Denis Duhamel
email: denis.duhamel@enpc.fr

A scaled wave finite element method for computing scalar wave radiation and scattering in exterior domains

Keywords: Finite element, scaled domain, wave, Helmholtz equation, radiation, scattering, infinite domain

The calculation of high-frequency wave radiation in exterior domains by finite element methods can lead to large computations. In this paper, it is shown that the solution in the exterior domain can be decomposed as a series expansion of functions with an analytical part made from the product of harmonic waves and polynomials in a scaled variable and a numerical part made of a finite element approximation vectors. The solution of the radiation or scattering problem can be found by solving a sparse linear system which is set from the dynamic stiffness matrices of several scaled layers around the radiating body. These dynamic stiffness matrices are classical finite element matrices obtained from any finite element software. Moreover, accurate results can be obtained from a small number of terms in the series expansion. Several examples are given to estimate the efficiency of the proposed method.

Introduction

Many approaches have been used in the past for computing the solutions of wave problems in unbounded media. Classical methods include the boundary element method described in numerous classical textbooks like [START_REF] Brebbia | Boundary Element Techniques in Engineering[END_REF][START_REF] Crouch | Boundary element methods in solid mechanics[END_REF][START_REF] Ciskowski | Boundary Element Methods in Acoustics[END_REF][START_REF] Chen | Boundary element methods, Computational mathematics and applications[END_REF][START_REF] Bonnet | Boundary Integral Equation Methods for Solids and Fluids[END_REF] and the fast multipole method developed by many authors like [START_REF] Rokhlin | Rapid solution of integral equations of classical potential theory[END_REF][START_REF] Rokhlin | Rapid solution of integral equations of scattering theory in two dimensions[END_REF][START_REF] Darve | The fast multipole method: Numerical implementation[END_REF][START_REF] Liu | The fast multipole boundary element method for potential problems: A tutorial[END_REF]. One can also find, for instance, the Dirichlet to Neumann (DtN) mapping proposed by [START_REF] Givoli | A finite element method for large domains[END_REF][START_REF] Keller | Exact non-reflecting boundary conditions[END_REF] which is also quite accurate but lead to non-local boundary conditions as for the boundary element method and full matrices on the boundary. Local methods, on the contrary, are more computationaly efficient as in these methods, the condition at a border node involves only a limited number of neighbouring nodes. Various classes of local absorbent boundary conditions were for instance developed long ago by [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF]. Infinite elements made with elements extending at infinity and satisfying the Sommerfeld radiation condition were proposed by [START_REF] Bettess | Infinite elements[END_REF][START_REF] Bettess | More on infinite elements[END_REF][START_REF] Bettess | Infinite Elements[END_REF][START_REF] Burnett | A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion[END_REF][START_REF] Astley | Infinite elements for wave problems: a review of current formulations and an assessment of accuracy[END_REF][START_REF] Gerdes | A review of infinite element methods for exterior helmholtz problems[END_REF]. Other absorbing boundary conditions involving differential operators of different orders on the boundary were proposed by different authors [START_REF] Engquist | Absorbing boundary conditions for numerical simulation of waves[END_REF][START_REF] Engquist | Absorbing boundary conditions for numerical simulation of waves[END_REF][START_REF] Clayton | Absorbing boundary conditions for acoustic and elastic wave equations[END_REF][START_REF] Reynolds | Boundary conditions for the numerical solution of wave propagation problems[END_REF] and then improved by Bayliss and Turkel [START_REF] Bayliss | Radiation boundary conditions for wave-like equations[END_REF][START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF] using sequences of local non-reflecting boundary conditions and modified by [START_REF] Higdon | Absorbing boundary conditions for difference approximations to the multidimensional wave equation[END_REF][START_REF] Higdon | Numerical absorbing boundary conditions for the wave equation[END_REF]. However, all these conditions are difficult to implement above the second order because of the high order derivatives involved in their formulations. Another possibility is the addition of variables on the exterior surface as in [START_REF] Collino | High-order absorbing boundary conditions for wave propagation models. straight line boundary and corner cases[END_REF][START_REF] Hagstrom | A formulation of asymptotic and exact boundary conditions using local operators[END_REF][START_REF] Givoli | High-order local non-reflecting boundary conditions: a review[END_REF]. They involve only second order derivatives of the auxiliary variables and so can be efficiently implemented. One can also surround the computational domain by absorbing layers, as proposed by [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF][START_REF] Berenger | Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[END_REF] in the perfectly matched layer, in which the wave equation is analytically continued into complex coordinates. With a correct choice of the size of the layer and the parameters of the absorbing layer, very efficient absorptions of waves can be obtained. More details on these methods are given in [START_REF] Duhamel | Finite element computation of absorbing boundary conditions for time-harmonic wave problems[END_REF] and a review on the different numerical methods for short wave scattering can be found in [START_REF] Bettess | Short-wave scattering: Problems and techniques[END_REF]. In all these local methods, a significant part of the exterior domain is meshed, and more or less accurate boundary conditions are put at the external surface of the mesh. Little or no information is given on the solution outside the meshed domain.

Among all the methods, we must especially notice the scaled boundary finiteelement method proposed by [START_REF] Song | Consistent infinitesimal finite-element cell method: threedimensional vector wave equation[END_REF][START_REF] Song | The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics[END_REF][START_REF] Song | The scaled boundary finite-element method: analytical solution in frequency domain[END_REF][START_REF] Wolf | Unit-impulse response of unbounded medium by scaled boundary finite-element method[END_REF]. It is based on a coordinate transform with a radial coordinate interpreted as a scaling factor with a reference to a center inside the bounded domain. The problem is formulated as a differential equation in the radial coordinate and can be solved analytically along this coordinate. The method was further developed by its authors in [START_REF] Wolf | The scaled boundary finite-element method -a primer: derivations[END_REF][START_REF] Song | The scaled boundary finite-element method -a primer: solution procedures[END_REF][START_REF] Wolf | The scaled boundary finite-element method -a fundamental solution-less boundary-element method[END_REF][START_REF] Song | Dynamic analysis of unbounded domains by a reduced set of base functions[END_REF]. The solution is obtained by solving an eigenvalue problem from matrices built on the boundary. In case of dynamic problems, it also needs expansions in the scaled variables. These points were developed in [START_REF] Yang | A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media[END_REF][START_REF] Song | Development of a fundamental-solution-less boundary element method for exterior wave problems[END_REF]. This subject is still active as evidenced by recent papers [START_REF] Birk | The scaled boundary finite element method for transient wave propagation problems[END_REF][START_REF] Yang | A scaled boundary finite element formulation for dynamic elastoplastic analysis[END_REF][START_REF] Gravenkamp | On mass lumping and explicit dynamics in the scaled boundary finite element method[END_REF][START_REF] Sepehry | Application of scaled boundary finite element method for vibration-based structural health monitoring of breathing cracks[END_REF]. The advantage of this method is that it gives analytical solutions in the scaled variable. Note that it can be easily coupled with FEM models of other fluid or structural domains as shown in [START_REF] Lehmann | Scaled boundary finite element method for acoustics[END_REF]. However, it needs special matrices defined on the boundary of the radiating body which are not the usual matrices obtained by classical finite element software.

Other interesting methods were developed for the analysis of waveguides. For example, the semi-analytical finite element (SAFE) method proposed by [START_REF] Nelson | Vibrations and waves in laminated orthotropic circular cylinders[END_REF][START_REF] Aalami | Waves in Prismatic Guides of Arbitrary Cross Section[END_REF][START_REF] Gavrić | Computation of propagative waves in free rail using a finite element technique[END_REF] and further developed by [START_REF] Shorter | Wave propagation and damping in linear viscoelastic laminates[END_REF][START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF][START_REF] Marzani | A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides[END_REF][START_REF] Treyssède | Investigation of elastic modes propagating in multiwire helical waveguides[END_REF][START_REF] Nguyen | Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods[END_REF][START_REF] Li | On the study of vibration of a supported railway rail using the semi-analytical finite element method[END_REF] separate the solution into an analytical harmonic function e ikx into the direction of the waveguide and a finite element approximation in the cross-section. This can only be applied to uniform waveguides and needs special matrices defined on the cross-section of the waveguide. So the approach has some common points with the scaled boundary finiteelement method: an analytical expression of the solution in radial/waveguide direction and special matrices in the boundary/cross-section. This method has been applied to various problems and some recent developments were proposed, for instance, by [START_REF] Hakoda | Weak form implementation of the semi-analytical finite element (SAFE) method for a variety of elastodynamic waveguides[END_REF][START_REF] Long | Numerical verification of an efficient coupled SAFE-3D FE analysis for guided wave ultrasound excitation[END_REF][START_REF] Khajah | Shape optimization of acoustic devices using the scaled boundary finite element method[END_REF].

For non uniform waveguides with truly periodic structures, the Wave Finite Element (WFE) method can be used. This consists in computing wave modes (propagation constants, wave shapes) of a periodic structure from the finite element (FE) model of a substructure and its related mass, damping and stiffness matrices which can be obtained from any FE software. Afterwards, these wave modes can be used to calculate the harmonic response of periodic structures in an efficient way, i.e., by computing small matrix systems for one substructure, or a few of them. The main steps of the method can be found in [START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF][START_REF] Duhamel | Finite element analysis of the vibrations of waveguides and periodic structures[END_REF][START_REF] Mencik | A wave-based model reduction technique for the description of the dynamic behavior of periodic structures involving arbitraryshaped substructures and large-sized finite element models[END_REF][START_REF] Mencik | Multi-mode propagation and diffusion in structures through finite elements[END_REF]. Also, some of its recent extensions and applications are reported in [START_REF] Mencik | A wave finite element-based approach for the modeling of periodic structures with local perturbations[END_REF][START_REF] Mencik | New advances in the forced response computation of periodic structures using the wave finite element (WFE) method[END_REF][START_REF] Mencik | A model reduction strategy for computing the forced response of elastic waveguides using the wave finite element method[END_REF][START_REF] Mencik | A wave finite element approach for the analysis of periodic structures with cyclic symmetry in dynamic substructuring[END_REF][START_REF] Ichchou | Wave finite elements for low and midfrequency description of coupled structures with damage[END_REF][START_REF] Hoang | Wave finite element method for waveguides and periodic structures subjected to arbitrary loads[END_REF][START_REF] Renno | On the forced response of waveguides using the wave and finite element method[END_REF][START_REF] Waki | Free and forced vibrations of a tyre using a wave/finite element approach[END_REF][START_REF] Singh | Stochastic wave finite element quadratic formulation for periodic media: 1D and 2D[END_REF][START_REF] Fan | Model reduction schemes for the wave and finite element method using the free modes of a unit cell[END_REF]. Compared to the SAFE method, the WFE can deal with more general structures and do not need the development of special matrices but can use matrices produced by commercial FE software. This Floquet's theory was applied to exterior problems by [START_REF] Hvatov | On application of the Floquet theory for radially periodic membranes and plates[END_REF][START_REF] Manconi | Wave propagation in polar periodic structures using Floquet theory and finite element analysis[END_REF] by dividing the exterior domain into layers and trying to apply the WFE on them with possible corrections on the solution to obtain constant energy flux through the layers. However, as the matrices in the different layers of the exterior domain are not constant, only approximate solutions have been obtained.

In this paper, we propose to improve these last attempts of [START_REF] Hvatov | On application of the Floquet theory for radially periodic membranes and plates[END_REF][START_REF] Manconi | Wave propagation in polar periodic structures using Floquet theory and finite element analysis[END_REF] by coupling the equilibrium equations between adjacent layers as in the WFE approach and which only needs classical finite element matrices, to the expansion of the solution suggested by the scaled boundary finite element. The solution in the exterior domain will be decomposed as a sum of elementary functions made of harmonic waves multiplied by polynomials and finite element approximation vectors. The solution can then be found by solving a sparse linear system which is set from the dynamic stiffness matrices of several scaled layers around the radiating body. These dynamic stiffness matrices are classical finite element matrices obtained from any finite element software, so eliminating the need of building special matrices. Accurate results can be obtained from a small number of terms in the series expansion. So a description of the solution in the exterior domain can be found for general geometries. The paper is divided into three sections and is outlined as follows. In the second section, the principles of the method are described in general. In the third section, examples are described to get a clear insight in the proposed approach before concluding remarks.

2 Solution in the exterior domain

Behaviour of a layer

We consider the general case of a convex body Ω surrounded by an infinite domain as in figure 1. To simplify, we consider here two-dimensional problems only. Linear waves are supposed to propagate in this exterior domain. They are described by the Helmholtz equation

∆q + k 2 q = 0 (1)
with k = ω/c the wavenumber, ω the circular frequency and c the velocity. Boundary conditions are applied on the boundary surface S 0 and at infinity such that

q = q 0 on S D 0 ∂q ∂n = r 0 on S N 0 ∂q ∂n -ikq = o( 1 √ r ) f or r → ∞ (2)
with S D 0 and S N 0 the parts of S 0 where respectively the Dirichlet and Neumann boundary conditions are applied and q 0 and r 0 given functions on S 0 . The last relation is the Sommerfeld radiation condition for two-dimensional problems.

A layer n, defined as the domain Ω n between surfaces S n and S n+1 , is described by a finite element model. The surface S n is defined by a constant value of the scaled parameter ξ = ξ n with ξ n = 1 + i=n-1 i=0 d i (as in the scaled boundary element method) where the d i are the variables giving the thickness in term of ξ of each layer and ξ = ξ 0 with ξ 0 = 1 defines the surface S 0 of the radiating body. A layer can be meshed with an arbitrary number of elements using the full possibilities of usual finite element software. This makes it possible to process structures of complex shapes directly from their dynamic d0 S 0 (x=1) stiffness matrix without having to go into the details of the implementation of the finite elements (shape functions, order of the elements, integration techniques, ...) and to be able to potentially generalize the proposed method to other situations without substantial modification of the software. The discrete dynamic behaviour of a layer n obtained from a finite element model at a circular frequency ω is thus given by

S 3 (x=1+d 0 +d 1 +d 2 ) S 2 (x=1+d 0 +d 1 ) S 1 (x=1+d 0 ) ξ W 1 W 0 q L 0 q R 0 =q L 1 q I 0 d 1 d 2 Ω
(K n -ω 2 M n )q n = f n (3)
where K n and M n are the stiffness and mass matrices respectively, f n is the vector of internal forces and q n the vector of the degrees of freedom. Introducing the dynamic stiffness matrix of the layer D n = K n -ω 2 M n , decomposing into degrees of freedom of the left boundary (L) associated to the dofs in S n , interior dofs (I) and right boundary dofs (R) associated to dofs in S n+1 , and assuming that there is no external force on the interior nodes, results in the following relation

       D n LL D n LI D n LR D n IL D n II D n IR D n RL D n RI D n RR               q n L q n I q n R        =        f n L 0 f n R        (4)
The assumption that there is no force on the interior degrees of freedom is satisfied for free waves inside the structure for which the forces on a layer are only produced by boundary forces from the adjacent layers.

Relations between adjacent layers

At the interface between layers n -1 and n (on surface S n ), there is continuity of the displacement and equilibrium of the forces, so that one has

q n L = q n-1 R (5) f n L + f n-1 R = 0 (6)
The equilibrium of the forces leads to the relation

D n-1 RL q n-1 L + D n-1 RI q n-1 I + D n-1 RR q n-1 R + D n LL q n L + D n LI q n I + D n LR q n R = 0 (7)
or, using (5) and denoting the vector of displacement q n on the surface S n , this can be written as

D n-1 RL q n-1 + D n-1 RI q n-1 I + (D n-1 RR + D n LL )q n + D n LI q n I + D n LR q n+1 = 0 (8)
This must be completed by the relations for the internal degrees of freedom in domains Ω n and Ω n+1 , given by

D n-1 IL q n-1 + D n-1 II q n-1 I + D n-1 IR q n = 0 (9) D n IL q n + D n II q n I + D n IR q n+1 = 0 (10)
So assembling relations (8), ( 9) and ( 10) yields

D n T Q n = 0 (11)
with

D n T =        D n-1 RL D n-1 RI D n-1 RR + D n LL D n LI D n LR D n-1 IL D n-1 II D n-1 IR 0 0 0 0 D n IL D n II D n IR        , Q n =               q n-1 q n-1 I q n q n I q n+1               (12)
Note that in case of no interior node, this simplifies to

D n T = D n-1 RL D n-1 RR + D n LL D n LR , Q n =        q n-1 q n q n+1        (13)
Note also that Q n is the vector made of the dofs associated to the domain Ω n-1 ∪Ω n and, if we denote by M n this number of dofs, the size of Q n is M n ×1 while the number of lines of D n T equals the number of dofs on the surface S n plus the number of internal degrees of freedom in Ω n-1 ∪ Ω n , the whole which is denoted m n , and so the size of

D n T is m m × M n with m m < M n .

Series expansion of the solution

As we deal with two-dimensional problems, we expect a large distance behaviour like e ikr √ r . So the displacement at a node of location r j of scaled parameter ξ j will be assumed given by the expansion,

q(r j ) = p=P p=0 a jp ξ -(2p+1)/2 j e ik|r j | (14)
in term of parameters a jp to be found. Note that this expansion includes both the distance r j of the node j to the center of the object and the parameter ξ j which locates the surface to which the node j belongs. For a node j s on the surface of the object, we therefore have ξ js = 1 and the expansion reduces to

q(r js ) =   p=P p=0 a jsp   e ik|r js | (15)
The form of the expansion ( 14) was suggested by the expansion of Hankel's functions for large arguments. Other expansion choices are probably possible.

Note also that we make the assumption that the domains are meshed in an identical way up to the scaled parameter ξ, see figure 2. So the nodes on the different surfaces S n are located on the same radial lines and the nodes on a surface S n can be obtained from the nodes on the surface S n-1 by scaling in ξ. So the value of q(r j ) is expressed in term of p + 1 parameters a jp with ξ

S n-1 S n +1 S n S n+2 r n -1,2 r n -1,1 r n -1, m
Nodes associated to the same j Fig. 2. Mesh of the surfaces S n .

0 ≤ p ≤ P associated to the same direction on different surfaces and so for a fixed value of j which defines the radial direction to which r j belongs. The vector of displacement Q n of the dofs of the domain Ω n-1 ∪ Ω n can thus be written as

Q n = T n a T n I    a q I    = T n    a q I    (16)
with

T n = T n a T n I
and where the vector of the parameters a jp is

a =               a 0 . . . a p . . . a P               , with a p =        a 1p . . . a mp        (17)
and

T n a = t n , Ξ n .t n , . . . , Ξ P n .t n ( 18
)
with

t n = ξ -1/2 n,j e ik|r n,j | d jj j=1...M,j =1...m (19)
Ξ n = diag( 1 ξ n,j ) j=1...M (20)
with diag meaning a diagonal matrix and the domain Ω n-1 ∪ Ω n is supposed meshed with M nodes of positions r n,j for j = 1...M with the associated scaled parameters ξ n,j and m is the number of nodes on a surface S n . Note that the ξ

-(2p+1)/2 j
factor can be found for the ξ -1/2 j part in the vector t n and for the ξ -p j part in the matrix Ξ p n . The parameter d jj equals 1 if the node number j in domain Ω n-1 ∪ Ω n is not an interior node and if node j is associated to node j on surface S n in the way shown in figure 2. In other words, nodes on the same radial line are associated to the same j . So the matrix t n is sparse and has only one non zero element on each line. The vector q I is the global vector of interior degrees of freedom associated to the considered domains. The matrix T n I is a matrix made of zeros and ones which extracts the subvectors q n-1 I and q n I of relation ( 12) from q I .

Combining with relation [START_REF] Keller | Exact non-reflecting boundary conditions[END_REF], the equation for the layer can be finally written as

S n    a q I    = (D n T T n )    a q I    = 0 (21)
where the precedent relation defines the matrix S n . The number of equations equals m n , the number of dofs on the surface S n plus the number of interior dofs in Ω n-1 ∪ Ω n . Note that according to the shape of the matrix T n seen in relations ( 16) and ( 18), the matrix S n is given by

S n = D n T t n , D n T Ξ n .t n , . . . , D n T Ξ P n .t n , D n T .T n I ( 22
)
and is also a sparse matrix as D n T , which is sparse, multiplies only other sparse matrices.

Global solution

Now, relations ( 16) and ( 21) are applied for the surface S 0 and layers n 1 , n 2 , ..., n P , see Figure 1. We consider first a Dirichlet problem with the given displacement q 0 on the boundary surface S 0 . Using relation ( 16) leads to

q 0 = T 0 r    a q I    (23)
with T 0 r obtained in the same way as ( 16) but keeping only the lines associated to the dofs of the surface S 0 . This is also the component q 0 of vector Q 1 in relations ( 12) and ( 16). So, the number of equations equals the number of dofs on surface S 0 , denoted m.

Then, we choose P central layers n 1 , ..., n P , and around each layer we apply the relation ( 21) so that for the other layers n i , the relation is

S n i    a q I    = 0 (24)
Assembling relations [START_REF] Bayliss | Radiation boundary conditions for wave-like equations[END_REF] and [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF], finally, the global system to solve is

          T 0 r S n 1 . . . S n P              a q I    =           q 0 0 . . . 0           (25)
The solution of this system gives the parameters a jp and the solution for any node can be obtained by the expansion ( 14) and the components of q I . To obtain a square system a parameter a jp must be associated to each dof of S n i and to each dof of S 0 .

In case of a Neumann problem where the external force f 0 is given on surface S 0 , relation ( 23) is changed to

   f 0 0    = S 0 r    a q I    (26)
with

S 0 r    a q I    = D 0 L T 1 0    a q I    (27)
with D 0 L the submatrix of D 0 with only the lines associated to the dofs of S 0 and the interior dofs of Ω 0 (first and second lines of relation ( 4)) and T 1 0 the submatix of T 1 with only the lines associated to the dofs of Ω 0 (first three lines of Q 1 in relation ( 12) and ( 16)). The right member of relation ( 25) is changed so that the final relation is now

          S 0 r S n 1
. . .

S n P              a q I    =           f 0 0 . . . 0           (28)
The number of equations in relation ( 25) equals m + P (m + 2n int ) and in [START_REF] Hagstrom | A formulation of asymptotic and exact boundary conditions using local operators[END_REF] it equals m + n int + P (m + 2n int ) with n int the number of interior dofs in a domain. Note that the linear system to solve is sparse.

Coupling with another FEM domain

The expansion ( 14) is efficient for large enough frequencies and for convex domains. We consider here the more general case of radiation by a body of any shape as in the figure 3. Note that here the radiating body is non-convex and therefore the previous method cannot be applied directly. So, we divide the domain outside the radiating body into two subdomains. The first subdomain is inside a convex surface noted S 0 and is described by a dynamic stiffness matrix of a classical finite element model obtained by any finite element software (the grid part in the figure). The second subdomain outside the surface S 0 can be described by the model presented previously since the surface S 0 is convex. We will then specify the equations satisfied in each of the two subdomains and the global relation obtained by coupling the two subdomains. The domain interior to the surface S 0 which is described by a classical finite element model satisfied the following relation.

       D bb D bi D b0 D ib D ii D i0 D 0b D 0i D 00               q b q i q 0        =        f b 0 f 0        (29)
with the subscript b describing the dofs belonging to the surface S b where the boundary conditions are prescribed (the surface of the radiating body), 0 the dofs belonging to the coupling interface S 0 between the domain described by classical FEM and the exterior domain described by the expansion ( 14) and i the dofs internal to the FEM domain, see figure 3.

ξ S 2 S 1 S 0 S b

FEM part

Fig. 3. Case of a non convex domain.

On the surface S 0 , one has the relations ( 23) and ( 26) so that

q 0 = T 0 r    a q I    (30)    f 0 0    = S 0 r    a q I    =    S 0 r0 S 0 ri       a q I    (31)
with the second line of (31) associated to the interior dofs of Ω 0 . Assembling relations ( 29), ( 30), [START_REF] Berenger | Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[END_REF] with the relations [START_REF] Clayton | Absorbing boundary conditions for acoustic and elastic wave equations[END_REF] for the other external surfaces leads to

                     D bb D bi D b0 T 0 r D ib D ii D i0 T 0 r D 0b D 0i D 00 T 0 r + S 0 r0 0 0 S 0 ri 0 0 S n 1 . . . . . . . . . 0 0 S n P                                q b q i a q I           =                      f b 0 0 0 0 . . . 0                      (32)
This system can be solved when q b or f b is known on the boundary S b and the solution gives the displacement in the FEM part and the coefficients of the expansion ( 14).

3 Numerical examples

Two-dimensional axisymmetric example

In this first simple example, we consider an infinite membrane under harmonic excitation with normal displacement q satisfying the Helmholtz equation

E∆q + ρω 2 q = 0 ( 33
)
The wavenumber is k = ω/c with c = E/ρ. This is also the case of acoustic radiation with a given value of the sound velocity c. The radiating body is supposed to be a disk and we are looking for axisymmetric solutions. The radial shape functions are

N 1 (ξ) = 1 2 (1 -ξ) N 2 (ξ) = 1 2 (1 + ξ) ( 34
)
The radius is interpolated by r(ξ) = r 1 N 1 (ξ) + r 2 N 2 (ξ) and the elements of the mass matrix for axisymmetric solutions are given by

m ij = 2π r 2 r 1 ρN i (r)N j (r)rdr = πρ(r 2 -r 1 ) 1 -1 N i (ξ)N j (ξ)r(ξ)dξ m = πρ(r 2 -r 1 )    r 1 2 + r 2 6 r 1 6 + r 2 6 r 1 6 + r 2 6 r 1 6 + r 2 2    (35)
The elements of the stiffness matrix are

k ij = 2πE r 2 r 1 ∂N i (r) ∂r ∂N j (r) ∂r rdr = 4πE r 2 -r 1 1 -1 ∂N i (ξ) ∂ξ ∂N j (ξ) ∂ξ r(ξ)dξ k = πE(r 1 + r 2 ) (r 2 -r 1 )    1 -1 -1 1    (36)
So that the dynamic stiffness matrix in the domain n is given by

D n = πE d (r n + r n+1 )    1 -1 -1 1    -ω 2 π 6 ρd    3r n + r n+1 r n + r n+1 r n + r n+1 r n + 3r n+1    (37)
with r n the radius of surface S n . In the following, one takes E = 1 and everything is given in term of kr. The analytical solution for a uniform displacement q 0 at r = r 0 is given by

q(ξ) = H 0 (kr) H 0 (kr 0 ) q 0 ( 38
)
with H 0 is the Hankel function of first type and order 0.

This problem was solved with the proposed method for different wavenumbers k such that 0.001 ≤ kr 0 ≤ 1000. First consider the case kr 0 = 1 with different values of the number of polynomial coefficients given by P . The radius of surface S n is such that kr n = 1 + nke with ke = 0.005 and ξ n = 1 + nd and d = ke/max(1, kr 0 ). Figure 4 presents the real and imaginary parts of the solution in the domain 50 ≤ kr ≤ 60 for different values of P . One clearly sees that P = 0, meaning taking only a constant for the polynomial, yields some errors. The solution is clearly improved with P ≥ 1. The relative error defined as |qnum-qana| |qana| is plotted in figure 5. Except for low values of kr, the error is almost constant for a given value of P and the value of the error decreases as P increases. For instance, taking P = 3, leads to an error of less than 1%. Now, we are interested in the influence of the wave number k on the solution of the problem. So, the solutions for different values of kr 0 are presented in figures 6 and 7 in the domain max(kr 0 , 1) ≤ kr ≤ max(kr 0 , 1) + 30. An important factor is the influence of the discretization of the FEM part on the solution for kr 0 < 1. We consider the case of one, five and ten layers of elements to discretize the FEM part. This is important for low frequencies as the solution H 0 (kr) has a singularity when k → 0. To improve the mesh, the FEM part discretizes the domain such that kr 0 ≤ kr ≤ 1 with thicknesses of layers in geometric progression of step 0.7 in this domain. In figures 5(a) and (b), for a very low frequency such that kr 0 = 1/1000, we see that a mesh with three layers is sufficient to get very good results. When kr 0 ≥ 0.1 only one layer is enough to obtain good results. So numerical solutions are very close to the analytical values with a simple mesh of the FEM part except for a very low value of kr 0 for which more element are needed in that domain to approach the singularity of the solution. For values of kr 0 larger than one, one sees in figure 7 that the expansion is sufficient to get good results and it is not necessary to have a classical finite element mesh of a domain as for kr 0 < 1. One must also note that it is possible to get accurate results for a very large range of values of kr 0 starting from very low frequencies at kr 0 = 0.001 to very high frequencies such that kr 0 = 1000.

Two-dimensional circular example

We consider now the radiation by a circle with a Dirichlet boundary condition and a solution satisfying the Helmholtz equation ( 1) in the exterior domain.

To be able to compare with analytical solutions, the boundary value is first built from the field created by a source at a position r s = (x s , y s ) inside the circle. So the boundary value is

q 0 (r b ) = H 0 (k|r b -r s |) (39)
with r b the position of a node on the boundary. We first study the influence of the parameter P giving the number of layers on the solution. In figure 8 we plot the L 2 relative error on the solution on the line 1m ≤ x ≤ 12m for the case of an axisymmetric solution obtained with r s = (0, 0) and a non axisymmetric case obtained with r s = (0.8m, 0m). In both cases, the boundary is the circle with r 0 = 1m, the frequency is f = 1000Hz and the velocity is 340m/s. Each layer is meshed first with linear four nodes elements, secondly with height nodes quadratic elements and the thickness is e = 0.001m. In all cases, the frequency is high enough so that no mesh of a FEM part is needed and only the expansion is used to build the solution. One can see that the optimal value of P is around 9 and that the error is reduced, as expected, by increasing the number of elements on the boundary. Taking second order elements clearly improves the solution for values of P around 9.

We now study the influence of the frequency and plot images of the solution on an annular domain. The source is at (0.8m, 0m) and kr 0 = 10. We plot the solution between radius r 0 = 1m and 7m in figure 9. One can observe a very good agreement between the numerical and analytical solutions.

We consider now the diffraction of a point source by a rigid cylinder of the same geometry. The source is located at (-2m, 0). P = 9 and 500 elements are used upto kr 0 = 10 while 1000 elements and P = 12 are used for kr 0 = 30. The solution is plotted on a semi circle of radius 2m between the angle -π and 0. Figure 10 presents the comparison between the present numerical solution and the analytical solution. Good agreement can be observed.

Two-dimensional square example

We consider now the case of a square with a side of length 2m to test a domain with corners. As for the circle, a point source is located inside the square to build an analytical solution on the boundary. This source is located at (0.5m, 0.5m). In figure 11, one compares the error for first and second order elements. One note a considerable improvement using elements of degree two.

Figure 12 presents the analytical and numerical solutions in the domain external to the square. The frequency is 500Hz, the elements are second order and each side is mesh with 200 elements. One can observe a very good agreement between the two solutions and that the corners do not create any particular problem.

Case of a resonant cavity

To finish, we consider the case of the circular and triangular resonant cavities presented on the left of figures 13 and 14. On the right of these figures, one has the meshes of the different exterior domains used for satisfying equations [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF]. A power four progression was found satisfactory such that in term of ξ we apply the relation on surfaces such that ξ = 1 + d × n 4 . For the circular resonator the interior radius is 0.06m, the medium radius is 0.09m and the exterior radius which also defines the surface S 0 has a radius of 0.1m. The opening connecting the interior cavity to the exterior domain has a width of 0.5cm. For the triangular resonator, the vertex points of the triangles are located on the same circles as for the circular case and the opening connection has the same size. The mesh, stiffness and mass matrices were created with FEniCS and then imported into Matlab. Any other finite element software could have been used. In the case of the circular cavity, a sound source was put at point (0.2m, 0) in front of the opening and the sound pressure was computed at the center of the cavity. The sound pressure versus the frequency is presented in figures 15a and 15b and compared to a BEM computation (obtained with a personal software of the author). A good agreement between the two solutions validates the accuracy of the present method. This can be observed both in term of modulus and real and imaginary parts. We see in particular that the resonance of the cavity is well reproduced.

Figures 16a and 16b consider the triangular resonator whose main difference is a non smooth geometry. The sound source is located at point (0, -0.2m).

In this case again the comparison between the BEM results and the present method is very good. Note that we see in figure 14 that only a small mesh in the exterior domain of the resonator is used because the scale domains follow the triangular geometry as can be seen in these figures.

Conclusion

A new numerical method has been presented for computing radiation and scattering of waves in exterior domains. The solution on the exterior domain is decomposed into the sum of products of polynomial and exponential functions multiplied by discrete FEM vectors. As in the scaled boundary element method there is no need of a fundamental solution but contrary to this method there is also no need of special matrices. The matrices involved in this problem are obtained from classical mass and stiffness matrices from any standard finite element software. One can note that the usual Scaled boundary finite element method (SBFEM) requires the solution of an eigenvalue problem while in the method proposed in this article, there are no modes to calculate but only the solution of a sparse linear system. For a large number of degree of freedom, this can certainly be more efficient. Examples show good agreements with analytical and BEM results. Future work should consider the optimisation of this method, the case of three-dimensional problems and other wave problems such as elastic wave propagation.

Fig. 1 .

 1 Fig. 1. Exterior domain divided into different layers.

Fig. 4 .

 4 Fig. 4. Real part (a) and imaginary part (b) of the solution for different values of P, -P = 0, --P = 1,. . . P = 2,-.-P = 3, analytical solution.

Fig. 5 .

 5 Fig. 5. Relative error on the solution for different values of P, -P = 0, --P = 1,... P = 2,-.-P = 3, ooo P = 4, xxx P = 5.

Fig. 6 .

 6 Fig. 6. Solutions for different values of kr 0 and nc (number of layers in the FEM part):nc = 0, -nc = 1,... nc = 3,-.-nc = 5, x analytical solution.

Fig. 7 . 2 Fig. 8 .

 728 Fig. 7. Solutions for different values of kr 0 and nc:nc = 0, -nc = 1,... nc = 3,-.-nc = 5, x analytical solution.

( a )

 a Numerical solution, real part. (b) Analytical solution, real part (c) Numerical solution, imaginary part. (d) Analytical solution, imaginary part. (e) Error on the solution.

Fig. 9 .

 9 Fig. 9. Numerical solution, analytical solution and error on an annular domain for kr 0 = 10.

(a) kr 0 = 0. 1 (b) kr 0 = 1 ( 30 Fig. 10 .

 113010 Fig. 10. Comparison of numerical and analytical solutions on a semi circle for different frequencies,real part analytical, o real part numerical, -.imaginary part analytical, + imaginary part numerical.

Fig. 11 .

 11 Fig. 11. Error for first (-) and second (--) order elements in case of a square.

( a )

 a Numerical solution, real part. (b) Analytical solution, real part (c) Numerical solution, imaginary part. (d) Analytical solution, imaginary part. (e) Error on the solution.

Fig. 12 .

 12 Fig. 12. Numerical solution, analytical solution and error on a square domain for the frequency 500Hz.

Fig. 13 .

 13 Fig. 13. Mesh of circular resonant cavity and its exterior domain.

Fig. 14 .

 14 Fig. 14. Mesh of triangular resonant cavity and its exterior domain.

Fig. 15 .

 15 Fig. 15. Results for the circular resonator.

  (a) Comparison between BEM (-.-) and the present method (-) for the modulus of the solution of the triangle resonator. (b) Comparison between BEM (-real part, ... imaginary part) and the present method (-real part, -.imaginary part).

Fig. 16 .

 16 Fig. 16. Results for the triangular resonator.


Michela Deni
email: michela.deni@unimes.fr

FOR A HISTORY OF SEMIOTICS OF DESIGN PROJECTS

Keywords: semiotics, project, design projects, methodology

Our work intends to investigate the emergence, development and contributions of semiotics to the vast and complex discipline of design, which has only recently sought semiotic methodology. In particular, in proposing to investigate these developments, I intend to focus on the two semiotic traditions I know best: the Paris school initiated by Algirdas-Julien Greimas; and the Peircean interpretative tradition, as developed by Umberto Eco. These are the two theoretical paradigms that I have utilized in my research on the methodologies used in the study of design and design projects.

INTRODUCTION 1

This work intends to investigate the emergence, development and contributions of semiotics to the vast and complex discipline of design, which has only recently sought semiotic methodology. In particular, in proposing to investigate these developments, we intend to focus on the two semiotic traditions we know best: the Paris school initiated by Algirdas-Julien Greimas; and the Peircean interpretative tradition, as developed by Umberto Eco. These are the two theoretical paradigms that we have utilized in our research on the methodologies used in the study of design and design projects.

DESIGN'S PLACE IN SEMIOTICS 2

We must distinguish the possible levels on which semiotics acts, before we introduce semiotic research on design. This will allow us to situate how semiotics contributed to the field of design research indicating how a historical perspective can account for what led to the emergence of a specific field of study on one hand, and highlight the reasons, the shortcomings and impasses that, in our opinion, slowed the progress and effectiveness of these research projects on the other.

Utilizing Umberto Eco's tripartite division, semiotic research contains three levels of study: general semiotics; specific semiotics; and applied semiotics. The first level addresses the codification and interpretation of signs (linguistic and non-linguistic), proposing a general classification that is derived from the way in which these signs are produced, and investigates meaning and signified. Specific semiotics addresses the specific characteristics of certain expressive manifestations by dividing general semiotics into sub-disciplines based on: the expression channel (visual, auditory, etc.); the type of discourse (legal, political, etc.); cultural practices (cinema, architecture, etc.); the media (semiotics of print, new media, etc.); relationships with other disciplines (semio-linguistics, socio-semiotics, etc.); or particular expressive systems (gestures, objects, etc.). Lastly, applied semiotics studies and analyses a specific case (an object, a film, a novel, etc.). Clearly, semioticians generally work in all three areas, as general semiotics provide the tools and methodology to conduct an analysis that addresses both the theoretical research of general semiotics and the more specific issues that characterize the field to which the object of analysis belongs (which must be addressed with more specific methodologies). In other words, one works in hierarchical process. For this reason, every semiotic research both draws on and enriches other possible levels of study at the same time.

Starting from this distinction, we will outline a brief reconstruction of the history of design semiotics, which focuses on France and Italy. It is therefore partial, for which I take responsibility, and will demonstrate, (after years of research in the field) the legitimacy of this reconstruction 1 .

Taking stock of the semiotics of design is not easy because, apart from a few exceptions (which will be discussed), until the mid-nineties there were only rare publications. In addition, these publications were often hard to find, in which any form of continuity or dialogue between researchers was absent. In most cases, these works were not essays on design object, but rather research in which the reflections on design and everyday objects represented one of the many possible examples, secondary issues of publications whose main theme was another (e.g. the connotation, value, etc. 2 ). For this reason, a real comprehensive theory of the semiotics of objects did not exist, because no author cited, and perhaps was not even aware of, the earlier work of other scholars. In other words, there was no research tradition, not even the term 'semiotics of design and objects', as a specific semiotics. This meant that on each occasion the scholar's reflections started from the beginning, without being able to rely on a wealth of shared knowledge that could have advanced studies in this field more rapidly.

Semioticians themselves, moreover, did not recognize the specificity of these contributions, nor the existence of semiotics of design or of design projects. This was true despite the fact that at the end of the sixties, on several occasions architects and designers had expressed a certain interest in this approach to design. Paradoxically, in the early seventies, the only people that had taken the study of semiotics of design seriously, at least in Italy, were architects and designers. They would spark a heated debate regarding semiotics, adopting it sometimes because it was fashionable, and testing it on real problems in their daily work. Such turmoil was potentially very productive because it ended up highlighting the limits of the discipline, closed and self-referential. It also highlighted theoretical needs that semiotics could have satisfied, if only it had taken the requests of the occasionally provocative debates from the world of design seriously. Clearly, the semiotic community did not think they had much to add to the debate, and had no interest in interacting on the same groundconceptual and metalinguistic -with other disciplines. Perhaps that is why designers' interest in the semiotics of design, as well as that of the semioticians, evaporated, and for fifteen years (until the mid-nineties) it was no longer heard of.

1 For a brief description of the historical tradition of design semiotics see Deni 2002 (with particular attention to the bibliography) and 2011.

2 See Barthes 1957; [START_REF] Greimas | Du sens II. Essais sémiotiques[END_REF][START_REF] Greimas | De l'imperfection[END_REF]Rossi-Landi 1968, 1972. For all these reasons it is difficult to coherently and thematically present the birth of semiotics of design. After a troubled beginning, this field of study was actually only really established at the beginning of the new millennium. A historical account of the semiotics of objects could therefore be presented through the chronological order of the publications referred to here as the most representative. It is precisely in placing the contributions to the semiotics of design in chronological order that one sees the redundancy as well as the problems that arose without being noted and that were later reintroduced. This can be seen both regarding reflections on the value that objects assume and communicate, as well as in questions regarding how they work in relation to the user.

THE PIONEERS 3

The analysis found in Roland Barthes' Mythologies is certainly one of the contributions that represent the origin of the semiotics of design. Barthes was one of the first to question the operation of different language meaning systems, showing that meaning also depends on other concrete manifestations, equally worthy of being studied and demonstrated, through a popular language that sought to move beyond the linguist community. Barthes proposed an analysis of everyday objects beyond their function (toys), in order to emphasize their ability to embody and reproduce meanings in line with precise cultural and gender stereotypes of bourgeois society in the late fifties. Similarly, Barthes also addressed more complex objects, such as the automobile in his well-known analysis of the Citroën DS. In addition (even before Baudrillard) Barthes raised the issue of the value and meaning of materials (e.g. plastic) with semiotic approach, different from the sociological prospective, teaching us to take a good look at the object as an actual ideological discourse framed by a cultural system. Barthes explained how objects come to produce, and reproduce, meanings, rules for use, and interpretation of the object itself precisely through its use. At the same time, the semiotician warned of the unusual mode these communication systems employ in comparison to language. In fact, everyday objects do not arbitrarily use communication rules and set meanings, that are characteristic of the language, but rather establish new rules at every opportunity. Barthes taught us to understand that we have to discover the communicative codes that allow us to interpret the objects that surround us each time. In addition Barthes can also be considered the precursor of the semiotics of practices3 in that his analysis also questioned everyday objects, moving beyond the concept of the decontextualized sign, to the types of use that we would now say transform the simple primary use into the practice of use (children's game for example), later becoming real ways of life that are meaningful and stereotypical.

While coming from a different discipline, another important author for the semiotics of design was Baudrillard (1968), probably the first to treat everyday objects and, with his The System of Objects, was also the first to emphasize how objects produce sense thanks to their mutual relationships during use. Baudrillard also emphasized how humans develop feelings of ownership and attachment in relation to objects, often due to interest in the materials, the colours and thanks to the personalization strategies that are paradoxically proposed by the market itself. The sociologist also stressed the importance of considering everyday objects, taking care always to interpret them in relation to the systems in which they are inserted (functional, gestural, social, cultural, as well as form, colour, etc.), proposing various classifications. In addition, despite having identified (like Barthes) the differences between language and other systems of signification, Baudrillard concluded, in any case, by stating the need for as many classification criteria as there are objects, starting from the fact that objects do not have a meaning of their own. In his work, the sociologist focused on the value of everyday objects that can be identified as symbolic, focusing on their functional use and, more importantly, on aesthetics and style. From here Baudrillard's analysis is mostly based on the values though which objects are placed into economic and cultural systems. This system includes mass produced objects typical of industrial production but Baudrillard points out that, through communication, the miracle of the system becomes, paradoxically, the transformation of standardized objects into objects that are perceived as unique. The result, according to the sociologist, is that individuals look alike while thinking they are all different from each other.

In any case, the first to suggest architecture and design as a preferred field of study for semiotics was Umberto Eco. In 1968 he proposed the founding act of transforming it into specific semiotics in the chapter "Function and Sign: Semiotics of Architecture". This chapter is part of Eco's well-known The Absent Structure, greatly discussed in cultural debates of the time. In this provocatively titled book Eco indicates some critical aspects of structuralism. In this discussion Eco affirms the legitimacy of the establishment of a semiotics of architecture and design. This legitimacy lies in the very definition of semiotics which, according to the semiotician is "the science that studies all communication phenomena as if they were a systems of signs -relying on the assumption that all cultural phenomena are systems of signs, namely that the culture is basically communication" (Eco 1968: 191, our tr.). In this chapter Eco has chosen to define the field, specifying that he was referring to the communicative properties of the function of everyday objects, but did not include aesthetic objects 4 . With 'Function and Sign: The Semiotics of Architecture', Eco was the first, and for a long time the only one, to question the role of theoretical and epistemological semiotics in treating architecture and design. He suggested a potential path by pointing out the need to reflect on the communication of object function. Eco indicated that not only everyday objects signify their function (the primary function, the denotation) and communicate values (the secondary function, the connotation). However, the historical variations that these aspects undergo must be recognized and described.

In my opinion, Eco contributions reflecting on the kind of epistemological approach that a semiotics of architecture and design should take, insisting on the need to investigate how objects produce semiosis, communicate and signify functions and values. In addition, in this chapter Eco became aware of the debate surrounding Functionalism, introducing the importance of the notion of codes 5 in order to better interpret it. He explained how Sullivan's famous formula "the form follows function" 6 was to be understood in the context of a 4 Some recent publications on design include semiotics of objects and practices as well as everyday objects and aesthetic objects. To this regard see Beyaert-Geslin 2012.

5 See Eco 1968. 6 See Sullivan 1896. theory of interpretation, which was still heavily discussed at that time. Even today, 'Function and Sign: Semiotics of Architecture' is one of the classic foundations of semiotics of design and, at least until the end of the 1980s, was the only text that identified an effective model for the interpretation of the communicative function of everyday and architectural objects.

CRITICAL ATTENTION OF DESIGN THEORISTS 4

Tomàs Maldonado is an Argentine design theorist that directed the Hochschule für Gestaltung (Ulm School of Design) from 1954 to 1966, where he introduced an innovative approach regarding training which included the teaching of semiotics to complete the scientific and rational preparation necessary in the formation of a designer. According to Maldonado, a designer's education need to be intellectually and historically rigorous as well as interdisciplinary. It is no coincidence, therefore, that the end of the sixties Tomàs Maldonado was the design theorist that argued the importance of design semiotics more than any other, filling it with many expectations but at the same time not sparing severe criticisms 7 . His criticisms most frequently involved what he called the failure of the semiotic approach to overcome a descriptive approach to design, the use of a language that was incomprehensible even to the designers and architects, a theoretical and methodological approach better suited to linguistic texts that did not allow new more suitable methods to develop. In particular Maldonado turned to Eco and Barthes research on several occasions, but did not appreciate (nor in our opinion know well) the semiotics of the Paris school, only belatedly recognizing some merit in Greimas 8 . In his view, semiotics has been useful for discussing the values and meanings of objects, but this had also entitled designers to abandon functionalism, justifying values and meanings that are not always present in the design. These criticisms circulated in the environment of the education of architects and designers for a long time and created, in my opinion, a prejudice against semiotics that further discouraged development until the end of the eighties.

Gillo Dorfles is another important figure in the twentieth century for his reflections on semiotics among theorists and critics of art and design. In particular, and unlike Maldonado, Dorfles strongly argued that semiotics was useful in enriching objects from a semantic point of view. In his opinion, it was precisely the objects' semantic layer that could reinforce and simplify the uses' function, thanks to semiotics' contributions 9 . In any case, Dorfles had a rather dated competence in semiotics, even for the time of he was writing in, albeit a representative knowledge for the debates that the he had with his colleagues. Dorfles investigated the problem of objects seen as complex and stratified signs, which, however, can be interpreted as the sum of the individual meanings of each sign. In any case, Dorfles' reflections have highlighted the urgent need to develop a semiotic methodology that could interpret objects' meaning. Such a method would help designers to, not just intuitively, navigate the maze of meanings.

7 See Maldonado 1970 e 1997. 8 See Maldonado 1992. 9 See Dorfles 1972.

BEYOND INTENTIONS, AN APPLIED SEMIOTICS 5

After the hiatus that lasted almost two decades, albeit not continuous, important semioticians resumed their consideration of objects. These were fundamental works for subsequent research that began at the end of the last decade of the previous century and transformed the semiotics of design into a research area that became a specific semiotic.

We can mention Luis Prieto's research among these contributions. In his essays he deliberated some of the basic concepts for the development of a semiotics of objects. Among these: the utility of a tool, defining it as the product of an industrial process; the identity of an object, especially that which Prieto defined as its specific identity, which consists in identification of the elements that make it possible to identify an object as a unit that is identical to others in the same collection (such as a pen); numerical identity that characterizes the object as unique, and allows one to recognize it among similar objects (my pen)10 ; the spatial-temporal limitation of objects which allows for the elaboration of classification systems. Prieto's other important observations for the semiotics of design and objects are those that address the concept of relevance, according to which it is necessary to address an object from a specific piece of analysis that will guide the entire interpretative process. In those same terms, the researcher stressed the importance of identifying distinctive elements that guide the exploration of an area of analysis in the object itself. Nevertheless, Prieto did not insist on focusing on one area of analysis instead of others, but rather on the need to maintain a consistent analytical path that acknowledges what the object itself suggests. Prieto is the first semiotician to explain the problem of relevance by referring to the analysis of the objects. This theoretical position was, and still is in my opinion, very useful to the semiotics of design field of studies. It is no coincidence, perhaps, that up until the 1990s semiotic papers on the semiotics of objects and design focused on the occasionally generic (everyday objects) general theoretical aspect, and that there was little interest in conducting specific research which could have developed a methodology for the analysis of specific objects.

Another scholar who gave a significant contribution to the semiotics of objects and design, without, however, coming from our discipline, is undoubtedly Bruno Latour. In the mid-nineties, at a time when semiotics had begun to question the legitimacy of their approach to the analysis of everyday objects again, Latour used the categories of semiotic analysis in various publications, with particular reference to the theory of actants and action programs considered from a narrative perspective 11 . In these terms, Latour introduced the concept of hybrid actants (between subjects and objects), syncretic actants, as intermediaries in the subject-object relationship for some everyday objects. The scholar also drew attention to the importance of interpreting objects in the context of their specific use, the interactions that objects suggest, and considering the places in which they are used. In addition, Latour described the objects in semiotic terms as if they were texts with the user and the designer incorporated, as if the roles were "distributed in advance" and " the designers' and users' instructions are very similar to those of authors and readers in a novel". 12 This concept is apparently very similar to that of Umberto Eco's Model Author and Model Reader 13 . For all of these reasons, the Latour's reflections have had an important role in the development of the concept of the factitiveness of objects; the issue which I have personally worked on for many years 14 . In addition, while not specifically talking about semiotics of the design projects, Latour emphasized the need to pre-inscribe the roles during the design phase, in order to avoid that there would be too large a difference between the prescribed user and actual in-the-flesh one in the final product. The scholar also proposed several important concepts, in my opinion, considered in view of design projects. Among them we can cite transposition, the process that allows one to provide a function for an object with the awareness of working on signification processes on the content (the function) and the expressive (the materiality of the object) level. A good example is the cement speed bump. Latour explained how the designers had several options (linguistic, iconic, object choice) with which to communicate the injunction to "slow down" to motorists. The final choice is, actually a form of translation that could implicate other forms of expression than just the verbal or visual (a road sign, a traffic cop, the speed bump). The choice of one object over another evidently has consequences, but it is also important to analyse the main purpose of a project to see if the result comes to fruition, beyond the reasons that are contextually assessed. The traffic signal, which indicates to motorists to slow down near a school, appeals to morality; the speed bump appeals to the selfishness of the motorists who wants to protect their car's suspension. In this case, Latour describes both how designers realized the need to evaluate the final result and go beyond the reasons why an injunction is respected (moral or egoism), and the effectiveness of awareness of the fact that a content (in this case a function, a value) may be transposed and translated through different expressive forms and substances.

and Tomàs Maldonado published a new book on the classification of technical and technological objects as prosthesis to the type of action, or the perception that these objects permit. In particular, Eco describes objects as prosthesis because they allow for actions that compensate for the shortcomings of the human body, based on this idea he proposes an articulated classification: the substitutive prostheses allow for actions that the body could perform, but which, for contingent reasons, it is not longer able to (eyeglasses); extensive prostheses amplify the body's natural functions (speakers, cutlery, clothes because they increase the body's resistance). Magnificative prostheses are improved extensive prostheses that allow us to do things we would not normally be able to do (microscope, telescope). They are subdivided into extensiveintrusive and magnificative-intrusive prostheses.

Between the end of the last century and the beginning of the new millennium, however, the Paris school had the leading role regarding the quantity of conferences, seminars and publications. As for the semiotic analysis of particular objects, Echian tradition somehow handed its place over to the school of Paris that began to publish the majority of papers.

Jean-Marie Floch is the most important reference for the semiotics of design. At the time, he was the first to address the usefulness of semiotics for design (1990), to focus attention on the enunciative traces present in the objects that guide the actions of the users, and to propose the analysis (the analysis of Opinel is the most famous, 1997) that follows the object description stages introduced by [START_REF] Greimas | Du sens II. Essais sémiotiques[END_REF][START_REF] Greimas | De l'imperfection[END_REF], becoming the classic model for any following analysis. These were also the years in which socio-semiotics developed from contributions that integrated the semiotic tools of the Paris school with a sociological perspective 19 .

Andrea Semprini dedicated several publications to the semiotics of objects. His L'Objet comme procès et comme action (The Object as a Process and an Action) 20 introduces the first systematization of the study of everyday objects. In the first part Semprini proposes a reflection on objects, tracing the contributions of disciplines that historically have treated this issue such as phenomenology, ethnomethodology, and anthropology. In the second part the scholar addresses the analysis of specific objects.

Most of the contributions in this period are reflections on the value of objects, their narrative potential and, to a lesser extent, the actions inscribed in their materiality. This theme includes work by Omar Calabrese, who addresses: the objects' embodied knowledge as well as their know-how; the mediating role of the objects with regard to relationships between human beings; the reasons for the success of objects that have revolutionized action; the concept of innovation in the design and production of objects of daily use 21 . In addition in the same period in Italy, some research on the semiotics of objects was published thanks to an enlightened industrialist, Alberto Alessi. Alessi, with the help of Alessandro Mendini, founded the Centro Studi Alessi (Alessi Research Center), ensuring a permanent collaboration between designers, semioticians and sociologists 22 . At the Research Centre there were workshops where the participants worked and wrote about the value of objects, the emotional and pathemic relationships that bind human beings to objects, use of materials, colours and shapes that act on the aesthesic dimension amplifying passions and emotions towards the objects.

Since the early 2000s there has been a lot of excitement in the field of the semiotics of design and objects, which, in particular in France and Italy, definitively took on the status of a specific semiotic through conferences, theses, and a growing number of publications 23 . From this moment the applied research fields of design, everyday objects, spaces, and interfaces specialized, and the research projects moved on to other problems such as the controversial issue of function. In many cases, the issue of function has been overlooked, and seen as an outdated issue. It was believed that there was not much more to add, or that the concept of affordance 24 had summed up any possible new discovery regarding the function and usability of the objects. Instead, I believe this was only the beginning of a possible reflection on function. On the contrary, this has resulted in a return to sociological themes (such as value), guilty of neglecting an accurate and timely semiotic analysis of the functions and morphology of objects. Based on this background, as a semiotician my goal was to reinitiate a reflection that would help understand object signification modalities, the ways they communicate and guide the actions they perform. Certain publications in this period return attention to the semiotic analysis of function communication (my first studies on factitiveness) 25 , interface, supports, commands 26 , organization of space, and the relationship between user and object.

CONCLUSIONS: FROM THE ANALYSIS TO THE METHODOLOGY OF 7 THE DESIGN PROJECTS

Most of the conferences, seminars 27 and publications on the semiotics of objects came to light in the late 1990s and the early twenty-first century. In addition, in the last few years, at least in Italy, design and architecture departments have established courses in which semiotics has an important role 28 . It is precisely this period in which design students indicate the need to go beyond the semiotic 22 See AA. VV.1996, in paricular Maffesoli 1996, Careri and Fabbri 1996 and Marsciani 1996. 23 See [START_REF] Marrone | C'era una volta il telefonino[END_REF]2001;Fontanille 2001Fontanille , 2001a[START_REF] Fontanille | Sémiotique des objets[END_REF][START_REF] Fontanille | Figure del corpo. Per una semiotica dell'impronta[END_REF][START_REF] Fontanille | Textes, objets, situations et formes de vie[END_REF][START_REF] Deni | Oggetti in azione[END_REF]Zinna 2004, Fontanille-Zinna (eds.) tools of analysis such as important improvements in order to fully understand how objects take on values for the subject, and how objects communicate their functions in precise use circumstances. But what is still missing at the beginning of the new millennia is a systematic reflection on the predictive capabilities of semiotics29 .

For various reasons such as teaching, consulting and research, many scholars are beginning to work with semiotics as the methodology of design projects. In particular, we are referring to both scholars and designers who teach in the applied arts and design departments in France30 , and a group of semioticians trained at the University of Bologna who teach in design schools, polytechnics, design institutes and work in marketing research31 . The primary aim of these scholars is: to provide theoretical and methodological tools to designers and planners that will allow them to organize the content and scope of a design project; to facilitate dialogue between the disciplines and skills involved in the same project (various specialized professional skills); to provide appropriate tools to foresee who the users of the final product will be through semiotics; to properly involve the participants as well as the project client. The methodological proposals stem from the Peircean semiotic tradition, from the School of Paris and Eco. The first is useful for investigating the problem, to classify the ways in which we can intervene or produce semiosis. The second assumes a separation between the expressive plane and the content plane, and goes through the generative process in the reverse direction of that of the analysis 32 . The third, based on Eco's work, is somewhat of a synthesis of the first two, in that it integrates the Hjelmslevian approach in which the expressive plane and the content plane are interdependent. It offers a reflection on enunciation strategies, a fundamental aspect in design, integrating the notion of the design project as a system that requires and, at the same time, expects an analysis of designer (singular or plural) and the user (singular, plural).

In all of the examples the proposed methods provide indications, depending on the case, for proceeding through a series of stages. The semiotic specificity does not refer to a lack of general instruments, but rather to adapting them to each specific project in a detailed manner.

The need to handle each project, and the complex phases of conception in particular is even more evident in contemporary design projects, which operate in the fields of product design, communication design, service design and social design. In all these last design fields, in fact, the information to be managed, as well as the project phases and communities involved, necessitate tools for both communication management and the overall project. In all of these situations, semiotics attempts to provide an adequate response for each specific case.

  2005, Mangano 2008, Beyaert-Geslin 2012. 24 Beginning with Gibson 1979 until the applications of[START_REF] Norman | The Psychology of Everyday Things[END_REF][START_REF] Nielsen | Web usability[END_REF]

	25 See Deni 2002 and 2005. For a discussion of the concept of factitiveness see Greimas and Courtés
	1979.
	26 See Zinna 2002 and 2011.

27 La semiotica degli oggetti [Semiotics of Objects], an annual doctoral seminar organized by Cid Jurado and Deni (directed by U.

Eco), 1995Eco), -1996

; the conferences "Les métiers de la sémiotique" organized by CeReS of Limoges (Centre de Recherches Sémiotiques), 6-8/11/97; 28-29/5/99, several presenters addressed the semiotics of objects.

28 For example at ISIA in Florence the students have 112 hours of Semiotics courses in the second year of the undergraduate program.

In any case semiotics of practices was introduced by Fontanille in 2008.

See Prieto 1975, 1989, 1991, 1995. Regarding the signs of belonging that allow us to recognize an object, seeFontanille 2001, on film.

See Landowski 1989;[START_REF] Semprini | L'objet comme procès et comme action[END_REF] Semprini ed. 1999,

2003; Landowski-Marrone eds. 2001. 20 See Semprini 1995.

In this regard A.Zinna (2002a) wrote an essay on doing semiotics, identifying its main features: description, production, comparison and design.

Belkhamsa-Darras eds. 2010; Darras ed. 2011; Beyaert-Geslin 2012, Beyaert-Geslin, Cardoso and Piponnier.

Such as the editors of the online magazine www.ocula.it, who also published in the two books on the semiotics of design projects; seeDeni-Proni eds. 2008 and Bianchi, Montanari and Zingale eds. 2010.