M Tayachi

A Rousseau

E Blayo

N Goutal

V Martin

Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

Keywords: dimensionally heterogeneous coupling, domain decomposition methods, multiscale analysis couplage multi-dimensionnel, décomposition de domaine, analyse muli-échelles

In the present work, we study and analyze an efficient iterative coupling method for a dimensionally heterogeneous problem . We consider the case of 2-D Laplace equation with non symmetric boundary conditions with a corresponding 1-D Laplace equation. We will first show how to obtain the 1-D model from the 2-D one by integration along one direction, by analogy with the link between shallow water equations and the Navier-Stokes system. Then, we will focus on the design of an Schwarz-like iterative coupling method. We will discuss the choice of boundary conditions at coupling interfaces. We will prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and the control of the difference between a global 2-D reference solution and the 2-D coupled one. These theoretical results will be illustrated numerically.

Introduction

Hydrodynamical phenomena can be described by a wide variety of mathematical and numerical models, spanning a large range of possible levels of complexity and realism. When dealing with the representation of a complex fluid system, such as an ensemble of rivers and channels or a human blood system, the dynamical behavior of the flow is often spatially heterogeneous. This means that it is generally not necessary to use the most complex model everywhere, but that one can adapt the choice of the model to the local dynamics. One has then to couple several different models, corresponding to different areas. Such an approach is generally efficient from a computational point of view, since it avoids heavy computations with a full complex model in areas where a simpler model is able to represent the dynamics quite accurately. Thus this makes it possible to build a hybrid numerical representation of an entire complex system, while its simulation with a unique model would be either non relevant with a simple one or too expensive with a complex one.

In such a hierarchy of models, the simplest ones are often simplifications of the more complex ones. Let mention for instance the so called "primitive equations", which are widely used to represent the large scale ocean circulation, and are obtained by making some assumptions in the Navier-Stokes equations. It is important to note that such simplifications may involve a change in the geometry and in the dimension of the physical domain, thus leading to simplified models which are m-D while the original one was n-D, with n > m. An obvious example is given by the shallow water equations, which are derived from the Navier-Stokes equations by integration along the vertical axis, see for instance [START_REF] Gerbeau | Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation[END_REF] for a rigorous mathematical derivation using asymptotic analysis techniques in the 2-D to 1-D case . Such a coupling between dimensionally heterogeneous models has been applied for several applications. Formaggia, Gerbeau, Nobile and Quarteroni [START_REF] Formaggia | On the coupling of 3D and 1D Navier-Stokes equations for flows problem in compliant vessels[END_REF] have coupled 1-D and 3-D Navier-Stokes equations for studying blood flows in compliant vessels. In the context of river dynamics, Miglio, Perotto and Saleri [START_REF] Miglio | Model coupling techniques for free-surface flow problems : Part I[END_REF], Marin and Monnier [START_REF] Marin | Superposition of local zoom models and simultaneous calibration for 1D-2D shallow water flows[END_REF], Finaud-Guyot, Delenne, Guinot and Llovel [START_REF] Finaud-Guyot | 1D-2D coupling for river flow modeling[END_REF], Malleron, Zaoui, Goutal and Morel [START_REF] Malleron | On the use of a high-performance framework for efficient model coupling in hydroinformatics[END_REF] have coupled 1-D and 2-D shallow water models. Leiva, Blanco and Buscaglia [START_REF] Leiva | Partitioned analysis for dimensionallyheterogeneous hydraulic networks SIAM Multiscale Model[END_REF] present also several such applications for Navier-Stokes equations.

Several techniques can be used to couple different models, either based on variational, algebraic, or domain decomposition approaches. In the context of dimensionally heterogeneous models, in addition to the previously mentioned references, there exists also a number of papers on this subject for purely hyperbolic problems (e.g. [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. The scalar case[END_REF][START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. The case of systems[END_REF], [START_REF] Bouttin | Mathematical and numerical study of nonlinear hyperbolic equations: model coupling and nonclassical shocks[END_REF]), but we will not elaborate on these studies since our focus is much more on hyperbolic/parabolic problems.

In our study, we will focus on the design of an efficient Schwarz-like iterative coupling method. The possibility of performing iterations between both models, i.e. of using a Schwarz method, is already considered in Miglio, Perotto and Saleri [START_REF] Miglio | Model coupling techniques for free-surface flow problems : Part I[END_REF] and Malleron, Zaoui, Goutal and Morel [START_REF] Malleron | On the use of a high-performance framework for efficient model coupling in hydroinformatics[END_REF]. This kind of algorithm has several practical advantages. In particular, it is simple to develop and operate, and it does not require heavy changes in the numerical codes to be coupled: each model can be run separately, the interaction between subdomains being ensured through boundary conditions only. These are important aspects in view of complex realistic applications.

Our final objective is to design an efficient algorithm for the coupling of a 1-D/2-D shallow water model with a 2-D/3-D Navier-Stokes model. As a first step in this direction, the present study aims at identifying the main questions that we will have to face, as well as an adequate mathematical framework and possible ways to address these questions. We will perform this preliminary stage on a very simple testcase, coupling a 2-D Laplacian equation with a corresponding simplified 1-D equation. Seemingly similar testcases were addressed by Blanco, Discacciati and Quarteroni [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF] and Leiva, Blanco and Buscaglia [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF], but with different coupling methodologies (variational approach in [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF] and Dirichlet-Neumann coupling in [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF]). Moreover we have chosen to use non symmetrical boundary conditions in our 2-D model, in order to develop a fully two dimensional solution, and our 1-D model is obtained by integration of the 2-D equation along one direction, by analogy with the link between the shallow water system and the Navier-Stokes system. The rigorous mathematical derivation of the 1-D model clearly highlights its validity conditions.

Section 2 is devoted to the presentation of the 2-D Laplacian model, and to the derivation of the corresponding reduced 1-D model. Then a Schwarz iterative coupling algorithm is presented in Section 3, and its theoretical convergence properties are analyzed. In particular, the influence of the interface location is discussed. Finally numerical tests are presented in Section 4, which fully validate the previous analytical results.

Derivation of the reduced model

We are interested in the following boundary-valued problem in a domain Ω ⊂ R 2 :

   -∆u(x, z) = F (x, z), ∀(x, z) ∈ Ω, α ∂u ∂n (x, z) + κu(x, z) = 0, ∀(x, z) ∈ ∂Ω, (1a) (1b)
with α and κ are nonnegative numbers that allows for Dirichlet, Neumann or Robin boundary conditions.

In this section, we want to take advantage of the shallowness of the domain Ω (or some subdomain of Ω) to derive a reduced model. As it is done for the derivation of the shallow water model (see e.g. [START_REF] Gerbeau | Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation[END_REF]), we want to replace the complete 2-D model (1) by a (simpler) 1-D equation wherever it is possible (and keep the original 2-D model everywhere else). We will finally obtain the coupled 1-D / 2-D system (15)-( 16) which we will analyze and simulate in the coming sections. In order to discriminate between 1-D and 2-D regions, we introduce the following definition: Definition 1. Let Ω 1D be the subset of Ω in which 2-D effects may be neglected, and Ω 2D = Ω\Ω 1D the subset of Ω in which 2-D effects cannot be neglected.

Remark 1. Naturally the definition of Ω 1D depends on several features, such as the domain aspect ratio, the considered system of equations, forcing terms (including boudary conditions), etc.

For the sake of clarity (see Figure 1), we will assume that there exists H and L 1 such that

Ω 1D = Ω ∩ {x < L 1 } = (0, L 1 ) × (0, H) and Ω 2D = Ω ∩ {x > L 1 }.
Let us now consider equation (1) with the following boundary conditions (see Figure 1 for the notations):

                     -∆u(x, z) = F (x, z), ∀(x, z) ∈ Ω, ∂u ∂n (x, z) = 0, ∀(x, z) ∈ Γ T , ∂u ∂n (x, z) + κu(x, z) = 0, ∀(x, z) ∈ Γ B , u(x, z) = γ 1 (x, z), ∀(x, z) ∈ Γ L , u(x, z) = γ 2 (x, z), ∀(x, z) ∈ Γ R . (2a) (2b) (2c) (2d) (2e)
Inria In order to derive the 1-D model in Ω 1D = Ω ∩ {x < L 1 }, we introduce the following dimensionless variables and numbers:

ε = H L 1 , (3) x
= x L 1 , z = z H , ũ(x, z) = u(x, z) U , F (x, z) = F (x, z) L 2 1 U and κ = κL 1 , (4)
where L 1 (resp H) is the characteristic length (resp height) of Ω 1D , ε is called the aspect ratio, and U is a characteristic value for u(x, z).

The nondimensional form of equations [START_REF] Formaggia | On the coupling of 3D and 1D Navier-Stokes equations for flows problem in compliant vessels[END_REF] in Ω 1D reads1 :

                         - ∂ 2 ũ ∂ x2 - 1 ε 2 ∂ 2 ũ ∂ z2 = F in Ω 1D ∂ ũ ∂ z = 0 on Γ 1 T = Γ T ∩ ∂Ω 1D - 1 ε ∂ ũ ∂ z + κũ = 0 on Γ 1 B = Γ B ∩ ∂Ω 1D ũ = γ 1 U on Γ 1 L = Γ L . (5a) (5b) (5c) (5d)
We assume (see [START_REF] Çengel | Introduction to thermodynamics and heat transfer[END_REF] for the scaling of κ) that

F = O(1), ∂ 2 ũ ∂ x2 = O(1) and κ = O(ε), (6)
which is a sufficient condition to ensure that the 2-D effects are negligible in Ω 1D . Indeed we deduce from equation (5a) that

∂ 2 ũ ∂ z2 = O(ε 2 ). (7)
By vertical integration on (z, 1), and accounting for the boundary condition (5b), we find:

∂ ũ ∂ z = O(ε 2 ) (8)
and finally:

ũ(x, z) = ũ(x, 0) + O(ε 2 ). (9)
Going back to original variables, we have:

u(x, z) = u(x, 0) + O(ε 2 ), ∀ (x, z) ∈ Ω 1D . ( 10
)
We now introduce the averaging operator in the vertical direction. For any function f of z, we set:

f = 1 H H 0 f (z) dz. (11)
We integrate equation ( 9) for z ∈ (0, H) and obtain:

ū(x) = u(x, 0) + O(ε 2 ), ∀x ∈ [0, L]. (12)
We now average equation (5a) in the vertical direction, taking into account the Robin boundary condition (5c) on Γ 1 B , and find:

- ∂ 2 ū ∂x 2 + κ H u(x, 0) = F , (13)
For every x ∈ [0, L 1 ] we may use approximation [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF] to introduce the new problem:

- ∂ 2 u 1 ∂x 2 + κ H u 1 = F in [0, L 1 ]. (14)
It will replace (1a) in Ω 1D . As evoked in Remark 1, the reader can be easily convinced that it is particularly awkward to guess the value of L 1 . Indeed one has to specify the criteria that define 2-D effects, and in practical situations we may only be able to define

L 2 which is such that Ω ∩ {x ≥ L 2 } ⊂ Ω 2D , or in other words L 2 ≥ L 1 .
In this work we consider two different situations:

-a funnel-shaped domain (see Figure 2) with a thin left part, so that we anticipate 2-D effects on the right (wide) part of the domain. In this case the definition of L 2 is based on a geometrical criterion.

-a rectangular domain (see Figure 3) with a small aspect ratio ε = H/L (so that we can anticipate weak 2-D effects), but with some 2-D forcing terms occuring in the right end of the domain. In that case the definition of L 2 is based on the support of forcing terms.

At this point we have defined an upper bound L 2 ≥ L 1 , but the exact value of L 1 remains unknown. From now on we choose an interface L 0 without any a priori information (other than L 0 < L 2 ) and decide to consider the model reduction [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF] on

Ω 1 = Ω ∩ {x < L 0 }, while we keep the 2-D model in Ω 2 = Ω ∩ {x > L 0 }.
Finally we have the two following systems 1-D model:

   - ∂ 2 u 1 ∂x 2 + κ H u 1 = F 1 in (0, L 0 ), u 1 (0) = γ1 . ( 15
)
Inria

             -∆u 2 = F 2 in Ω 2 , ∂u 2 ∂n = 0 on Γ 2 T = Γ T ∩ ∂Ω 2 , ∂u 2 ∂n + κu 2 = 0 on Γ 2 B = Γ B ∩ ∂Ω 2 , u 2 = γ 2 on Γ R . (16)
where

F 1 = F and F 2 = F |Ω2 .
Two cases may occur:

-Favourable case: L 0 < L 1 , so that Ω 1 ⊂ Ω 1D and the following model reduction is relevant.

In particular, hypothesis (6) holds so that Theorem 1 applies, -Unfavourable case: L 0 ≥ L 1 , so that Ω 1 ⊂ Ω 1D and the 1-D model will not be able to reproduce the 2-D reality (in particular, hypothesis [START_REF] Malleron | On the use of a high-performance framework for efficient model coupling in hydroinformatics[END_REF] does not hold). We now want to evaluate this model reduction in the favourable case L 0 < L 1 .

Coupling algorithm

Let us consider the two models ( 15) and ( 16) to be coupled respectively through the interfaces x = L 0 and Γ as shown in Figure 4.

In coupling problems, the first difficulty lies in defining the coupling notion by itself, i.e. defining the quantities or values to be exchanged between the two models through the coupling interfaces.

In our case and from a physical point of view, one may propose the following conditions, see [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF], [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF]:

         u 1 (L 0 ) = 1 H H 0 u 2 (L 0 , z)dz ∂u 1 ∂x (L 0 ) = 1 H H 0 ∂u 2 ∂x (L 0 , z)dz (17a) (17b)
which correspond to the conservation of u and its flux through the interface.

Unfortunately these two constraints do not allow the well-posedness of the 2-D model and of the coupled problem. They are called defective boundary conditions in the literature, [START_REF] Formaggia | On the coupling of 3D and 1D Navier-Stokes equations for flows problem in compliant vessels[END_REF], [START_REF] Formaggia | Numerical treatment of defective boundary conditions for the Navier-Stokes equations[END_REF], [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF]. One should then rather apply a coupling method ensuring the following points:

(i) the well-posedness of the 1-D and 2-D models (ii) the physical constraints are satisfied (iii) the control of the difference between the coupled solution and the reference one (corresponding to the 2-D model over the whole domain Ω). Indeed, due to the nature of the problem, the reader can be easily convinced that one does not expect to end with a solution of the 2-D model equal to the restriction of the reference one on Ω 2 .

The coupling problem with (17a) and (17b) has been studied in [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF] and [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF] using variational and algebraic approaches. In this section, we propose an iterative coupling method based on classical Schwarz algorithms. These iterative methods were used for the first time in the context of dimensionally heterogeneous coupling in [START_REF] Formaggia | On the coupling of 3D and 1D Navier-Stokes equations for flows problem in compliant vessels[END_REF] and [START_REF] Miglio | Model coupling techniques for free-surface flow problems : Part I[END_REF] to study a nonlinear hyperbolic coupling Inria We will prove the convergence of these algorithms given an appropriate choice of boundary conditions at x = L 0 and on Γ. Then we will study the solutions obtained after convergence and compare them to the global reference solution u defined by ( 2). Finally we will give some results regarding the choice of the coupling interface position.

Schwarz algorithms

Let us introduce first the Schwarz algorithms in the context of dimensionally homogeneous coupling. Consider the two systems, defined on Ω 1 and Ω 2 shown in Figure 5:

L u u = f u in Ω 1 ⊂ R n B out u u = g u on ∂Ω out 1 and L v v = f v in Ω 2 ⊂ R n B out v v = g v on ∂Ω out 2 (18)
The operators L u and L v are different. We assume that u and v have to satisfy the following constraints derived from the physics :

C 1 u = C 2 v C ′ 1 u = C ′ 2 v (19a) (19b)
through the interface Γ, where C 1 , C ′ 1 , C 2 and C ′ 2 are differential operators. To couple these two models we can implement the following iterative algorithm: For a given v 0 and at each iteration k ≥ 0, solve:

   L u u k+1 = f u in Ω 1 B out u u k+1 = g u on ∂Ω out 1 B u u k+1 = B v v k on Γ then    L v v k+1 = f v in Ω 2 B out v v k+1 = g v on ∂Ω out 2 B ′ v v k+1 = B ′ u u k+1 on Γ.
Once convergence is achieved, the physical constraints (19a) and (19b) have to be satisfied. Then care should be taken to choose the operators B u , B ′ u , B v and B ′ v in order to ensure convergence toward the unique solution defined by ( 18), (19a) and (19b), see [START_REF] Quarteroni | Domain Decomposition Methods for Partial Differential Equations[END_REF] and [START_REF] Martin | Méthodes de décomposition de domaine de type relaxation d'ondes pour des équations de l'océanographie[END_REF]. This method can be generalized to the dimensionally heterogeneous coupling case. Let assume that we have to solve the following 1-D model/2-D model coupled problem:

L 1 u 1 = f 1 in Ω 1 ⊂ R B out 1 u 1 = g 1 on ∂Ω out 1 and L 2 u 2 = f 2 in Ω 2 ⊂ R 2 B out 2 u 2 = g 2 on ∂Ω out 2
and we suppose that we have the following coupling constraints to satisfy at x = L 0 and on Γ:

C 1 u 1 (L 0 ) = C 2 (Ru 2 ) (L 0 ) C ′ 2 u 2 (L 0 , z) = C ′ 1 (Eu 1 ) (L 0 , z) on Γ (20a) (20b)
Ru 2 is a restriction of u 2 at x = L 0 and Eu 1 is an extension of u 1 (L 0 ) all along Γ. More generally we can define the operators R and E as in [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF] by:

R : Λ -→ Λ 0 u 2|Γ -→ Ru 2|Γ
and

E : Λ 0 -→ Λ u 1|x=L0 -→ Eu 1|x=L0
The spaces Λ 0 et Λ are the trace spaces on the interface x = L 0 for 1-D functions and on the interface Γ for 2-D functions. As mentioned in [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF], these two operators are not invertible. One may thus implement the following algorithm: For a given u 0 2 and at each iteration k ≥ 0, solve:

   L 1 u k+1 1 = f 1 in Ω 1 B out 1 u k+1 1 = g 1 on ∂Ω out 1 B 1 u k+1 1 = B 2 Ru k 2 at x = L 0 then    L 2 u k+1 2 = f 2 in Ω 2 B out 2 u k+1 2 = g 2 on ∂Ω out 2 B ′ 2 u k+1 2 = B ′ 1 Eu k+1 1 on Γ.
In practice, we do not have conditions such as (20a) (at x = L 0 ) and (20b) (for (x, z) on Γ), but only conditions at x = L 0 such as (17a) and (17b). This leads us to make a choice of the operators R, E, C 1 , C 2 , C ′ 1 and C ′ 2 . In general the choice of the restriction operator R is more straightforward than the choice of the extension operator E one. In this study, since the 1-D model is obtained after some approximations and by averaging the 2-D model, it is reasonable to define R as the vertical average. On the other hand, the question of the choice of the operator E remains open. In [START_REF] Blanco | Modeling dimensionallyheterogeneous problems: analysis, approximation and applications[END_REF] and [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF], authors proposed a constant extension of (17a) and (17b) along Γ, and impose the following strong coupling constraints:

   u 2 (L 0 , z) = u 1 (L 0 ) on Γ ∂u 1 ∂x (L 0 ) = 1 H H 0 ∂u 2 ∂x (L 0 , z)dz or      u 1 (L 0 ) = 1 H H 0 u 2 (L 0 , z)dz ∂u 2 ∂x (L 0 , z) = ∂u 1 ∂x (L 0 ) on Γ
It is a choice among many others. One may also choose a multitude of operators C 1 , C 2 , C ′ 1 and C ′ 2 ensuring the physical constraints to be satisfied. The strategy that we adopt here is to choose, due to relations [START_REF] Bouttin | Mathematical and numerical study of nonlinear hyperbolic equations: model coupling and nonclassical shocks[END_REF] and [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF], a constant extension of u 1 along Γ and then to implement a family of Schwarz algorithms with appropriate boundary Inria conditions at x = L 0 and on Γ. In this case Schwarz coupling algorithm reads: For a given u 0 2 and at each iteration k ≥ 0, solve :

               - ∂ 2 u k+1 1 ∂x 2 + κ H u k+1 1 = F 1 in (0, L 0 ) u k+1 1 (0) = γ1 B 1 u k+1 1 (L 0 ) = B 1 ūk 2 (21)
and then solve

                                   -∆u k+1 2 = F 2 in Ω 2 ∂u k+1 2 ∂n = 0 on Γ 2 T ∂u k+1 2 ∂n + κu k+1 2 = 0 on Γ 2 B u k+1 2 = γ 2 on Γ R B 2 u k+1 2 = B 2 u k+1 1 on Γ (22)
The linear operators B 1 and B 2 will be defined such that the points (i), (ii) and (iii) (see introduction of Section 3) are satisfied and such that the algorithm converges. We will first study the convergence of the coupling algorithm. Subsequently we move on the point (iii). To ensure the convergence of Schwarz algorithms in the case of classical domain decomposition without overlapping, it is proposed in [START_REF] Lions | On the Schwarz alternating method. III. A variant for nonoverlapping subdomains[END_REF] to use Robin operators. We will extend the use of these operators to our coupling problem. We define the operators B 1 and B 2 for a given λ > 0 as follows:

B 1 = ∂ ∂n 1 + λId (23)
and

B 2 = ∂ ∂n 2 + λId (24)
where n 1 and n 2 are the outward unit normal to the 1-D and 2-D domains respectively. We note that the operators B 1 and B 2 ensure the well-posedness of the problem at each iteration. Let us study the convergence of Schwarz algorithm with this family of operators.

Algorithm convergence

Proposition 1. For each λ > 0, the

(u k 1 , u k 2 ) Schwarz algorithm converges in H 1 (Ω 1 ) × H 1 (Ω 2 ) to (u λ 1 , u λ 2 )
that satisfies the physical constraints (17a), (17b).

Proof:

Let define the differences between two successive iterations:

e k+1 1 (x) = u k+1 1 (x) -u k 1 (x), ∀x ∈ (0, L 0 ) RR n°8182 and e k+1 2 (x, z) = u k+1 2 (x, z) -u k 2 (x, z), ∀(x, z) ∈ Ω 2
These functions satisfy the following systems:

                 - ∂ 2 e k+1 1 ∂x 2 + κ H e k+1 1 = 0 in (0, L 0 ) e k+1 1 (0) = 0 ∂e k+1 1 ∂x (L 0 ) + λe k+1 1 (L 0 ) = ∂ē k 2 ∂x (L 0 ) + λē k 2 (L 0 ) (25)
and

                                     -∆e k+1 2 = 0 in Ω 2 ∂e k+1 2 ∂n = 0 on Γ 2 T ∂e k+1 2 ∂n + κe k+1 2 = 0 on Γ 2 B e k+1 2 = 0 on Γ R - ∂e k+1 2 ∂x (L 0 , z) + λe k+1 2 (L 0 , z) = - ∂e k+1 1 ∂x (L 0 ) + λe k+1 1 (L 0 ) on Γ. (26)
The first two equations of (25) lead to:

e k+1 1 (x) = α k+1 sinh(ax), ∀x ∈ (0, L 0 ) (27)
where α k+1 ∈ R and a = κ H .

If we take the vertical average of the boundary condition on Γ in (26), and due to the boundary condition at x = L 0 , we obtain :

         - ∂ē k+1 2 ∂x (L 0 ) + λē k+1 2 (L 0 ) = - ∂e k+1 1 ∂x (L 0 ) + λe k+1 1 (L 0 ) ∂ē k+1 2 ∂x (L 0 ) + λē k+1 2 (L 0 ) = ∂e k+2 1 ∂x (L 0 ) + λe k+2 1 (L 0 ).
This implies:

ēk+1 2 (L 0 ) = 1 2λ (Aα k+2 + Bα k+1 ) (28)
and

∂ē k+1 2 ∂x (L 0 ) = 1 2 (Aα k+2 -Bα k+1 ) (29)
Inria where A = a cosh(aL 0 ) + λ sinh(aL 0 ) and B = -a cosh(aL 0 ) + λ sinh(aL 0 ). Now by multiplying the first equation of ( 26) by e k+1 2 and by integrating in Ω 2 , we obtain:

Ω2 |∇e k+1 2 | 2 dxdz - ∂Ω2 ∂e k+1
2 ∂n e k+1 2 dσ = 0 then using the boundary conditions on Γ 2 T and Γ R , we obtain:

Ω2 |∇e k+1 2 | 2 dxdz + Γ 2 B κ|e k+1 2 | 2 dx = - Γ ∂e k+1 2 ∂x (L 0 , z)e k+1 2 (L 0 , z)dz. ( 30
)
We replace ∂e k+1 2 ∂x (L 0 , z) in ( 30) by its value obtained by using Robin boundary condition on Γ:

Ω2 |∇e k+1 2 | 2 dxdz + Γ 2 B κ|e k+1 2 | 2 dx = Γ - ∂e k+1 1 ∂x (L 0 ) + λe k+1 1 (L 0 ) -λe k+1 2 (L 0 , z) e k+1 2 (L 0 , z)dz = Bα k+1 H ēk+1 2 (L 0 ) -λ H 0 |e k+1 2 | 2 (L 0 , z)dz = Bα k+1 H 2λ (Aα k+2 + Bα k+1 ) -λ H 0 |e k+1 2 | 2 (L 0 , z)dz. (31)
We now replace e k+1 2 (L 0 , z) in (30) using the same Robin boundary condition:

Ω2 |∇e k+1 2 | 2 dxdz + Γ 2 B κ|e k+1 2 | 2 dσ = - Γ 1 λ - ∂e k+1 1 ∂x (L 0 ) + λe k+1 1 (L 0 ) ∂e k+1 2 ∂x (L 0 , z)dz - 1 λ H 0 ∂e k+1 2 ∂x (L 0 , z) 2 dz = - Bα k+1 H λ ∂ē k+1 2 ∂x (L 0 ) - 1 λ H 0 ∂e k+1 2 ∂x (L 0 , z) 2 dz = - Bα k+1 H 2λ (Aα k+2 -Bα k+1 ) - 1 λ H 0 ∂e k+1 2 ∂x (L 0 , z) 2 dz.
Due to the fact that λ > 0, we deduce that:

Bα k+1 H 2λ (Aα k+2 + Bα k+1 ) ≥ 0 and - Bα k+1 H 2λ (Aα k+2 -Bα k+1 ) ≥ 0.
Thus:

A 2 α 2 k+2 -B 2 α 2 k+1 ≤ 0. Then we obtain: α 2 k+2 α 2 k+1 ≤ B 2 A 2 = -a cosh(aL 0 ) + λ sinh(aL 0 ) a cosh(aL 0 ) + λ sinh(aL 0 ) 2 < 1 RR n°8182
and finally

α k+2 α k+1 < B A < 1. ( 32
)
So that the sequence (α k ) k∈N converge to zero.

Let us now remark that for all k ≥ 0, n ≥ 0, we have:

u k+n 1 -u k 1 = n-1 p=0 e k+p+1 1 and ∂ u k+n 1 -u k 1 ∂x = n-1 p=0 ∂e k+p+1 1 ∂x
Using the relation ( 27) and the fact that the sequence (α k ) k∈N converges, we can prove that (u k 1 ) k∈N and (

∂u k 1 ∂x ) k∈N are Cauchy sequences in L 2 (Ω 1 ). So that (u k 1 ) k∈N is a Cauchy sequence in H 1 (Ω 1 ).
In the same way we observe that for all k ≥ 0, n ≥ 0, we have:

∇(u k+n 2 -u k 2 ) = n-1 p=0 ∇e k+p+1 2
Using (31), we deduce that:

Ω2 |∇e k+1 2 | 2 dxdz ≤ Bα k+1 H 2λ (Aα k+2 + Bα k+1 )
and then we can prove that (∇e k 2 ) k∈N is a Cauchy sequence in L 2 (Ω 2 ) and due to the Poincaré inequality we have also (u k

2 ) k∈N is a Cauchy sequence in L 2 (Ω 2 ). So that (u k 2 ) k∈N is a Cauchy sequence in H 1 (Ω 2 ). To conclude we have prove that

(u k 1 , u k 2 ) Schwarz algorithm converges in H 1 (Ω 1 ) × H 1 (Ω 2 ). Moreover, at convergence the limit (u λ 1 , u λ 2 ) verifies B 2 u λ 2 = B 2 u λ 1 and B 1 u λ 1 = B 1 ūλ 2 .
Taking the vertical average on Γ gives two linear combinations of the constraints (17a), (17b).

Remark 2.

• The Schwarz algorithms converge for all λ positive, but we remark that for λ = a coth(aL 0 ), we have exact convergence in two iterations. Indeed we have in this case:

- ∂e k+1 1 ∂x + a coth(aL 0 )e k+1 1 = 0 ∀k ≥ 0,
and then:

B 2 e k+1 2 = - ∂e k+1 2 ∂x + a coth(aL 0 )e k+1 2 = - ∂e k+1 1 ∂x + a coth(aL 0 )e k+1 1 = 0 ∀k ≥ 0,
The operator ∂ ∂n 1 + a coth(aL 0 )Id corresponds to the absorbing operator of the 1-D model.

We denote by λ opt this value of λ, see for example [START_REF] Japhet | The best interface conditions for domain decomposition methods: absorbing boundary conditions, in Absorbing boundaries and layers, domain decomposition methods[END_REF].

Inria • If we take λ 1 = λ 2 , we have a priori (u λ1 1 , u λ1 2 ) = (u λ2 1 , u λ2
2 ). This is in accordance with the ill-posedness of coupling problem defined by ( 15), ( 16), (17a) and (17b).

For the sake of clarity we will denote (u 1 , u 2 ) the limit of Schwarz algorithm instead of (u λ 1 , u λ 2 ).

Control of the difference between the coupled solution and the global reference solution

Unlike the case of domain decomposition, at convergence of Schwarz algorithm, we have u 2 = u| Ω2 due to the model reduction. But as mentioned above, we have chosen the family of Robin operators in order to get some control of the difference between u 2 and u| Ω2 . In fact we have the following result:

Theorem 1. for each λ > 0, let (u λ 1 , u λ 2 ) denotes the limit of the Schwarz algorithm. If L 0 < L 1 then there exists M (λ) > 0 such that:

u |Ω2 -u 2 H 1 (Ω2) ≤ M (λ)ε 1 + δ 2 (33)
where δ = L 1 L 1 -L 0 .
Proof The function u| Ω2 -u 2 is the solution of the system:

                         -∆(u -u 2 ) = 0 in Ω 2 ∂(u -u 2 ) ∂n = 0 on Γ 2 T ∂(u -u 2 ) ∂n + κ(u -u 2 ) = 0 on Γ 2 B u -u 2 = 0 on Γ R .
By multiplying the first equation by u -u 2 and using the boundary conditions on

Γ 2 T ∪ Γ 2 B ∪ Γ R , we obtain: Ω2 |∇ (u -u 2 ) | 2 dxdz + Γ 2 B κ|u -u 2 | 2 dx - Γ ∂(u -u 2 ) ∂n (u -u 2 ) dz = 0. ( * )
The integral term on Γ is reformulated using the boundary condition -

∂u 2 ∂x + λu 2 = - ∂u 1 ∂x + λu 1 satisfied by the limit u 2 : Γ ∂(u -u 2 ) ∂n (u -u 2 ) dz = - H 0 ∂(u -u 2 ) ∂x (L 0 , z) (u -u 2 ) (L 0 , z)dz = - H 0 ∂u ∂x (L 0 , z) (u -u 2 ) (L 0 , z)dz - H 0 - ∂u 1 ∂x (L 0 ) + λu 1 (L 0 ) -λu 2 (L 0 , z) (u -u 2 ) (L 0 , z)dz = H 0 - ∂u ∂x (L 0 , z) + λu(L 0 , z) (u -u 2 ) (L 0 , z)dz - H 0 - ∂u 1 ∂x (L 0 ) + λu 1 (L 0 ) (u -u 2 ) (L 0 , z)dz -λ H 0 (u -u 2 ) 2 (L 0 , z)dz.
• The first term reads:

H 0 - ∂u ∂x + λu (L 0 , z) (u -u 2 ) (L 0 , z)dz = H 0 - ∂u ∂x + λu (L 0 , z) (u(L 0 , z) -ū(L 0 )) dz + H 0 - ∂u ∂x + λu (L 0 , z) (ū(L 0 ) -u 1 (L 0 )) dz + H 0 - ∂u ∂x + λu (L 0 , z) (u 1 (L 0 ) -u 2 (L 0 , z)) dz.
Due to the relations [START_REF] Bouttin | Mathematical and numerical study of nonlinear hyperbolic equations: model coupling and nonclassical shocks[END_REF] and [START_REF] Leiva | Iterative strong coupling of dimensionallyheterogeneous models[END_REF] and to the fact that -∂u ∂x (L 0 , z) + λu(L 0 , z) = O(1), we deduce:

H 0 - ∂u ∂x + λu (L 0 , z) (u(L 0 , z) -ū(L 0 )) dz = O(ε 2 )
In the same way, if we assume that L 0 < L 1 , so that 2-D effects are insignificant in Ω 1 ∩ {L 0 ≤ x ≤ L 1 }, and applying a similar asymptotic analysis as in the first section to the 2-D model defined in Ω 2 , we can deduce that:

u 2 (L 0 , z) = ū2 (L 0 ) + O(δ 2 ε 2 ) = u 1 (L 0 ) + O(δ 2 ε 2 ), ∀z ∈ [0, H] (34)
So that:

H 0 - ∂u ∂x + λu (L 0 , z) (u 1 (L 0 ) -u 2 (L 0 , z)) dz = O(δ 2 ε 2 )
Finally:

H 0 - ∂u ∂x + λu (L 0 , z) (ū(L 0 ) -u 1 (L 0 )) dz = H - ∂ ū ∂x + λū (L 0 ) (ū(L 0 ) -u 1 (L 0 )) (35) Inria • Since u 1 (L 0 ) = u 2 (L 0 )
, the second term reads:

- H 0 - ∂u 1 ∂x (L 0 ) + λu 1 (L 0 ) (u -u 2 ) (L 0 , z)dz = -H - ∂(u 1 -ū) ∂x + λ(u 1 -ū) (L 0 ) (ū -u 1 ) (L 0 ) -H - ∂ ū ∂x + λū (L 0 ) (ū -u 1 ) (L 0 ) ( * * )
We reformulate the first term of the right. Note that the function u 1 -ū satisfies the equation:

- ∂ 2 (u 1 -ū) ∂x 2 (x) + a 2 (u 1 -ū)(x) = a 2 (ū(x) -u(x, 0)), ∀x ∈ (0, L 0 )
So that by multiplying this equation by u 1 -ū, after integration on (0, L 0 ) and use of the boundary condition u 1 (0) = u(0), we obtain:

L0 0 ∂(u 1 -ū) ∂x 2 (x)dx + a 2 L0 0 (u 1 -ū) 2 (x)dx - ∂(u 1 -ū) ∂x (L 0 )(u 1 -ū)(L 0 ) = L0 0 a 2 (ū(x) -u(x, 0))(u 1 (x) -ū)(x)dx thus: - ∂(u 1 -ū) ∂x (L 0 )(u 1 -ū)(L 0 ) = L0 0 a 2 (ū(x) -u(x, 0))(u 1 (x) -ū(x))dx -A 1 (u 1 -ū, u 1 -ū)
where

A 1 (u 1 -ū, u 1 -ū) = L0 0 ∂(u 1 -ū) ∂x 2 (x)dx + a 2 L0 0 (u 1 -ū) 2 (x)dx.
And then ( * * ) becomes:

- H 0 - ∂u 1 ∂x (L 0 ) + λu 1 (L 0 ) (u -u 2 ) dz = H L0 0 a 2 (ū(x) -u(x, 0))(u 1 (x) -ū(x))dx -HA 1 (u 1 -ū, u 1 -ū) + λH (ū -u 1 ) 2 (L 0 ) -H - ∂ ū ∂x + λū (L 0 ) (ū -u 1 ) (L 0 ) (36)
• To recap, the boundary term on Γ in ( * ) becomes:

Γ ∂(u -u 2 ) ∂n (u -u 2 ) dz = O(ε 2 ) + O(δ 2 ε 2 ) + H L0 0 a 2 (ū(x) -u(x, 0))(u 1 (x) -ū)dx -HA(u 1 -ū, u 1 -ū) + λH (ū -u 1 ) 2 (L 0 ) -λ H 0 (u -u 2 ) 2 dz
We first observe that:

L0 0 a 2 (ū(x)-u(x, 0))(u 1 (x)-ū(x))dx ≤ a 2 2 L0 0 (ū(x)-u(x, 0)) 2 dx+ a 2 2 L0 0 (u 1 (x)-ū(x)) 2 dx.

RR n°8182

It follows that:

Γ ∂(u -u 2 ) ∂n (u -u 2 ) dz ≤ C(1 + δ 2 )ε 2 ) + H a 2 2 L0 0 (ū(x) -u(x, 0)) 2 dx -HA 1 (u 1 -ū, u 1 -ū) + H a 2 2 L0 0 (u 1 (x) -ū(x)) 2 dx +λH (ū -u 1 ) 2 (L 0 ) -λ H 0 (u -u 2 ) 2 (L 0 , z)dz
where C is a positive constant.

Then we have:

-A 1 (u 1 -ū, u 1 -ū) + a 2 2 L0 0 (u 1 (x) -ū(x)) 2 dx ≤ 0
and finally, using the definition of u(L 0 ) and the relation u 1 (L 0 ) = u 2 (L 0 ), we obtain:

λH (ū -u 1 ) 2 (L 0 ) -λ H 0 (u -u 2 ) 2 dz = λH 1 H H 0 (u -u 2 )dz 2 -λ H 0 (u -u 2 ) 2 dz = λ 1 H H 0 (u -u 2 )dz 2 -λ H 0 (u -u 2 ) 2 dz ≤ λ 1 H H 0 1dz H 0 (u -u 2 ) 2 dz -λ H 0 (u -u 2 ) 2 dz ≤ 0.
We now come back to ( * ), which gives:

Ω2 |∇ (u -u 2 ) | 2 dxdz + Γ 2 B κ|u -u 2 | 2 dx ≤ M (1 + δ 2 )ε 2
and thus:

Ω2 |∇ (u -u 2 ) | 2 dxdz ≤ M (1 + δ 2 )ε 2
Where M denotes a positive constant. Finally, due to the fact that u -u 2 = 0 on Γ R , and by using Poincaré inequality we can deduce the inequality (33).

Remark 3.

• This proposition fails if we choose the interface position in a zone where 2-D effects are significant. In this case relation (34) is no more available.

• The right term of (33) is also an upper bound of u λ1 2 -u λ2 2 H 1 (Ω2) for all λ 1 and λ 2 positive.

Inria

Convergence of the Schwarz algorithm

In this section we provide numerical results to assess the theoretical results of §3. We are interested in illustrating the optimal convergence of Schwarz algorithm for the parameter λ = λ opt . Figure 10a shows the difference between the iterates of the Schwarz algorithm in L ∞ norm for the two test cases. As demonstrated in §3, the Schwarz algorithm converges in two iterations for the optimal parametrer λ = λ opt . It is important to notice that this result is independent of the interface location.

Difference between the coupled solution and the full 2-D solution

One important point in the analysis of the accuracy of the coupling procedure is the comparison of the coupled solution with the reference solution as a function of the interface location x = L 0 . Contrarily to the classical domain decomposition problems, here there is a difference between the (converged) coupled solution and the full 2-D solution; this difference is due to the model reduction that is performed in the 1-D part of the domain. This difference depends on the location chosen to discriminate between 1-D and 2-D regions. Figures 11a and11b resenting the error as a function of L 0 (see Figure 11, left column). This discontinuity occurs both for the numerical difference between u 2 and u Ω2 (black curve), and the theoretical curve (in red) corresponding to the right-hand-side of estimate (33). Indeed, if L 0 is greater than a certain threshold, the error grows very rapidly (and δ -→ ∞ in estimate (33)). This could be an indication of the real (a priori unknown) value of L 1 (see discussion at the end of Section 2 above).

Conclusion

In this paper we studied a linear boundary valued problem set in a 2-D domain, and assume that the solution may be approximated by a 1-D function in some part of the computational domain. We thus derive a reduced model that consists coupling a 1-D model (wherever we think it is legitimate) together with the original 2-D system (everywhere else). The model reduction is performed thanks to a small aspect ratio hypothesis, with an integration in the shallow direction (we mimic the derivation of the shallow water equations). After this derivation we introduce an iterative method that couples the 1-D and 2-D systems and we prove some convergence results. One original aspect of this work is the particular attention that is paid to the location of the 1-D/2-D interface. These theoretical results are illustrated with numerical simulations that underline the importance of the interface position, but also the way 1-D and 2-D models are coupled (boundary conditions at this interface). All these aspects, that have been studied here with a linear model, will be considered in a forthcoming study of dimensionally heterogeneous modelling in fluid dynamics.

Figure 1 :

 1 Figure 1: Typical computational domain Ω, including a zone with a true 2-D behavior (x ≥ L 1 ) together with a shallow zone where we intend to use a 1-D model (x < L 1 ).

Figure 2 :

 2 Figure 2: Funnel-shaped computational domain. The domain is shallow for x < L 2 .

Figure 3 :

 3 Figure 3: Rectangular computational domain Ω. The domain is shallow: H/L ≪ 1, and we assume that the forcing terms are supported in {x > L 2 }.

and 2 -

 2 D model:

Figure 4 :

 4 Figure 4: Computational domains for the 1-D/2-D reduced model

Figure 5 :

 5 Figure 5: Computational domains for the dimensionally homogeneous coupling problem

  (a) Computational domain for the 2-D reference model (b) Computational domain for the 1-D/2-D reduced model

Figure 7 :

 7 Figure 7: Computational domains for both the reference and reduced models in test case #1. For the reduced model (b), the 1-D/2-D interface Γ is located in x = L 0 .

  left show the H 1 error between coupled and reference solutions as a function of the interface location for the two test cases. Figures 11a and 11b right show the H 1 error in Ω 2 between coupled and reference solutions as a function of ε = H L for the two test cases. It is interesting to notice that for both test cases there is a discontinuity in the curve rep-Inria (a) Computational domain for the 2-D reference model (b) Computational domain for the 1-D/2-D reduced model

Figure 8 :

 8 Figure 8: Computational domains for both the reference and reduced models in test case #2. For the reduced model (b), the 1-D/2-D interface Γ is located in x = L 0 .

Figure 9 :Figure 10 :

 910 Figure 9: 2-D reference solution for the second test case where L = 2, H = 0.05, l = 3 and κ = 0.001

  error Log10 theoretical upper bound error (a) Test case # 1: (left): L = 20, H = 0.5 and κ = 0.001. (right): L = 20, L0 = 14 and κ = 0.01. error Log10 theoretical upper bound error (b) Test case # 2: (left): L = 2, H = 0.05 and κ = 0.001. (right): L = 2, L0 = 1.5 and κ = 0.001.

Figure 11 :

 11 Figure 11: Relative error as a function of L 0 (left) and ε (right) between the coupled solution and the 2-D reference solution in test case #1 (top) and #2 (bottom). In the left column, the red curves correspond (for both test cases) to the RHS of estimate (33).

RR n°8182

Since Ω 1D = (0, L 1 ) × (0, H) we have n = ±ez in (5b) and (5c).RR n°8182

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgments

This work was supported by the research department of the French national electricity company, EDF R&D.

Numerical results

The test cases presented in this section illustrate the coupling method of 1-D and 2-D elliptic equations based on Schwarz algorithm. All the computations have been done using the software package Freefem++ [START_REF] Hecht | FreeFem++ manual[END_REF], with a P 2 finite element discretization. In the first part of this section, the two test cases will be described in details. In the second and third parts, we will focus on one hand on the Scharwz algorithm convergence and on the other hand on the comparison of the coupled solution with the reference solution in order to enlight the theoretical results obtained in the previous paragraphs.

Description of the test cases

Test #2:

In this second test case, the 2-D effects are due to the funnel-shaped geometry of the domain (see Figure 8a), and the forcing term is constant (= 1).

The reference solution in the whole domain is displayed in Figure 9. The coupled model is defined by splitting the domain in two parts. The interface Γ is located at x = L 0 , 0 < L 0 < L 2 as shown in Figure 8.


A H Fakra
email: fakra@univ-reunion.fr

F Miranville

H Boyer

S Guichard

Development of a new model to predict indoor daylighting : integration in CODYRUN software and validation

Keywords: Indoor daylighting, building simulation software, experimental validation, CODYRUN

Many models exist in the scientific literature for determining indoor daylighting values. They are classified in three categories: numerical, simplified and empirical models. Nevertheless, each of these categories of models are not convenient for every application. Indeed, the numerical model requires high calculation time; conditions of use of the simplified models are limited, and experimental models need not only important financial resources but also a perfect control of experimental devices (e.g. scale model), as well as climatic characteristics of the location (e.g. in situ experiment).

In this article, a new model based on a combination of multiple simplified models is established. The objective is to improve this category of model. The originality of our paper relies on the coupling of several simplified models of indoor daylighting calculations. The accuracy of the simulation code, introduced into CODYRUN software to simulate correctly indoor illuminance, is then verified.

Besides, the software consists of a numerical building simulation code, developed in the Physics and Mathematical Engineering Laboratory for Energy and Environment (P.I.M.E.N.T) at the University of Reunion.

Initially dedicated to the thermal, airflow and hydrous phenomena in the buildings, the software has been completed for the calculation of indoor daylighting. New models and algorithms -which rely on a semidetailed approach -will be presented in this paper.

In order to validate the accuracy of the integrated models, many test cases have been considered as analytical, inter-software comparisons and experimental comparisons. In order to prove the accuracy of the new modelwhich can properly simulate the illuminancea confrontation between the results obtained from the software (developed in this research paper) and the major made at a given place is described in details. A new statistical indicator to appreciate the margins of errors -named RSD (Reliability of Software Degrees)is also be defined.

The objective is not only to develop an efficient research tool to improve visual comfort and reduce energy consumption, but also to transfer the knowledge through these decision-making aids tools to praticians and decision makers.

Index i
Positioning index of a point on a horizontal plan surface indoor the building. i varies from 0 to N.

Introduction

CODYRUN software is a dynamic building simulation code developed at the Physics and Mathematical

Engineering Laboratory for Energy and Environment (P.I.M.E.N.T). The software uses a multiple-model approach to simulate the energy behaviour of buildings, including thermal, aeraulic and hydrous aspects. It was in its first release created in 1993 [START_REF] Boyer | Airflow and thermal design of multizone buildings : proposal of a tool are the multiple choice models[END_REF] for the needs of two types of users:

 Building designers and operators

 Building physics researchers

Thereafter, several works have been investigated to improve the computer tool. Many physical models were tested and introduced into the software to make it more powerful and more diversified.

Recently, a new model has been developed and integrated to allow the simulation of indoor daylighting. The objective is to simulate illuminance at any point of a work plane by using a new concept to simplify the combination of photometric models. This model is original because it is based on the combination of three simplified types of models to determinate indoor illuminance.

It is indeed important to provide a tool capable of simulating daylighting, simultaneously coupled with thermal phenomena. This allows studying cases involving interactions between several energetic phenomena and determining the global behaviour of buildings.

Literature review

The literature review will be only conducted on the quantity of indoor daylighting coming from sidelights toward a point located on a specified useful work plane.

History

The CIE (International Commission of Illumination) was created in 1928 [2] in order to answer to the problem associated with daylighting and lighting. The objective of the organization is to develop norms, measurement methods and models to characterize luminous environment [3]. It is recognized by International Standardisation Organisms due to its know-how and professionalism in illuminance analysis applications.

From 1960, Kittler [4] introduced artificial skies with the aim of simulating indoor daylighting of scale models. Thus, more complex indoor daylighting configurations could be studied under normalised experimental conditions.

In 1973 [START_REF] Cie | Standardization of luminious distribution on clear sky[END_REF], the CIE published a new methodology to assess indoor daylighting from Daylight Factor (DF).

Some fifty other methods are presented in this same report.

Tregenza and Waters [START_REF] Tregenza | Daylight coefficients[END_REF] introduced the notion of Daylight Coefficient (DC) in 1983.

Collins [START_REF] Collin | The developpement of daylighting -a british view[END_REF] marked the history and discoveries in the evaluation of indoor daylighting. In fact, in his article, published in 1984, he reminded us that the very first research on quantitative study of daylighting was present since 1865. The studies were made at that time from a graphical approach to be used with simple rectangular sidelight (without considering complex cases).

In the same year, numerical calculation codes in daylighting and lighting have progressed in computers technology, more precisely, with the development of computers and software: Superlite [START_REF] Grynberg | Comparison and validation of Radiance and Superlite[END_REF], Radiance [START_REF] Ward | Application of RADIANCE to architecture and lighting design[END_REF],

Genelux [START_REF] Mitanchey | Synthesis imaging applied to indoor lighting, ENSM high school of Saint-Etienne[END_REF], Lightscape 3.2 [START_REF] Khodulev | Physically accurate lighting simulation-computer graphics software[END_REF], Adeline [START_REF] Erhorn | Adeline 2.0 -Using computer tools to Evaluate Daylightong and Electric lighting Applications in Building[END_REF], etc.

In 1988, Clarke and Janak [START_REF] Clarke | Simulating the thermal effects of daylight-controlled lighting[END_REF] improved RADIANCE and ESP-r to determine indoor daylight from Daylight Factor (or DF).

In 1999, Michel [START_REF] Michel | Experimental method for evaluating illuminance performance of buildings[END_REF] introduced a new factor for the calculation of indoor daylighting, called the Partial DF.

In the same year, Paule [START_REF] Bernard | Application of Logic Fuzzy in indoor daylighting decision[END_REF] created a simulation code (DIAL) for study of indoor daylighting and lighting.

The characteristic of this software is that it allowed preliminary draft diagnosis of a building using simplified and accurate models (minimal calculation time).

In 2004, Maamari [START_REF] Maamari | The numerical simulation of lighting : advantages and potential[END_REF] defined a series of test cases to verify the reliability of numerical simulation software applicable to indoor natural and artificial lighting.

Currently, many laboratories and organizations across the world are working to improve numerical simulation tools and modelling dedicated to indoor illuminance. This is the case, for example, of the models developed by Enrique Ruiz in 2002 [START_REF] Iea | Daylight for the 21St century : new opportunities with daylighting techniques[END_REF], Jenkins and al. in 2005 [18], and Danny in 2007 [START_REF] Danny | Daylight and energy implications for CIE standard skies[END_REF], or the task 31 of IEA in 2007 [START_REF] Ruiz | Statistical assessment of a model for global illuminance on inclined surfaces from horizontal global illuminance[END_REF].

State of the art -Daylighting, choice and selected models

Characterising indoor daylighting is very difficult because various parameters must be considered (view factor, aperture size, depth of the local, etc.).

Generally, there are three types of methods to determine quantitative indoor illuminance:

 Experimental (scale models, full-sized buildings, etc.);

 Numerical (Radiosity, Ray tracing, etc.);

 Simplified (Lumen, DF, etc.).

Experimental models require important financial means (scanning sky simulator, for example, in the case of studies of daylighting from scaled models) and the perfect observation of experimental phenomena, through measurements, on scale models or full-sized buildings. The second drawback in this method is that the time delay to obtain exploitable data is relatively long and depends on the climatic constraints of the site. Concerning scale models, errors are very high when compared to results obtained by full-scale measurements. This approach is therefore to be avoided.

In the case of numerical detailed models, calculation time is considerable. In addition, these models are generally used for visual rendering (quality of the illuminance) rather than for quantitative characteristics of daylight in a room.

Finally, the major inconvenience of simplified models is that they are only applicable under certain conditions (sky conditions, limited to a certain position, etc.). However, they have enormous advantages: easy to understand and to implement, simple to use and less time consuming, give acceptable results, etc. These models are more adaptable to the code source of CODYRUN. Therefore, this category has been selected for integration of a daylighting model in CODYRUN. Our contribution in the field has been to combine three types of models normally used in different conditions (overcast sky or not, only direct sunlight, etc.). This allowed not only to circumvent the conditions calculations, but also to improve their applications range.

Modelling of indoor daylighting

In this paragraph, new models and algorithms for the numerical simulation of daylighting in CODYRUN are presented.

Brief description of CODYRUN software

CODYRUN is a multi-zone software integrating natural ventilation, thermal and moisture transfers, developed with a friendly interface.

The software is a tool dedicated to researchers and professionals. Physical models taken into account in the Details of the multiple-model characteristics of CODYRUN are given in [START_REF] Boyer | A multimodel approach to building thermal simulation for design and research purposes[END_REF], the thermal model constitution in [START_REF] Boyer | Thermal building simulation and computer generation of nodal models[END_REF], the pressure airflow model in [START_REF] Boyer | Building ventilation : a pressure airflow model computer generation and elements of validation[END_REF], the outdoor illuminance model in [START_REF] Miranville | A combined approach for determining performance of radiant barriers field conditions[END_REF], the datastructuration and the description of the tool in [START_REF] Fakra | A simple evaluation of global and diffuse luminous efficacy for all sky conditions in tropical and humid climate[END_REF] and CO 2 model in [START_REF] Boyer | Modelisation methods and data structuration induced for building thermal simulation codes[END_REF].

Many validation tests of CODYRUN code were successfully applied to the software. Most of them are part of the BESTEST procedure [START_REF] Calogine | Identification of Parameters in Building Concentration Dispersion Model[END_REF] and led to the validation of the results.

Implementation of a new model

Introduction

All these models allow the software to address not only the various issues related to building design but also to make a detailed study of the physical variables of the surrounding.

However, the initial model does not study other aspects related to the interior environment, such as visual comfort. An improvement of the software was necessary to overcome this.

Studies have thus been conducted to integrate indoor daylighting models in the software and ensure that the software correctly simulates illuminance received at any point of a given work plane (see Figure 1).

This study is original because many research laboratories are only interested in the qualitative study of the relationship between thermal transfer and luminous aspects (for example, the association of a temperature for each luminous colour) as opposed to CODYRUN, which is able to examine quantitative values.

Improvement of the software

Many improvements have been made to the software CODYRUN to enable it to properly simulate indoor daylighting. Figure 2 summarizes the various computer codes that have been developed in this study. The objective is to develop a 3-D toolkit for building environments dedicated to the daylighting simulation. The tools to represent this environment are algorithms and data structures in space vector (vector, polygon and projection).

These algorithms are often used in programming of any sort of 3D computer graphics.

For this, the computer code (program) of CODYRUN has been amended at four different levels:

 In the simulation code: it was necessary to add indoor daylighting model, to create a work plane and to make the mesh of the plane from an algorithm based on Euclidean geometry (polygons, vertex, vector, etc.) and finally integrate a model capable of reproducing solar illuminance from a meteorological database containing only irradiance (global, diffuse and direct beam).

 In the result files: CODYRUN creates an Excel file where it usually stores all the information on the results of simulations. This file is not suitable in the photometric study. So, two improvements have been made at this level: first, the existing result file has been modified to take into account the value of illumination on the working plane. On the other hand, new file storage information, much more detailed in the field of lighting, was added. This file (output variable of the software) can be opened with a notepad application.

 In the weather file: the numerical simulation of a building cannot be done without prior knowledge of the weather conditions. Indeed, CODYRUN needs to be informed about climatic conditions (input variables of the software). Regarding the simulation of the interior illuminance, the software needs to have information on the illumination normally provided by external data format files such as TMY2.

CODYRUN accepts such files. Sometimes, we do not have a similar database. Knowing that the software has its own type of weather file but does not give any information on illuminance variable, an improvement has therefore been made to provide information on outdoor illuminance. In the lack of weather data, it is possible to simulate indoor illuminance. Indeed, at a given time it is sufficient to indicate an overall value of outdoor illuminance for the calculations of the indoor illuminance at any point on a work plane.

 In the building description: another input variable should be given to the software: the description of the studied building. However, the indoor illuminance strongly depends on the geometric position in space of openings, building orientation, etc. Previously, the software could not take into account all these parameters. This problem was solved by adding the cartesian coordinates in the declaration of each component (wall, floor, ceiling, windows, etc.) of the building. Each component is constituted of two sides and each side is characterized by one vertex. sidelight. The principle of calculation for each one of these two methods will be given in the following paragraphs. In addition, a new specific mathematical relation for the determination of the diffuse illuminance on a sun patch will be described.

Calculation of diffuse illuminance in the absence of sun patch: DF method

The DF (Daylight Factor) method will be used to calculate diffuse illuminance. This method was elaborated by BRE (Building Research Establishment) [START_REF] Soubdhan | Use of BESTEST procedure to improve a building thermal simulation program[END_REF] and published by CIBSE [START_REF] Bre | Estimating daylight in building : Part 1 and 2 / Diggest 309 and 310[END_REF]. We can consider the light falling on a point in the rooms as being composed of three distinct components. Light that comes directly from the sky

Equation.3 (3)
The general equation of diffuse illuminance at an inside point (of the room) on the sun patch becomes:

Equation.4 (4)

Calculation of the sunlight

For this part, the Siret's method is used [START_REF] Cibse | Window design, application manual[END_REF]. The relation is written as followed:

Equation.5

(5)

Recapitulation: new method of indoor illuminance calculation

Relations (2), ( 3) and ( 5) allow establishing a new combination model of simplified categories to calculate indoor illuminance for every condition:

(6)
It is important to note that this simple new relation is original since (2), ( 3) and ( 5) are usually used separately (for each according to different conditions of application). Whereas the new relation found ( 6) is the combination of the three models. Thanks to this new combination, it is possible to calculate the illuminance values for any sky conditions. Indeed, (2) is applied only in overcast sky condition, while (3) and ( 5) are applied if and only if a light flux -coming from the sun -enters the room. The scientific literature does not give information on the combination of relations (2), (3) and ( 5), which are combined together (i.e. relation ( 6)).

Validation

The validation step is very important to know the precision of the simulation results. Especially when dealing with daylighting simulation programs, the achievement of the accuracy of the predictions is difficult. It is thus necessary to refer to rigorous procedures, recognized worldwide.

There are few documents on the procedures to follow. In most cases, laboratories implement their own experimental database to serve as a reference for comparisons between model predictions and measurements.

The study of Maamari [START_REF] Siret | A geometrical framework for solving sunlighting problems within CAD systems[END_REF] perfectly responds to our needs concerning the validation step. Indeed, his thesis is related to the establishment of methods to check the reliability of simulation codes of indoor daylighting and artificial lighting, from analytical and experimental tests cases. This work is used as reference in the task TC3-33 of the CIE [START_REF] Maamari | Application of the CIE test cases to assess the accuracy of lighting computer programs[END_REF].

Many other test cases found in the scientific literature (papers from CSTB, BRE, Task 21 of IEA and experimental test case of CIBSE, etc.) have been applied to the simulation software CODYRUN. An intersoftware comparison has also been done and finally, database references for the local study in dynamic conditions (from a cell called LGI) have been established. This last case will only be presented here. The software only considers sidelight. In addition, it does not take into account internal obstructions (furnishing, occupants, etc.), bidirectional transmissions (BRDF) and lightwell. Therefore, simulation results will be confronted to reference test cases, taking into account the above-said hypothesis.

New approach to determine the percentage of reliability

It is important to consider a square percentage deviation to calculate the error between measurements and used models, when studying outdoor or/and indoor daylighting.

Outdoor errors

We have recently taken into account used models for the study of luminous efficacy [START_REF] Cie | Test Cases to assess the acuracy of lighting computer programs[END_REF]. It concerns the references [START_REF] Dumortier | Measurement, analysis and modeling outdoor daylighting and application to the assessment of illuminance performance into the building[END_REF] and [START_REF] Gabriel | Clear-sky solar luminous efficacy determination using artificial neural networks[END_REF]. In order to increase accuracy, some statistical indicators also need to be defined as follows:

Equation.8 (8)

Indoor errors

Concerning indoor daylighting study, the estimate is made from relative errors  at the point i defined by: Equation.9

Reliability of the Software Degrees (RSD): new concept of quantification of the accuracy

In this study, a new percentage indicator that verifies the capacity of the software to correctly simulate indoor daylighting is defined as RSD (or Reliability of Software Degree). It is the ability of the software to simulate correctly and realistically a physical variable. In our case study, the variable is indoor daylighting.

Photometric phenomena are very complex, so we can consider that results of numerical simulation software are correct compared to reality (or reference values) when its RSD is greater or equal to 50%. This gives a precise idea of the limits, strengths and weaknesses of the software in its application field. As a preliminary, this method requires the knowledge of reference values (or reference test case study).

Two situations are presented for the determination of RSD:

 On the one hand, the reference values are given at each measurement. For example, positioning reference point A 1 at A 4 (or reference calculation) in form of margin (upper and lower limits of the curve below) of acceptable values for the results (results given by CODYRUN on the curve below) of the simulation. These margins are given by measured values more or less the total estimated error with and without errors linked to the simulation (see figure 3). In this case, the RSD is the percentage of simulated values situated between references values.

Summary and results of test cases submitted to CODYRUN

Table 1 summarises all test cases applicable to CODYRUN and associated errors for each case. CODYRUN successfully passed selected test cases (the errors were acceptable).

Globally, results for CODYRUN are comparable to those of other codes for daylighting. This shows that the model included in CODYRUN is well implemented and gives acceptable results.

Results of analytical test cases confirm the aptitude and the limitations of the software to simulate daylighting. Unlike the approximate values of results concerning the calculation analytical test case, the simulation of CODYRUN code gives exact values. Consequently, this difference of calculation increases the errors in the comparison between the simulation in CODYRUN and the analytical test case.

Concerning the references of experimental test cases, it is important to know that they are very complicated to implement, requiring a lot of logistics, technical and financial means. For example, accuracy for spatial location of the sensors is very important; it is also important to ensure that the measureable limits of the sensors corresponds to the range of measured values.

As a conclusion, this work allowed us to assess the strengths and weaknesses of the software. Thus, the improvement of the software could be done on the illuminance calculation at the furthest and nearest point to the apertures.

Test

Example of dynamic validation: experimentation on LGI cell

Table 1 shows the details of test cases submitted to the software CODYRUN to check its level of reliability to simulate indoor daylighting.

The highest error between the reference values and those from the simulations were observed during the dynamic study performed locally using the LGI cell (about 32%) of relative error [START_REF] Pattanasethanon | All sky modeling daylight availability and illuminance / irradiance on horizontal plane for mahasarakhan[END_REF]. The following paragraph will present a detailed study to explain the results.

Experimental test cell LGI

An existing experimental full-scale test cell has been instrumented in order to verify the reliability of the software CODYRUN to correctly simulate daylighting in transient conditions during an entire day. The database collected is used as a reference for the comparison between simulation and experimental values.

The experimental test cell LGI is located at Saint-Pierre (see figure 5). This test building was built for experimental validation reference on physical models introduced in CODYRUN [START_REF] Mitanchey | Synthesis imaging applied to indoor lighting, ENSM high school of Saint-Etienne[END_REF] and ISOLAB [START_REF] Iea | Daylight Responsive lighting Control-Cases[END_REF]. The LGI cell is representative of a typical room or office in Reunion Island. The door is made up of aluminium, with a 6 mm glass window.

Instrumentation and experimental procedure

The meteorological data are measured every minute. The station measures irradiation (global and diffuse) and wind speed and direction (respectively at a height of 2 m and 10 m from the ground). We added an outdoor daylight sensor that measures more than 14,000 values each day. The data logger is of CAMPBELL type. The measurements of indoor and outdoor daylighting are made from instruments of AHLBORN type. All the measuring instruments are synchronised. To simplify our study, we have positioned an exterior luxmeter (sensor) on the roof of the LGI cell (see figure 8). Indoors luxmeters (sensors) were aligned on the slabs as shown in the figure (see figure 9). These sensors are equidistant by 0.5 m. The horizontal distance between position of the sensor A 1 and aperture is 0.23 m. Globally, three categories of days (refer to [START_REF] Miranville | A combined approach for determining performance of radiant barriers field conditions[END_REF] for more information) are used as a reference experiment: it consists of intermediate day, clear day and overcast day conditions. The reference Table 2 gives the subjective and objective measurements errors. Total error associated with the experimental measurements (outdoor and indoor measurements) is 13.22% and the global error is 15%.

Measurement errors

Table 2 indicates the values of subjective and objective errors of our experiment. This approach to study the measurement error is highly recommended by the C.I.E [START_REF] Maamari | Application of the CIE test cases to assess the accuracy of lighting computer programs[END_REF].

These values will allow us to give margins of lower and upper limits for each measurement. The mean error (measurements errors) is 13.4% and global error (measurements and software hypothesis of simulation errors) is 15%. Table 3 gives the characteristics of the interior measurements sensors.

Error Identifications

Error value (en %)

Objective

Sensor accuracy 1 (indoor) + 0.03 (datalogger) + 9.9 (outdoor) Cosine correction 2.9 (indoor sensor) + 2, Operating/storage temperature 0°C to +60°C / -10°C to +80°C The simulation conditions of the LGI cell in CODYRUN are:

 Albedo = 70%
 Interior reflectance of the walls and ceiling (white colour) = 60%

 Interior reflectance of the floor (grey-coloured slabs) = 20%

 The height from the slabs to the workplane is at 0.01 m.

 Dimension of the elementary surface grid = 0.1m x 0.1m  Correction factor of DF: MG = FR = 0.8 and MF = 0.9 (see [START_REF] Boyer | Airflow and thermal design of multizone buildings : proposal of a tool are the multiple choice models[END_REF] for the definition of these coefficients)

Comparison of results

This paragraph presents the simulation results and the comparison with those of measured indoor daylighting obtained for the three days conditions.

 Clear day condition We obtained a RSD value of more than 50%. The simulation results obtained -without considering the direct sunlight flux -better correspond to the values of measurements (see Figure 8). There is obviously no inertia in photometric phenomena (i.e. abrupt variations of daylighting with time). So, during our measurements we took a time step of one minute to take into consideration these abrupt variations. The inconvenience is that this increases the errors between the simulation results and reference measurements. To verify this hypothesis, we plotted the same curve with hourly averaged values and we got the following figure (refer to Figure 9, for the same point A 3 ). Reference values obtained hourly present more realistic results than minutely simulated results. the morning, very high illuminance measurements have been observed, due to direct sun penetration. Indeed, the interior sensors were not able to measure the direct illuminance flux of values which were superior to 12 500 lux because of their measurements range, which does not go beyond this value (See Table 3). This explains why the software seems to simulate incorrectly the global available illuminance in the morning. In fact, simulation results in the evening better correspond to actual values.

 Overcast day

In this study, we present simulation results obtained from reference positions points A 5 . Figure 12 illustrates the results of simulation. In most cases, simulated values in CODYRUN between 10:30 am and 12:45 pm are higher compared to measured values (for example see the highlighted zone R 2 in the Figure 12). The explanation of this result is that the studied cell is equipped with an overhang on the northern façade. This prevents a major part of the sunlight R 1 R 2 from penetrating into the room. At this time, this particular device is not taken into account by the software. We noted that (for the five reference positions) a significant difference between simulation and measurement values exists in the morning (highlighted region R 1 ) compared to the evening (highlighted region R 3 ):

 In fact, from sunrise till 12:00 am (corresponding to highlighted R 1 zone in the Figure 12)

CODYRUN better simulates the indoor daylighting. This is certainly due to the recognition of obstacles by the software (buildings in the neighbourhood of the north-east façade corresponding to the sun rising façade). This has been taken into account for indoor daylighting by the software;

 Concerning the evening, we noticed that (for example, the highlighted R 3 zone in A 5 -Figure 12) the simulated values are lower than the measured values. The fact is that South and West parts of vertical façades of the test building are opaque. Then, the North-facing overhang does not influence skylight at these hours of the day. We obtain more coherent values between the simulation and the measurements.

As previously, we were interested in comparing these reference values with those simulated values for each hour. The results (see the result comparison of figure 13 at reference point A 1 ) show a good accuracy of the software in the calculation of indoor daylighting, with the exception of close and far positions from the opening.

To conclude, in the case of an overcast day we can note that reliability accuracy is higher or equal to 68%.

Globally, CODYRUN has the tendency to underestimate indoor daylighting.

Conclusion of the transient study

The case of a clear day study demonstrated the difficulty of simulating transient indoor daylighting. Indeed, the photometric effect is very complex (it is difficult to try to draw the shape of the illuminance for a given day and at a given point). For this reason, many other research laboratories prefer to conduct a time step study, but not during an entire day (see [START_REF] Boyer | Modelisation methods and data structuration induced for building thermal simulation codes[END_REF] and [START_REF] Calogine | Identification of Parameters in Building Concentration Dispersion Model[END_REF]). However, we found that the simplified models introduced in the software were able to simulate fairly well the dynamic photometric variable. Similar situations were presented in an intermediate day study.

For an overcast day, we noted that the impact of the overhang in the simulation was important. Indeed, the model introduced in CODYRUN does not consider, for the moment, the influence of overhangs. The fact that the overhangs are not considered in the simulation increases the errors of comparisons in the experimental validation. We also noted that the software made higher errors at reference points located close and far from the opening.

During an entire day, the RMSD of the overall simulation is about 0.61. Considering the relative errors of theses hourly simulations, we obtain an average error of 32.7%.

Conclusions

The sun is a renewable energy source often used in interior lighting (inside buildings).

This allows a decrease in the electrical energy consumption for artificial lighting in buildings. But, we do not have enough information for the study concerning the needs of illuminance in tropical and humid climate regions.

In the building, the integration of daylighting coming directly or indirectly from the sun is a variable, which is more and more used in development tools to determine the illuminance availability. However, these tools must satisfy the needs of architects and building design offices. According to them, the tool should not only quantify daylight, but should also elaborate method tools, which are easily usable. CODYRUN tries to satisfy this requirement.

As for any other software, the use of CODYRUN to simulate indoor daylighting presents advantages and drawbacks.

Concerning the advantages, we can indicate:

 Reduced calculation time. For example, the software takes 2 minutes to simulate daylighting on a workplane of 39x35 position points inside a building and for a time step, it takes one minute all long year);

 The software is simple, user-friendly, and is perfectly adapted to design office (engineering consulting firm) and to individuals;

 This software is a tool used in research laboratories. We were able to show that CODYRUN was reliable at 86.25%;

 It is possible to reconstitute the outdoor illuminance from a meteorological file, which only has irradiance information;

 It is possible to simulate the effects of sunlight coming from a secondary light source;

 We can monitor (or follow) the sun patch in the workplane;

 CODYRUN can separately simulate the direct, diffuse and global daylighting for every sky condition,

through a new simplified model that is defined in this paper;

 The software is capable of dynamical studies.

The limitations of the model introduced in the software are:

 It does not consider overhangs and shadow masks in daylight simulation;

 No daylight calculation on vertical or inclined walls are considered;

 The cartesian coordinates of the building are manually inserted. This makes the description of the building (in CODYRUN) a time-consuming task;

 The are no graphical user interface, and no 3D visual representation of the building or visual rendering;

 The used models are simplified categories, so they are not applicable to complex architectural configurations;

 It does not consider obstructions inside the room (furniture, occupants, etc.) when simulating daylighting;

 The opening should only be sidelights.

The conditions, which can be used in the software, are those of simple (the volumes are rectangular, triangular, cubic, or L-shaped) closed systems (or sub-systems). We cannot simulate complex systems (for example, inclined walls of buildings).

Future research

The future research in this field can be classified in three categories as follows:

 The integration of an indoor artificial lighting model and the improvement of the new indoor daylighting model in CODYRUN tool;

 Complementary validation results, e.g. performed once again the experimental study for intermediate and clear condition days in LGI test cell by using sensors capable of taking into account indoor direct sunlight;

 Using the tool to optimise the electrical energy consumption and to consider models taking into account simultaneously the temperature and the daylight.



  Airflow and humidity transfer  Outdoor convection  Indoor convection  Outdoor long wave exchanges  Indoor long wave exchanges  Indoor short wave repartition  Heat conduction  Conductive exchanges with the ground  HVAC system  Outdoor illuminance  CO 2 transport

Figure 1 .

 1 Figure 1. Objective of the study

Figure 2 .

 2 Figure 2. Improvements to the software CODYRUN

1 ( 1 )

 11 and called the Sky Component (SC), light that comes from external surfaces such as buildings called the Externally Reflected Component (ERC), and light that is reflected from internal surfaces called the Internally Reflected Component (IRC). The sky component normally refers to the diffuse sky: it is not used to describe direct sunlight. The DF is simply calculated as: Equation.If a DF value (from the relation (1)) is known, it is possible to calculate the global inside illuminance at the point of interest from the relation (2): .3.2. Calculation of diffuse illuminance part in the presence of the sun patch : new method It is considered that the diffuse illuminance at a point (which is inside the sun patch) is the sum of the diffuse illuminance generated by the DF and that produced by the sun patch itself in the neighbourhood of the point. A simple relation originally determines the diffuse part coming from the sun patch:

1 .

 1 Limitations of models used in CODYRUN for indoor daylighting simulation CODYRUN only provides daylight values for a given horizontal surface (i.e. no daylighting values are calculated on vertical walls).

  Generally, three statically indicators are considered to characterise the accuracy of the models associated to outdoor and indoor daylight: the coefficient of determination R 2 , the Mean Bias Deviation (MBD) and the Root Mean Square Deviation (RMSD). MBD demonstrates the model's tendency to underestimation or to overestimation. RMSD offers a deviation measure from the predicted values in relation to the measured values. The use of the correlation coefficients R 2 , MBD, and RMSD determines the accuracy of the models. The following relations give the definitions of these statistical estimators:

( 9 )

 9 Global relative () errors are thus written:Equation.[START_REF] Mitanchey | Synthesis imaging applied to indoor lighting, ENSM high school of Saint-Etienne[END_REF]

Figure 3 .

 3 Figure 3. Illustration of RSD evaluation when reference points are margins (upper and lower limits)  On the other hand, reference values are given at each measurement point (or reference calculation)

Figure 4 .

 4 Figure 4. Illustration of RSD evaluation when reference points are given without margins

Figure 5 .

 5 Figure 5. Experimental site for L.P.B.S. Laboratory Two types of luxmeters were used for the global illuminance: one to characterize the outdoor daylighting and another to characterize the penetration of indoor daylight. The experimental platform includes a meteorological station and a technical room. The internal and external walls were initially painted white. Absorption coefficient of the interior surfaces is about 60 %. The roof is constituted of corrugated iron and insulated with a radiant barrier. The vertical walls are made of insulating board. The floor is composed of concrete slabs, of polystyrene on the top (4 and 5 cm thick). The dimensions of the LGI cell are illustrated in the figure 8.

Figure 6 .

 6 Figure 6. LGI cell test of L.P.B.S.

Figure 7 .

 7 Figure 7. Schematic position of the five interior luxmeters

3 Table 2 .

 32 Subjective and objective errors of experimentation of the LGI cell test

Figure 8

 8 Figure 8 indicates the comparison between measured values and simulated values at point A 3 . The trend at points A 1 , A 2 , A 4 , and A 5 are practically the same, but with different amplitudes.

Figure 8 .

 8 Figure 8. Clear day: comparison between measured and simulated values at the point A3 (without direct sunlight and for one-minute time step)

Figure 9 .

 9 Figure 9. Clear day: comparison between measured and simulated values at the point A3 (for one-hour time step)

Figure 10 .

 10 Figure 10. Intermediate day: comparison between measured and simulated values at the point A1 (for oneminute time step) Figure 11 compares simulation results with results for an hourly time step at point A 1 . That day was divided into two equal parts: a relatively clear morning and a cloudy evening. During some precise moments of

Figure 11 .

 11 Figure 11. Intermediate day: comparison between measured and simulated values at the point A1 (one-hour time step)

Figure 12 .

 12 Figure 12. Indoor daylighting : Overcast day condition (LGI cell)

Figure 13 .

 13 Figure 13. Overcast day: hourly comparison between measured and simulated values at the points A1

Table 1 . Reference test cases applied to CODYRUN

 1

	Type	Reference	Procedure	Relative	RSD	Observation
				error (%)	(%)
		[30]	II.10	3.87	91.44	Side opening
		[30]	II.11	4.54	93.65	Different types of skies CIE
	Analytical
		[30]	II.14.4	1.96	98.04	Diffuse (form factor)
		[30]	II.14.9	15.61	84.34	Solar irradiance (outdoor)
	Comparative	[37]	CSTB	5.92	95	Daylight Factor (DF)
		[30]	III.2.5	3.38	93.49	Artificial sky
		[36]	I.E.A	9.31	74.46	Scale model E.P.F.L.
	Experimental	[38]	I.E.A	11.31	70.92	Real Building H.U.T.
		[35]	LGI test	32	88.95	Transient (or dynamical
						aspect)
			Minimum =	1.96	84.34
			Maximum =	32	98.04
			Average =	9.76	87.81

Table 3 . Characteristics of the interior measurements sensors (FLA603VL2), of AHLBORN type

 3

	4.3.4. Simulation hypothesis in CODYRUN