Eric Benoit

Gilles Mauris

Laurent Foulloy

Fuzzy sensor aggregation: application to comfort measurement

This paper focuses on the acquisition of abstract information, i.e. information which are not analytically related to conventional physical quantities as for example the comfort. In these complex cases, we propose to use fuzzy sensors which compute and report linguistic assessment of numerical acquired values. Two methods are proposed to realize the aggregation from basic measurements. The first one performs the combination of the relevant features by means of a rule based description of the relations between them. With the second one, the aggregation is realised through an interpolation mechanism that make a fuzzy partition of the numeric multi-dimensional space of the basic features.

Introduction

When attempting to qualify complex phenomena, especially those related to human perception, one is often led to use words of the natural language [START_REF] Finkelstein | Representation by symbol systems as an extension of the concept of measurement[END_REF], [START_REF] Zingales | The role of artificial intelligence in measurement[END_REF]. This linguistic description is less precise than the numerical one obtained through usual sensors. This description is also subjective because it depends on the observer.

Several advantages may be found in such an approach. In particular, the linguistic description is easily understood by human beings even if the concepts are abstract or if the context is changing. For example, everybody can figure out what is the concept of danger and can qualify it even if the context is unknown. Everybody is also able to classify colours without having an explicit knowledge of the origin of colours and of the perception mechanisms. Even for simple problems such as temperature measurements, human can easily integrate a priory knowledge. Imagine two people, one living close to the north pole and the other living close to the equator, speaking of a comfortable outdoor temperature in their respective countries. they are both able to abstract the concept of comfort even if the related temperatures are not the same. The representation of measurements by means of linguistic scales provides abstract information which can be integrated in decision, diagnosis or control especially with systems using symbolic coding.

Integrating capabilities of symbolic representation directly at the sensor level has led to the concept of symbolic sensor [START_REF] Benoit | Symbolic sensors : one solution to the numerical-symbolic interface[END_REF], [START_REF] Benoit | Symbolic sensors[END_REF]. The numeric-symbolic conversion has been studied in particular when the symbols of the linguistic scale have their meaning represented by means of fuzzy subsets. We have proposed to call these new sensors fuzzy symbolic sensors or simply fuzzy sensors [START_REF] Foulloy | Applications of fuzzy sensors[END_REF], [START_REF] Mauris | Fuzzy symbolic sensors : From concept to applications[END_REF].

Two ways could be considered in order to obtain a linguistic description of properties or attributes from the features linked to them: defining the linguistic description by means of a rule formalism that involves a linguistic description of every feature returned by the corresponding fuzzy sensor or building the linguistic description directly on the numerical product space of the independent features by means of an interpolation method. After having recalled our formalism of the numeric-linguistic conversion, these two strategies will be investigated in this paper and then applied to a linguistic description of comfort.

The numeric to linguistic conversion

To perform a symbolic measurement, it is necessary to clearly specify the relation between symbols and numbers. Let E be the set of all possible measurements. Let L(E) be the set of symbols associated to this universe. Denote P(E) the set of subsets of E. An injective mapping, called a meaning and denoted τ : L(E) ¡ P(E) associates any symbol with a subset of measurements. Injectivity means that two symbols with the same meaning should be considered as identical. Symbolic measurement is now obtained from a mapping, called a description and denoted ι: E ¡ P(L(E)). It associates any measurement with a subset of the symbolic set L(E).

There is a fundamental relation between description and meaning. If a symbol belongs to the description of a measurement, then the measurement belongs to the meaning of the symbol, i.e: The descriptions of measurements comes directly from the definitions. For example:

L 1 ∈ ι(x) ⇔ x ∈ τ(L 1 )
ι(22 o C) = {mild} ι(14 o C) = {cold, cool} ι(18 o C) = ∅
In the general case, the description of a measure can contain any number of symbols. In a natural language, a measure is usually described by only one symbol. To apply this condition we impose the set of the meanings to be a partition of the measurement set (fig. 3).

The fuzzy numeric-symbolic conversion

The previous approach, based on subsets, obviously leads to sharp transitions in the sensor response. Fuzzy subset theory, developed by Zadeh, provides a nice solution to this problem [START_REF] Zadeh | Fuzzy sets[END_REF], [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning[END_REF]. The extension of the characteristic function of a crisp subset (values in {0,1}) to the membership function of a fuzzy subset (values in [0,1]) can be used to model gradual transitions between symbols.

The fuzzy numeric to symbolic interface depends on the extension to the fuzzy case of the definitions of the previous section. So fuzzy meaning and fuzzy description have to be defined [START_REF] Zadeh | Quantitative fuzzy semantics[END_REF]. The fuzzy meaning is a mapping from the symbolic set L to the set of the fuzzy subsets of measurements. The fuzzy meaning of a symbol L is characterized by its membership function denoted µ τ(L) (x).

In a same manner, a fuzzy description can be defined as a mapping from the measurement set into the set of the fuzzy subsets of symbols, so the fuzzy description is characterized by its membership function denoted µ ι(x) (L).

The relation between the membership functions of a fuzzy description and the corresponding fuzzy meaning comes directly from the fundamental relation between the meaning and the description, i.e:

µ ι(x) (L) = µ τ(L) (x)
In order to conserve a link between the sensor description and the human feeling, we impose that the set of symbols meaning is a fuzzy partition of the measurement set in the sense of Bezdek [START_REF] Bezdek | Pattern recognition with fuzzy objective functions[END_REF]. Then, the triplet <E,L(E), ι> is a fuzzy nominal scale [START_REF] Mauris | Fuzzy symbolic sensors : From concept to applications[END_REF].

Σ a ∈L(T) µ ι(t) ( a) = 1 L(E) E ι L 3 • x • • • L 2 L 1 τ t ( o C) 0 1 0 3 0 4 0 τ(cold) τ(cool) τ(mild) τ(warm) τ(hot)

Rule based symbolic aggregation

With multi-dimensional measurements, the definition of symbolic meanings leads to a combination between basic measurements. For example, the description of comfort needs the knowledge of the relative humidity and of the temperature. Then a linguistic definition of the comfort can be modelized by a set of rules. Let us describe by this example the mechanism used to define the semantic of comfort.

The temperature and humidity measurements take their respective values in the sets denoted T and H. The temperature is described by symbols in the set L(T) = {cold, cool, mild, warm, hot}. The humidity is described by symbols in the set L(H) = {very_low, low, medium, high}. It also assumed that the meanings of the symbols generate a partition of the respective numerical sets.

The problem is now to aggregate both measurements to obtain information about the feeling of comfort. Such a feeling could be symbolically defined as follows.

atmosphere is comfortable when : the temperature is mild and the humidity is medium atmosphere is acceptable when : the temperature is cool or mild or warm and the humidity is low or the temperature is cool or warm and the humidity is medium atmosphere is uncomfortable when : atmosphere is not (comfortable or acceptable)

One solution is to consider that two symbols connected by the and operator are in fact one symbol whose meaning is defined on the cartesian product of the numerical sets. For example, "the temperature is mild and humidity is medium" should be no more considered as the two expressions connected by the operator and, but as one expression such as "the temperature_humidity is mild_medium". The new symbolic variable temperature_humidity takes its value in a new set of symbols, for example denoted L(TH). The symbol mild_medium belongs to L(TH), and its meaning is defined on the cartesian product TxH. Then the meaning of comfortable is defined by

τ(comfortable) = τ(mild_medium) = τ(mild) x τ(medium)
The meanings of the or and the not operators are respectively defined by the union operator and by the negation operator.

Fig. 6 : Graph representing the meanings of comfortable, acceptable and uncomfortable

Rule based fuzzy sensor aggregation

Let t and h be the temperature and humidity measurement. The fuzzy description of the couple (t, h) can be expressed in terms of the fuzzy descriptions of t and h by means of a triangular norm T.

µ ι(t, h) ( a_b) = µ ι(t) ( a) T µ ι(h) ( b)
As a fuzzy partitioning, in the sense of Bezdek, was chosen for the temperature and the humidity, we have :

Σ a ∈L(T) µ ι(t) ( a) = 1 and Σ b ∈L(H) µ ι(h) ( b) = 1
Therefore, if we want to obtain a fuzzy partition for the new symbolic set L(TH), the triangular norm has to be distributive with respect to the addition (for example the product).

Σ a_b ∈L(HT) µ ι(t, h) ( a_b) = Σ a ∈L(T) Σ b ∈L(H) µ ι(t) ( a) T µ ι(h) ( b) = Σ a ∈L(T) µ ι(t) ( a) T Σ b ∈L(H) µ ι(h) ( b) =Σ a ∈L(T) µ ι(t) ( a) T 1 =Σ a ∈L(T) µ ι(t) ( a) = 1
Now, we have to define the fuzzy meaning of the or operator. Let E be a set of measurements and L(E) its associated set of symbols. Let L 1 and L 2 be two symbols of L(E). Let ⊥ be the triangular conorm which defines the fuzzy meaning of or:

µ τ(L1_or_L2) (x) = µ τ(L1) (x) ⊥ µ τ(L2) (x) = µ ι(x) (L 1 ) ⊥ µ ι(x) (L 2 ) Denote L * (E) = L(E) -{L 1 , L 2 }, and L + (E) = L(E) ∪ {L 1 _or_L 2 }
The triangular conorm must be chosen such that the meaning of the symbols of L + (E) forms also a fuzzy partition:

Σ L ∈L(E) µ ι(x) (L) = Σ L ∈L * (E) µ ι(x) (L) + µ ι(x) (L1) + µ ι(x) (L2) =1 Σ L ∈L + (E) µ ι(x) (L) = Σ L ∈L * (E) µ ι(x) (L) + µ ι(x) (L1_or_L2) =1 Therfore we have µ ι(x) (L 1 ) ⊥ µ ι(x) (L 2 ) = µ ι(x) (L1) + µ ι(x) (L2) ≤ 1
Thus, the triangular conorm should verify x ⊥ y = x+y if x+y ≤ 1 for example : x ⊥ y = min(x+y, 1). This is the triangular conorm chosen to define the or operator.

The meaning of the negation operator not is defined by: µ The fuzzy sensor is now able to describe the temperature and humidity for the symbol comfortable by a grade of membership which τ(very_low) τ(low) τ(medium) τ(high)

τ(not L) (x) = 1 -µ τ(L) (x)
0 2 0 40 6 0 80 100 1 0 h(%) µ H τ(medium_or_low) h (%) t ( o C) µ τ(comfortable) µ τ(acceptable) h (%) t ( o C)
qualify the comfort feeling. Three results are given below for three different values of the temperature in °C and humidity in %.

µ ι(23, 50 ) ( comfortable) = 1 µ ι(25, 60 ) ( comfortable) = 0.5 µ ι(5, 80 ) ( comfortable) = 0

Multi-component numeric-symbolic conversion

When a sensor uses several transducers, the measure is a vector of numerical values, and the measurement set is a multi-dimensional volume. An alternative way to the preceding method is to define directly the meaning of each symbol on this multi-dimensional volume. Furthermore the set of meanings has to be a fuzzy partition of the measurement set.

In this section, we consider an initial knowledge about the measurements. This knowledge is materialized by the meaning of symbols on a small subset V of the measurement set. Then the measurement set is partitioned in nsimplexes with the Delaunay triangulation method. A n-simplex in a n-dimensional space is a polyhedra with n+1 vertices. For example, a 2simplex is a triangle and a 3-simplex is a tetrahedron. The points used to perform the triangulation are the elements of the subset V.

The membership function of the meaning of each symbol is defined by piece on the nsimplexes. A multi-linear interpolation is used to define this function on each n-simplex. We suppose the restriction on a n-simplex of the membership function of the meaning of a symbol s is :

µ τ E (s) (v)= µ τ E (s) (x 1 , ..., x n )= a 1 x 1 + ... +a n x n +a n+1
The value of this function is known for the n+1 vertices of the n-simplex. Indeed, the vertices are elements of the subset V. So the n+1 factors a i can be computed by resolving the system of n+1 equations with n+1 unknowns.

Where v i is the i th vertex of the n-simplex, and x i j is its j th component. This process is performed on each n-simplex and for each symbol. Then we have a fuzzy nominal scale defined on E. This scale is an extension of the fuzzy nominal scale on V.

With this method, the knowledge needed to configure the sensor is very compact. It can be acquired during a learning phase by a communication with a system called teacher which can be a man or an expert system. During the learning phase, the teacher and the sensor analyse the same phenomenon, and the teacher gives its description to the sensor. The sensor increases its knowledge with its measure associated to the teacher description. Then it owns a crips meaning of the symbols on the subset of the measurement set. The sensor can now build the fuzzy nominal scale on the measurement set. This technique was succefully used to implement a fuzzy color sensor [START_REF] Benoit | Capteurs symboliques et capteurs flous: un nouveau pas vers l'intelligence[END_REF]. If we consider the example of comfort measurement, the meaning of comfortable, acceptable, uncomfortable under a subset V of TxH are defined as follow: τ V (comfortable) = {(20,50)} τ V (acceptable) = {(26,35), (16,35), (24,65), (18,65), (28,50), (15,50), (20,25), (20,80)} τ V (uncomfortable) = {(0,0), (20,0), (40,0), (0,100), (20,100), (40,100), (0,50), (40,50)}

The following figure shows the meaning of comfortable and of acceptable under TxH.

A M 1 -B = B µ τ V s ( ) v 1 ( ) µ τ V s ( ) v 2 ( ) ... µ τ V s ( ) v n 1 + ( ) = M x 1 1 ... x 1 n 1 ... ... ... … x n 1 1 + ... x n 1 + n 1 = A a 1 … a n

Conclusion

This paper has been concerned by the aggregation of complementary information using fuzzy sensors which compute and report linguistic assessments of numerical acquired values. When the aggregation cannot be represented by a numerical model, we have proposed to use a linguistic model of the aggregation or to use a set of characteristic examples associated with a fuzzy interpolative method. The rule based method is particularly efficient when the basic features are of a linguistic type as it is often the case at the reasoning level. In this approach, all the numerical operations are made at the low level (i.e. inside the sensor or the actuator). So the central unit has a reduced computation load, because it works on compact information. With the interpolative method, the computation load is higher because it works on a multidimensional numerical space. But it provides a richer structure for the output information. The first proposed method seems more adapted to complex systems that handle a lot of information and in which the perception is only one of the different considered tasks. The second proposed method seems more adapted to cases where the perception is the principal objective.

Fig. 1 :Fig. 2 :

 12 Fig. 1 : Relation between the description and the meaning Here is an example for a sensor that returns the ambient temperature. Let the measurement set be T = [0 o C, 40 o C] and the symbolic set be L(T) = {cold, cool, mild, warm, hot} The following figure shows an example of symbol meanings:

Fig. 3 :

 3 Fig. 3 : Meanings satisfying the partition

Fig. 4 :Fig. 5 :

 45 Fig. 4 : Meanings of the temperature lexical set

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7 : Meanings of the humidity lexical set, and meaning of a compound symbol

1 +=Fig. 10 :Fig. 11 :

 11011 Fig. 10 : Meaning of comfortable


Ahmad Mobasher Amini

David Dureisseix

Patrice Cartraud

Multi-scale domain decomposition method for large scale structural analysis with a zooming technique : Application to plate assembly

Keywords: Domain Decomposition Method, FETI-DP, Multi-scale, homogenization, structural heterogeneities, Mortar method

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

INTRODUCTION

The structural design of complex structures now often relies on finite element simulations. In the case of a large structure with small-size structural details, the finite element analysis is a difficult problem. To obtain a solution for the whole structure as well as a good accuracy near the structural details, a model with a fine mesh is required; this approach leads to a huge global finite element model with a large number of unknowns that is difficult to solve.

In the last years, many researches have been conducted to develop efficient numerical methods those are capable of solving large-scale problems. The direct sparse (out-of-core) solvers are robust, efficient and are already employed in several commercial finite element codes. These solvers need large memory resources and have a limited parallel scalability. The classical iterative solvers (Jacobi, Gauss Seidel, Conjugate Gradient, etc. [START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF]) are excellent from the memory usage point of view and can be easily parallelized [START_REF] Farhat | Implicit parallel processing in structural mechanics[END_REF]. But, their efficiency depends on the type of problem considered, and are not the most suited methods for mechanical engineering problems that are often ill-conditioned. Multigrid methods, see [START_REF] Parsons | The multigrid method in solid mechanics: Part I, algorithm description and behaviour[END_REF][START_REF] Parsons | The multigrid method in solid mechanics: Part II, practical applications[END_REF], take advantage of using different description levels within an iterative approach (sometimes, coarse levels can be built automatically, as in smoothed aggregation methods [START_REF] Vanek | Convergence of algebraic multigrid based on smoothed aggregation[END_REF]). Usually the coarsening ratio between two successive levels is somehow small; this leads to the use of many different levels to bridge the gap between microscopic and macroscopic scales.

An alternative choice is the domain decomposition methods (DDM), which combine advantages of both direct and iterative solvers. The main idea is based on the splitting of a large-scale domain into several subdomains with either overlapping or non-overlapping interfaces. For domain decomposition methods with non-overlapping interfaces, three approaches have been intensively studied. Based on parameters chosen on the interfaces, to enforce the continuity between the neighboring subdomains, these approaches are named as the primal, dual and mixed domain decomposition method, respectively [START_REF] Gosselet | Non-overlapping domain decomposition methods in structural mechanics[END_REF]. The primal method chooses displacements as interface unknowns (Balancing Domain Decomposition method, or BDD, [START_REF] Mandel | Balancing domain decomposition[END_REF]). The dual method chooses interface forces as unknowns (classically dealt with as Lagrange multipliers in the FETI, [START_REF] Ch | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF] family of algorithms). In the mixed method where the interfaces play a major role, both displacements and forces are unknowns (LATIN method, [START_REF] Champaney | Large scale applications on parallel computers of a mixed domain decomposition method[END_REF], and in a less extend [START_REF] Park | An algebraically partitioned FETI method for parallel structural analysis: Algorithm description[END_REF][START_REF] Rixen | Theoretical comparison of the FETI and algebraically partitioned FETI methods, and performance comparisons with a direct sparse solver[END_REF]).

Dual-primal FETI (FETI-DP) [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF][START_REF] Ch | FETI-DP: A dual-primal unified FETI method -Part I: A faster alternative to the two-level FETI method[END_REF] is the latest generation of the FETI methods that preserves the numerical and parallel scalability of the original FETI and FETI-2 [START_REF] Ch | The two-level FETI method -Part II: Extension to shell problems, parallel implementation and performance results[END_REF] methods for second and fourth-order problems. This method also uses Lagrange multipliers to satisfy the continuity constraints on the interfaces. Indeed, instead of introducing the coefficients for rigid body modes in the original FETI and the second set of Lagrange multipliers in FETI-2, it chooses some 'corner node' degrees of freedom as basic unknowns so that each subdomain is non-singular [START_REF] Lesoinne | A FETI-DP corner selection algorithm for three-dimensional problems[END_REF]. The coarse problem of FETI-DP, which is essential for reaching scalability properties, is also sparser than the coarse problem of the method FETI or FETI-2. These features make FETI-DP method much more robust and applicable for implementation than its previous versions. The numerical experiments show that it also delivers better computational performance for most cases [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF][START_REF] Ch | FETI-DP: A dual-primal unified FETI method -Part I: A faster alternative to the two-level FETI method[END_REF].

In the domain decomposition method based iterative solvers, a direct sparse solver is used as the local solver in each subdomain. During this step, local forward and backward substitutions consume somehow a large percentage of the total CPU time. The total number of local resolutions depends almost linearly on the number of subdomains when the Dirichlet optimal preconditioner is used.

Therefore, the computational cost can be optimized by reducing the time spent in the local resolution step. In this article, we propose a method using the macro (homogenized) description of the subdomains which are not directly in the zones of interest. This algorithm requires a preliminary local homogenization step on the fine mesh of the subdomains, and removes unnecessary local factorizations on every subdomain in the coarse zones. Additionally, this algorithm reduces the size of the interface condensed problem and removes the unnecessary forward and backward substitution steps in the coarse zone, to save computational cost. Nevertheless, an overhead due to the numerical homogenization step is introduced by this strategy. This overhead is problem dependent, but in general, for large scale problems, it is acceptable and does not prevent savings of overall CPU time costs.

In the framework of domain decomposition method with incompatible meshes, several approaches have already been proposed in the literature, see e.g. [START_REF] Ch | Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: A numerically scalable algorithm[END_REF], [START_REF] Bavestrello | Incorporation of linear multipoint constraints in domaindecomposition-based iterative solvers -Part II: Blending FETI-DP and mortar methods and assembling floating substructures[END_REF], [START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF] and references herein. However, in these works, the point was the coupling of different discretizations across subdomain interface. In this article, starting with a domain decomposition with matching interface, non-matching interfaces arise from the homogenization of subdomains. This method is called FETI-DP micro-macro throughout this article, and is developed in an elasticity framework.

For this purpose, this article is organized as follows. In Section 2, the general framework of the FETI-DP method is briefly recalled. In Section 3, the main steps of the FETI-DP micro-macro approach are presented. Section 4 introduces the essentials of two connection methods (collocation and Mortar) and Section 5 concerns the determination of the macro (homogenized) stiffness of the macro zone. Section 6 presents the new interface problem of the FETI-DP micro-macro approach with two discretization scales for the subdomains. In Section 7, the method is validated with patch tests, and several examples and results are discussed. Finally, Section 8 concludes the article and suggests several future works.

DOMAIN DECOMPOSITION APPROACH

Selection of a domain decomposition method

Domain decomposition methods are both efficient and flexible tools for structural analysis [START_REF] Tallec | Domain decomposition methods in computational mechanics[END_REF][START_REF] Farhat | Implicit parallel processing in structural mechanics[END_REF]. When the size of the model increases, the iterative resolution methods, e.g. conjugate gradient (CG) method, can be used to solve the problem. They often allow a parallel treatment of the resolution phase. This article is not concerned with the parallelization of the resolution, but focus on the modularity of the methods, especially for coupling different subdomains that may have been modeled with different discretization levels.

The different versions of the FETI method (FETI-1 [START_REF] Ch | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF], FETI-2 [START_REF] Ch | The two-level FETI method -Part II: Extension to shell problems, parallel implementation and performance results[END_REF], FETI-DP [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF][START_REF] Ch | FETI-DP: A dual-primal unified FETI method -Part I: A faster alternative to the two-level FETI method[END_REF]) belongs to the family of the non-overlapping domain decomposition methods with Lagrange multipliers. These methods have been developed as iterative solvers for large-scale systems of equations in structural analysis, obtained by using the finite element method. Among all the domain decomposition methods that were presented by several authors, the FETI-DP method is chosen here as a basis of development for the following reasons. Usually with a multi-scale domain decomposition method, the reference problem is split into subdomains and the coarse space problem is numerically built from this fine scale description. The coarse problem can be discrete by nature, and is not always related to any finite element model.

Within a micro-macro approach, as detailed in 3, the coarse nodes of the FETI-DP method may be used to define a coarse subdomain, which constitutes a structural homogenization of the detailed subdomain. In that way, the coarse compatibility of displacements are ensured automatically between neighboring coarse subdomains.

Basic FETI-DP method

In this Section, the FETI-DP method, as presented in [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF][START_REF] Ch | FETI-DP: A dual-primal unified FETI method -Part I: A faster alternative to the two-level FETI method[END_REF] is briefly reviewed to keep this article selfcontained, and to define the notations that will be used hereafter.

Let us consider the domain Ω, partitioned into a set of N s non-overlapping subdomains Ω s . The subdomain-related nodes are classified into three groups: corner nodes (c), non-corner interface nodes (b) and the remaining internal nodes (i), as well as their related degrees of freedom, see Figure 1. The assembled stiffness matrix K s , the solution vector u s and the loading vector f s on each subdomain can be split as follows:

corner node (c) interface node (b) internal node (i) Ω 4 Ω 5 Ω 6 Ω 1 Ω 2 Ω 3
K s =   K s ii K s ib K s ic K s bi K s bb K s bc K s ci K s cb K s cc   , u s =   u s i u s b u s c   , f s =   f s i f s b f s c   (1)
Denoting with u c the global vector of corner degrees of freedom, the global vector of degrees of freedom, u, and the subdomain-related vector of non-corner degrees of freedom, u s r , are defined as:

u = u r u c =         u 1 r . . . u Ns r u c         , u s r = u s i u s b (2)
Using these notations, one can also split the subdomain stiffness matrix into:

K s = K s rr K s rc K s T rc K s cc (3)
Moreover, the continuity of the global solution is enforced at corner nodes. Two mapping matrices, B s r and L s c , are introduced to locate the corresponding degrees of freedom. Here, B s r are signed boolean matrices locating the non-corner degrees of freedom of the subdomain Ω s that belongs to the global interface, and L s c are localization matrices that give the corner degrees of freedom of each subdomain, from the global vector of corner degrees of freedom: u s c = L s c u c . Expressing the equilibrium equation for each subdomain in a global variational form leads to:

K s rr u s r + K s rc L s c u c + B s T r λ = f s r for s=1, ... , N s Ns s=1 L s T c K s T rc u s r + Ns s=1 L s T c K s cc L s c u c = Ns s=1 L s T c f s c = f c (4)
with Lagrange multipliers λ on the global interface, that traduce forces used to enforce the interface continuity condition:

Ns s=1 B s r u s r = 0 (5)
To guaranty the non-singularity of the K s rr , it is important that each subdomain contains at least three non colinear corner nodes, [START_REF] Lesoinne | A FETI-DP corner selection algorithm for three-dimensional problems[END_REF].

With the above equations and after some algebraic transformations, the following dual-primal problem is obtained (using Lagrange multipliers λ, and primal corner displacement u c , as unknowns):

F Irr F Irc F T Irc -K cc λ u c = d r -f c (6)
where

F Irr = Ns s=1 B s r K s -1 rr B s T r F Irc = Ns s=1 B s r K s -1 rr K s rc L s c d r = Ns s=1 B s r K s -1 rr f s r f c = f c - Ns s=1 L s T c K s T rc K s -1 rr f s r K cc = Ns s=1 L s T c K s cc L s c K cc = K cc - Ns s=1 (K s rc L s c ) T K s -1 rr (K s rc L s c ) (7)
By condensing u c on λ in the equation ( 6), the following symmetric positive definite dual interface problem is obtained:

F I λ = D r (8)
with:

F I = F Irr + F Irc K -1 cc F T Irc D r = d r -F Irc K -1 cc f c (9)
The interface problem (8) is usually not solved by a direct method. Since in this case, the matrix F I and the vector D r should be assembled and for large-scale problems, assembling the interface problem can require unaffordable storage and computational resources. Moreover, assembling and solving the interface problem in a direct way would be a bottleneck for the parallel performance of the method.

The FETI-DP method can be viewed as the transformation of the global problem Ku = f , into the interface problem [START_REF] Ch | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF], and a resolution with a iterative, e.g. conjugate gradient (CG), algorithm. This method does not need to explicitly assemble the interface operators [START_REF] Champaney | Large scale applications on parallel computers of a mixed domain decomposition method[END_REF]. The main cost lies in the factorization of the subdomain matrices K s rr , and the use of matrix-vector products. Each matrix-vector product can be efficiently performed using subdomain by subdomain sparse matrix-vector product and local forward and backward substitutions.

From an implementation point of view, solving the interface problem (8) leads to the following steps:

• step 1:

δ k = F Irr λ k = Ns s=1 B s r K s -1 rr B s T r λ k (10)
• step 2:

δ k ← δ k + F Irc K -1 cc F T Irc λ k (11)
Step 1 is the same as for the FETI method (local computation on each subdomain and assembling).

Step 2 can be presented in three sub-steps as follows:

• Step 2.1:

y k = F T Irc λ k = Ns s=1 L s T c K s T rc K s -1 rr B s T r λ k (12) • Step 2.2: Solve K cc x k = y k to get x k • Step 2.3:
z k = F Irc x k = Ns s=1 B s r K s -1 rr K s rc L s c x k (13)
Steps 2.1 and 2.3 correspond to local computations in each subdomain. The product K s -1 rr B s T r λ k is already performed in Step 1 and the product K s -1 rr K s rc is pre-computed in a preliminary step, when constructing K cc .

Step 2.2 is the coarse space resolution of the mutilevel domain decomposition. This problem is solved at each iteration, for which it propagates globally the information among all the subdomains. Such a global problem is mandatory to reach scalability [START_REF] Bramble | The construction of preconditioners for elliptic problems by substructuring, I[END_REF]. Note that for 3D problems arising for second-order partial differential equations, to recover scalability, an enrichment of the coarse space of FETI-DP method is required [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF]. The matrix K cc is sparse; its pattern is that of a stiffness matrix obtained by considering only the coarse super-elements defined on coarse nodes on each subdomain.

FETI-DP MICRO-MACRO DESCRIPTION

Depending on the problem and on the design process, if a global fine mesh is available, it can be splitted artificially into subdomains. In such a case, if a classical domain decomposition method is applied, each subdomain possesses its own fine mesh, which are compatible on the interface. When the local effects (stress concentration, discontinuity, etc.) are limited to few zones, this approach can be improved, since a quite large percentage of the CPU time is consumed by the local resolution on subdomains, i.e. computation of K s -1 rr q k where q k is a vector in equations ( 10), [START_REF] Ch | A scalable dual-primal domain decomposition method[END_REF] and [START_REF] Ch | FETI-DP: A dual-primal unified FETI method -Part I: A faster alternative to the two-level FETI method[END_REF]. The detailed analysis of the CPU time profiles depends on the number of subdomains, the size of the coarse problem, and the selection of the preconditioner. Some CPU time reports can be found in [START_REF] Ch | The two-level FETI method -Part II: Extension to shell problems, parallel implementation and performance results[END_REF] and [START_REF] Sun | A fast implementation of the FETI-DP method: FETI-DP-RBS-LNA and applications on large scale problems with localized non linearities[END_REF]. According to these references, 55 % to 65 % of the total CPU time is consumed by local resolutions. The reduction of these computations will therefore significantly improve the computational performance.

A multi-scale domain decomposition approach with different levels of discretization in subdomains is proposed herein: one considers the fine mesh only into the subdomains with local effects that the designer decided to check. The subdomains in zones of interest are therefore meshed finely (microscopic level) and subdomains located in the remaining areas of the structure are described with a coarse mesh only (macroscopic or homogenized level), see Figure 2. In order to develop a multi-scale domain decomposition method, one has to tackle three issues: (i) the choice of the micro and macro zones, (ii) the connection of fine and coarse meshes (incompatible mesh connection) on the subdomain interfaces and (iii) the determination of the macro stiffness.

It must be mentioned that in this work, the choice of the fine and coarse mesh is not automated, but rely on the expertise of the user. An important feature, not developed herein, is the possibility of switching the zoomed areas of interest on-the-fly during the analysis. Since the process is iterative, it won't need to be restarted from scratch, but would be designed in an adaptive framework. This aspect of the approach is currently under development. The issues (ii) and (iii) will be addressed in next Sections.

FINE AND COARSE MESH CONNECTION METHOD

The classical FETI-DP method is based on the perfect continuity of the displacement field on the interface, [START_REF] Vanek | Convergence of algebraic multigrid based on smoothed aggregation[END_REF]. When different discretizations in the subdomains are considered, the problem of incompatible connection of meshes appears at the micro-macro interface and the interface connections have to be reformulated, see Figure 2.

Let us consider two neighbouring subdomains Ω f and Ω c (fine and coarse, respectively) that are connected on a local interface Γ, see Figure 3. Along this interface, the coarse subdomain shares only two corner nodes, which are common to the finely meshed subdomain. fine mesh side (f) coarse mesh side (c) As a general rule, the mesh connection on the interface Γ, consists in satisfying the following continuity equation:

Ω f Ω c Γ } n f nodes
∀x ∈ Γ = Ω f ∩ Ω c , u f b (x) = u c b (x) (14)
where u f b (x) and u c b (x) are the continuous displacement fields on the fine and coarse sides of the interface Γ. On the fine side, the continuous displacement field, u f b (x), is defined from the underlying fine mesh and the shape functions of the corresponding elements. On the coarse side, the coarse element is built from an homogenization process, and is defined with its coarse nodes. In order to define a continuous displacement field on this coarse element, the shape functions corresponding to a classical finite element are used. In this work, since plate assemblies are considered, discrete Kirchhoff elements will be used, i.e. DKT or DKQ, see Appendix 1. With these shape functions, the displacement vectors u c r and u c b are denoted as:

u c r = C c u c c u c b = L c r C c u c c ( 15
)
where C c is the matrix of shape function values (interpolation matrix) on the coarse subdomain at the location of the fine mesh nodes, u c c is the displacement vector of subdomain Ω c at the coarse nodes and L c r is the localization matrix that restrict the degrees of freedom on the global incompatible interface, the corner nodes being excluded.

Several methods were presented in the literature to connect two incompatible meshes, such as collocation [START_REF] Rixen | Substructuring and dual methods in structural analysis[END_REF], Mortar [START_REF] Bernardi | A new nonconforming approach to domain decomposition: The mortar element method[END_REF], interface element method [START_REF] Kim | Interface element method (IEM) for a partitioned system with non-matching interfaces[END_REF] as well as other approach discussed in [START_REF] Guidault | On the L2 and H1 couplings for an overlapping domain decomposition method using Lagrange multipliers[END_REF]. The weak relation for continuity of the displacement field at the subdomain interface can be written as follow:

Γ w(x).(u f b (x) -u c b (x))dΓ = 0 ( 16
)
where w is a weighting function. Different choices of w correspond to different types of connection. A micro-macro interpretation of the displacement fields is presented hereafter to guide the modeling choice. For this purpose, the displacement field on the fine mesh is divided in two parts: a macroscopic part (u f M b ) and a microscopic part (u f m b ), as follow:

u f b (x) = u f M b (x) + u f m b (x) (17)
A classical choice consists to define the macro part (u f M b ) as the interpolation of the coarse field at the nodes of the fine mesh on the interface (u c b ).

u f M b (x) = u c b (x) (18)
From Equations ( 17), [START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF], it can be seen that the connection between fine and coarse meshes amounts to define locality constraints for the microscopic part of the displacement. Several approaches are discussed in [START_REF] Hund | Locality constraints within multiscale model for non-linear material behaviour[END_REF], where macro and micro scales are superposed in the localization zone. In the work presented in this paper, micro and macro scales are separated. Moreover, the approach is embedded in the FETI-DP framework. As a consequence, due to the strong continuity at the corner nodes, the microscopic field u f m b has to be zero on these nodes. Two types of connection are discussed here, which are based on the displacements (kinematic) and Mortar (static) approach.

Collocation method

The simplest way to satisfy the equation ( 16) consists of using the weighting function, w, as Dirac (δ) functions on the all nodes of the fine mesh side of the interface. Such a procedure will be called 'collocation'.

This approach enforces the equality of the fine displacement field on one side, with the interpolation of the coarse displacement field on the other side:

u f b (x i ) = u c b (x i ) for i=1 to n f ( 19
)
where n f is the number of interior interface nodes on the fine mesh side, see Figure 3.

Taking into account ( 17) and ( 18), it yields for the microscopic displacement vector:

u f m b = 0 (20)
i.e. for all interior interface nodes on the fine mesh side, the micro displacement is null. Finally, from u f b = B f r u f r and using [START_REF] Lesoinne | A FETI-DP corner selection algorithm for three-dimensional problems[END_REF], the continuity equation ( 19) on the interface can be written in matrix form as:

A f u f r = D c u c c (21)
where:

A f = B f r D c = L c r C c (22)
and u r , u b and u c are subdomain, interface and coarse nodes displacement vectors, respectively.

Mortar method

The Mortar method introduced in [START_REF] Bernardi | A new nonconforming approach to domain decomposition: The mortar element method[END_REF] is one rigorous and popular approach to enforce a weak continuity of the displacement field ( 16) at non-matching discrete interfaces. Within the framework of Mortar methods, the weighting function, w, plays the role of a Lagrange multiplier field, and a key point of the method is the space of these Lagrange multipliers which has to be chosen carefully [START_REF] Herry | An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis[END_REF][START_REF] Hauret | BV estimates for mortar methods in linear elasticity[END_REF]. A usual choice is a modified finite element trace space of the coarsest mesh [START_REF] Herry | An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis[END_REF], with modifications concerning the end nodes of the interface [START_REF] Bernardi | A new nonconforming approach to domain decomposition: The mortar element method[END_REF], which are here the coarse nodes shared by the two subdomains, see e.g. [START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF]. In the case studied here with a coarse subdomain interface with only two nodes, if the shape functions are linear, the Lagrange multiplier has to be taken as a constant.

For plate elements used in this work, the shape functions used for membrane degrees of freedom are linear, while for bending the interpolation can be linear, quadratic, or cubic, see Appendix 1. However, for sake of simplicity, the Lagrange multiplier is chosen as a constant for each type of degree of freedom, on each interface between two subdomains. It leads us to an average displacement connecting method on the interface.

In this case, the displacement fields on the two sides of the interface satisfy the following relation:

Γ (u f b (x) -u c b (x))dΓ = 0 ( 23
)
Here the same assumption are made as in [START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF], i.e. the macro displacement part of the fine side equals the interpolation of the coarse displacement fields on the coarse interface side. Therefore, with considering equation [START_REF] Bavestrello | Incorporation of linear multipoint constraints in domaindecomposition-based iterative solvers -Part II: Blending FETI-DP and mortar methods and assembling floating substructures[END_REF], one obtains:

Γ u f m b (x)dΓ = 0 (24)
i.e. the average of the micro displacement part, u f m b , is null on the interface. It is recalled here that the micro displacement u f m b is null at the corner node locations. The discretization of the gluing condition [START_REF] Bernardi | A new nonconforming approach to domain decomposition: The mortar element method[END_REF] leads to the linear constraint of the degrees of freedom on two sides of the interface. The following notation is used again:

u f b (x) = N (x)u f b = N (x)B f r u f r (x) u c r = C c u c c u c b (x) = N (x)L c r u c r = N (x)L c r C c u c c ( 25
)
where for sake of simplicity, the continuous displacement fields, u f b (x) and u c b (x), are approximated using piecewise linear finite element shape function, N (x), defined on the interface fine mesh side.

Note that once the coarse displacement field has been interpolated (with matrix C c ), the same discretizations are obtained for the coarse and fine sides of the interface. Therefore, the shape functions N f and N c on the two sides of the interface are the same and are denoted N .

By substituting [START_REF] Guidault | On the L2 and H1 couplings for an overlapping domain decomposition method using Lagrange multipliers[END_REF] in the equation ( 23), one obtains the following equation:

Γ [N (x)B f r u f r -N (x)L c r C c u c c ]dΓ = 0 ( 26
)
And in the matrix form, it yields:

A f u f r = D c u c c ( 27
)
with

A f = HB f r D c = HL c r C c H = Γ N (x)dΓ (28)
where H denotes the integral of the shape function on the interface. Disregarding the choice of the connection method, collocation or Mortar, the relation between the degrees of freedom on the two sides of the interface can be put in similar forms [START_REF] Sun | A fast implementation of the FETI-DP method: FETI-DP-RBS-LNA and applications on large scale problems with localized non linearities[END_REF] and [START_REF] Herry | An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis[END_REF]. This relation is used for managing the incompatible interface constraints by domain decomposition method in the next Section.

Numerical homogenization of a finely meshed subdomain

In the previous section, two connection methods between incompatible displacement fields on the interface between non-matching subdomains were presented. In this section, we are concerned with the determination of an homogenized stiffness for coarse subdomains.

It is recalled there that the starting point of the method is a domain decomposition of the structure from subdomains with a fine mesh, compatible on the interface. Then, macro zones are chosen, and their corresponding subdomains are described only from their coarse nodes, see Figure 2. When a subdomain is discretized at a macroscopic (homogenized) scale, a macroscopic stiffness should be associated to it. This can be deduced from its fine discretization in a preliminary stage.

The critical step of any homogenization method lies in the localization step and concerns the way a macroscopic field loading is applied to the microscopic level. A main difficulty is the definition of suited boundary conditions to be applied to the microscopic field on the micro domain. These boundary conditions correspond to a modeling of the loadings applied by the neighboring subdomains. Therefore, to ensure consistency of the FETI-DP micro-macro approach, the previous assumptions used for connection (collocation or Mortar) are applied again. These assumptions can be explained in the framework of homogenization methods. Thus, for the collocation approach, since the micro displacement field is supposed to be zero on the subdomain boundary, this approach amounts to the strain approach of the Hill-Mandel method [START_REF] Suquet | Elements of homogenization for inelastic solid mechanics[END_REF]. For the Mortar method, the micro displacement field is constrained to be zero at the corner nodes. The other boundary degrees of freedom are associated with a Lagrange multiplier which is constant. Thus there are similarities with the stress approach of the Hill-Mandel method [START_REF] Suquet | Elements of homogenization for inelastic solid mechanics[END_REF], in which uniform stress boundary conditions are considered. For a subdomain s, these boundary conditions were given by equations ( 15) and ( 19) (collocation) and equation ( 26) (Mortar) respectively, when fine and coarse mesh connection was treated. Since in the homogenization process only one domain with a fine mesh is considered, they can be written in the general form of the equation ( 21) and ( 27) as follows:

A s u s r = D s u s c (29)
Moreover, the internal degrees of freedom, which are free of any loading, are condensed statically on the coarse ones. In this way, it is possible to derive a macro stiffness only defined on the coarse degrees of freedom. This homogenized stiffness is thus of a kinematic or static type, depending on the assumption used for the localization process. For each subdomain s, omitting the superscript s for simplification in the following, and considering the boundary conditions ( 29) on all the boundary of the subdomain, the equilibrium equation of this subdomain is:

∀ũ i , ũb , ũc , Aũ b = Dũ c ,   ũi ũb ũc   T (   K ii K ib K ic K bi K bb K bc K ci K cb K cc     u i u b u c   -   f i f b f c   ) = 0 ( 30
)
where i, b and c still represent the subdomain internal, interface (non-corner) and corner degrees of freedom, respectively. One can enforce the boundary conditions by Lagrange multipliers in the equilibrium equations, which leads to:

    K ii K ib K ic 0 K bi K bb K bc A T K ci K cb K cc -D T 0 A -D 0         u i u b u c µ     -     f i f b f c 0     = 0 (31)
The macroscopic stiffness on the corner degrees of freedom of the subdomain is obtained by condensing the internal degrees of freedom u i , then in a second stage, u b and µ onto the coarse degrees of freedom u c . As a result, one obtains the following expression for the macroscopic (homogenized) stiffness K H and the macroscopic (homogenized) consistent generalized force f H from the equilibrium equation [START_REF] Rixen | Extended preconditioners for the FETI method applied to constrained problems[END_REF]:

K H u c = f H (32)
where:

K H = ( K cc + K cb K -1 bb K bc ) f H = f c + K cb K -1 bb f b (33)
with:

K cc = K cc -K cb K -1 bb K bc K cb = D T + K cb K -1 bb A T K bb = AK -1 bb A T f c = f c -K cb K -1 bb f b f b = AK -1 bb f b (34)
and the terms obtained from the first condensation are:

K cc = K cc -K ci K -1 ii K ic K cb = K cb -K ci K -1 ii K ib K bb = K bb -K bi K -1 ii K ib f c = f c -K ci K -1 ii f i f b = f b -K bi K -1 ii f b (35)
These notations will be used in next Section to derive the new interface problem of the FETI-DP micromacro approach.

FETI-DP MICRO-MACRO FORMULATION

Up to this point, three types of subdomain connections on the interface can be encountered, see Figure 2: • connection of two subdomains with compatible fine meshes,

• connection of two subdomains with compatible coarse meshes,

• connection of the subdomains with incompatible fine and coarse meshes.

The first one is classical in FETI-DP method and the second one is satisfied automatically, since the coarse nodes are common between several subdomains. For the third one, the two methods (collocation and Mortar) presented previously, can be used. The different connections on the interfaces, once assembled for the whole problem, are:

Ns s=1 B s r u s r = 0 micro-micro connections Ns s=1 (A s u s r -D s u s c ) = 0 micro-macro connections (36)
Here, the formulation of the FETI-DP micro-macro method is presented by using the connections (36), the consistent macro stiffness matrix K H , and the generalized force vector f H , [START_REF] Amini | Analyse d'un navire à passagers avec une méthode de décomposition de domaine à plusieurs échelles[END_REF].

This can be achieved by introducing a second set of Lagrange multipliers, µ, for micro-macro interface, in addition to the classical Lagrange multipliers, λ, for micro-micro interface.

The unconstrained equilibrium equation for all the subdomains now reads:

∀ũ s r , ũs c = L s c ũc , s∈N sf ũs r L s c ũc T K s rr K s rc K s cr K s cc u s r L s c u c - f s r f s c + s∈Nsc (L s c ũc ) T (K s H L s c u c -f s H ) = 0 (37)
where N sf and N sc are the sets of subdomains with fine and coarse mesh, respectively. It should be noted that u s r contains all degrees of freedom of a subdomain, other than the corners degrees of freedom. So, for the macro subdomains, the degrees of freedom are restricted to u s c . Using the interface continuity condition [START_REF] Guidault | A two-scale approach with homogenization for the computation of cracked structures[END_REF] in equation [START_REF] Alart | A scalable multiscale LATIN method adapted to nonsmooth discrete media[END_REF], it leads to:

K s rr u s r + K s rc L s c u c = f s r -B s T r λ -A s T µ for s=1, ... , N sf (38)
and:

s∈N sf L s T c K s cr u s r +   s∈N sf L s T c K s cc L s c + s∈Nsc L s T c K s H L s c   u c =   s∈N sf L s T c f s c + s∈Nsc L s T c f s H   + s∈N sf L s T c D s T µ (39)
From equation [START_REF] Ladevèze | Mastering Calculations in Linear and Nonlinear Mechanics[END_REF] it follows that:

u s r = K s -1 rr (f s r -B s T r λ -A s T µ -K s rc L s c u c ) (40)
Substituting equation [START_REF] Batoz | Modélisation des structures par éléments finis, volume 2 -Poutres et Plaques[END_REF] into equation ( 39) leads after some algebraic transformations to the new dualprimal interface problem with three unknowns (two Lagrange multipliers, λ and µ, and a primal displacement u c ):

  -K cc F cl F cm F lc F ll F lm F mc F ml F mm     u c λ µ   =   -f c d l d m   (41)
where:

K cc = s∈N sf L s T c K s cc L s c + s∈Nsc L s T c K s H L s c - s∈N sf L s T c K s cr K s -1 rr K s rc L s c F ll = s∈N sf B s r K s -1 rr B s T r F mm = s∈N sf A s K s -1 rr A s T and F cl = s∈N sf L s T c K s cr K s -1 rr B s T r F cm = s∈N sf L s T c (K s cr K s -1 rr A s T + D s T ) F lm = s∈N sf B s r K s -1 rr A s T
and finally

f c = s∈N sf L s T c f s c + s∈Nsc L s T c F s H - s∈N sf L s T c K s cr K s -1 rr f s r d l = s∈N sf B s r K s -1 rr f s r d m = s∈N sf A s K s -1 rr f s r

SOLUTION METHOD AND NUMERICAL EXAMPLES

Several approaches, in the context of substructure based iterative solver, already dealt with additional constraints with additional Lagrange multipliers, such as the incorporation of linear multipoint constraints [START_REF] Ch | Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: A numerically scalable algorithm[END_REF][START_REF] Bavestrello | Incorporation of linear multipoint constraints in domaindecomposition-based iterative solvers -Part II: Blending FETI-DP and mortar methods and assembling floating substructures[END_REF], and for FETI-DP formulation with Mortar constraints [START_REF] Dokeva | Scalable mortar methods for ellipic problems on many subregions[END_REF][START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF]. Numerous techniques (outer iterations on µ, simultaneous iterations on λ and µ and outer iterations on λ) presented in [START_REF] Bavestrello | Incorporation of linear multipoint constraints in domaindecomposition-based iterative solvers -Part II: Blending FETI-DP and mortar methods and assembling floating substructures[END_REF][START_REF] Ch | Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: A numerically scalable algorithm[END_REF][START_REF] Rixen | Extended preconditioners for the FETI method applied to constrained problems[END_REF], are used for solving the resulting problem (41).

In this article, the topic concerns mainly the modularity of the method. Therefore, the method of simultaneous iterations on λ and µ is chosen, which is the direct extension of the FETI-DP method, to solve the interface problem (41) of the FETI-DP micro-macro method.

In a first step, the consistency of the FETI-DP micro-macro method is checked with patch tests. After that, in the second step, several numerical examples (cantilever beam in bending, infinite plate with a hole in traction and bending, etc.) are presented. Throughout the numerical examples, the results are compared either to an analytical solution or to a numerical reference solution (obtained with a classical finite element method, with a fine mesh in all subdomains).

Two categories of FETI-DP micro-macro methods are considered in the following, according to the assumptions used for the connection of the fine and coarse mesh and the homogenization of the coarse mesh. A static approach where the Mortar method is used, and a kinematic one if the collocation approach is chosen. It is recalled at this point that collocation corresponds to the strain approach of the Hill-Mandel homogenization theory, while the Mortar method presents similarities with the stress approach, see section 5.

The first results of this approach were presented in [START_REF] Amini | Multi-scale domain decomposition method for structural analysis of a passenger ship[END_REF] and [START_REF] Amini | Analyse d'un navire à passagers avec une méthode de décomposition de domaine à plusieurs échelles[END_REF].

Patch tests

Before applying the FETI-DP micro-macro method to different examples, one has to validate the two main features of the method, i.e. the connection between fine and coarse subdomains and the homogenization used to derive the coarse subdomain stiffness. To this end, patch tests are now considered, with a uniform state of stress and strain, and a membrane or bending loading. The examples are presented in Figure 4, with a pure traction and a pure bending case, in 2D and 3D respectively, and displacement boundary conditions in order to prevent any rigid body displacement.

Exemple in bending In a first step, the problem is solved using only one coarse domain. Its macro stiffness is computed from the static or kinematic homogenization of a fine mesh, as detailed in section 5. In both cases, and for the membrane and bending patch tests, the analytical displacement solution is recovered, up to the machine precision for floating point computations. In a second step, the same problems are considered, using the FETI-DP micro-macro proposed in this article. Thus, an heterogeneous domain decomposition is studied, with fine and coarse subdomains, as shown in Figure 5. Fine mesh is used in the subdomain located in the center, while the others are described as a coarse subdomain with the macro stiffness obtained from homogenization of a fine mesh similar to that of the central domain. Once again, the analytical displacement solution is obtained, for membrane and bending loadings, and for both types of methods, kinematic or static.

It can be noticed that the analytical solution of the problems studied in this section exhibits a uniform state of stress (generalized stress for the bending case) and a zero micro displacement on the interface, according to the definition of equations ( 17) and [START_REF] Kim | A FETI-DP formulation for compressible elasticity with mortar constraints[END_REF]. These situations are fully compatible with assumptions used for the homogenization and connection between micro and macro subdomains, see equations ( 20) and [START_REF] Kim | Interface element method (IEM) for a partitioned system with non-matching interfaces[END_REF]. Therefore, it is logical that the the FETI-DP micro-macro approach passes the patch test.

More complicated examples will be considered in the following sections, in order to assess the approximations of the FETI-DP micro-macro method, due to the homogenization of coarse subdomains and to the connections between coarse and fine subdomains.

End loaded cantilever beam

In this subsection, an end-loaded cantilever beam, depicted in Figure 6, is studied. The loading consists in a parabolic shear force P and a linear traction force repartition at the end of the beam as follows:

• the shear forces (-P ) and (+P ) in y direction, with a parabolic distribution defined by (43), are imposed at two ends of the beam (at x = 0 and x = L, -H 2 < y < H 2 ),

• at x = L, a linear force density in the x direction leads to a resultant bending moment (-M = -P L) in z direction, to balance the structure.

Displacement boundary conditions are also imposed at few nodes in order to prevent any rigid body motion, see Figure 6. The overall beam equilibrium provides the reactions which correspond to the force repartition shown in Figure 6. The goal of this example is to study the global response of structure by FETI-DP micro-macro method. This problem is bi-dimensional (plane stress) and the analytical solution is given e.g. in [START_REF] Timoshenko | Theory of elasticity[END_REF]. The displacement is:

u(x, y) = - P y EI 1 2 (L 2 -x 2 ) + (2 + ν) 6 y 2 - H 2 4 v(x, y) = - P EI L 3 3 - L 2 x 2 + x 3 6 + (4 + 5ν) 24 H 2 (L -x) + ν 2 xy 2 (42)
The stresses are given by: For the computations, the data used are, for the material parameters: Young's modulus E = 1 GPa and Poisson's ratio ν = 0.3. For the beam geometry, a unit thickness is considered, H = 4 m and L = 8 m. The loading parameters are P = 7.5 × 10 5 N and M = 6 × 10 6 N.m, respectively. Finally the moment of inertia is I = H 3 /12.

In this example, the stress state is not uniform on the interface and in addition the micro displacement is not zero on the micro-macro interface. These two conditions will check the homogenization and micro-macro connection in the situation more severe than the patch test.

Let us consider the deflection in vertical direction at the point A, see Figure 6, for the two types of mesh connection and homogenization. For different number of subdomains, one consider half of them equipped with a fine mesh, see Figure 7. The results are compared with the solution obtained with the uniform fine discretization (with a displacement -2.78 × 10 -2 m), see Figure 8, while the analytic reference value is -2.81 × 10 -2 m.

It can be seen that the FETI-DP micro-macro solution is in very good agreement with the analytical solution, even with a very small number of subdomains (one coarse and one fine). In the case of 8 subdomains (4 coarse and 4 fine subdomains) the relative difference on the global displacement of the point A is 3 % for the kinematic case and 1.5 % for the static case. The FETI-DP micro-macro method has been assessed in the previous example regarding overall results. The objective of the next example is to study the ability of the method to provide a good approximation of local results. To this end, the example of a plate in traction with a circular hole is considered, see Figure 9. Large stress gradients are expected around the hole, and the method will be used limiting the subdomains with a fine discretization in the vicinity of this stress concentration zone, other zones being modeled with coarse (homogenized) mesh. The plate is subjected to a uniform tension, P , in the x direction. The analytical solution in the case of an infinite plate is presented in [START_REF] Timoshenko | Theory of elasticity[END_REF]. The stresses are given by the following relations:

σ rr = P 2 [(1 - a 2 r 2 ) + (1 - 4a 2 r 2 + 3a 4 r 4 ) cos(2θ)] σ θθ = P 2 [(1 + a 2 r 2 ) -(1 + 3a 4 r 4 ) cos(2θ)] σ rθ = P 2 [(1 + 2a 2 r 2 - 3a 4 r 4 ) sin(2θ)] (44)
where r and θ are the polar coordinates and the angle θ is measured from the positive x axis in the counterclockwise direction. The maximum values of stress occur on the hole boundary and are given by:

σ θθ (B) = σ θθ (θ = π 2 ) = 3P σ θθ (A) = σ θθ (θ = 0) = -P (45)
This analysis is based on the plane stress assumption. The isotropic material parameters are: Young's modulus E = 210 GPa, and Poisson's ration ν = 0.3. The following parameters are used for the plate geometry: L = 2.1 m , H = 2.1 m, r = 0.07 m and a unit thickness. The load value is p = 100 kN.

The parameters for the FETI-DP micro-macro method are 7 × 7 subdomains, which means that the hole diameter is close to half the subdomain length, see Figure 9. As a first step, the subdomain containing the hole is the only subdomain with a fine mesh. Since the studied plate is not infinite, a numerical reference solution is computed with a fine mesh in all the subdomains. The FETI-DP micro-macro solution is then compared to numerical and analytical reference solutions.

In order to assess the difference between the solution of the FETI-DP micro-macro method (with subscript num) and numerical reference solution (with subscript ref), we can define a relative energy norm of the error

r Ω = E dif E ref (46)
where:

E dif = Ω ε(u dif ) : σ(u dif )dΩ E ref = Ω ε(u ref ) : σ(u ref )dΩ u dif = u num -u ref (47)
This error is also computed at the finite element level, i.e. by integrating on each element, to get a local error and is displayed in Figure 10, for the central subdomain with a fine mesh. It can be seen that the static method provides more accurate results than the kinematic one. Moreover, for the static method, the error is localized on the subdomain boundary, i.e. in the vicinity of the connection zone with surrounding coarse subdomains, with a concentration at the corner nodes. However, the error decreases quickly when one moves away from the boundary. An other comparison point of view would consist in using the local contribution to the error. Since the elements of the mesh are approximatively of the same size, both plots lead to the same conclusion. These results are consistent with those obtained in [START_REF] Guidault | Une stratégie de calcul pour les structures fissurées : analyse locale-globale et approche multiéchelle pour la fissuration[END_REF][START_REF] Guidault | A two-scale approach with homogenization for the computation of cracked structures[END_REF], where a static connection approach was used, without using corner continuity (note that such an additional constraint can be also prescribed in this last approach, as in [START_REF] Alart | A scalable multiscale LATIN method adapted to nonsmooth discrete media[END_REF]). On the other hand, the error obtained with the kinematic approach is lower in the vicinity of the connection zone, but increases as one gets closer to the central hole.

The deformed shape of the same subdomain is presented in Figure 11. Clearly, with the kinematic approach the interfaces are straight lines, while for static approach, the average of the displacement is null on each interface.

Kinematics Static

Figure 11: Deformation of the central subdomain

To compare local stress values, the orthoradial stress at the points A and B (see Figure 9) are reported, as a function of the mesh density, see Figures 12 and13. As mentioned previously, a significant error is obtained with the kinematic approach, while satisfactory results are given by the static approach, even if the size of the micro zone is only twice the hole diameter. It can also be checked that for both methods, the error rapidly decreases when the size of the micro zone increases. Thus, for this problem and 7 × 7 subdomains again, the micro zone is now extended to 3 × 3 subdomains at the center of the plate (instead of 1 in the previous example). The errors obtained with the FETI-DP micro-macro method are given in Figure 14, and are significantly smaller than those obtained with only one micro subdomain, see Figure 10. For the kinematic method, the error still reaches its maximum value in the vicinity of the hole boundary.

Stress at node A (MPa

Plate with a hole in bending

The same example is reused, but with a bending loading, prescribed in the same way as for the beam-like structure of Section 7.2. For the case where the micro zone is restricted to only one subdomain, the relative energy error is shown in Figure 15, which has to be compared to Figure 10 for the traction loading (membrane solution). In the bending case, the error is lower than in the traction case. The spatial distribution of the error is also different, since maximum values are obtained on the micro zone boundary. However, in the vicinity of the hole and with a closer examination of the error levels, the static approach is still slightly more accurate than the kinematic one.

Application to a plate assembly

In this Section, a more complicated example, with plate assembly and structural heterogeneities (holes) is considered, see Figure 16.

The structure is subjected to an overall bending and traction loading, through a linearly varying traction force applied at one end, the other end being clamped. The material is considered as an isotropic one with the following parameters: Young's modulus E = 200 GPa and Poisson's ration ν = 0.3. The following data are used for the geometry: length L = 3.0 m, breadth B = 2.0 m, height H = 2.0 m, with a uniform thickness t = 0.005 m and a hole radius r = 0.1 m. For this study, the structure was divided into 36 subdomains, and fine mesh is considered only in the subdomains embedding the holes, see Figure 16. Once again, a reference numerical solution is obtained with a fine mesh in every subdomain. Using this solution, the relative energy error of the FETI-DP micro-macro solution is computed, on the micro subdomains, and is given on Figure 17. As previously, the accuracy of both static and kinematic approaches is satisfactory. For the static approach, the error is mainly located close to the coarse nodes, and decreases rapidly to a small value in the vicinity of the hole. The reverse is obtained for the kinematic approach.

CONCLUSIONS

In this article, a multi-scale extension of the FETI-DP method was presented for large-scale structural analysis. With this method, called FETI-DP micro-macro, different discretization scales can be used in the subdomains, thanks to an homogenization step. Starting from a classical domain decomposition with matching interfaces, most of the subdomains are homogenized, and the original mesh (fine mesh) is kept only in the zones where local phenomena with high stress gradients are expected. One can then optimize the computation time.

Figure 1 :

 1 Figure 1: Classification of the subdomain nodes

Figure 2 :

 2 Figure 2: Different levels of description: a) fine meshes in all the subdomains b) mixing fine and coarse descriptions

Figure 3 :

 3 Figure 3: Interface of fine and coarse mesh

Figure 4 :

 4 Figure 4: Patch test with one subdomain

Figure 5 :

 5 Figure 5: Patch test with an heterogeneous domain decomposition

Figure 6 :

 6 Figure 6: 2D cantilever beam in bending

σ

  xx (x, y) = P I xy σ yy (x, y) = 0 σ xy (x, y)

Figure 7 :

 7 Figure 7: Decomposition in subdomains and micro-macro mesh (beam bending)

Figure 8 :

 8 Figure 8: 2D beam with bending: displacement of the node A

Figure 9 :

 9 Figure 9: Plate with a hole in traction and decomposition in subdomains

Figure 13 :

 13 Figure 13: Infinite plate: stress at node B (σ θθ )

Figure 15 :

 15 Figure 15: Relative energy norm error, 1 micro subdomain, bending case

Figure 16 :

 16 Figure 16: Plate assembly subjected to an overall bending and traction loading (left), subdomains with fine and coarse meshes (right)

Figure 17 :

 17 Figure 17: Relative energy norm of error, plate assembly

  Figure 14: Relative energy norm of error, 3 × 3 micro subdomains

		2.29E-08
		1.68E-03
		2.51E-03
		3.35E-03
		4.19E-03
		5.03E-03
		5.86E-03
		6.70E-03
		7.54E-03
		8.38E-03
		1.01E-02
		1.09E-02
	Kinematic approach	Static approach

Acknowledgements. The first author gratefully acknowledges the support of Iran's Ministry of Science, Research and Technology and SFERE exchange program in France for financial supporting during his thesis.

Two methods have been proposed for the connection of micro and macro subdomains on the interface: a collocation approach and a method which can be viewed as a limit case of the Mortar method. The interface continuity is written in a weak sense, while corner continuity is enforced in a strong sense, due to the FETI-DP framework. These methods contrast with other connection approaches where the continuity is written on macro interface quantities, which are not classical degrees of freedom, see e.g. [START_REF] Guidault | Une stratégie de calcul pour les structures fissurées : analyse locale-globale et approche multiéchelle pour la fissuration[END_REF][START_REF] Guidault | A two-scale approach with homogenization for the computation of cracked structures[END_REF]. In the same way, two homogenization methods of the macro subdomains have been presented, with a localization process consistent with the assumptions made for the connection process. Therefore, two FETI-DP micro-macro approaches can be defined. Both were validated by using the patch tests.

The other numerical examples have shown that accurate results can be obtained from the proposed methods. For a given problem, it turns out that the results are sensitive to the size of the micro zone. In the case of a problem with structural heterogeneities, it means that the micro zone has to be large enough around them. Both static and kinematic approaches exhibit errors in the vicinity of the incompatible interfaces. However, for the static approach, this can be considered as a boundary layer effect, since the error decreases away from the interfaces. The reverse is obtained for the kinematic approach. Thus, the static approach provides more accurate results and appears to be more efficient than the kinematic one.

Concerning outlooks on this study, from a user point of view, automatic assessment of the error due to the discretization [START_REF] Ladevèze | Mastering Calculations in Linear and Nonlinear Mechanics[END_REF] and to the scale description [START_REF] Oden | Hierarchical modelling of heterogeneous solids[END_REF] would be of particular interest, as an indication to decide to change the scale modeling. This decision could also be done as an interactive way, during iterations of the scheme to reach an adequate error level on the areas of interest. Obviously, the scale description is not limited to two levels. A third level of description could also be used, for instance for a very local analysis of fatigue or crack propagation risk, which could also be performed with coupling the plate model with a 3D model at the finest level.

Another interesting aspect to pursue this work is to adapt a specific preconditioner for the method of simultaneous iterations on λ and µ to solve the interface problem (see section 7). Several procedures have been proposed in the literature [START_REF] Bavestrello | Incorporation of linear multipoint constraints in domaindecomposition-based iterative solvers -Part II: Blending FETI-DP and mortar methods and assembling floating substructures[END_REF], that could be tested on the problem we are interested herein.

Appendix 1

The stiffness matrix of the coarse elements are obtained from homogenization of a fine mesh. However, in order to connect coarse element with surrounding micro subdomain, a continuous displacement field has to be defined on its boundary. To this end, shape functions corresponding to a classical finite element (coarse element) are chosen.

For the membrane behavior, a linear interpolation, is a good approximation for the in-plane displacements. But for the bending case with a Discrete Kirchhoff Triangle (DKT) and/or a Discrete Kirchhoff Quadrilateral (DKQ) plate element, the interpolation functions are cubic for the out-of-plane displacement w, quadratic for the rotation denoted with θ n , and linear for the rotation denoted with θ t (t is the in-plane direction of the element edge, and n is the in-plane normal to the edge) [START_REF] Batoz | Modélisation des structures par éléments finis, volume 2 -Poutres et Plaques[END_REF], see Figure 18.