFine(u F Γ F ,j ; f F )        for i spanning all Global degrees of freedom on Γ A λ F j,i = Ω A σ F h,j : ε(φ G i ) -f • φ G i dx - ∂nΩ A g • φ G i dS
Residual: r j = -(λ F j + p jλ A j ) Update: p j+1 = p j + r j ; end Even if complex, this computation is feasible in certain software. Anyhow in the nonlinear case, σ F h is only known at Fine Gauss points and the integral can only be approximated.

There is another conceptual difficulty which is common to many methods with overlap [START_REF] Dhia | Multiscale mechanical problems: the Arlequin method[END_REF]: the definition of a monolithic reference problem. Indeed the domain between the interfaces (the overlap) is a buffer zone Ω B (Γ A ∪ Γ F = ∂Ω B ) where the Complement and the Fine model coexist. At convergence they are equivalent in the sense that they have the same value (up to transfer error) on Γ F (Dirichlet) and Γ A (Neumann) and they solve the same problem in the overlap.

Illustration of the coupling with overlap

The overlap was used in [START_REF] Guguin | Nonintrusive coupling of 3d and 2d laminated composite models based on finite element 3d recovery[END_REF] for the coupling between a Global laminate plate model and a Fine 3D model. Various plate-to-3D transfer techniques were tried but it appeared that 3D edge effects were impossible to avoid completely. Figure 6 shows the variation of the peeling stress in the Fine 3D model direction orthogonal to the interface for various lifting of the plate displacement (named "Lagrangian" and "warping" in the figure). We see that edge effects are important on the boundary of the Fine domain (Γ F ) but they fade quickly so that the boundary of the Auxiliary domain (Γ A ) can be positioned not too far inside the zone of interest. The domain between Γ F and Γ A was called the buffer zone and a width of two times the thickness was sufficient.

Note that the overlap also allows to apply any type boundary conditions on the Fine domain. For instance if the zone of interest bears Dirichlet conditions, it is possible to use Neumann boundary conditions on Γ F . As said earlier, in the case of non-matching meshes, the difficulty for the coupling with overlap is the computation of the fine reaction on Γ A written λ F j,i in Algorithm 5. In [START_REF] Guguin | Stratégie non-intrusive de couplage plaque/3D pour l'application aux plaques composites stratifiées[END_REF], in the more complex case of plate/3D coupling, it was proposed to extract a band of Auxiliary elements connected to Γ A and project on it the Fine stress (defined at the Gauss point of the Fine mesh). This was implemented in Code Aster using existing routines (PROJ CHAMP() with keyword ECLA PG).

Figure 7 presents a simple application of an isotropic plate in flexion where the Global model is a solid plate with unstructured mesh with triangular elements and the 3D Fine model bears a hole and is meshed with structured hexahedral elements. At convergence, the continuity is ensured on Γ F as well as the equilibrium on Γ A .

Conclusion

The global/local non-invasive coupling technique is a convenient way to enrich a global coarse model, handled by a commercial software, with local features, handled by most the adapted software. In this paper we proposed to interpret the method as an alternate non-overlapping Schwarz domain decomposition method. In this framework the coarse representation of the zone of interest is a clever way to build an approximation of the Dirichlet-to-Neumann operator of the Fine model, which includes the effects of the imposed load. Belonging to the Schwarz family of domain decomposition method allows to benefit many theoretical results and practical shrewdness. We then derive a conjugate gradient solver in the linear and nonlinear cases, in that later case the line search is realized by a sampling which can be conducted in parallel in order not to penalize the wallclock time. Finally we show that an overlapping version can also be applied which enables to connect non-matching meshes.

Algorithm 2 :

 2 Non-invasive SR1 Quasi-Newton iterations Arbitrary initialization p 0

Figure 1 :

 1 Figure 1: 2D test case

δ 2 (

 2 b) Mises stress error wrt Reference on the most loaded element

Figure 2 :

 2 Figure 2: Convergence of 2D case with linear Global model

  Influence of the linesearch sampling

Figure 3 :

 3 Figure 3: Study of variants of nonlinear conjugate gradient

δ 2 (

 2 b) Mises stress error wrt Reference on the most loaded element

Figure 4 :

 4 Figure 4: Convergence of 2D nonlinear case

Figure 5 :

 5 Figure 5: Technique with overlap for non-conforming meshes

Figure 6 :

 6 Figure 6: Plate/3D coupling: use of the overlap to avoid edge effects.

Figure 7 :

 7 Figure 7: Exploded view of the plate/3D coupling with non-matching meshes. The interior of the Auxiliary model is not drawn in order to better see the intermediate mesh used for the stress projection.

Table 1 :

 1 CPU time for various methods in the case of a linear Global model

	0	2	4	6	8	10

Acknowledgement: this work was partially funded by the French National Research Agency as part of project ICARE (ANR-12-MONU-0002-04). The authors wish to thanks Guillaume Guguin for the figures related to the non-conforming overlaping meshes.


Elise Jacquier

Pierre-Henri Jouneau

Denis Falconet

Denis Mariolle

Emmanuel Thévenon

Si Grégory

Raluca Larbi

François Tiron

Pierre-Henri Parcy

Renaud Elchinger

Dumas

Parcy

Pierre-Henri Elchinger

Renaud Dumas

Self-Assembly of a Ginkgo Oligomerization Domain Creates a Sub-10-nm Honeycomb Architecture on Carbon and Silicon Surfaces with Customizable Pores: Implications for Nanoelectronics, Biosensing, and Biocatalysis

Keywords: helical polymer, tunable pore, nanomaterial, protein nanostructure, nanopatterning, green technology

published or not. The documents may come

L'archive ouverte

Introduction

Nanopatterning of surfaces using biological macromolecules such as DNA, peptides and proteins has become a highly active field of research yielding many technological breakthroughs.

Due to their self-assembling ability, biological macromolecules have been widely used as scaffold for the construction of nanomaterials [START_REF] Thiruvengadathan | Nanomaterial Processing Using Self-Assembly-Bottom-up Chemical and Biological Approaches[END_REF][START_REF] Sun | Nanostructures Based on Protein Self-Assembly: From Hierarchical Construction to Bioinspired Materials[END_REF][START_REF] Fan | Create Nanoscale Patterns with DNA Origami[END_REF][START_REF] Liu | Using Self-Assembling Peptides to Integrate Biomolecules into Functional Supramolecular Biomaterials[END_REF] . Among these macromolecules, proteins with their 20 different amino acids, offer flexibility in self-assembled structures with nanometric specifications.

The initial objective in protein self-assembly was to obtain regular, robust and wide 2D layers with controlled height and organization. Nowadays, these protein lattices are studied to evaluate their usefulness for nanotechnological applications.

Over the years, different types of bottom-up processes have been implemented to generate protein self-assemblies on surfaces (Table S1). Pioneering work began with self-assembly of Tlayer [START_REF] Aebi | A Study of the Structure of the T-Layer of Bacillus Brevis[END_REF] and S-layer 6 proteins [START_REF] Gyorvary | Self-Assembly and Recrystallization of Bacterial S-Layer Proteins at Silicon Supports Imaged in Real Time by Atomic Force Microscopy[END_REF][START_REF] Moll | S-Layer-Streptavidin Fusion Proteins as Template for Nanopatterned Molecular Arrays[END_REF][START_REF] Baneyx | Self-Assembled Two-Dimensional Protein Arrays in Bionanotechnology: From S-Layers to Designed Lattices[END_REF] which are still used as biomaterial [START_REF] Baneyx | Self-Assembled Two-Dimensional Protein Arrays in Bionanotechnology: From S-Layers to Designed Lattices[END_REF][START_REF] Charrier | Engineering the S-Layer of Caulobacter Crescentus as a Foundation for Stable, High-Density, 2D Living Materials[END_REF] . So far, the creation of selfassembled 2D protein networks for nanotechnological applications has been achieved mainly through the design and engineering of protein interfaces from known 3D crystallographic structures (Table S1), with new interfaces created by different types of interactions such as disulfide bridge [START_REF] Zhou | Disulfide-Mediated Reversible Two-Dimensional Self-Assembly of Protein Nanocages[END_REF][START_REF] Nguyen | Construction of Supramolecular Nanotubes from Protein Crystals[END_REF][START_REF] Du | Precise Fabrication of De Novo Nanoparticle Lattices on Dynamic 2D Protein Crystalline Lattices[END_REF][START_REF] Zhang | Self-Assembly of Protein Crystals with Different Crystal Structures Using Tobacco Mosaic Virus Coat Protein as a Building Block[END_REF][START_REF] Suzuki | Self-Assembly of Coherently Dynamic, Auxetic, Two-Dimensional Protein Crystals[END_REF] , metal-induced interactions [START_REF] Zhang | Self-Assembly of Protein Crystals with Different Crystal Structures Using Tobacco Mosaic Virus Coat Protein as a Building Block[END_REF][START_REF] Suzuki | Self-Assembly of Coherently Dynamic, Auxetic, Two-Dimensional Protein Crystals[END_REF][START_REF] Heyman | Float and Compress: Honeycomb-like Array of a Highly Stable Protein Scaffold[END_REF][START_REF] Brodin | Chemically Tunable Assembly of One-, Two-and Three-Dimensional Crystalline Protein Arrays[END_REF][START_REF] Brodin | Exceptionally Stable, Redox-Active Supramolecular Protein Assemblies with Emergent Properties[END_REF][START_REF] Zhang | Self-Assembly of Glutathione S-Transferase into Nanowires[END_REF][START_REF] Yang | Diverse Protein Assembly Driven by Metal and Chelating Amino Acids with Selectivity and Tunability[END_REF][START_REF] Matthaei | Designing Two-Dimensional Protein Arrays through Fusion of Multimers and Interface Mutations[END_REF] , hydrophobic interactions [START_REF] Suzuki | Self-Assembly of Coherently Dynamic, Auxetic, Two-Dimensional Protein Crystals[END_REF][START_REF] Lo | Fungal Hydrophobin Proteins Produce Self-Assembling Protein Films with Diverse Structure and Chemical Stability[END_REF][START_REF] Zhou | On-Axis Alignment of Protein Nanocage Assemblies from 2D to 3D through the Aromatic Stacking Interactions of Amino Acid Residues[END_REF] , as well as the addition of fused amyloid domains [START_REF] Zheng | Designed Two-and Three-Dimensional Protein Nanocage Networks Driven by Hydrophobic Interactions Contributed by Amyloidogenic Motifs[END_REF] . Complex processes have been also used to tune protein network based on ligand-receptor models [START_REF] Sakai | Protein Crystalline Frameworks with Controllable Interpenetration Directed by Dual Supramolecular Interactions[END_REF][START_REF] Yang | Highly Ordered Self-Assembly of Native Proteins into 1D, 2D, and 3D Structures Modulated by the Tether Length of Assembly-Inducing Ligands[END_REF][START_REF] Liu | Fabrication of Pascal-triangle Lattice of Proteins by Inducing Ligand Strategy[END_REF] , streptavidin-Streptag affinity [START_REF] Sinclair | Generation of Protein Lattices by Fusing Proteins with Matching Rotational Symmetry[END_REF] or covalent coupling by enzymatic catalysis [START_REF] Zhao | Enzyme-Triggered Defined Protein Nanoarrays: Efficient Light-Harvesting Systems to Mimic Chloroplasts[END_REF] . Finally, promising de novo approaches have also been used to predict and generate different protein/protein [START_REF] Gonen | Design of Ordered Two-Dimensional Arrays Mediated by Noncovalent Protein-Protein Interfaces[END_REF][START_REF] Chen | Self-Assembling 2D Arrays with de Novo Protein Building Blocks[END_REF] and protein/inorganic surface interfaces [START_REF] Pyles | Controlling Protein Assembly on Inorganic Crystals through Designed Protein Interfaces[END_REF] for tunable protein self-assemblies.

Overall, current studies have shown that self-assembly of proteins on surfaces allows the creation of mesoporous nanomaterials with different pore shapes and sizes. These self-assembled pores can be used as grafting sites for nanotechnology applications [START_REF] Du | Precise Fabrication of De Novo Nanoparticle Lattices on Dynamic 2D Protein Crystalline Lattices[END_REF][START_REF] Zhao | Enzyme-Triggered Defined Protein Nanoarrays: Efficient Light-Harvesting Systems to Mimic Chloroplasts[END_REF] . However, most current self-assemblies do not offer the possibility to modulate both type and number of grafting sites which depend on i) the amino acid sequence and ii) the number of protein blocks composing the pore, often limited to a small number (4-6) of monomers.

Here, to overcome this challenge, we propose to use the Sterile Alpha Motif (SAM) domain of the LEAFY (LFY) protein from the Ginkgo biloba (GbLFY) tree. In Nature, the LFY protein is a transcription factor, studied for its role in controlling plant meristem division, differentiation, and flower development [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF][START_REF] Denay | A Flower Is Born: An Update on Arabidopsis Floral Meristem Formation[END_REF] . The SAM domain of GbLFY (GbLFY-SAM) self-assembles to form a helical polymer through head-tail interactions. This polymer interlocks with six others leading to a honeycomb structure with an 8 nm pitch (distance between two neighboring pores) (Fig. 1). Each helical polymer has a 5-nm pore (Fig. 1) in which a disordered and modular C-terminal extension of each monomer is located. In solution, self-assembly leads to exceptionally robust micrometer hexagonal crystals that can be functionalized inside the pore [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF][START_REF] Chiari | LEAFY Protein Crystals with a Honeycomb Structure as Platform for Selective Preparation of Outstanding Stable Bio-Hybrid Materials[END_REF] . The robustness of the honeycomb relies on the tight interaction between the edge of each helical polymer and the groove of its six neighbors (Fig. 1). Whether this 3D modular honeycomb architecture can occur on surfaces has remained elusive. In this paper, we have successfully formed a honeycomb on different surfaces thank to the addition of a N-terminal extension. Each helical polymer of the honeycomb is built by about 40 stacked functionalizable proteins. Moreover, the pore lumen of each helical polymer can be modified by N-or C-terminal extension modifications without affecting the self-assembly.

The concept of specific grafting into the honeycomb pores is illustrated here by differential metallization of the pores with nickel or gold.

Methods

Protein synthesis and purification.

GbLFY-SAM is composed of an 87-amino-acid core SAM oligomerization domain surrounded by 24 N-and 23 C-terminal residues. The N-terminal extension contains a 6-histidine tag used for affinity purification and a cleavage site whereas the C-terminal extension contains a disordered sequence. These residues remain invisible by X-ray crystallography (Fig. 1) because of their flexible structure, but their presence was confirmed by mass spectrometry [START_REF] Chiari | LEAFY Protein Crystals with a Honeycomb Structure as Platform for Selective Preparation of Outstanding Stable Bio-Hybrid Materials[END_REF] . GbLFY-SAM proteins were expressed in Escherichia coli BL21(DE3) strain. First, cells were transformed with pETM11 plasmid coding for a N-terminal histidine-tag and a cleavage site for TEV (referred as N-terminal extension) followed by the sequence corresponding to GbLFY-SAM wild type protein [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . Transformed cells were used to inoculate 1 L Luria-Bertani (LB) medium supplemented with 1 mL of kanamycin at 50 mg.mL-1 and 1 mL of chloramphenicol at 34 mg.mL -1 . Cells were grown at 37°C under agitation up to an optical density of 0.6 at 600 nm. Temperature is then shifted to 18°C for 1 hour. IPTG (0.4 mM) is added to the culture and growth continues at 18°C overnight before cells pelleting. Pellets corresponding to 0.25 L culture are sonicated in 50 ml buffer A (20 mM Tris-HCl pH 8.0, 1 mM TCEP) supplemented with one protease inhibitor tablet (ThermoFischer). After centrifugation (45 min at 15 000 g), the supernatant is loaded on a Ni-Sepharose High Performance column pre-equilibrated in buffer A. The column is then washed with buffer A containing 20 mM imidazole and proteins are eluted with buffer A containing 300 mM imidazole. SDS-PAGE was performed to evaluate elution fraction purity. Then, eluted fractions were dialyzed overnight against buffer A and concentrated using Amicon Ultra Centrifugal filters (Merck) until 5 mg.mL -1 . Proteins were then flash frozen and stored at -80°C.

Cysteine mutant K110C, GGSGGSCHCHCHC mutant and shorter C-terminal extension were obtained by gene synthesis (ShineGene) from the initial GbLFY-SAM sequence cloned in pETM11 [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . For the shorter N-terminal extension mutants, the nucleotide sequence corresponding to seven (MHHHHHH) or the first nine (MKHHHHHHP) amino acids of the pETM11 tag was inserted in the 5' side of the sequence of GbLFY-SAM in a pET-30a (+) plasmid. The main problem with mutants modified on their N-terminal or C-terminal extension is that their ability to oligomerize via head-tail interactions has been increased, leading to the aggregation of large oligomers. To solve this problem during purifications, mutated proteins are purified in buffers containing 40-200 mM CAPS-NaOH pH 10.5 in the presence of 50-500 mM NaCl and 1 mM TCEP. The proteins are then dialyzed in a 40 mM CAPS-NaOH pH 10.5 buffer containing 50 mM NaCl and 1 mM TCEP before being used to form self-assemblies. Shorter protein without His-tag was obtained by cleavage overnight with the TEV protease (5% w/w) and the cleaved protein was purified on a Ni-Sepharose High Performance column [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . Amino acid sequence of the different proteins used in this work are described in the Table S2.

Self-assembly onto TEM grids.

To perform protein self-assembly, vapor diffusion sitting drop method was used. A 18 µL drop composed by GbLFY-SAM solution and crystallization buffer is deposed in a well chamber while the reservoir chamber contains only buffer crystallization. Optimum conditions were obtained by mixing 6 µL of a 2.5 mg.mL -1 GbLFY-SAM solution with 12 µL of the reservoir solution containing 20 mM Tris HCl pH 8.0, 1 mM TCEP and 315 mM ammonium sulfate. A 200 mesh gold TEM grid coated with a pure carbon film (TED Pella) or SiO2 grid (SIMPore) was hydrophilized using a plasma glow discharge (Cordouan Technologies) at 2 mA for 40 seconds under 2.10 -1 mbar and then put on the top of the drop. For the experiments with SiN grid (TED Pella) either hydrophilic (coated with hydroylated alumina) or hydrophobic (coated with alumina and fluoro-methyl-silane), no glow discharge was performed.

After closing of the crystallization chamber, self-assemblies were allowed to grow at 20°C for 24 h. The same protocol was used for experiments with shorter N-or C-terminal extension.

Negative staining for STEM experiments.

After 24h crystallization, the grid is incubated with a 18 µL drop of 2% uranyl acetate for 2 minutes. Liquid excess was then removed by capillarity with a filter paper and grid was air-dried. Self-assemblies were imaged with STEM mode using a Zeiss Merlin microscope. Images were collected using both bright field and dark field detectors at 30 kV with a beam current of 240 pA with magnification between 2 KX and 135 KX.

Metallization and chemical analysis of the self-assembly. After 24h crystallization, the

grid is incubated with a 18 µL drop of Ni 2+ salt (Hampton Research) or gold salt (Sigma-Aldrich).

For nickel salt (NiCl2), a range of concentration from 0.18 mM (i.e. 10 equivalents relative to protein amount) to 1.79 mM (i.e. 100 equivalents relative to protein amount) was tested. In the same way, a range from 0.23 mM (i.e. 10 equivalents relative to protein amount) to 2.30 mM (i.e. 100 equivalents relative to protein amount) of gold salt (HAuCl4) has been investigated. Moreover, time incubation from 2 to 10 min has been considered. The optimal conditions are 0.53 mM (i.e. 30 equivalents relative to protein amount) for nickel salt and 0.23 mM for gold salt with an incubation time of 5 min in both cases. Then, for reduction step, a 18 µL drop of NaBH4 at a final concentration of 32 mM was made and the grid was put on top of this drop allowing reduction to occur for 5 min. Liquid excess was then removed by capillarity with a filter paper and grid was air-dried. STEM images were collected in the same microscope than the negative staining experiments at 30 kV but using only dark field detector. EDX spectrum of the metallized sample was acquired on a FEI Osiris microscope operated at 200 kV using a Super X Detector (Bruker).

AFM imaging.

For AFM experiments, self-assemblies were imaged directly on the TEM grid or silicon wafer using PeakForce Tapping mode in air with ScanAsyst Air probes (Bruker) on a Dimension XR AFM (Bruker). The ScanAsyst Air probes used in all the experiments have a spring constant of 0.25 N/m. Images were collected with a resolution of 1024 lines and a scan rate between 0.5 and 0.2 Hz. Scan analysis was performed using Gwyddion software (http://gwyddion.net). Scans were base flattened and zero was fixed before height measurements using the profile tool. Then, an adjusting function is fitting on each self-assembly profile to extract its height. Average height (Hn) was determined by measuring individual heights (Hi) of at least 30 self-assemblies (ni) such as (1). Standard deviation (σn) is determined by (2).

Image analyses.

Thanks to Fiji software (http://imagej.net/Fiji), STEM image contrast was enhanced by removing the background through background subtraction. Surface analysis was performed by applying a threshold on the image, then particle analysis was performed to determine the overall coverage of the self-assemblies. To detect the self-assembly pores, a threshold and a Mexican Hat Filter were used to reduce noise in the images. The number of pores detected was counted by finding the maximal edges on the image. Lattice parameters of the self-assemblies were measured using the Fast Fourier Transform calculation. This analysis yields the power spectrum of the STEM images, whereby each frequency is associated to angle and radii related to the lattice parameters of the self-assemblies.

Results and Discussion

3.1 Orientation of the honeycomb on a carbon surface. GbLFY-SAM honeycomb oligomerization has been previously observed to form 3D crystals [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . It is a head-tail interaction involving mainly ionic and hydrogen bonds, as determined using mutations affecting either the head or the tail side [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . Based on the same crystallographic method, we aimed to reproduce the GbLFY-SAM honeycomb self-assembly on a surface. We opted for a carbon grid surface that facilitates the characterization of protein assemblies by scanning transmission electron microscopy (STEM) after negative staining. During the first attempts, the cleaved GbLFY-SAM domain [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] , devoid of its N-terminal extension, formed supramolecular helices lying parallel to the surface grid without forming an apparent honeycomb (Fig. 2 and Fig. S1).

To determine whether the N-terminal extension initially present on the protein could have a role in the orientation of the self-assembly, we repeated the experiments with the uncleaved protein.

STEM images of the self-assembly of GbLFY-SAM with its N-terminal extension revealed black dots corresponding to the pores of the honeycomb oriented perpendicular to the carbon grid (Fig. 2, Fig. 3 and Fig. S1). To further check whether residual Ni 2+ from the Ni-NTA column was not responsible for the self-assembly by promoting interaction between the N-terminal extensions (e.g. the histidine tag), we performed the same experiment in the presence of EDTA, a Ni 2+ chelator.

The STEM images (Fig. S2) showed that the honeycomb architecture was not affected by EDTA supporting the absence of Nickel effect on the self-assembly.

Since the N-terminal extension was not present in the X-ray structure of the 3D crystals growing in solution [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] we used the I-TASSER modelling server [START_REF] Yang | I-TASSER Server: New Development for Protein Structure and Function Predictions[END_REF] to elucidate its position (Fig. S3). Based on this model, the N-terminal extension of each monomer is not found in the same pore as the C-terminal extension but in an adjacent pore (Fig. S3). However, in the honeycomb, both N-and Cterminal extensions are located within the pores (Fig. S3).

Next, we determined which part of the 23 amino acids of the N-terminal extension was key for perpendicular honeycomb orientation. By successively deleting the amino acids of the N-terminal extension starting from the ones closest to GbLFY-SAM core, we established that the first six histidine residues are sufficient for this orientation (Fig. 3 and Fig. S4). The presence of the Nterminal extension likely allows interaction with the grid and perpendicular growth of the honeycomb. Alternatively, the absence of the N-terminal extension could induce a large formation of 3D crystals in solution, depleting the protein available for surface crystallization. In solution, the head-tail interactions extend to several monomers to create an helical oligomers of a limited sized of about 8 monomers at a protein concentration of 4.5 mg/ml as demonstrated by SEC-MALLS experiments [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . Under crystallization conditions, the helices can grow to larger sizes and each helix interacts with six other helices to form a honeycomb structure [START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] . The 3D crystals can reach 500 µm in length and 50 µm in width. However, the self-assemblies grown on carbon grid are thinner. Correlative microscopy between STEM and Atomic Force Microscopy (AFM) was performed and the AFM (Fig. 3) measurements indicated a height of 31 (+/-5) nm, corresponding to a stacking of around 40 oligomerization domains (Fig. 3 and Fig. S5). We also determined that the height of the shorter N-terminal mutant is similar to GbLFY-SAM (Fig. 3).

Overall, the protein honeycomb covers around 10 % of the grid, with individual crystals of up to 0.3 µm 2 . For nanotechnological applications, large scale nanopatterning is required [START_REF] Sreenivasan | Status of UV Imprint Lithography for Nanoscale Manufacturing[END_REF] . Moreover, amorphous carbon used to coat the EM grid is not a suitable substrate for these applications as it does not conduct electrons. In order to overcome these limitations, we adjusted our protocol. First, we attempted to increase the surface covered by the self-assembly, by glow discharging the grids to hydrophilized them. The total area covered by the honeycomb increased by up to 54%, with individual self-assembly surface up to 4.9 µm 2 (Fig. 4). These observations suggest that the honeycomb binds on the grid surfaces through weak polar interactions. Whatever the protein concentration (0.625 to 2.5 mg.mL -1 ), a mixture of shapes is obtained of which the trapezium (Fig. 3a) and its elongated form (Fig. 4c) in one dimension are the most representative.

Then, we changed the carbon substrate to silicon wafers widely used in nanoelectronic applications. We succeeded in reproducing such assemblies on silicon grids and wafers and obtained a similar honeycomb architecture as revealed by STEM (Fig. 5a). It is noteworthy that the change in the surface (silicon wafer or carbon grid) does not affect the height of the GbLFY-SAM lattice (Fig. 5b) neither the lattice parameters. To further investigate the influence of the surface properties on the self-assembly, we also tried hydrophilic and hydrophobic commercial silicon nitride grid (Fig. S6). These grids are coated respectively with hydroxylated alumina and alumina plus fluoro-methyl-silane. There is no difference between the hydrophilic and the hydrophobic coating on the GbLFY-SAM self-assembly. In addition, the lattice parameters are also the same as the ones on carbon and silicon surface. Altogether, these results show that selfassembly can take place on a wide variety on surfaces.

Polymeric honeycomb structure metallization.

Finally, one of the key points for the use of the protein self-assemblies in the field of nanotechnology is the functionalization of the pores.

To this end, we investigated whether it was possible to selectively metallize the polymeric honeycomb structure. As the N-terminal extension is composed of at least 6 histidine and is facing inside the lattice pores, we aimed at using the specific affinity of histidine for Ni 2+ to metallize the pores of the self-assembly (Fig. S3). After the crystallization of the GbLFY-SAM self-assembly on the TEM grid, Ni 2+ salt was added onto the self-assembly for 5 min. Then, NaBH4 was used to reduce Ni 2+ salt. It is noteworthy that in this experiment, nickel was the only element allowing the staining of the self-assembly in STEM. As expected, metallization of the GbLFY-SAM honeycomb with Ni 2+ salt reduced occurs inside the pores demonstrating that the N-terminal extension of GbLFY-SAM is inside the lattice pores (Fig. 6a) as previously modeled. To fully characterize the metallized honeycomb, we also acquired an Electron Dispersive X-ray Spectroscopy spectrum of the self-assembly corroborating the presence of nickel (Fig. 6a). Thus, despite the reduction of Nickel by NaBH4, the honeycomb structure remains intact and FFT calculations of the STEM images showed lattice parameters similar to those measured in STEM images of negative stained GbLFY-SAM self-assembly. Then, we did these experiments in presence of 1 mM EDTA before the salt reduction to check the specific effect of Ni 2+ . We showed that the chelation of Ni 2+ by EDTA can block the metallization of the self-assembly which appears unstained in STEM with no characteristic nickel X-ray in the related EDX spectrum (Fig. S7).

From these results, it appears that a specific metallization of the pores of the honeycomb lattice is achievable using the advantageous affinity of a metal for amino acid located inside the pores. To further investigate this point, we metallized the honeycomb structure with another metal showing different binding properties. Affinity of gold salt for amino acid have been studied previously [START_REF] Tan | Uncovering the Design Rules for Peptide Synthesis of Metal Nanoparticles[END_REF] showing rather broader interactions and different affinities than Ni 2+ for histidine.

As expected, the in situ reduction of gold salt in the self-assembly does not lead to a specific staining of the inner part of the pores but to the coloration of their outlines (Fig. 6b). This labeling is explained by a higher number of amino acids able to interact with gold in the core oligomerization domain compared to the N-and C-terminal extensions. As in the nickel experiments, gold was the only element allowing the staining of the self-assembly in STEM.

Presence of gold on the self-assembly was confirm by acquiring an EDX spectrum (Fig. 6b). These results demonstrated that the honeycomb structure can be metallized either inside or outside of the pore depending on the specific affinity of the amino acid present in the pore for metal salt.

Honeycomb lattice engineering for futures nanotechnological applications.

Metallization of the GbLFY-SAM can be specifically driven inside the honeycomb pores thanks to the affinity of the N-terminal extension histidines for a metal such as Ni 2+ . Besides, it has been shown that the C-terminal extension of GbLFY-SAM is also facing inside the pore [START_REF] Chiari | LEAFY Protein Crystals with a Honeycomb Structure as Platform for Selective Preparation of Outstanding Stable Bio-Hybrid Materials[END_REF] . We investigated if it was possible to modify the amino acids of the C-terminal extension without affecting the oligomerization capacity of the monomer and the honeycomb structure. We aimed to explore a modification of GbLFY-SAM involving a complete deletion of its C-terminal extension.

This modification had an impact on the solubility and stability of the GbLFY-SAM in solution.

The deletion of the C-terminal extension makes GbLFY-SAM oligomerization faster and leads to protein precipitation in the classic purification protocol. Nevertheless, after optimization of the purification and surface crystallization protocols, we obtained honeycomb self-assemblies identical to GbLFY-SAM (Fig. S4b).

We also made others C-terminal constructions to emphasize the versatility of the GbLFY-SAM self-assembly. In a first construction, we mutated a lysine into a cysteine residue (GbLFY-SAM K110C) (Fig. S4c) in order to insert a unique cysteine residue for specific grafting. In a second construction, we deleted all the C-terminal part and replace it by the amino acid sequence GGSGGSCHCHCHC (Fig. S4d) containing a linker and three histidine and cysteine residues for specific grafting. For these two mutants, the modification has no impact either on the honeycomb formation or on the lattice parameters.

Thus, both the N-and C-terminal extension of GbLFY-SAM can be modified without affecting the honeycomb self-assembly ability. Engineering these extensions by inserting amino acid residues or sequences capable of reacting with specific organic, inorganic or metallic compounds would make this honeycomb architecture a modular grafting platform. Moreover, this platform could accommodate a large number of grafting sites. Indeed, a surface of 1 µm 2 allows to obtain approximately 12 000 pores. Having determined an average pore height per AFM of 31 nm corresponding to a stacking of 40 monomers, we can estimate a capacity of at least 480 000 specific grafting sites per µm 2 (based on one grafting per monomer) (Fig. S8). Selective grafting on the protein can be easily obtained and many amino acid residues can be used [START_REF] Taylor | A Protein Functionalization Platform Based on Selective Reactions at Methionine Residues[END_REF][START_REF] Gunnoo | Chemical Protein Modification through Cysteine[END_REF] as demonstrated with the grafting of a ruthenium-based complex onto modified residues of the pores of 3D-GbLFY-SAM crystals in solution [START_REF] Chiari | LEAFY Protein Crystals with a Honeycomb Structure as Platform for Selective Preparation of Outstanding Stable Bio-Hybrid Materials[END_REF] . The pores of the honeycomb could hold specific grafting of catalysts and chromophores at different positions. Hence, the GbLFY-SAM self-assembly on surface could be easily used as a biocatalysis platform.

In addition, great efforts have been made to produce nanostructures with a sub 10 nm-pitch for microelectronics applications [START_REF] Jung | Recent Progress in Simple and Cost-Effective Top-Down Lithography for ≈10 Nm Scale Nanopatterns: From Edge Lithography to Secondary Sputtering Lithography[END_REF] . Indeed, the production of masks for lithography below 10 nm remains difficult and expensive [START_REF] Hasan | Promising Lithography Techniques for Next-Generation Logic Devices[END_REF][START_REF] Liddle | Nanomanufacturing: A Perspective[END_REF] . Among all the lithographic techniques, block copolymers remain one of the main technologies currently studied to overcome this challenge [START_REF] Bates | Block Copolymer Lithography[END_REF] . The use of the GbLFY-SAM honeycomb as a hard mask (Fig. S8) represents a promising opportunity that could compete with the technology of block copolymers [START_REF] Park | Macroscopic 10-Terabit-per-Square-Inch Arrays from Block Copolymers with Lateral Order[END_REF][START_REF] Li | Nanoporous Films with Sub-10 Nm in Pore Size from Acid-Cleavable Block Copolymers[END_REF] . Combining photo-etching or chemical etching (e.g. hydrofluoric acid) with a modified GbLFY-SAM with short N-and C-terminal extension making the honeycomb pore emptier could lead to an etching of sub 10 nm-pitch motif.

Nanoelectronics is not the only field in which the 5 nm modifiable pores of GbLFY-SAM could be useful. Honeycomb pores could also be exploited to build nanopillars by grafting and stacking metallic nanoparticles or quantum dots for use in photovoltaic devices [START_REF] Kapadia | Nanopillar Photovoltaics: Materials, Processes, and Devices[END_REF][START_REF] Saadi | Recent Developments and Applications of Nanocomposites in Solar Cells: A Review[END_REF] . In addition, the graftable pores of GbLFY-SAM, which are almost 10 times smaller in diameter than those currently being tested, could offer an interesting opportunity to increase photovoltaic yield [START_REF] Zhou | Optimization of the Optical Properties of Nanostructured Silicon Surfaces for Solar Cell Applications[END_REF] .

Closely and evenly spaced metallic nanopillars would also create a dense network that could also improve the signal of biosensor devices for the detection of lower molecular concentrations [START_REF] Oh | Engineering Hot Spots on Plasmonic Nanopillar Arrays for SERS: A Review[END_REF][START_REF] Lee | Dual Sensing Arrays for Surface Plasmon Resonance (SPR) and Surface-Enhanced Raman Scattering (SERS) Based on Nanowire/Nanorod Hybrid Nanostructures[END_REF][START_REF] Wang | Nanostructured Gold Films for SERS by Block Copolymer-Templated Galvanic Displacement Reactions[END_REF] .

Whether these 8-nm pitch arrays of 5 nm nanopillars increase efficiency or provide new properties -due specifically to quantum effects -is an open question.

Conclusion

In summary, we present here the natural self-assembly capabilities of a Ginkgo biloba protein oligomerization domain, building a robust honeycomb architecture with a pitch of 8 nm and 3D pores of 5 nm diameter. Our results show that the orientation of this honeycomb can be controlled on different surfaces. Furthermore, we have shown that the N-and C-terminal extensions can be modified without altering the self-assembly and that the amino acids present on these extensions can be used to specifically metallize the inside or the outside of the pores. Engineering the extensions of the oligomerization domain by inserting amino acid sequences capable of reacting with specific organic, inorganic or metallic compounds will make this honeycomb architecture a modular grafting platform for an extensive range of applications covering biocatalysis, biosensing, or photovoltaic. Moreover, the pores formed by 40 stacked oligomerization domains lead to a huge number of grafting sites above 500 000 per µm [START_REF] Sun | Nanostructures Based on Protein Self-Assembly: From Hierarchical Construction to Bioinspired Materials[END_REF] . In addition to the specific grafting possibilities, the nanometric dimensions of the honeycomb architecture are a promising opportunity to produce masks for lithography processes that could compete with current technologies such as block copolymers.

In conclusion, the possibility of modifying the pore composition for etching or specifically grafting a large number of molecules makes this Ginkgo self-assembled honeycomb a new and unique green nanomaterial pioneering major biotechnological breakthroughs previously impossible with current nanomaterials.

ASSOCIATED CONTENT

Supporting Information

Overview on protein self-assembly, amino acid sequence of N-and C-terminal extensions, superposition of GbLFY-SAM crystallographic structure on STEM image, STEM image with

Figure 1 .

 1 Figure 1. Polymeric honeycomb structure of the SAM oligomerization domain of the GbLFY protein mediated through head tail interactions. (a) GbLFY-SAM asymmetric unit (PDB: 4UDE) showing the interaction of two monomers. (b) Each helical polymer interacts with six neighbors leading to the honeycomb architecture. Each helical polymer has a 5-nm pore and the distance between two neighboring pores is 8 nm. (c) and (d) detail of one helical polymer. (e) and (f) interactions between two helical polymers. In the schemes (b-f), only 22 monomers are shown for each helix (instead of 40 determined by AFM).

Figure 2 .

 2 Figure 2. Orientation of the GbLFY-SAM honeycomb depends on the N-terminal extension. (a) Self-assembly of GbLFY-SAM domain without N-terminal extension. (b) Self-assembly of GbLFY-SAM domain with a N-terminal extension. Each black dot corresponds to a pore of the self-assembly. The Fast Fourier Transform shown in inset indicates that GbLFY-SAM with the Nterminal extension forms an architecture (8 nm pitch and 5 nm lumen diameter) similar to the ones measured in the X-Ray crystal structure of the GbLFY-SAM crystallized in solution 33 without the N-terminal extension. The self-assembly on surfaces uses the same mechanism as 3D crystals. A double mutant (one mutation in the head face and another one in the tail face) is not able to create the honeycomb structure. The scale bars are 30 nm on both images.

3. 2

 2 Characterization of the honeycomb growth on carbon and silicon surfaces. Selfassembly of molecules onto surfaces often lead to miss folding or defect in the lattice 37 . Here, Fast Fourier Transform (FFT) calculations, performed on STEM images of the GbLFY-SAM honeycomb array, reveals lattice parameters (a = b = 8.2 (+/-0.1) nm et γ = 120 (+/-1) °) identical to those found on the crystal by X-ray diffraction[START_REF] Sayou | A SAM Oligomerization Domain Shapes the Genomic Binding Landscape of the LEAFY Transcription Factor[END_REF] (Fig.2and 3, Fig S1, S2 and S4).

Figure 3 .

 3 Figure 3. The N-terminal extension of GbLFY-SAM can be reduced to 6 Histidine residues. (a) STEM image of the GbLFY-SAM with N-terminal extension. (b) STEM image of the GbLFY-SAM with a N-terminal extension corresponding to the first 9 amino acids containing the first six histidine residues. Similar organization was obtained with a shorter N-terminal extension containing only the N-terminal methionine and six histidine residues. (c) Height of the selfassembly shown in (a) using AFM. (d) Height of the self-assembly shown in (b) measured using AFM. Fourier transform calculation show that the GbLFY-SAM with its N-terminal extension and the mutant with a shorter N-terminal extension disclose similar lattice parameters. AFM measurements indicate that both self-assemblies have an average height of 31 nm. The scale bars are 200 nm for all four images.

Figure 4 .

 4 Figure 4. Surfaces covered by GbLFY-SAM with its N-terminal extension after carbon grid treatment using glow discharge. (a) The use of a threshold in ImageJ to detect self-assemblies on STEM images reveals a cover up to 54 % of the carbon surface. This image is a 20 x 20 µm 2 area.

Figure 5 .

 5 Figure 5. Self-assembly of GbLFY-SAM with its N-terminal extension on silicon surface. (a) STEM image of the self-assembly on a silicon grid. Fourier transform image shown in inset indicates dimensions similar to those observed on carbon surface. (b) Height of a GbLFY-SAM self-assembly on a silicon wafer measured using AFM. The scale bars are 100 nm on both images.

Figure 6 .

 6 Figure 6. Specific metallization of the self-assembly with Ni 2+ salt (a-c) or Au 3+ salt (d-f). In these images, the contrast is only provided by the reduced metal. (a) STEM image in dark field mode of a self-assembly with Ni 2+ salt reduced by NaBH4 inside the pore. (b) Zoom of the STEM image. Scale bar: 30 nm. (c) EDX spectrum of the self-assembly emphasizing the presence of nickel. (d) STEM image in dark field mode of a self-assembly with Au 3+ salt reduced by NaBH4 outside of the pore. (e) Zoom of the STEM image. Scale bar: 30 nm. (f) EDX spectrum of the selfassembly emphasizing the presence of gold. The inset show a zoom of the pics from 2 to 3 keV to illustrate the overlapping between Au-M and the Mo-Lα. The scale bars are 100 nm on both images.

ACKNOWLEDGMENT

We thank Lucile Chiari, Olivier Hamelin, Jean-Luc Putaux, the Nanoprot 3D and A3DN work group (Didier Gasparutto, Xavier Baillin, Robert Baptiste, Caroline Fontenoye, Guillaume Nonglaton, Pascal Mailley), Patrice Rannou and Luc Federzoni for helpful discussions.

Funding Sources

This work was supported by a CEA program Nanoprot 3D to F.P. and R.T. with a post-doctoral fellowship to P.H.E., a PhD fellowship from the CEA [E.J.], a DRF impulsion grant [to Olivier Hamelin, P.H.E and R.D.], the LabEx Grenoble Alliance for Cell and Structural Biology [ANR-10-LABX-0049 to F.P. and R.D.] and the LabEx Arcane and CBH-EUR-GS [ANR-17-EURE-0003 to P.H.E].

EDTA, localization of the N-and C-terminal extension in the honeycomb, STEM images of engineered N-and C-terminal, AFM scan of multiple honeycomb, STEM images on hydrophilic and hydrophobic surfaces, desorption of Nickel salt with EDTA, self-assembly as a platform for specific grafting of a huge number of ligands.

AUTHOR INFORMATION

Corresponding Authors

Renaud Dumas -Univ. Grenoble Alpes, CNRS, CEA, INRAE, IRIG-DBSCI-LPCV, 38000 Grenoble, France. Email: renaud.dumas@cea.fr Pierre-Henri Elchinger -Univ. Grenoble Alpes, CEA, CNRS, IRIG-DIESE-SYMMES, 38000 Grenoble, France. Email: pierre-henri.elchinger@cea.fr