Serge Dumont
email: serge.dumont@unimes.fr

Frédéric Lebon
email: lebon@lma.cnrs-mrs.fr

Raffaella Rizzoni
email: raffaella.rizzoni@unife.it

Michele Serpilli
email: m.serpilli@univpm.it

Derivation of Imperfect Interface Laws for Multi-Physic Composites by a Multiscale Approach: Theoretical and Numerical Studies

Keywords: Composites, Interfaces, Multi-physic materials, Asymptotic analysis

In the present study, we focus our attention to a specific type of composite, constituted by two media, called the adherents, bonded together with a thin interphase layer, called the adhesive. We assume that the composite constituents are made of different multi-physic materials with highly contrasted constitutive properties. The study considers a generic multi-physic coupling in a very general framework and can be adapted to well-known multi-physic behaviors, such as piezoelectricity, thermo-elasticity, as well as to multifield microstructural theories, such as micropolar elasticity.

Introduction

Structural bonding assembly has become an important technological solution over the past few years and is increasingly replacing bolting assembly [START_REF] Ascione | Strength and stiffness of adhesively bonded GFRP beam-column moment resisting connections[END_REF] as shown in Fig. 18.1). The resulting structure has many advantages, such as weight savings or the elimination of stress concentration. Similarly, in nature there are many living or natural structures that are composed of substructures, cells or soils for example, glued together. There are also many other examples of glued structures in the field of bioengineering [START_REF] Breschi | Dental adhesion review: Aging and stability of the bonded interface[END_REF] as illustrated in Fig. 18.2. Understanding and modelling the bonding process then becomes a necessity. An obvious common point between all these bonded composite structures is the thinness of the adhesive compared to those of the substrates or adherents. This is true for both industrial structures and living or natural structures. This thinness will obviously lead to numerical modelling difficulties. Indeed, the mesh size of the glue will mechanically lead to computations with a very large number of degrees of freedom and therefore very expensive computations. These costs will increase further if the adhesive surface is irregular and has a high roughness. Similarly, in the presence of kinematic or behaviour non-linearities, in the presence of cracks, etc., the costs become prohibitive. "Direct" calculations are then limited to academic cases [START_REF] Dumont | An asymptotic approach to the adhesion of thin films[END_REF]. There are two very different possibilities, either to develop suitable numerical methods [START_REF] Alart | Solution of contact and friction problems using ILU and coarse-fine preconditioners[END_REF][START_REF] Alart | Solution of frictional contact problems by an EBE preconditioner[END_REF][START_REF] Barbie | An automatic multilevel refinement technique based on nested local meshes for nonlinear mechanics[END_REF] or to set up macroscopic models of the adhesive's behaviour. In this chapter, we will focus on the second family of solutions.

There are at least two methodologies in the literature for obtaining constitutive laws of adhesive (interface) behaviour. The most classic is the introduction of phenomenological models, usually based on experimental results, such as Coulomb models, compliances, etc. In this chapter, we will prefer to focus on deductive models. The idea of this methodology is to start from a micromechanical study of the adhesive (interphase) and to deduce, using mathematical methods, an equivalent macroscopic behaviour (interface).

In this chapter, we focus our attention to a bonded composite, constituted by two adherents, bonded together with a thin adhesive. We assume that the composite constituents are made of different multi-physic materials with highly contrasted constitutive properties. The study considers a generic multi-physic coupling in a very general framework and can be adapted to well-known multi-physic behaviors, such as piezoelectricity, thermo-elasticity, as well as to multifield microstructural theories, such as micropolar elasticity (see, e.g. [START_REF] Chatzigeorgiou | Multiscale modelling for composites with energetic interfaces at the micro-or nanoscale[END_REF]. Several works have suggested a generalization of the classical interface models, including the effects of other physical (thermal, piezoelectric, etc.) interactions (dell'Isola and Romano, 1987;[START_REF] Chen | Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects[END_REF][START_REF] Wang | An xfem/level set strategy for simulating the piezoelectric springtype interfaces with apparent physical background[END_REF][START_REF] Firooz | Understanding the role of general interfaces in the overall behavior of composites and size effects[END_REF][START_REF] Saeb | Designing tunable composites with general interfaces[END_REF], and within the framework of linear multifield theories, such as higher order continua theories [START_REF] Placidi | Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials[END_REF][START_REF] Eremeyev | Two-and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids[END_REF].

The analysis has been carried out by means of the asymptotic expansions method, using the thickness as a small parameter. This technique is based on the fact that the thickness of the adhesive can be considered as a small parameter (intended to tend towards zero) and denoted by ε in the following. The asymptotic analysis has been applied to the rigorous derivation of simplified models for complex assemblies, presenting thin interphases, in the field of linear elasticity [START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF]; [START_REF] Dumont | Soft and hard interface models for bonded elements[END_REF]; [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces[END_REF]; [START_REF] Serpilli | An overview of different asymptotic models for anisotropic three-layer plates with soft adhesive[END_REF]) as well as in piezoelectricity, taking into account other physical interactions, micropolar elasticity and poroelasticity [START_REF] Serpilli | An asymptotic strain gradient Reissner-Mindlin plate model[END_REF]; [START_REF] Serpilli | Mathematical modeling of weak and strong piezoelectric interfaces[END_REF][START_REF] Serpilli | Asymptotic interface models in magneto-electro-thermo-elastic composites[END_REF][START_REF] Serpilli | On modeling interfaces in linear micropolar composites[END_REF][START_REF] Serpilli | Classical and higher order interface conditions in poroelasticity[END_REF]). As mentioned above, the asymptotic methods allow to replace the adhesive layer with a two-dimensional surface, the so-called imperfect interface, with non-classical transmission conditions between the two adherents. By defining the small parameter and constitutive properties of the middle layer, we perform an asymptotic analysis. We assume that the multi-physic stiffness ratios between the adherents and the adhesive depend on ε p . As proposed by [START_REF] Caillerie | The effect of a thin inclusion of high rigidity in an elastic body[END_REF], we identify three critical exponents p, corresponding to different imperfect interface models: p =1 , the soft (also called lowly-conducting) multi-physic interface model; p =0 , the hard (also called moderately-conducting) multi-physic interface model; p = -1, the rigid (also called highly-conducting) multi-physic interface model. Following the approach introduced by Rizzoni et al (2014), we characterize the order zero and the order one transmission problems. Finally, a general multi-physic interface model is developed, and numerically tested through the finite element method. In particular, in the framework of piezoelectricity, we compare the results obtained by modeling the adhesive as an interphase, having a thin finite thickness, with the results obtained with the general multi-physic interface model.

Statement of the Problem

We consider the composite assembly constituted of two solids Ω ε ± ⊂ R 3 , called the adherents, bonded together by an intermediate thin layer B ε := S × (-ε 2 , ε 2 ) of thickness ε, called the adhesive, with cross-section S ⊂ R 2 . In the following B ε and S will be called interphase and interface, respectively. Let S ε ± be the plane contact surfaces between the adhesive and the adherents and let

Ω ε := Ω ε + ∪ B ε ∪ Ω ε
denote the composite system comprising the interphase and the adherents (cf. Fig. 1.3a). We suppose that the composite is constituted by a multi-physic material, in which different physical behaviors interact together, such as in piezoelectricity. Its equilibrium state is determined by a collection of order parameters s ε := (u ε 1 ,...,u ε N ,ϕ ε 1 ,...,ϕ ε M ): N vector state variables, namely u ε i , and M scalar state variables, namely ϕ ε k . With the multi-physic state s ε , we associate its conjugated physical quantity t ε = t ε (∇ ε s ε ), where ∇ ε s ε denotes the gradient of s ε . The vector field t ε := (σ ε 1 ,...,σ ε N , D ε 1 ,...,D ε M ) represents a generalized stress field. We also consider the following homogeneous and linear constitutive law:

t ε = K ε ∇ ε s ε ,
where K ε is a generalized linear constitutive matrix. The constitutive tensor K ε satisfies suitable symmetry and positivity properties.

We assume that the adherents are subject to a generalized system of body forces

F ε : Ω ε ± → R 3N ×M and surface forces G ε : Γ ε g → R 3N ×M , where Γ ε g ⊂ (∂Ω ε + \ S ε + ) ∪ (∂Ω ε -\ S ε -).
Body and surface forces are neglected in adhesive layer. On

Γ ε u := (∂Ω ε + \ S ε + ) ∪ (∂Ω ε -\ S ε -) \ Γ ε
g homogeneous boundary conditions are prescribed, so that s ε = 0 on Γ ε u . We assume that on Γ lat := ∂S × (-ε 2 , ε 2 ) homogeneous Neumann boundary conditions are applied. The differential formulation of the governing equations has the following structure:

⎧ ⎨ ⎩ -div t ε = F ε in Ω ε , t ε n ε = G ε on Γ ε g , s ε = 0 on Γ ε u , (18.1)
where

t ε n ε := (σ ε 1 n ε ,...,σ ε N n ε , D ε 1 • n ε ,...,D ε M • n ε
) represents the generalized traction vector on the boundary Γ ε g and n ε the outer normal unit vector to

Γ ε g . Let us introduce the functional space V (Ω ε ):={s ε ∈ H 1 (Ω ε ; R 3N ×M ); s ε = 0 on Γ ε u }.
The variational formulation of problem (18.1) defined on the variable domain Ω ε can be written as follows:

Find s ε ∈ V (Ω ε ) such that Āε -(s ε , r ε )+ Āε + (s ε , r ε )+ Âε (s ε , r ε )=L ε (r ε ), (18.2) for all r ε := (v ε 1 ,...,v ε N ,ψ ε 1 ,...,ψ ε M ) ∈ V (Ω ε )
, where defined by

Āε ± (s ε , r ε ):= Ω ε ± Kε ∇ ε s ε • ∇ ε r ε dx ε , Âε (s ε , r ε ):= B ε Kε ∇ ε s ε • ∇ ε r ε dx ε , L ε (r ε ):= Ω ε ± F ε • r ε dx ε + Γ ε g G ε • r ε dΓ ε .

Method of Asymptotic Expansion

In order to perform an asymptotic analysis of problem (18.2) when ε tends to zero, we rewrite the problem on a fixed domain Ω independent of ε. By using the approach of [START_REF] Ciarlet | On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface[END_REF], we consider the change of variables

π ε : x ∈ Ω → x ε ∈ Ω ε given by π ε : πε (x 1 ,x 2 ,x 3 )=(x 1 ,x 2 ,x 3 ∓ 1 2 (1 -ε)), for all x ∈ Ω ± , πε (x 1 ,x 2 ,x 3 )=(x 1 ,x 2 ,εx 3 ), for all x ∈ B,
where, after the change of variables, the adherents occupy

Ω ± := Ω ε ± ± 1 2 (1 -ε)e 3 and the interphase B = {x ∈ R 3 :( x 1 ,x 2 ) ∈ S, |x 3 | < 1 2 }. The sets S ± = {x ∈ R 3 :( x 1 ,x 2 ) ∈ S, x 3 = ± 1
2 } denote the interfaces between B and Ω ± and Ω = Ω + ∪ Ω -∪ B is the rescaled configuration of the composite. Lastly, Γ g and Γ u indicate the images through π ε of Γ ε g and Γ ε u (cf. Fig. 1.3b). Consequently,

∂ ∂x ε α = ∂ ∂xα and ∂ ∂x ε 3 = ∂ ∂x3 in Ω ± , ∂ ∂x ε α = ∂ ∂xα and ∂ ∂x ε 3 = 1 ε ∂ ∂x3 in B.
We assume that the constitutive coefficients of Ω ε ± are independent of ε, Kε = K, while the constitutive coefficients of B ε depend on ε, Kε = ε p K, with p ∈ {-1, 0, 1}. Finally, we assume that the forces are such that L ε (r ε )=L(r). By virtue of the previous hypothesis, the rescaled problem can be written in the following form:

Find s ε ∈ V (Ω), such that Ā-(s ε , r)+ Ā+ (s ε , r)+ε p-1 â(s ε , r)+ε pb (s ε , r)+ε p+1 ĉ(s ε , r)=L(r), (18.3) for all r ∈ V (Ω):={s ∈ H 1 (Ω; R 3N ×M ); s = 0 on Γ u }, where Ā± (s ε , r):= Ω± K∇s ε • ∇rdx, â(s ε , r):= B K33 s ε ,3 • r ,3 dx, b(s ε , r):= B K3α s ε ,3 • r ,α + Kα3 s ε ,α • r ,3 dx, ĉ(s ε , r):= B Kαβ s ε ,β • r ,α dx,
and Kij denote the sub-matrices of K, defined by

K = Kαβ Kα3 K3α K33 , ( Kij ) T = Kji .
We can now apply the asymptotic expansions method to the rescaled problem (18.3), whose fundamental assumption relies in considering the solution s ε of the problem as a series of powers of ε:

s ε = s 0 + εs 1 + ε 2 s 2 + ..., sε = s0 + εs 1 + ε 2 s2 + ..., ŝε = ŝ0 + εŝ 1 + ε 2 ŝ2 + .... (18.4)
where sε = s ε • πε and ŝε = s ε • πε . By injecting (18.4) into the rescaled problem (18.3), and by identifying the terms with identical power of ε, we obtain, as customary, a set of variational problems to be solved in order to characterize the limit multi-physic state s 0 , the first order corrector term s 1 and their associated limit problem, for p ∈{-1, 0, 1}.

Following the approach described in [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces[END_REF]; [START_REF] Dumont | Soft and hard interface models for bonded elements[END_REF], we introduce the matching conditions based on the continuity of the generalized traction t ε e 3 and multiphyisic state s ε at the interfaces S ε ± in the initial configuration and on the continuity of the traction and state tε e 3 , sε , tε e 3 , ŝε at the interfaces S ± in the rescaled configuration. Hence, one has

[[s ε ]]=[s ε ] -ε s ε ,3 + o(ε), s ε = sε -ε 4 [[s ε ,3 ]], [[t ε e 3 ]] = [ tε e 3 ] -ε t ε ,3 e 3 + o(ε), t ε e 3 = tε e 3 -ε 4 [[t ε ,3 e 3 ]], (18.5)
where

f (x):= 1 2 (f (x, (1/2) + )+f (x, -(1/2) -), x := (x α ) ∈ S, [f ](x):=f (x, (1/2) + ) -f (x, -(1/2) -), f (x):= 1 2 (f (x, 0 + )+f (x, 0 -)), [[f ]](x):=f (x, 0 + ) -f (x, 0 -),
denote, respectively, the mean value and the jump functions at the interfaces.

Multi-Physic Interface Models

In this section we present the asymptotic models for multi-physic interfaces obtained for the soft, hard and rigid cases at order 0 and order 1. For the sake of brevity, we will skip all the mathematical computations carried out in the deduction of the limit models.

The Soft Multi-Physic Interface

The transmission problems at order 0 and order 1 can be summarized as follows:

• Order 0 Governing equations ⎧ ⎪ ⎨ ⎪ ⎩ -div t0 = F in Ω ± , t0 n = G on Γ g , s0 = 0 on Γ u , Transmission conditions on S ± [s 0 ]=( K33 ) -1 t0 e 3 , [ t0 e 3 ]=0.
• Order 1

Governing equations

⎧ ⎪ ⎨ ⎪ ⎩ -div t1 = 0 in Ω ± , t1 n = 0 on Γ g , s1 = 0 on Γ u , Transmission conditions on S ± [s 1 ]=( K33 ) -1 t1 e 3 -Kα3 s0 ,α , [ t1 e 3 ]=-K3α [s 0 ] ,α .
The transmission problems for a soft multi-physic interface at order 0 and order 1 represent a formal generalization of the soft interface models obtained by means of the asymptotic methods in linear elasticity (see, e.g., [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces[END_REF][START_REF] Dumont | Soft and hard interface models for bonded elements[END_REF] and in other multifield frameworks, such as poroelasticity (see [START_REF] Serpilli | Classical and higher order interface conditions in poroelasticity[END_REF]. At order 0, the interface behaves from a mechanical point of view as a series of linear springs, reacting to the discontinuity of the multi-physic state between the upper and bottom faces, while the generalized traction vector remains continuous. At order 1, the interface conditions maintain a similar structure, but both the multiphysic state and the traction vector are discontinuous through the interface. Moreover, they depend on the in-plane derivatives of the jump and mean values of s0 , which can be considered a known source term, identified in the order 0 problem.

The Hard Multi-Physic Interface

The hard interface transmission problems at order 0 and order 1 take the following expressions:

• Order 0 Governing equations ⎧ ⎪ ⎨ ⎪ ⎩ -div t0 = F in Ω ± , t0 n = G on Γ g , s0 = 0 on Γ u , Transmission conditions on S ± [s 0 ]=0, [ t0 e 3 ]=0.
• Order 1

Governing equations

⎧ ⎪ ⎨ ⎪ ⎩ -div t1 = 0 in Ω ± , t1 n = 0 on Γ g , s1 = 0 on Γ u , Transmission conditions on S ± ⎧ ⎨ ⎩ [s 1 ]=( K33 ) -1 t0 e 3 -Kα3 s0 ,α , [ t1 e 3 ]=-K3α [s 1 ] ,α + Kαβ s0 ,αβ .
It is interesting to notice that the hard multi-physic interface problems is equivalent to the ones derived in the case of linear elasticity in [START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF]; [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces[END_REF]; [START_REF] Dumont | Soft and hard interface models for bonded elements[END_REF]. At order 0, we recover the classical continuity conditions for both the multi-physic state and generalized traction vector. Thus, the adherents are perfectly bonded together. At order 1, we encounter a mixed interface model with a jump of the state and traction vector depending on the values of the multi-physic state and traction vector at order 0. These order 0 terms, being known from the solution of the previous problem, can be viewed as external source terms.

The Rigid Multi-Physic Interface

The differential formulations of the rigid interface problems at order 0 and order 1 take the following form:

• Order 0 Governing equations ⎧ ⎪ ⎨ ⎪ ⎩ -div t0 = F in Ω ± , t0 n = G on Γ g , s0 = 0 on Γ u , Transmission conditions on S ± [s 0 ]=0, [ t0 e 3 ]=-Lαβ s0 ,αβ .
• Order 1

Governing equations

⎧ ⎪ ⎨ ⎪ ⎩ -div t1 = 0 in Ω ± , t1 n = 0 on Γ g , s1 = 0 on Γ u , Transmission conditions on S ± [s 1 ]=-( K33 ) -1 Kα3 s0 ,α , [ t1 e 3 ]=-K3α ( K33 ) -1 t0 e 3 ,α -Lαβ s1 ,αβ ,
where Lαβ := Kαβ -K3α ( K33 ) -1 Kβ3 . The rigid multi-physic interface problems

show the same features of the rigid interface asymptotic models obtained in different frameworks in [START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF]; [START_REF] Serpilli | Mathematical modeling of weak and strong piezoelectric interfaces[END_REF][START_REF] Serpilli | Asymptotic interface models in magneto-electro-thermo-elastic composites[END_REF][START_REF] Serpilli | On modeling interfaces in linear micropolar composites[END_REF][START_REF] Serpilli | Classical and higher order interface conditions in poroelasticity[END_REF]. Concerning the order 0 model, we obtain a continuity of the multi-physic state at the interface level, while the traction vector is discontinuous and depends on the divergence of a generalized membrane stress vector N 0 α := Lαβ s0 ,β . The interface behaves as a multi-physic membrane. The order 1 presents a discontinuity on both the multiphysic state and traction vector. Analogously to the order 0 model, we obtain a generalized equilibrium membrane problem defined on the plane of the interface.

The General Multi-Physic Interface

The approach of [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces[END_REF] can be extended in order to obtain a condensed form of the transmission conditions summarizing both the orders 0 and 1 of the soft, hard and rigid cases in only one couple of equations in terms of the jump of the multi-physic state and generalized tractions at the interface.

Therefore, we denote by sε := s0 + εs 1 + ε 2 s2 and tε := t0 + ε t1 , two suitable approximations for sε and tε . Let us consider the rigid multi-physic interface conditions, as starting point. After rescaling back the constitutive coefficients K = ε Kε in B ε , we can write [s ε ] and [ tε e 3 ]. Indeed, one has

[s ε ]:=[s 0 ]+ε[s 1 ]+ε 2 [s 2 ]=-ε( Kε 33 ) -1 Kε α3 sε ,α -tε e 3 + o(ε 2 ), tε e 3 := [ t0 e 3 ]+ε[ t1 e 3 ]=-ε Kε 3α ( Kε 33 ) -1 tε e 3 ,α -ε Lε αβ sε ,αβ + o(ε 2 ).
An alternative expression of the above transmission conditions can be given in terms of tε e 3 and tε e 3 , which will be useful to write the variational formulation of the interface multi-physic problem:

tε e 3 = 1 ε Kε 33 [s ε ]+ Kε α3 sε ,α + o(ε 2 ), tε e 3 = -Kε 3α [s ε ] ,α -ε Kε αβ sε ,αβ + o(ε 2 ). (18.6)
It is easy to prove that this interface law is general enough to describe the interface laws at order 0 and order 1 prescribing the multi-physic state jump and traction jump in the cases of the soft and hard interfaces, by choosing the following scalings for the constitutive matrices: Kε = ε K, for the soft case, and Kε = K, for the hard case.

The relations (18.6) can be transformed into interface equations defined on S,by making use of the matching relations (18.5), up to higher order terms:

te 3 = 1 ε K33 [[s]] + Kα3 s ,α , [[te 3 ]] = -K3α [[s]] ,α -ε Kαβ s ,αβ .
(18.7)

Finite Element Implementation and Numerical Test

In order to derive the variational form of the multi-physic problem, which will be numerically tested through a finite element procedure, one can write the variational form of the equilibrium problem on each sub-domain Ω + and Ω -:

Ω± K∇s • ∇rdx - S t(x, 0 + )n(x, 0 + ) • rdΓ - S t(x, 0 -)n(x, 0 -) • rdΓ = = Ω± F • rdx + Γg G • rdΓ,
which can be rewritten as

Ω± K∇s • ∇rdx + S [[te 3 • r]]dx = L(r),
letting e 3 = n(x, 0 -)=-n(x, 0 + ) and dΓ = dx. Then, using the property

[[ab]] = a [[b]] + [[a]
] b , relations (18.7) and after an integration by parts, we obtain Find s ∈ W ( Ω), such that Ā-(s, r)+ Ā+ (s, r)+A(s, r)=L(r), (18.8)

for all r ∈ W ( Ω):={s ∈ H 1 ( Ω; R 3N ×M ), s| S ∈ H 1 (S; R 3N ×M ), s = 0 on Γ u }, with Ω := Ω + ∪ S ∪ Ω -and A(s, r):= S 1 ε K33 [[s]] • [[r]] + Kα3 s ,α • [[r]] + K3α [[s]] • r ,α + +ε Kαβ s ,α • r ,β dx.
A standard finite element method is employed to solve this equation. In order to take into account the jumps in the displacements across the interface, a 'flat" finite element is considered on the interface S that has all nodes on S, the first ones related to Ω -, and the other ones related to Ω + . It is then possible to write a stiffness matrix of this problem that is invertible and with standard error estimates (for more details, see for example [START_REF] Nairn | Numerical implementation of imperfect interfaces[END_REF].

Numerical Study: The Piezoelectric Composite Plate

The aim of this study is to numerically test the general interface law, expressed in (18.8), comparing it to a three-dimensional analysis of the problem. As preliminary test, we consider the piezoelectric case. The piezoelectric state at the equilibrium is determined by the pair s := (u,ϕ), where u and ϕ represent the displacement field and the electric potential. The generalized stress vector is given by t := (σ, D), where σ and D denote, respectively, the Cauchy stress tensor and the electric displacement.

Let us consider a piezoelectric three-phases composite plate, which occupies a 3D domain defined by

Ω =[ 0 ,L 1 ] × [0,L 2 ] × [-h/2,h/2],
with L 1 =1 0 h and L 2 =5h (see Fig. 1.4). The adhesive thickness is set to be ε. The adherents are constituted by PVDF (Polyvinylidene fluoride), a monoclinic piezoelectric material with poling axis e 3 , while the adhesive is made of PZT-4, a transversally isotropic piezoelectric material with poling axis e 3 . This constitutive sub-matrices (K ij ) are defined as follows: The elastic, dielectric and piezoelectric coefficients are listed in Table 18.1. The piezoelectric composite plate is subject to surface uniform load equal to p =1kN/m 2 on the top face, as shown in Fig. ??. We assume that no voltage is applied on the upper and lower faces. In this case, the composite plates behaves as a sensor (see [START_REF] Bonaldi | An asymptotic plate model for magnetoelectro-thermo-elastic sensors and actuators[END_REF]. The finite element simulations (made with GetFEM) are carried out using Q1 elements (linear on a cube), with 7280 nodes (29203 degrees of freedom) for the three phases problem and 5824 nodes (23379 degrees of freedom) for the problem with the interface law.

K 33 = ⎛ ⎜ ⎜ ⎝
First, the influence of the relative thickness of the interphase ε L is investigated in order to evaluate the accuracy of the various modeling proposed in the previous sections. In particular, the quality of the solutions is evaluated considering the L 2relative error e = s ε -s model s ε

, where s ε denotes the reference solution computed using the three-phases problem with a fine finite element mesh, while s model indicates the solution of the interface model (18.8). The convergence of the general interface model towards the three-phases one with respect to the thickness ratio ε L is presented in Fig. 18.5. From the plot, it can be observed that, by reducing the thickness of the adhesive, the relative error has a drastic reduction and so, the proposed general interface model provides an acceptable solution and it is able to correctly approximate the solution s ε . The convergence rate is ε 3 . Besides, even if the relative thickness is of 1%, the relative error is equal to 7.65 • 10 -2 % for the displacement field and 9.06 • 10 -4 % for the electric potential, meaning that the general interface model can also be used for moderately thick adhesive layers. Now, let us fix the relative thickness ε L =0 .02. The numerical results for the variables are provided using the dimensionless units. We set:

(U i ,Φ)= E 0 V (u i , ϕ E 0 )( T ij , D k )= hE 0 C 00 V (σ ij ,E 0 D k ),
where we have chosen, for numerical convenience, V =50V , E 0 =10 9 Vm -1 and C 00 =1GP a. The results are represented in Fig. 18.6, 18.7 and 18.8. Figure 18.6 represents the trend of the displacement field and electric potential, evaluated in x = L 1 /2,y = L 2 /2,z/h ∈ [-0.5, 0.5]. The plot shows a good agreement between the solution of the general interface problem (dotted line) and the solution of the three-phases problem (solid line). The composite plate behaves mostly as a Kirchhoff-Love single-layer plate, taking also into account the transversal deformation of the adhesive. From the electric point of view, the electric potential is constant through the adhesive: this is due to the fact that the intermediate layer (PZT-4) has a higher electrical conductivity with respect to upper and lower bodies (PVDF), see Table 1, and, hence, it behaves as a highly conducting interface.

Figure 18.7 and Fig. 18.8 represent the trend of the jumps of the displacement and electric potential and the jumps of the stress vector and normal electric displacement along the x-axis, namely (x ∈ [0,L 1 ],y = L 2 /2,z =0 ) , and y-axis, namely (x = L 1 /2,y =∈ [0,L 2 ],z =0 ) . The numerical simulations highlight that the proposed model is able to describe the mechanical behavior of the composite. Few solution oscillations can be found close to the lateral boundaries, due to the presence of edges, which produce expected stress concentrations and singularities.

Concluding Remarks

General imperfect contact conditions have been proposed, simulating the behavior of a thin interphase undergoing linear coupled multi-physic phenomena. These con- ditions link the generalized traction vector field and its jump to the multi-physic state vector field and its jump at the interface, which is the geometric limit of the interphase as its thickness parameter ε goes to zero. Three interface models (soft, hard and rigid) have been deduced by means of the asymptotic methods, by varying the rigidity ratios between the adhesive and adherents and considering the order 0 and order 1 corrector terms. Furthermore, these three different models have been condensed in one general imperfect interface model and its variational formulation has been presented. The weak formulation represents a key step towards the FEM simulation. A numerical example has been presented considering a piezoelectric composite plate, subject to an electric potential difference at the top and bottom

Fig. 18 . 1

 181 Fig. 18.1 An example of bonding, for civil engineering structures.

Fig. 18 . 2

 182 Fig. 18.2 An example of adhesion, for dental structures.

Fig. 18 . 3

 183 Fig. 18.3 Initial (a), rescaled (b) and limit (c) configurations of the composite.

Fig. 18 . 4

 184 Fig. 18.4 The geometry of the piezoelectric composite plate in the plane (x, z).

  For the transversally isotropic material with poling axis e 3 , one has c 11 = c 22 , c 13 = c 23 , c 55 = c 44 , c 66 =( c 11 -c 12 )/2, e 15 = e 24 , e 31 = e 32 and H 11 = H 22 .

Fig. 18 . 5

 185 Fig. 18.5 Convergence results with respect to the thickness ε L .

Fig. 18 . 6

 186 Fig. 18.6 Representation of the displacements and the electric potential on a section along the z-axis.

Fig. 18 . 7

 187 Fig. 18.7 Representation of the jumps in the displacements and the electric potential across the interface on a section along the x-axis and y-axis.

Fig. 18 . 8

 188 Fig. 18.8 Representation of the jumps of the stress vector and normal electric displacement across the interface on a section along the x-axis and y-axis.

Table 18

 18

	.1 Piezoelectric material properties
	Moduli	PZT-4	PVDF
	c 11 ,GPa	139	238.24
	c 22 ,GPa	139	23.6
	c 33 ,GPa	115	10.64
	c 12 ,GPa	77.83 .98
	c 13 ,GPa	74.32 .19
	c 23 ,GPa	74.31 .92
	2c 44 ,GPa	25.62 .15
	2c 55 ,GPa	25.64 .4
	2c 66 ,GPa	30.66 .43
	e 31 , C/m 2	-5.2	-0.13
	e 32 , C/m 2	-5.2	-0.145
	e 33 , C/m 2	15.1	-0.276
	e 24 , C/m 2	12.7	-0.009
	e 15 , C/m 2	12.7	-0.135
	H 11 , nF/m	13.06	0.111
	H 22 , nF/m	13.06	0.106
	H 33 , nF/m	11.51	0.106

faces. The numerical results show the convergence of the solution of the three-phases model towards the solution of the proposed model as ε tends to zero, highlighting the accuracy and "goodness" of the general interface conditions.


Julie Jovignot
email: julie.jovignot@hepvs.ch

Thomas Hausberger²

Transitions in abstract algebra throughout the Bachelor: the concept of ideal in ring theory as a gateway to mathematical structuralism

Keywords: teaching and learning of linear and abstract algebra, transition to, across and from university mathematics, structuralist praxeologies, anthropological theory of the didactic, ideal in ring theory

. To do so, we will place ourselves in the framework of the Anthropological Theory of the Didactic (Chevallard, 2000) and provide salient results of praxeological analyses of our corpus. The latter is made up of the teaching material of three French post-secondary teachers. These analyses will allow us to study continuities and ruptures in the praxeologies of the abstract algebra track throughout the Bachelor in France, but also to shed light on the way in which the professors manage the transitions in the development of structuralist praxeologies.

INTRODUCTION

The issue of transitions has received increasing attention in research on mathematics education, resulting in a recent topical ICME survey [START_REF] Gueudet | Transitions in Mathematics Education[END_REF]. The state of the art of the literature reviewed in the survey underlines different facets (cognitive, epistemological, socio-cultural, institutional,...) of the transitions. The latter are investigated in terms of continuities, discontinuities, ruptures that occur at different transition points (e.g. from school to university) and research-based devices are proposed to accommodate them. Even more recently, [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF] report on the works on this theme carried out within INDRUM. This paper focuses on transitions across university mathematics courses. Such transitions have been mainly investigated in the context of the analysis path (loc. cit., p. 203-204). In particular, several studies adopt the institutional perspective offered by the Anthropological Theory of the Didactic (ATD; [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF] and refer to a model of the calculus to analysis transition introduced by Winsløw (2006). In their concluding section, [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF] raise the question of the situation for the other fields, and, in reference to Winsløw, "what does the observed jump from the first to the third stage of the model mean in a longer-term perspective, e.g. taking into account what students learn in more advanced mathematics studies?" (p. 210).

Our goal is to investigate these questions in the context of the abstract algebra path, with an epistemological and institutional lense. It is based on the perspective opened up by [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF]: the concrete to abstract transition identified by Winsløw in analysis is generalized in the form of the wider perspective of the teaching and learning of mathematical structuralism at large -in ATD terms, the development of structuralist praxeologies. A new three stage model has been proposed by Hausberger and applied to abstract algebra in a small-scale pioneering study. A larger-scale study, centered on second and third year post-secondary teaching practices in France and Switzerland, is carried out by Candy (2020a) in her PhD project with a focus on the concept of ideal in ring theory. This choice is motivated by the role played by the concept of ideal (Corry, 2004, p. 15): its central importance for the theory of abstract rings and, even more, for the "rise of structures" due to its strong interconnections with other algebraic concepts (fields, modules, groups, etc.). A model for such praxeologies taught at the second year of post-secondary studies in France has been presented at INDRUM2020 [START_REF] Candy | Etude de l'enseignement du concept d'idéal dans les premières années post-secondaires : élaboration de modèles praxéologiques de référence[END_REF]. This paper reports on the results of the PhD project that connect to the issue of transitions, in France. It addresses the following research questions: What continuities and ruptures can be observed in the praxeologies of abstract algebra that involve the concept of ideal, as they are taught throughout the Bachelor in France? How are transitions in the development of structuralist praxeologies handled by abstract algebra teachers? Both questions are related since our methodology is based on the analysis of teaching material provided by selected teachers, under the light of Hausberger's model. It also refers implicitly to didactic transposition processes [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF], but the discussion of conditions and constraints that explain the observed states of equilibrium within institutions are out of the scope of the paper.

We begin by presenting our theoretical framework and the model, and then outline the methodology for analyzing the data. We illustrate the methodology through its application to selected excerpts from exercise sheets. Then, we discuss the results obtained in relation to the research questions, before concluding with the highlights of the study and prospects for further development. [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF] highlight the following main features of ATD that justify its frequent use in research on transitions: the consideration of knowledge as living within institutions, the institutionalization of knowledge seen as the result of complex processes of didactic transposition subject to a set of conditions and constraints at various levels, and finally the central 4T-model of praxeologies (task, technique, technology, theory) that allows researchers to build reference models of the knowledge to be taught for application to teaching-learning phenomena. We direct the reader unfamiliar with ATD to the mathematics education encyclopedia [START_REF] Chevallard | Anthropological Theory of the Didactic (ATD)[END_REF] for an introduction to these notions and will focus the rest of our discourse on the additional tools, specific to the point of view of mathematical structuralism, that have been developed.

THEORETICAL FRAMEWORK

Structuralist praxeologies and their levels. The starting point is the consideration of mathematical structuralism as a methodology, which consists of reasoning in terms of classes of objects, relations between these classes and stability properties for operations on structures [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF]. The general view of structures thus allows particular properties of objects to be demonstrated by making them appear as consequences of more general facts (theorems about structures). Dually, generealizations are put to the test of objects, hence a dialectical relationship between objects and structures. In praxeological terms, the simplification produced lies in the passage from a praxeology Р=[T/?/?/Ө particular ] where it is unclear which technique to apply, to a structuralist praxeology P s =[T g /τ/θ/Ө structure ] where, modulo generalization of the type of task (T g ), the theory of a given type of structure guides the mathematician in solving the problem. Furthermore, [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF] distinguishes several levels of structuralist praxeologies: at level 1, structures act as a vocabulary and appear mainly through definitions (e.g., the type of tasks "prove that a ring A is a principal ideal domain (PID)" is solved by showing, by hand, that the definition is satisfied, i.e. that any ideal is monogeneous); at level 2, the technique used mobilizes general abstract results about structures (on our example, one shows the existence of an Euclidean algorithm, which invokes in the logos of the praxeology the structuralist theorem that any Euclidean ring is a PID).

Transitions in the development of structuralist praxeologies. Following Winsløw, Hausberger (2018, p. 89) proposes a three-phase model (Figure 1): while the first type of transition amounts to going from P to P s , the second type leads to praxeologies whose entirety of praxis and logos lies in the abstract. To unfold our example, the student then encounters tasks like "show that a Noetherian integral domain such that any maximal ideal is principal is a PID". Contextualization and decontextualization of structuralist praxeologies in relation to the dialectic of objects and structures. In the example given, the ring A plays the role of a didactic variable of the type of tasks: the structuralist praxeology is thus contextualized to domains of mathematical objects, whose variation is crucial to lead -in fine -to a decontextualization (the ring A is defined abstractly). We will be particularly attentive to the choices made by university teachers in relation to these didactic variables which are essential to operate the objects-structures dialectic.

METHODOLOGY

In order to shed light on didactic choices that concern transitions in abstract algebra throughout the Bachelor, we conducted a case study of 9 university teachers considered representative of 5 teaching levels, in France and Switzerland (Candy, 2020a). In this article, we will rely on data from three teachers: MP1, EC2 and EC4. MP1 teaches mathematics in the second year of the Classes Préparatoires aux Grandes Ecoles Mathématiques-Physiques (CPGE-MP; these are classes reserved for the best students who are destined to enter the French Grandes Ecoles), EC2 teaches in the second year of the Bachelor's degree in a 7.5 ECTS course called "Linear Algebra" and EC4 teaches in the third year of the Bachelor's degree in a 5 ECTS course called "Elements of Ring and Field Theory". EC4 teaches in the same university as EC2; moreover, some of the students of the third year of the Bachelor's degree come from a CPGE-MP. Thus, in this article, we can study the transitions through two possible curricula experienced by those third year Bachelor students.

In our study, we began by conducting an ecological analysis of the official sylabi in order to bring to light the places where the concept of ideal lives. Then, our praxeological analysis of the course documents (lecture notes and tutorial sheets) consisted in highlighting the praxeologies that mobilize the concept of the ideal within the exercise sheets of the corpus. When the exercises were not corrected, we used the correction of exercises of the same type of tasks present in the institution. We took care to link the exercises to the contents of the lectures, which allow to identify the global organization of the praxeologies (their unification by common technologies or theories, within themes or sectors of study) and to provide certain technological and theoretical elements that remain partially implicit in tutorials.

Finally, the structuralist level of the praxeologies has been carefully noted, as well as the choice of the didactic variables of contextualization of the structuralist praxeologies. The aim is to analyze the continuity and rupture that can be observed in relation to the two types of transitions described in the model, under the hypothesis that the type I (epistemological) transition would be situated at the level of the (institutional) transition between the second and third year of the Bachelor's degree, while the type II transition would be linked to that of the Bachelor's to the Master's degree.

PRAXEOLOGICAL ANALYSES

In this section, we illustrate our analytical tools on salient excerpts from the corpus while reporting on the main findings of our analyses. The discussion of the results in order to answer the research questions will be the subject of the next section.

CPGE-MP: the MP1 corpus

The analysis of the official program of CPGE-MP 1 allows to identify three main habitats of the concept of ideal. The latter is introduced by its general definition in the sector "common algebraic structures", at the level of the theme "ideals of a commutative ring" where it is linked to the notion of ring homomorphism (as kernel) and to the notion of divisibility (inclusion of ideals), then illustrated in the case of the ring Z. It is then taken up again at the level of the theme "rings of one-dimensional polynomials" of the same sector, through the item "ideals of K[X]". One can note that the program does not mention the principality property (of an ideal, of a ring) and that it does not underline the analogy between the arithmetic of Z and that of K[X] (which follows from the principality). Nevertheless, the implicit organization of the contents is based on this analogy. Finally, the concept of ideal is mentioned in the 1 https://cache.media.education.gouv.fr/file/special_1_MEN_ESR/42/4/MP-mathematiques_287424.pdf theme "polynomials of an endomorphism, of a square matrix" of the sector "reduction of endomorphisms and square matrices" where the properties of ideals previously studied allow to justify the existence of the minimal polynomial.

The study of the objects at stake in the exercise sheets shows a diversity of contextualizations, although limited to numbers and polynomials. For the principal ideal domains (PID), the classical examples Z and K[X], quoted in the official program, are mainly worked on through their arithmetic (definition of gcd and lcm in terms of ideals) and the consequences in linear algebra of the principality of K[X]. The ring Z[X] is studied as a non-example of a PID. Finally, MP1 has chosen to introduce Z[i] (the Gaussian ring of integers, whose historical importance in the development of abstract algebra is well known) and the set D of decimal numbers to work on the principality of Euclidean rings on less classical examples.

The analysis of the tasks shows that the students' work is mainly situated at level 1 of the structuralist praxeologies. Thus the type of task T 1 (to demonstrate that a subset I of a ring A is an ideal), present in 2 occurrences, gives rise to a praxeology whose technology is based on the definition of an ideal and the type T 2 (to demonstrate that a given Euclidean ring A is a PID, 4 occurrences), in spite of the genericity of the technique, proceeds by hand from the definition of principality. It is at the level of a meta-discourse that the teacher underlines the analogy between the two contexts and the generality of the method, without going so far as to quote a structuralist theorem (figure 2). Indeed, the theorem in question is not on the syllabus; its status is that of a cultural element and the definition of a Euclidean ring is not formalized. Only one occurrence of T 1 leads to a level 2 structuralist praxeology where the technique uses the structuralist theorem characterizing ideals as kernels of ring homomorphisms. The abstract tasks, 3 in total, are situated within the same exercise devoted to the notion of radical √I of an ideal I of a ring A: it is proved to be an ideal (T 1 ) then appears the type of task T 7 (to prove properties of operations on ideals). This last type, introduced during the lectures on the gcd and lcm (addition and intersection of ideals), is carried out here in an abstract context, about a new operation whose behavior with respect to the two previous ones is studied (e.g. √I ∩ J=√ I ∩ √J), to finally be contextualized to Z through the task of determining the radical of an ideal of Z. These tasks show a dialectic between contextualization and decontextualization, since the general formula may be used to reduce to computing the radical of prime ideals of Z. It is a local implementation of the dialectic between objects and structures in the sense of [START_REF] Hausberger | Structuralist praxeologies as a research program on the teaching and learning of abstract algebra[END_REF], but the notion of radical remains weakly motivated.

Second year of the Bachelor: the EC2 corpus

We notice that the concept of ideal does not appear explicitly in the official syllabus of EC2. However, the same niches as in the case of MPSI-MP are likely to be invested, since the arithmetic of polynomials and the reduction of endomorphisms are part of the study program. Although he starts his course with a chapter "small panorama of algebraic structures" (like MP1), EC2 chooses to introduce the ideal concept at the level of the theme "arithmetic of K[X]" of the sector "the algebra K[X]", which constitutes its main habitat, with as niche the principality of K[X] and the reformulation of the gcd (defined from the divisibility relation) in terms of ideals. Not surprisingly, these results are subsequently applied to the theme "polynomials of endomorphisms" of the sector "reduction of endomorphisms".

Of the 7 tasks on ideals contained in the tutorial sheets, only two are contextualized (to K[X], one to prove principality and the other to prove the existence of the lcm of two elements a and b of K[X], via the introduction of a generator of the ideal (a)∩(b)). They appear as isolated tasks aimed at proving theoretical elements stated in the course. We identify a single proper praxeology in this corpus, generated by the type of task denoted previously T 1. Its 4 occurrences are all decontextualized: it consists in proving that the sum and intersection of ideals is still an ideal, starting with the case of principal ideals. This shows a deficit of the objects-structures dialectic. A last abstract task is given in connection with the reduction of endomorphisms: it is to prove that the kernel of a homomorphism of algebras is an ideal. It shall be noted that the definition of a ring homomorphism (and its intimate link with the notion of ideal) are not mentioned in the course.

Third year of the Bachelor: the EC4 corpus

The syllabus of EC4 is quite detailed: it includes both a large panel of concepts (ideal, ideal generated by a subset, quotient rings, prime/maximal ideal, PID/euclidean ring/unique factorization domain) and the study of their properties in a structuralist perspective (behavior of ideals under homomorphisms, isomorphism theorems), from which structuralist theorems result (e.g. characterization of prime and maximal ideals by quotient properties), but it also mentions specific examples that must be treated ( Z,K[X],Z[X],Z[i]). We thus find the paradigmatic examples used by MP1 and the perspective of unification between the contexts of numbers and polynomials, which gives meaning to the abstract theorems. Unlike MP1, the learning goals are organized around the structures, it is no longer the numbers or the polynomials that are put to the fore.

EC4 reintroduces in his course all the basic notions related to ideals even if EC2 had already introduced some of them. The structure of the course allows to link the 12 exercises dealing with ideals to the 4 following sectors: "ideals and quotients", "polynomials and ideals", "prime and maximal ideals", "operations on ideals". Considering their complexity (we have identified 41 different tasks), we can only sketch the corresponding praxeologies and refer the reader to the thesis manuscript (Candy, 2020a, appendix A5, p. 400-410) for a detailed description.

Of these 41 tasks, 18 use a definition and 23 use a theorem about structures. This highlights EC4's didactic intention to reach a structuralist level 2. More precisely, the praxeologies mobilizing prime or maximal ideals (for example, the praxeology generated by T 18 : show that an ideal I of a ring A is prime, figure 3) mostly use as technology the isomorphism theorems or the Chinese theorem; they thus work at a level 2. On the other hand, those concerning the principality of ideals (e.g. T 13 : show that an ideal I of a ring A is principal) have as technology the definition of a principal ideal and are thus mostly done "by hand". Indeed, they are mostly situated in the sector "polynomials and ideals" and are contextualized to K[X]. We do not note any subsequent development of the praxeology at the structuralist level 2, when the principality of Euclidean rings is known. Of these 41 tasks, 10 are abstract tasks. Within the corpus, we find occurrences of task types that are first contextualized and then decontextualized in order to demonstrate a general result (but not a generalization of the former contextualized occurrences). This is the case, for example, of T 18 , contextualized twice in Z[X] in exercise 7 (figure 3), before being posed to the case of the inverse image of any prime ideal by a ring homomorphism, and then to the case of an ideal P defined abstractly by a condition that may be interpreted as a rewriting, in the set-theoretic terms of ideals, of the definition of a prime ideal (figure 4). The corresponding praxeology is of structuralist level 1, unlike the one applied to Z[X]; we can regret that the usefulness of these results is not highlighted by new contextualizations, in the spirit of the dialectic between objects and structures. The other abstract tasks are related to the discussion of the consequences of the presence of invertible elements in an ideal, to the determination of the ideals of a Cartesian product of rings and its consequences for a quotient of such a product. These are therefore theoretical results which are intended to feed contextualized structuralist praxeologies of level 2. For example, the question of the determination of the ideals of Z 2 is posed in application of the principality of Z and the results on the Cartesian product. A praxeology generated by T 13 is thus obtained, at structuralist level 2. As suggested by the analysis of the syllabi, the type 1 transition does not appear as a learning goal in the CPGE-MP and second year Bachelor institutions under study. Accordingly, when tasks are contextualized to object domains, mainly K[X] and Z, the technique consists in applying the definitions of concepts without relating the properties at stake to general results. Thus, the structuralist praxeologies involved are all level 1 and the objects-structures dialectics remains largely invisible in these institutions.

DISCUSSION

Nevertheless, MP1 chose to introduce a relative diversity of examples of PIDs which are all Euclidean, and he uses meta discourse to allude to the underlying general principle (a structuralist theorem). This didactic gesture may be considered an intention to facilitate the type 1 transition. At university, EC2 introduces structures as a "vocabulary" and assigns abstract tasks on basic formal properties of ideals. The type 1 transition is therefore envisaged from a top-down perspective: although the course content is organized around domains of objects (polynomials, matrices, endomorphisms), concepts are introduced beforehand and taken as a given. This strategy may be questioned since it may be argued that the resulting level 1 structuralist praxeologies will tend to be weakly motivated.

It is from the third year of the Bachelor's degree onward that the type 1 transition appears as a real objective. We have seen, in the EC4 course, that the syllabus is organized around structures. Moreover, EC4 deploys praxeologies, in numerous contexts, which mobilize the concept of ideal at a structuralist level 2, notably around prime/maximal ideals and the isomorphism theorems or the Chinese theorem. The objects-structures dialectic is at play and assigned abstract tasks punctually serve to subsequently develop contextualized level 2 structuralist praxeologies.

Type 2 transition does not appear as a learning objective in the institutions under study. Abstract tasks are rare in CPGE-MP and restricted to the first basic properties of ideals in the second year of Bachelor. At the third year, EC4 assigns tasks to explore structuralist properties of ideals (e.g. behavior under inverse image, Cartesian product). Nevertheless, the main application is the enrichment of contextualized level 2 structuralist praxeologies. A body of abstract knowledge in which ideals play a major role (e.g. Elimination Theory, Algebraic Geometry) is not in the horizon. However, the scope of exercise 8 (figure 4) may be connected to Algebraic Geometry. The teacher probably had this connection in mind, but it remains invisible to the students. Thus, the type 2 transition is left to the Master's degree.

Our analyses are limited to the case of three professors chosen as representatives of their respective institutions. However, Candy (2020a) provides results on a larger corpus that support these analyses. In the context of this article, we can provide some initial answers to our research questions.

To answer the question of continuities and ruptures which can be observed in the abstract algebra track around the concept of ideal through the Bachelor in France, it seems important to us to recall that the students who take the course of EC4 could have first followed the course of EC2 or that of MP1. However, the treatment between EC2 and MP1 appears different. In both cases, the praxeologies involved are of structuralist level 1. But, if EC2 chooses to work on praxeologies in a decontextualized way, MP1 chooses to work on contextualized praxeologies and to accompany the type 1 transition by meta-discourse. Thus, students from the MP1 course could be better prepared for the upcoming transition since the structuralist perspective is pointed out as an horizon.

Structuralist praxeologies at the third year of Bachelor are either of level 1 or 2. However, discontinuities in the type 1 transition can be noted since structuralist praxeologies are most of the time elaborated from theoretical elements provided a priori, in a top-down perspective. The analogy between the arithmetic of numbers and polynomials, carefully developed by MP1 in a bottom-up perspective, remains a missed opportunity to develop a structuralist praxeology in the EC4 course, since all students do not share such a background. Finally, the transition of type 2 is not worked out; it would be necessary to analyze a corpus of teaching material at the Master's degree to measure the epistemological gap in praxeological terms.

As for the management of transitions by teachers, our study tends to show, for both type 1 and type 2 transition, that it is the objects-structures dialectics which is central in the management of transitions by teachers. For the type 1 transition, this can be done through meta-discourse which deals with the identification of a technique present in the structuralist level 1 praxeologies which could be generalized to a structure class in order to create a technological element of level 2 (figure 2, MP1).

The type 1 transition can also be managed through the choice of variables contextualizing the tasks which favour the use of level 2 praxeologies and make level 1 praxeologies costly. This is the case, for example, of EC4 in the framework of prime and maximal ideals (figure 4). Finally, for the type 2 transition, we have seen that it can be punctually favored by a set of contextualization then contextualization as in figures 3 and 4 of the EC4 corpus.

GENERAL CONCLUSION AND PERSPECTIVES

This analysis allows us to highlight the objects-structures dialectic as a nodal point of the management of transitions in abstract algebra. The interplay of contextualizations and decontextualizations allows the passage from structuralist praxeologies of level 1 to 2 by recognition of a unifying technology on contextualized praxeologies (type 1 transition); then, decontextualization offers an opportunity to engage in abstract tasks with familiarity gained from contextualized praxeologies (type 2 transition).

The phenomena observed in this reduced corpus would benefit from being tested in the context of a larger corpus, as is the case with Candy (2020a), so as to be able to argue for the presence of dominant praxeological models within institutions. They are also to be related to the ecological study of the conditions and constraints that are exercised at the level of the different institutions, in order to shed light on the states of equilibrium reflected in these dominant models. Moreover, our results suggest that this equilibrium is likely to be unstable with respect to the management of transitions in structuralist algebra, and this instability should also be investigated.

Figure 1 :

 1 Figure 1: a model for transitions in the development of structuralist praxeologies

Figure 2 :

 2 Figure 2: example of a structuralist theorem that remains implicit

Figure 3 :

 3 Figure 3: illustration of T18 in EC4's corpus

Figure 4 :

 4 Figure 4: one occurrence of T 18 , decontextualized

Table 1 :

 1 summary of the main results

		Summary of main characteristics	Type transition	1	Type transition	2
	MP1	Praxeologies mainly contextualized and limited to the structuralist level 1; a relative diversity of contexts (Z, K[X], Z[i], D)	Spotted meta discourse) preparation (via	Absent
	EC2	A single type of tasks dealt with at structuralist level dialectic is extremely weak 1 and in an abstract context; the objects-structures	Absent		Absent
	EC4	Work at structuralist levels 1 and 2; decontextualized instances of praxeologies are introduced to establish structuralist properties of contextualized praxeologies of structuralist level 2 ideals and punctually serve to develop	At the center of the course	Spotted preparation (implicit)


Thomas Hausberger
email: thomas.hausberger@umontpellier.fr

Heidi Strømskag
email: heidi.stromskag@ntnu.no

TWG1: Transitions to, across and from University Mathematics

INTRODUCTION

A specific thematic working group dedicated to the topic of transitions first appeared at the INDRUM conference in 2018. To attend to the growing number of papers, the programme committee introduced this theme-in line with increasing research on transitions in mathematics education research [START_REF] Gueudet | Transitions in mathematics education[END_REF]-as a new TWG transversal to mathematical domains, alongside students' and teachers' practices. Although it disappeared in 2020, it continued to thrive through a dedicated chapter in the ERME volume on INDRUM research from the first two conferences [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF]. The INDRUM2020 keyword "transition to and across university mathematics" was then modified in the INDRUM2022 call for papers to encompass a larger spectrum of transitions and TWG1 was named accordingly.

In fact, the school to university transition (Klein's first discontinuity) is still dominant: it is the focus of 5/8 papers and 3/4 posters which were assigned to our group. Dually, 2 papers and 1 poster deal with the transition from university to secondary education (Klein's second discontinuity). A single paper considers the "across university" transition (with a focus on the teaching of a mathematical concept throughout the Bachelor in the Abstract Algebra track). To complete the perspective, papers assigned to other TWGs but mentioning transitions as a keyword shall also be counted; hence there are 2 additional papers on the school-university transition in TWG3 (focusing on proof), 2 in TWG6 (on students' learning), as well as 3 papers on Klein's second discontinuity attached to the new TWG5 on teacher education. It is worth noting that papers which investigate the case of engineering students do not use the lens of transitions, so that the transition from university to the workplace remains underresearched except in the context of pre-service teacher education.

Altogether, the theme of transitions overlaps with several TWGs and the core of the idea of transition that grants the unity to our TWG is still an open research question. Moreover, various facets of transitions may be studied using a diversity of theoretical/methodological frameworks. In what follows, we restrict our account to the 8 papers and 1 poster which have been presented and discussed during the group sessions, hence the figures in parentheses. We thus note the following facets: epistemological (7), cognitive (2), affective-emotional (1), socio-cultural, institutional (7); and the frameworks used: the Anthropological Theory of the Didactic, ATD (6), Commognition (1), concept image/concept definition (1), person-environment fit (1), and mathematical content analysis (3).

As [START_REF] Hochmuth | Transitions to, across and beyond university[END_REF] already pointed out, a large number of authors use the institutional perspective of ATD to study transitions, which led us to group those papers in the first parallel presentation session. By contrast, a diversity of perspectives (facets of transitions and theoretical/methodological tools) were offered in the second presentation session. After an in-depth discussion of each paper, discussions opened up to examine the topic of transitions in the light of all the papers and finally envisage opportunities for collaborations and avenues for further research. We will begin with an account of the contributions and then highlight some of the main points raised during our discussions. We conclude with a few ideas on the topic of transitions that may inspire future research.

HIGHLIGHTS FROM THE CONTRIBUTIONS

We asked authors to produce a highlight of their research in the form of a question/problem and its answer. In this section, we use these highlights-which were communicated in the group report at the conference-as a means to summarize striking features of the contributions and introduce readers to these works.

The school-university transition

Sarah Khellaf and Jana Peters raise the following questions: In what way can praxeological analysis inform the creation of study materials for first-year mathematics (teacher) students, that aim to make apparent to them differences between the institutions of school mathematics and university mathematics? What type of empirical questions about the implementation of these tasks could be asked and answered in the framework of ATD? As an answer, a task-design rationale has been explained in the paper. The reference model discussed can be used to identify 'unusual' (personal) praxeologies in student solutions. These can be compared with known dominant epistemological models from school and university, to generate hypotheses about their possible origin.

Tobias Mai and Rolf Biehler put the following problem in the foreground: School textbooks tend to introduce vectors as a mixture of the notions of n-tuples, translations, and sets of arrows-there is a need to explicitly and mathematically work out and integrate these settings in order to analyse and untangle interwoven approaches in school textbooks. As an answer, in the reference model presented in the paper, all three approaches to vectors are explained and finally discussed regarding their isomorphy. In the end, the most ostensive (illustrative) approach via arrows turns out to be the most complex approach of the three.

Jelena Pleština and Željka Milin Šipuš ask: How do polynomial-related praxeologies develop and differentiate through secondary school and a first-year bachelor programme in mathematics? In secondary school textbooks, the algebraic and analytical approaches to polynomials induce two disjoint praxeological organizations. In a first-year bachelor programme, specific and reduced praxis blocks align with the general logos blocks of praxeologies whose object of knowledge is the notion of polynomial. As a consequence, the relation first-year undergraduate students have to formal polynomials is marked by almost empty logos blocks.

Sarah Schlüter and Michael Liebendörfer explore: Which strategies do students use to cope with difficulties in "borderline cases" when their concept image seems to contradict the definition? Even if students apply the definition correctly, they do not trust the formal argumentation and tend to rely on intuition and their concept image. In addition to strategies based on informal reasoning, they manage to argue on a metalevel themselves, for instance by using transfers to similar "borderline cases". Katharina Kirsten and Gilbert Greefrath ask: What are the characteristics of university students who choose on-campus or distance learning courses? Students with weaker connections to mathematics (e.g., in terms of self-efficacy and final math grade) and a higher digital readiness are more likely to choose a distance learning course. By contrast, students with strong math prerequisites tend to choose an on-campus course. Learning types based on self-regulation and peer learning do not play a significant role in course decision-at least in preparatory courses.

Finally, the poster by Ana Katalenić, Aleksandra Čižmešija and Željka Milin Šipuš tackles the following question: How can the discourses on asymptotes change and develop in the transition from upper secondary to university education? As a result, discourses can develop from colloquial narratives supported by iconic representation through working on techniques of evaluating function values and finding asymptotes, towards the formal definition using distance between points on a curve and the line and expressions with a function limit.

Other transitions

Thomas Hausberger and Julie Jovignot investigate: How can students' difficulties in acquiring a structural sense be understood in terms of institutional gaps in the Abstract Algebra track throughout the bachelor programme in mathematics? As a result, the study of structuralist levels of structuralist praxeologies and the values of their didactic variables in relation to the dialectic of contextualisation and decontextualisation points towards a huge gap at the 3rd year of the bachelor programme in France. It seems to be reinforced by the compartmentalisation of knowledge in small teaching units that hinders the vitality of the dialectic.

Heidi Strømskag and Yves Chevallard examine: What transformations has the notion of concavity of functions undergone during the didactic transposition process from the knowledge taught at university to the knowledge to be taught in upper secondary school? Praxeological analyses of a university textbook and a Grade 12 textbook show that while in the university presentation, the graphical notion of concavity is mathematised, in the school presentation, it remains non-mathematised: concavity is to be seen on the graph of the function-where the theorem proved at university becomes now a mathematically unfounded definition of concavity.

Finally, Max Hoffmann and Rolf Biehler study the following question: What prior knowledge do student teachers have on the geometric concept of congruence before taking a geometry course at university? As a fact, this multi-faceted concept is treated rather "one-dimensionally" in German schools. Taking at university the resulting preformal and superficial prior knowledge not into account to focus on formal aspects is likely to perpetuate Klein's second discontinuity. There is a risk that prior mathematical knowledge from school will coexist with the academic mathematics learned, rather than being studied, corrected, and updated.

HIGHLIGHTS FROM THE DISCUSSIONS

Common themes emerging

Definitions in mathematics were debated in relation to their role in acquiring concepts, solving problems, and proving theorems. The notion of borderline/challenging cases was treated as such examples play an important role in complementing an incomplete predominant concept image. Polynomials in school mathematics, in abstract algebra, and in analysis appeared as examples in the considerations. Another aspect that came up was that of theorems and examples used as definitions in school mathematics, most notably in textbooks-possibly with the intention of making the knowledge at stake available to a larger group of students-, a transformation that likely simplifies and distorts the mathematical knowledge.

Attention was also given to the various transitions that occur in the education of teachers and that teacher education should address. Studying mathematics in view of teaching it requires developing other, new relations to mathematics compared to relations a mathematics student must develop. The topos changes, for instance, a mathematics teachers will have to choose examples and design tasks related to particular mathematics content in order to create opportunities for others to study it.

Theoretical frameworks and methodologies

The concept of praxeology-an analytic tool provided by ATD to model any human activity in terms of praxis (the type of tasks and the technique to solve them) and logos (the way to explain the technique and the theory to justify the explanation)-was used in six papers. Praxeological analysis was discussed on a general basis and linked to the notion of reference epistemological model (REM) to be used, for example, in didactic design, trying to remedy ruptures identified in dominant epistemological models, as well as overcoming didactic phenomena caused by such dominant models.

In ATD, Klein's double discontinuity can be expressed in terms of transpositive processes: When one goes from a level n to a level n + p in a curriculum (e.g., from secondary school to university), one generally faces an increasing rate of mathematisation, and conversely, in the opposite direction, there is generally a demathematisation of the mathematical content. This was discussed and related to the formalization developed by [START_REF] Winsløw | Klein's double discontinuity revisited: Contemporary challenges for universities preparing teachers to teach calculus[END_REF].

On methodology, the problem of standardized methods for elaboration of REMs was raised and related to three dimensions of the questioning of any object: its structure, its functioning, and its utility. Networking of theories, ATD and Commognition or ATD and Stoffdidaktik (subject matter didactics), was mentioned as a promising research methodology to cross perspectives and promote collaborations but not really discussed in depth due to lack of time. Finally, an understanding was reached that when communicating research to non-specialists of ATD (especially in oral presentations), it is appropriate to avoid excessive formalism.

CONCLUSION

With a focus on transitions, researchers are aiming at the investigation of didactical phenomena in terms of continuities/discontinuities/ruptures. They may be pursuing different goals: their endeavour may be to identify difficulties related to epistemological/cognitive/institutional discontinuities, to suggest ways to smoothen ruptures or assess existing measures (to respond to institutional and societal demands), to contribute to teacher education (since most researchers are teacher educators), to refine theoretical constructs (such as models of transitions), or to study the effects of the didactic transposition.

Avenues for further research are wide. At the level of the school-university transition, collaboration among researchers should entitle a shift from small-scale local studies (centred on a concept or a single institutional context) to wider perspectives and contexts, including comparative or longitudinal studies. Research on ruptures across university studies, in particular towards advanced mathematics, is still rare. With the intensification of research on Klein's second discontinuity, we expect reports on curricular innovation to account for strategies developed to tackle institutional constraints and to provide means to cooperate with mathematicians. Finally, transitions from university to the workplace for other careers than teachers (e.g., engineers) need also greater attention. INDRUM looks forward to receiving contributions in these directions at the next conference.


Abdelkrim Ali
email: abdelkrim.ali-zazou@ensma.fr

Zazou Srd

Ensma Poitiers

Jean-Paul Gaubert
email: jean.paul.gaubert@univ-poitiers.fr

Emilie Chevrier
email: emilie.chevrier@srd-energies.fr

Emmanuel Grolleau
email: grolleau@ensma.fr

Pascal Richard
email: pascal.richard@univ-poitiers.fr

Ladjel Bellatreche
email: bellatreche@ensma.fr

Distribution network reconfiguration problem for energy losses minimization

This paper presents a study on different method of modeling the network reconfiguration problem to take into account load variation during a period of time. This work aim to determine if optimization process needs to take into account this fact in order to be used by a real distribution network operator. It presents a way to optimize an integer objective function.The optimization problem of type Mixed Integer Quadratic Constrained Program (MIQCP) and Mixed Integer Non-Linear Program (MINLP) is presented. Three different process of optimization on a define period are presented, two avoiding the use of power flow calculation and one using it. Methods are computed using ***three*** different load scenarios on the famous Baran & Wu test case. And results tends to show that it is not useful to take into account load variation and worst case optimization is the best method.

I. INTRODUCTION

This paper present a joint work between the DSO (Distribution System Operator) of the french department Vienne called SRD, and LIAS research lab. The aim of this document is to present results for loss minimization on a period of time in regards to future implementation in the real network. Several method for loss reduction have already been proposed such as capacitor placement, distributed generation allocation or feeder reconfiguration [START_REF]Loss minimization techniques used in distribution network: bibliographical survey[END_REF]. Feeder reconfiguration by switch state modification have been well studied and different optimization methods have been used to solved this problem [START_REF]Loss minimization techniques used in distribution network: bibliographical survey[END_REF]. The majority of the work focused on instaneous power only, and neglected the variability of load demand as stated in [START_REF] Ghofrani-Jahromi | Distribution switches upgrade for loss reduction and reliability improvement[END_REF]. In fact the power demanded is not the same at each moment of the day. Also focusing on the energy instead of power is more representative of the reality of DSOs. Because calculating configuration based on power means that this proposed configuration is valid for only the power taken into account and maybe is to conservative or it can not handle larger consumption. But it is not possible to reconfigure the network oftenly during a short period of time due to two reason. Firstly remotely controlled switches will be damage to quickly and secondly it is not imaginable do send an agent to modify the manual switches several time during a short period. The first reason can be view as only an investment problem and maybe the gain in term of losses can compensate the switches renewing. And the second reason have already been studied [START_REF] Ghofrani-Jahromi | Distribution switches upgrade for loss reduction and reliability improvement[END_REF] to determine which switch has to be modified in a network to become remotely controlled. And the switch modified can become inappropriate if the network evolve during his life. Also automatic switching for network reconfiguration have been studied in [START_REF] Zidan | Network reconfiguration in balanced and unbalanced distribution systems with variable load demand for loss reduction and service restoration[END_REF]. automatically changing network configuration is practically infeasible because security device has to be modified to handle the new load of the feeders so it could not be done so oftenly. In our study we only focused on finding fixed configuration for certain period of time. In other words finding a way to optimize an integer objective function. And this model aimed to be used on real network for long-term (season) or for shortterm (day, week) period. Also the proposed method have the objective to be use one large scale networks, so it has to be developed in order to avoid computation problems. The optimization model is define in section II. The way of taking into account the energy instead of the power and detail about the variable load demand are presented in section III. Section IV present the results computed over Baran & Wu test case [START_REF] Baran | Network reconfiguration in distribution systems for loss reduction and load balancing[END_REF] with different scenarios.

II. OPTIMIZATION MODEL

The model presented bellow aimed to be used in an industrial context with a large network containing a large number of nodes and edges. In order to handle this size of problem some simplification are made. This is also due to the lack of information available in real distribution network. As reactive consumption and phase (ϕ) are unknown at each node of the network. Also u is considered constant. In fact in this study the model optimize a power flow only based on real power. Furthermore the energy losses on the line to be minimized can be deduced with the active power alone.

A. Power optimization problem definition 1) Glossary:

x(i, j) : Real value representing the power flow from vertex i to j ch(i) : Real value representing relative voltage drop at node I b(i, j) : Binary value representing line state (1=edges (i,j) active, 0= edge (i,j) not active) kp(i, j) : Constant loss coefficient from i to j kc(i, j) : Constant voltage drop coefficient from i to j up(i, j) : Upper bound of x(i,j) up ch (i) : Upper bound of ch(i) nb source : Number sources src(p) : Maximum power that a source p can deliver a) objective function:

Z = kp(i, j) * x(i, j) * b(i, j) ∀(i, j) (1) b) Constraint: n i=0 x(i, j) × b(i, j) - n k=0 x(j, k) × b(j, k) = P (j) (2)
∀j with ∀k consecutive to j and ∀i incident to j b(i, j) ≤ 1 ∀j incident to i (3)

ch(i) = (i, j) * (x(i, j) × kc(i, j)) + ch(j)) (4)
∀ j incident to i b(i, j) + b(j, i) ≤ 1 ∀(i, j), ∀(j, i) (5) x(i, j) ≤ up(i, j) ∀(i, j) (6) x(i, j) ≥ -up(i, j) ∀(i, j) (7) ch(i) ≤ up ch (i) ∀i (8) ch(i) ≥ -up ch (i)∀i (9)
b(p, i) = nb source ∀(p, i) i consecutive to p (10)

x(i, p) ≤ src(p) ∀(i, p) i incident to source p (11)

B. calculation of energy loss

In this section, we show how to compute the energy loss between two nodes i and j of the power network. And this calculation defines the objective function to minimize. P loss (i, j) = R(i, j) × i(i, j) 2 i(i, j) = P (i,j) u×cos ϕ so P loss (i, j) = kp(i, j) × P (i, j) 2 (12)

with kp(i, j) = R(i,j) u 2 ×(cos ϕ) 2 ) (13)
where R(i, j) is the resistance of the line connecting node i to j and P (i, j) the power flowing between i and j. The aim of this model is to be used to minimize the total amount of loss of the network caused by the loss of each individual line. normally it should be added to the real network flow, because in reality power loss indirectly increase the value of intensity in the line but in our study this fact is not taken into account.

P loss total = P loss (i, j) ∀ (i, j) (14)
The distribution network is modeled by a graph composed by two type of nodes: All those nodes are connected by edges having a fixed cost, the kp(i, j) value (see Eq. 13). Each electrical line is represented by two edges to represent the two possible directions of the electrical flow. Each connected component composed of one source and all the consumption which get power from this source are called feeder in reality.

Note that a consumption node may also be a renewable energy source and in this case its consumption has a negative value.

We can not count renewable sources as normal source because the power delivered by this source can be add to the power for the source. And we set in the model constraint that avoid powering on consumption by two different sources. Figure 1 show the graph representation of a simple distribution network. We assume that their is only one possible direction for the flow at the HV/MV transformers. But when in a feeder or all the feeders connected to this transformers the mean of the power produced is greater than the mean of the power consumed during the minimum period of measured time (10min). It is considered as a negative flow due to the fact that productions are computed as negative consumption.

C. Electrical flow computation

In section II-B the objective function is described and in the following sections the way to compute the power flowing through each line is presented. Also we explain how take into account the operability problem of the uniqueness of a source for a consumption. Graph theory says that the flow coming into a node is equal to the flow going out, this is a rule applicable to active power flow or reactive power flow but not apparent power (Kirschofs law) [START_REF] Gondran | Graphes et algorithmes[END_REF]. Also the graph has to be conservative, so the graph of Fig. 1 is defined as follows: With equations 15-18 we know the flow going through the line of the network, but we still have to constraint this flow to be realizable. That is the aim of the operational constraints

D. Operability constraint

As stated above because of the fact that distribution network are used in radial configuration we have to find a way to constrained the search of a configuration in that way. That means simply that a consumption node cannot be powered by two different sources. In order to do that we have to introduce into the model a binary variable b(i, j) representing whether an edge connects (i, j) or not. That is the set of b values that gives a network configuration by defining where the network should be opened and closed.

Based on figure 1 we can define the following equations: Also another important rule has to be implemented, concerning the fact that two opposite edges cannot be active at the same time, which means that the power can only go in one direction through a line.

Node 1: b(s2, 1) + b(2, 1) = 1 (19)
In the example figure 1 this applies to only two pairs of edges:

b(1, 2) + b(2, 1) ≤ 1 (22) b(2, 3) + b(3, 2) ≤ 1 (23)
When the model is used for a real application, equations ( 22) and ( 23) are slightly modified. In fact in reality not all lines can be opened, because their isn't a switch at each side of each line. So the condition for allowing a line to be opened is that a switch belongs to the edge considered. If this condition is respected, the constraints are written as in 22 and 23. Whereas in the other case the inequality becomes an equality meaning that at least one of the two edges have to be connected. Note that for computation problems b(s1, 2), b(s2, 1) and b(s3, 3) are forced to one. With these constraints described in above sections and knowing equation (12) the optimization model defined in section II-A can be defined with the variables x(i, j) = P (i, j). Two different equations for energy loss calculation are used to simplify the computation, one taking into account the square of x(i, j) and the other not. By taking into account the square of the power in the objective function, the Mixed Integer Quadratic Constrained Problem (MIQCP) becomes a Mixed Integer Non-Linear Problem (MINLP), and due to the non-convexity it is harder to solve. We will see in the next sections how it affect the result of the optimization.

III. OPTIMIZATION PERIOD MODELING

The model described in the above sections is presented for power optimization, but in reality we must think in terms of energy. In general every cost associated with the electricity market is set in energy (kW/h), and power doesn't represent the total amount of energy transited and lost into a network during a period of time. So in order to realize the optimization on a defined period of time (day, week, season...), we have to find a way to model the integral of the power loss (Equation 12 presented in previous section II-A.

In real networks load curves are known only for some Usually load curves are obtain by measuring power every ten minutes (light grey continuous curve), and the mean of this values is calculate using 6 points to defined an energy in kWh (dark grey bar graph). This particular value (dark grey bar) are called unitary-period value of the load curve in our study and are consistent with the usual energy unit (kWh). Their is a usual pattern in the daily load curve that can be used so we defined typical load curve of consumption for summer and winter days. This curve representing the power consumption during 24 hours has to be integrate but we do not have the exact function of this curve. Therefore integration by part is used. Three methods are presented to handle that kind of integration into the optimization problem presented in previous section: (a) Integrate all parts directly in the optimization problem, and deduce a configuration from this solution (b) Solve an optimization problem for each unitary-period of the load curve and externally calculate the loss for this given solution over all the period (c) solve the optimization problem for only one characteristic value of the load curve (min, mean, max)

A. Methods description

In this subsection we explain how the previous model of power loss optimization described in section II-A is modified to optimize energy loss for each methods (a), (b) and (c).

1) Method (a):

This seems to be the best way of modeling the integer behaviour of optimization in term of energy instead of power, because it calculate an configuration taking into account all unitary-period of the load curve. Indeed the objective function will sum all losses generated by each unitary-period into every line. So the result of the objective function should be equal to the energy losses.

• objective function (Eq. 1) modification for a h hours period

Z = Σ n p=0 kp(i, j) × x p (i, j) × b(i, j) ∀(i, j) (24)
x p (i, j) represent the energy in the line (i, j) for the period k. But the fact is that every x p (i, j) have to be calculated • Kirschofs law (Eq. 2) modification for h hours period

n i=0 x p (i, j) × b(i, j) - n k=0 x p (j, k) × b(j, k) = P p (j) (25)
∀j with ∀k consecutive to j and ∀i incident to j This set of equations is defined for every p to compute every x p (i, j). So with this method the optimization model is defined with m × h variables and n × h Kirschofs equations (m number of edges, n number of node).

2) Method (b):

In this method the model described in section II-A is unchanged. And one solution of the optimization problem is found for each unitary-period of the load curve. But it needs the use of an external power flow software. The only thing that change for every search is the value P (j) in Equation ( 2) that is replaced by the unitary-period value. Also for each obtained solution a loss calculation is computed using an state estimator software (MATPOWER [START_REF] Zimmerman | Matpower: Steadystate operations, planning, and analysis tools for power systems research and education[END_REF]) to get the energy losses generates by this configuration for the whole period. And when we found as much configuration as unitary-period we can deduce the configuration with the lesser losses in term of energy. This process is described in Figure 3.

Power flow calculation could be avoided by adding the same set of equations (2) with replacing x(i, j) by another variable y(i, j) and the maximum of the p(j) value. It will give us the flow for the maximum consumption for the given configuration. And we can just compute in an another variable Z the sum of the loss. This metric Z will be used to compare directly each solution without the use of a powerflow tool. But it will add a lot of variable and computation time in a real network.

3) Method (c):

The aim of this method is to check if the fact to take only an extreme value or a particular one (the mean) of the load curve is relevant to define a configuration that minimize the energy loss. As in the method (b) only P (j) in the Equation ( 2) is modified by the min, max or mean of the load curve. And for each of these configuration the energy loss of the entire period are computed using MATPOWER.

IV. METHODS COMPUTATION RESULTS

The proposed model has been tested over Baran & Wu test case [START_REF] Baran | Network reconfiguration in distribution systems for loss reduction and load balancing[END_REF] composed of 33 nodes. In order to also test the impact of the load modeling on the solutions found we defined 4 different scenario test:

1) Two 24 hours load curve are used, one representing a typical summer day and on winter day. Each unitaryperiod is equal to the mean of 1 hour of load. 2) A total year load curve is dived in 2 seasons (winter and summer), each unitary-period is equal to the mean of load of 1 week. To use this load curve as entry data for our optimization model we normalize the two load curve (winter and summer) by dividing the power by the maximum value of both summer and winter load curve. And this coefficient is applied to the nominal power of each consumption of the Baran & Wu test case. In those two previous test case all consumption in the network have the same load profile. We assume that they consume the same amount of load proportionally to their nominal one. But we know that in reality every consumer of different kind (household, industry, farmers) have different load curve during the same day. So to try to model this behaviour we attribute to each consumption a particular load curve, in accordance to their nominal consumption value. The profile used in the study are taken from DSO data base. The wave form of these curves can be discuss but are used to test how the optimization algorithm react with it. Results of these two scenario are found in Table I and II. It should be noted that in case where simplification of the square on the power to compute losses is made, all three methods give the exact same solution.

With typical summer weekly and daily load curve the two scenarios shows that method (b) gives the best results in term of power losses. This method give this results twice during the search and can not be related to any specific value of the load curve. Also method (c) gives a result not so far from this previous value and correspond to the optimum value given by the literature. In fact the specific value that give the best result in method (c) is the maximum of the load curve so it corresponds to the basic test case. Whereas method (a) gives the worse solution.

In the case where typical summer daily and weekly load curve are taken the same observation can be made as with winter load curve. Whereas with method (c) the configuration found with the max value is better and don't correspond to the optimum configuration of the literature. Also it should be noted that with method (a) in winter and summer in scenario (2) the best value is given with simplification on the square of the power for loss calculation. But it don't affect the comparison between the three methods.

In those two scenario we assume that all load of the network have the same load curve. In other word they consume the same amount proportionally to their nominal value. But we know that in reality it never occur so in next subsection we assign load profile to each node. Also we will focus only on daily load curve because the period chosen don't affect the comparison between the three methods.

B. Scenario 3) results

In this third scenario the behaviour is slightly different from previous scenario 1) and 2). In fact configuration given by method (b),(c) and the optimum configuration are the same. For method (c) this behaviour is normal because it is the max value that give this configuration which is the same as the case used in the literature.

C. Comparison of three methods

The results presented above tend to show that the best way to take into account the variability of the load is to find one configuration for each level of the load curve and choose the one that minimize the losses on the rest of the period (method (b)). But this approach is good only in a small test case like the Fig. 6. Losses in function of load (winter day case) one presented here with small computation time. Because if you want to optimize a network for one year or for a season and still use hourly dived load curve computation time can become a limitation. In fact it create as many optimization problem as unitary-period of the load curve. Also it should be noted that it need the use of an additional tool to compute the loss. And in an industrial context this sort of tools are not always available. Also we would think that method (b) could be used to define the best level of load curve to use to find the best solution. But Figure 6 shows no linearity or relation between load and losses. Method (c) has more advantages because it's easiest to implement. It doesn't need any load curve only the maximum value of the load curve can be used to define the best configuration for the period. It is also easy to compute in term of time taken by the optimization solver. And by taking the maximum of the load curve you are sure to be able to handle the worst case of consumption. Method (a) seems to be to complicated to be used because it hasn't good performance in term of losses optimization and computation time. Whereas it's the only method that test in one optimization procedure all the load constraint of the period.

V. CONCLUSION This study proofs that in a specific industrial context it is more suitable to simplify the load modeling in the optimization model. It is also shown that it is not necessary to model the integral on the power loss calculation to have good results on energy loss reduction. This study used a specific optimization model and specific profile and load curves that could be different for another kind of network (more urban than rural). In fact even if best results are found with method (b) it is the hardest to implement in term of scaling to large scale problem. Also taking into account individual profile for each node consumption has not show particular importance. However these observations are valid for small network but may be different for large scale networks. But other works have shown the same results [START_REF] Lopez | Online reconfiguration considering variability demand: applications to real networks[END_REF].

Fig. 1 .

 1 Fig. 1. Simplified network graph

Node 1 : 2 P

 12 P (s2, 1) + P (2, 1) = P (1, 2) + P 1 P (s2, 1) + P (2, 1) -P (1, 2) = P 1 (15) Node 2: P (1, 2) + P (s1, 2) + P (3, 2) = P (2, 3) + P (2, 1) + P (1, 2) + P (s1, 2) + P (3, 2) -P (2, 3) -P (2, 1) = P 2 (16) Node 3: P (s3, 3) + P (2, 3) = P (3, 2) + P 3 P (s3, 3) + P (2, 3) -P (3, 2) = P 3 (17) Conservation : P (s1, 2) + P (s2, 1) + P (s3, 3) = P 1 + P 2 + P 3 (18)

Node 2 :

 2 b(1, 2) + b(s1, 2) + b(3, 2) = 1 (20) Node 3: b(s3, 3) + b(2, 3) = 1 (21)

Fig. 2 .

 2 Fig. 2. typical daily load curve

Fig. 3 .

 3 Fig. 3. method(b) process

Fig. 4 . profile 3 )

 43 Fig. 4. profile

Fig. 5 .

 5 Fig. 5. Baran and Wu network

TABLE II METHODS

 II RESULTS COMPARISON FOR SUMMER VALUE