J Ridoux

L Monasse

F Coulouvrat

Extension of Geometrical Shock Dynamics for blast waves propagation

Keywords: Blast waves, Geometrical Blast Dynamics, Geometrical Shock Dynamics, Fast running method, Lagrangian scheme

The direct numerical simulation of blast waves (accidental or industrial explosions) is a challenging task due to the wide range of spatial and temporal scales involved. Moreover, in a real environment (topography, urban area . . . ), the blast wave interacts with the geometrical obstacles resulting in reflection, diffraction and waves recombination phenomena. The shape of the front becomes complex, which limits the efficiency of simple empirical methods. This work aims at contributing to the development of a fast running method

Introduction

A blast wave results from the sudden release of a finite amount of energy from an impulsive source. A precise prediction of airborne blast waves propagation is required in numerous applications such as the pyrotechnics industry, explosion hazards or noise annoyance among others. The direct numerical simulation of blast waves from the detonation neighborhood to long range propagation is a challenging task due to the wide range of spatial and temporal scales involved. Moreover, the blast wave propagation is affected by numerous physical conditions such as the source shape, the height of burst, the interaction with obstacles or topography as well as the atmospheric conditions. This results in numerous physical phenomena such as diffraction, regular reflection, Mach stem formation or waves recombination. The shape of the front becomes complex, which limits the efficiency of simple empirical methods restricted to basic configurations.

As examples of canonical cases, we can cite the use of free field abacus such as Kinney and Graham's ones [START_REF] Kinney | Explosive shocks in air[END_REF], coupled with empirical reflection [START_REF]Structures to resist the effect of accidental explosions[END_REF] or with numerical diffraction [START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF] coefficients. Other examples include the energy concentration factor [START_REF] Silvestrini | Energy concentration factor. a simple concept for the prediction of blast propagation in partially confined geometries[END_REF] for predicting blast propagation in partially confined geometries, or empirical laws for Mach stem evolution [START_REF] Kinney | Explosive shocks in air[END_REF][START_REF] Needham | Blast wave propagation[END_REF][START_REF]Structures to resist the effect of accidental explosions[END_REF][START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF] in the case of shock reflection over the ground. More sophisticated methods exist for simple structures such as the image burst method [START_REF] Needham | Blast wave propagation[END_REF] for multi-reflected shock waves or the ray tracing method [START_REF] Miller | Towards the modelling of blast loads on structures[END_REF] for diffraction around structures.

An example of the coupling of these methods can be found in [START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF] for blast-wave mitigation by a prismatic blast wall. For more complex three-dimensional configurations, the FLASH (Fast Lethally Assessment for Structures and Humans) code [START_REF] Lapébie | Flash : Fast lethality assessment for structures and humans[END_REF], designed to rapidly estimate explosion effects in urban areas, compiles these different geometrical methods. However, all these semiempirical models are limited to simple geometries and their extension to arbitrary three-dimensional configurations seems difficult. As an alternative way, Flood [START_REF] Flood | A new method for very fast simulation of blast wave propagation in complex built environments[END_REF] proposed to solved full Euler equations over a rough mesh, and then to use a neuronal network to refine results. The neuronal network is coupled with a data base previously provided by fine simulations. However, few details are provided about this approach. Moreover, the far field propagation turns out challenging given the important numerical diffusion occurring for rough meshes.

Consequently, the development of alternative simplified models able to describe precisely blast waves propagation in complex environment and over long ranges is required.

Geometrical Shock Dynamics (GSD) model [START_REF] Whitham | A new approach to problems of shock dynamics. Part I: Two-dimensional problems[END_REF][START_REF] Whitham | A new approach to problems of shock dynamics. Part II: Three-dimensional problems[END_REF][START_REF] Whitham | Linear and Nonlinear Waves, chapter 8: Shock Dynamics[END_REF] is identified as a good alternative model for blast waves propagation. It is able to estimate at a moderate cost, but with a reasonable accuracy, the propagation of a shock interacting with geometrical elements. This model is based on the decomposition of the shock front into elementary ray tubes. Assuming small changes in the ray tube area and neglecting the influence of the post-shock flow on the shock itself, a simple relation can be established [START_REF] Best | A generalisation of the theory of Geometrical Shock Dynamics[END_REF], linking the local curvature and the front velocity, known as the A-M rule. It is well known that GSD is quite accurate for sustained shock propagation problems. It has been investigated by numerous authors in the past, including among others cases of converging flows [START_REF] Schwendeman | On converging shock waves[END_REF][START_REF] Schwendeman | On converging shock waves of spherical and polyhedral form[END_REF], propagation through nonuniform media [START_REF] Catherasoo | Shock dynamics in non-uniform media[END_REF], and outdoor propagation [START_REF] Besset | Propagation of vertical shock waves in the atmosphere[END_REF] . Nevertheless, the model suffers from an intrinsic limitation for the issue of shock diffraction over a convex wall.

For sufficiently weak shocks, no solution of GSD model exists up to the wall above a given deflection angle. This is in contradiction with experimental studies [START_REF] Skews | The shape of a diffracting shock wave[END_REF] showing that the diffracted shock front should still exist at the wall, even for weak shocks and at large deflection angles. Theoretical approach of nonlinear acoustics indicate that weak shock diffraction is dependent on the overall waveform, this means is influenced by post-shock flow [? ]. Some modifications of GSD model, such as its extension to post-shock flow [START_REF] Best | A generalisation of the theory of Geometrical Shock Dynamics[END_REF], or a modified treatment of the wall condition [START_REF] Bazhenova | Change in the shape of the diffracting shock wave at a convex corner[END_REF], are able to recover the inflection point experimentally observed for strong shocks, but do not remove the restriction. The more recent Kinematic model [START_REF] Sharma | On one-dimensional planar and nonplanar shock waves in a relaxing gas[END_REF][START_REF] Sharma | Three dimensional shock wave propagation in an ideal gas[END_REF][START_REF] Pandey | Kinematics of a shock wave of arbitrary strength in a non-ideal gas[END_REF] is no more efficient to remove this limitation [START_REF] Ridoux | Comparison of Geometrical Shock Dynamics and Kinematic models for shock wave propagation[END_REF]. Based on the original Oshima's idea [START_REF] Oshima | Diffraction of a plane shock wave around a corner[END_REF][START_REF] Oshima | Propagation of spacially non-uniform shock waves[END_REF] of transverse flow along the shock, an ad-hoc modification of the A-M relation for two-dimensional configurations has recently be proposed in [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF]. The closure is designed to systematically remove the limitation of the model. GSD model is quite accurate for sustained shock propagation problems, for which post-shock flow has little influence on the shock propagation. In such a problem, the source energy is indeed infinite and continuously supplies the shock front. Consequently, the shock propagation is only driven by the local changes of shock geometry. It is no longer the case for blast waves propagation with a finite source of energy. Taking into account post-shock effects in the A -M relation yields an infinite sequence of Ordinary Differential Equations (ODE) with higher-order post-shock flow terms [START_REF] Best | A generalisation of the theory of Geometrical Shock Dynamics[END_REF]. Truncation at zeroth-order corresponds to the original GSD model. The dynamical influence of post-shock flow over the shock front is introduced as of first-order truncation. Nevertheless, an analysis of flow non-uniformities effects for blast waves reveals the complexity of choosing the order of truncation [START_REF] Peace | On the propagation of decaying planar shock and blast waves through nonuniform channels[END_REF]. As previously proposed [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF], GBD model also includes additional transverse flow effects designed to remove the limitation of GSD model for the case of a shock diffracting over a convex wall.

The present paper is organized as follows. First, section 2 recalls the phenomenology of blast waves and presents some semi-empirical models selected to evaluate GBD model. Secondly, the derivation of GSD model and its ad-hoc modification to incorporate transverse flow is reviewed in section 3. The simple extension of GSD to blast waves is then fully described in section 4.

The two-dimensional axisymmetrical conservative Lagrangian scheme, designed to solve GBD model, is also briefly recalled in this section. In section 5, GBD model is evaluated for several cases of increasing complexity and compared to semi-empirical models, Eulerian simulations and experimental data. Finally, section 6 provides a conclusion and suggests ways for further improving.

Problem set up

A blast wave results from a sudden and local release of a finite amount of energy from an impulsive source.

As a consequence, a shock wave develops in air with a dramatic reduction in its strength as it propagates from away the source. For the ideal case of a spherical source in free field, Figure 1 sketches the typical temporal pressure signal. The shock, characterized by the peak overpressure ∆P + , reaches the observation point (a gauge for example) at time t a , called arrival time. The shock is continuously overtaken by an unsteady rarefaction wave resulting in an alternation between positive phase, where the pressure is up to ambient pressure p 0 , and negative phase where it is down to p 0 . The positive phase is fully described in terms of t a , ∆P + , the positive phase duration t + and the positive phase impulse I + . The negative phase duration t -, the pic underpressure ∆P -and the negative phase impulse I -characterizes the negative phase. The negative phase is usually not considered for most considerations such that the load evaluation on structures.

Free field laws

For a spherical or hemispherical charge of TNT 1 , the empirical Kinney's laws [START_REF] Kinney | Explosive shocks in air[END_REF] provides an approximation for main characteristics of blast waves. In particular,

1 TNT: Trinitrotoluene, C 7 H 5 N 3 O 6 .
Fig. 1 Typical temporal pressure signal for an ideal blast wave according to [START_REF] Kinney | Explosive shocks in air[END_REF].

the evolution of the non-dimensional peak overpressure,

ζ K = ∆p + /p 0 , is expressed as ζ K (Z) = 808 " 1 + " Z 4.5 « 2 « s " 1 + " Z 0.048 « 2 « " 1 + " Z 0.32 « 2 « " 1 + " Z 1.35 « 2 « , (1)
where Z = r/W 1/3 is the scaled distance, r the distance at the source and W the mass of TNT expressed here in kilogram. The blast wave intensity, characterized by the peak overpressure, rapidly decreases with the distance as seen in figure 2. It degenerates to an acoustical wave, with an overpressure bellow to 20 mbar, at only 43 meters from the source. Another empirical laws for peak overpressure can be found in the literature, referred as Kingery's law [START_REF] Michael | Simplified kingery airblast calculations[END_REF] or the Needham's law [START_REF] Needham | Blast wave propagation[END_REF] for example. In this paper, only Kinney's law is considered for condensed explosives. For another nature of explosive such that gaseous charges, let us cite the empirical law and [START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF].

for a stoichiometric propane-oxygen mixture [START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF]:

log (ζ E (Y )) = 0.0895 -1.7633 ln (Y ) + 0.1528 ln (Y ) 2 -0.0066 ln (Y ) 3 -0.0021 ln (Y ) 4 , (2)
with ζ E = ∆p + /p 0 , and where Y is the scaled distance reported to the released energy: Y = r/E 1/3 (in m/MJ 1/3 ).

Blast wave reflection

Outdoor blast waves are affected by numerous physical conditions such as source shape, height of Burst (HoB)

or interaction with obstacles. For a spherical explosion above the ground as sketched in figure 3, the shock wave interaction with the ground first results in a Regular Reflection (RR): incident and reflected shock waves are Point (TP). The Mach stem may be regarded as the fusion of the incident and reflected shock waves resulting in a rise of the peak overpressure [START_REF] Ben-Dor | Oblique Shock Wave Reflections[END_REF]. The regime of reflection depends on past history of the shock and two main parameters [START_REF] Kinney | Explosive shocks in air[END_REF]: the incident shock overpressure ∆P i , or equivalently the incident Mach number M i (6), and the angle of incidence β. The empirical Kinney's formula [START_REF] Kinney | Explosive shocks in air[END_REF] provides an estimation of the critical angle of transition, β max , between RR and MR: Fig. 4 Reflected pressure coefficients from UFC [START_REF]Structures to resist the effect of accidental explosions[END_REF].

β max = 1.75 M i -1 + 39 (in degree). ( 3
Several empirical laws allows to estimating the distance of transition between RR and MR as well as the Height of Triple Point (HTP) [START_REF] Kinney | Explosive shocks in air[END_REF][START_REF]Structures to resist the effect of accidental explosions[END_REF][START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF][START_REF] Mckinzie | The US nuclear war plan: A time for change[END_REF]. Boutillier et al. [START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF] defined an empirical law from experimental data for C4. The scaled ground distance, SR 0 = R 0 /W 1/3 , and the scaled height of burst, SHT P = HT P/W 1/3 , are expressed for TNT as:

SR 0 = 1.99 × 10 -3 E 1/3 q SHoB 2 + 0.601SHoB, (4)
SHT P = 0.07 SHoB SR 2 0 SD 2 -2SR 0 SD + SR 2 0 , (5)
where E q is the TNT equivalency taken to 1.28 here.

SD is the scaled distance at the ground: SD = D/W 1/3 . These laws are valid for SHOBs ranging from 22.7 cm/kg 1/3 to 159.7 cm/kg 1/3 .

The reflected overpressure at the ground, ∆P r , can be estimated with empirical reflection coefficients from UFC [START_REF]Structures to resist the effect of accidental explosions[END_REF], shown in figure 4. This reflection table depends on β and ∆P i , where ∆P i can be evaluated with a free field law such as Kinney's one [START_REF] Kinney | Explosive shocks in air[END_REF]. The reflection table intrinsically takes into account the regime of reflection.

Blast wave diffraction

In the case of diffraction of a spherical blast wave as sketched in figure 5(a), [START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF] proposes a numerical abacus inspired from reflection table (see figure 5(b)). This table gives the diffraction coefficient, All empirical and semi-empirical laws are alternative approaches to the direct simulation of blast waves for simple geometries. Nevertheless, they are unable to estimate blast waves propagation in interaction with complex environment. An intermediate model, able to account for the shock interaction with structures of any shape, is the Geometrical Shock Dynamics model.

C d = ∆P d ∆P ff ,

Review of GSD model

In this section, we recall the classical derivation of the Whitham's Geometrical Shock Dynamics (GSD) model [START_REF] Whitham | Linear and Nonlinear Waves, chapter 8: Shock Dynamics[END_REF].

Its limitation for expansive shock waves is discussed, followed by the presentation of a simple correction aiming at removing this restriction.

We consider the propagation of a shock wave in a uniform, quiescent, and calorically perfect gas. We denote by ρ, p, and v the density, pressure, and fluid velocity respectively. For a perfect gas, the sound speed c We introduce the shock Mach number:

M = U c 0 ≥ 1,
where U denotes the shock velocity. The Rankine-Hugoniot relations [START_REF] Courant | Supersonic flow and shock waves[END_REF] links this quantity to the shock overpressure ∆P + :

M = 1 + γ + 1 2γ ∆P + p 0 . ( 6
)
The shock position, x, is a solution of the Ordinary Differential Equation (ODE):

dx dt = U n = c 0 M n, ( 7
)
where t is the arrival time of the shock, and n denotes the unit outward normal vector at the front. At time t, the shock front is identified as the zero level set of a scalar function Φ :

Γ (t) = {x ∈ Ω ⊂ R 3 /Φ(x, t) = 0}. (8)
In the neighbourhood of the shock, we suppose that Φ is differentiable and that ∇Φ is not identically null.

From (8), the unit normal vector at the shock front is defined by

n = ∇Φ |∇Φ| , (9)
and Φ verifies the equation:

∂Φ ∂t + U n • ∇Φ = 0. ( 10
)
Injecting ( 9) into (10), we get:

U = - ∂ t Φ |∇Φ| . (11)
Assuming a single-pass front, the level-set function Φ may be defined as

Φ(x, t) = α(x) -c 0 t, (12)
where α describes the shock position. Since c 0 is constant, α = c 0 t is also called a pseudo-time for the sake of simplicity. The unit normal vector at the shock front, [START_REF] Flood | A new method for very fast simulation of blast wave propagation in complex built environments[END_REF], is then

n = ∇α |∇α| , (13)
and the normal velocity of the shock, [START_REF] Whitham | A new approach to problems of shock dynamics. Part II: Three-dimensional problems[END_REF], leads to the local eikonal equation, equivalent to [START_REF] Miller | Towards the modelling of blast loads on structures[END_REF]:

M |∇α| = 1. ( 14
)
GSD model consists in splitting the shock front into elementary areas A propagating along ray tubes in which the cross-flow is neglected. The flow in a ray tube can be expressed mathematically by the equation [START_REF] Whitham | A new approach to problems of shock dynamics. Part II: Three-dimensional problems[END_REF] ∇

• n A = 0. ( 15
)
By considering a ray tube as a channel with rigid walls, a simple law linking A to M closes the system. This relation, called A -M rule, is obtained from the 1D

Euler system with varying cross-section [START_REF] Best | A generalisation of the theory of Geometrical Shock Dynamics[END_REF]:

1 A dA dα + M λ(M ) M 2 -1 dM dα + h(M )Q = 0, (16)
where

λ(M ) = 1 + 2 γ + 1 1 -µ 2 µ 1 + 2µ + 1 M 2 , (17)
h(M ) = γ + 1 2 µ(µ -1) M 2 -1 , ( 18
)
and µ is the post-shock Mach number :

µ = (γ -1)M 2 + 2 2γM 2 + 1 -γ . ( 19
) Q = (∂ t p + ρc∂ t v) - p 0 c 3 0
, indexed by -for quantities just behind the shock, contains all the post-shock flow terms.

Q is an unknown of the problem. As lim

M →1 h(M ) = -0.6 and lim M →+∞ h(M ) = 0,
Whitham chose to neglect the term of truncation under the assumption of the smallness of post-shock effects Q, resulting in the simple form:

1 A dA dα + M λ(M ) M 2 -1 dM dα = 0. ( 20
)
This approximation appears to work remarkably well in a large number of configurations [START_REF] Whitham | Linear and Nonlinear Waves, chapter 8: Shock Dynamics[END_REF][START_REF] Henshaw | Numerical shock propagation using geometrical shock dynamics[END_REF][START_REF] Ridoux | Comparison of Geometrical Shock Dynamics and Kinematic models for shock wave propagation[END_REF] where the post-shock effects have little influence on the shock propagation as for sustained shock. Finally, GSD model is composed of geometrical system ( 13)-( 14)-( 15) and the A -M relation [START_REF] Sharma | On one-dimensional planar and nonplanar shock waves in a relaxing gas[END_REF].

The function λ is a bounded increasing function, varying from 4 at M = 1 to

λ ∞ = 1 + 2 γ + 2γ γ -1 ≈ 5.074, for γ = 1.4, (21)
as M → +∞.

From [START_REF] Sharma | On one-dimensional planar and nonplanar shock waves in a relaxing gas[END_REF], A depends only on M and can be consider as dimensionless: has been noticed that the Mach stem height is small enough to assimilate the reflection as regular [START_REF] Whitham | Linear and Nonlinear Waves, chapter 8: Shock Dynamics[END_REF]. In any event, considering the simplicity of the model, a good agreement with data can be observed as shown in [START_REF] Ridoux | Comparison of Geometrical Shock Dynamics and Kinematic models for shock wave propagation[END_REF].

A(M ) = exp   - M M0 mλ(m) m 2 -1 dm   , (22)
In presence of an obstacle, wall boundaries coincide with rays in Whitham's theory. Consequently, the front is orthogonal to the wall. The hyperbolicity property of GSD insures taking into account of the shock interaction with structures. As explained in [START_REF] Whitham | Linear and Nonlinear Waves, chapter 8: Shock Dynamics[END_REF][START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF], the solution for GSD model is not insured for an expansive sufficiently weak shock.

Limitation of GSD model

α = 0 α > 0 M 0 , θ 0 =0 M 0 , θ 0 =0 M w , θ
In 2D, this default is highlighted with the Riemann problem of a planar shock diffraction over a convex wedge as sketched in figure 7. In this elementary problem, the shock travels from the left to the right at initial Mach number M 0 , and diffracts over a convex wall of deflection angle θ w < 0 at time α = 0. The first state for the Riemann problem is given by the incident shock:

(M 0 , θ 0 = 0). After diffraction, the wall Mach number, M w , and the deflection angle, θ w , represent the second state. It can be shown that a rarefaction wave links the states (M 0 , θ 0 ) to (M w , θ w ), diminishing the Mach number along the shock from M 0 to M w . In particular, it results at the wall in [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF]:

Mw M0 λ(m) m 2 -1 dm = θ w -θ 0 < 0. (23)
From ( 23), one can easily show that a solution such that M w ≥ 1 exists if and only if M 0 ≥ M lim > 1, with M lim the minimal reachable value of the incident Mach number:

M lim 1 λ(m) m 2 -1 dm = -θ w . (24)
This means that, for a given incident Mach number M 0 , no solution of GSD model exists up to the wall above a given deflection angle. Equivalently, for a given deflection angle θ w < 0, the condition M 0 ≥ M lim must be verified to have a solution up to the wall.

The restriction [START_REF] Oshima | Diffraction of a plane shock wave around a corner[END_REF] is plotted in figure 8. For any pair of variables (M 0 , θ w ) in the area below the curve, there is no solution up to the wall for the diffraction of a planar shock over a convex corner. For instance, we have M lim ≈ 1.310 for θ w = -90 • and M lim = 1.5 for θ w ≈ -113.7 • . The restriction of GSD model is in contradiction with Skew's experimental observations [START_REF] Skews | The shape of a diffracting shock wave[END_REF] as seen in figure 9. Indeed, experiments show that the diffracted shock front still exists at the wall, even for weak shocks

(M 0 = 1.
2) and at large deflection angles (θ w up to -165 • ). Consequently, a modification of GSD is necessary.

Ad-hoc modification of the A -M relation

More extensions of GSD model have been proposed taking into account the post-shock flow [START_REF] Best | A generalisation of the theory of Geometrical Shock Dynamics[END_REF] or transverse flow along the shock [START_REF] Oshima | Diffraction of a plane shock wave around a corner[END_REF][START_REF] Oshima | Propagation of spacially non-uniform shock waves[END_REF] during the derivation of the A -M relation. A modification of the treatment of the wall condition has been studied in [START_REF] Bazhenova | Change in the shape of the diffracting shock wave at a convex corner[END_REF]. However, all these modifications seem unable to completely remove the restriction. More recently, an ad-hoc correction of the A -M rule [START_REF] Sharma | On one-dimensional planar and nonplanar shock waves in a relaxing gas[END_REF], designed to systematically remove the limitation of the model, has been developed in [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF].

In 2D, it consists on modelling the interaction between neighbouring ray tubes by the variation of the Mach number along the shock:

1 A ∂A ∂α + M λ(M ) M 2 -1 ∂M ∂α + H (κ) f (M ) ∂M ∂s = 0, ( 25
)
where s is the curvilinear abscissa along the shock. This modification is only active in expansive regions of the shock, namely region where the curvature of the shock curve, κ, is positive:

H(κ) =        0 if κ ≤ 0 1 if κ > 0.
The function f is determined from an empirical law defined with Skews' experimental data [START_REF] Skews | The shape of a diffracting shock wave[END_REF]:

f (M ) = kλ(M ) 2 - 2M 2 k(M 2 -1) , k = 0.985. ( 26
)
The reader is referred to [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF] for technical details. The system of geometrical equations ( 13)-( 14)-( 15) with the modified A -M relation ( 25) is referred as GSDT, T standing for Transverse variation along the shock, in the remainder of the article. Some studies in [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF] have proven the ability of this new model to estimate in a better way the shock front position compared to GSD. In particular, GSDT model has a solution up to the wall for the problem of diffraction over a convex wedge as shown in figure 9.

GSDT model, without restriction for weak shocks contrary to GSD, is a good candidate to blast waves propagation extension. This point is developed in section 4.

In this idealized configuration, the local variation of the area expresses in term of r and there is no transverse variation along the shock:

A (r) A(r) = d -1 r and ∂ ∂s ≡ 0. ( 28
)
Consequently, GSDT model reduces in GSD model. From [START_REF] Sharma | On one-dimensional planar and nonplanar shock waves in a relaxing gas[END_REF],

the Mach number only depends on r:

M (r) = - M 2 (r) -1 M (r)λ(M (r)) d -1 r . ( 29
)
For a given initial condition M (r 0 ) = M 0 , where r 0 is the initial shock position, the equation ( 29) can be solved with a high-order algorithm. For example, figure 2 presents the solution obtained with a fourth-order Runge-Kutta scheme (RK4) with r 0 = 0.25 and M 0 ≈ 10.124.

Simple extension of GSDT to blast waves and numerical integration

GSDT model are quite accurate for sustained shock propagation problems [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF] for which the local changes of front curvature dominate post-shock effects. For blast waves, the modelling of post-shock contribution is necessary as seen in figure 2. In this section, a simple ad-hoc extension of GSDT model to blast waves is proposed for spherical or hemispherical charges.

Modelling of blast effects

Considering the post-shock flow in [START_REF] Oshima | Propagation of spacially non-uniform shock waves[END_REF], the A -M rule is expressed as

1 A ∂A ∂α + M λ(M ) M 2 -1 ∂M ∂α +H (κ) f (M ) ∂M ∂s +E = 0, ( 30
)
with

E = h(M ) (∂ t p + ρc∂ t v) - ρ 0 c 3 0 ,
where h is given by [START_REF] Skews | The shape of a diffracting shock wave[END_REF]. We propose here to model

E
M 2 r + M λ(M ) M 2 -1 dM dr + E (α) = 0, (31)
considering ( 27) and [START_REF] Michael | Simplified kingery airblast calculations[END_REF]. For the sake of generality, let us denote r → ζ a (r) the law overpressure/distance.

For example, we have ζ a (r) = ζ K r/W 1/3 for Kinney's law [START_REF] Kinney | Explosive shocks in air[END_REF]. From [START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF], the Mach number expressed as

M a (r) = 1 + γ + 1 2γ ζ a (r). ( 32
)
Injecting ( 32) into [START_REF] Courant | Supersonic flow and shock waves[END_REF], the blast effects term may expressed as

E (α) = E r (r a (α)), (33)
with

E r (r) M a (r) = - 2 r - λ 1 + γ+1 2γ ζ a (r) 2 ζ a (r) ζ a (r) , (34)
where M → λ(M ) is defined by [START_REF] Besset | Propagation of vertical shock waves in the atmosphere[END_REF]. The mapping between α and r a is expressed from the ODE [START_REF] Peace | On the propagation of decaying planar shock and blast waves through nonuniform channels[END_REF], which results in the following integral form:

r a (α) = r 0 + α α0 M a (r a (τ ))dτ , (35)
where r 0 is the shock position at initial time α 0 . For a condensed explosive, the shock position at α 0 = 0 corresponds to the charge radius:

r 0 = r c = 3W 4πρ c 1/3 , (36)
where ρ c is the density of the explosive. For instance, ρ c = 1650 kg/m 3 for TNT and r c = 0.053 m for W = 1 kg. Finally, blast effects are modeled with relations ( 32)- [START_REF] Gottlieb | Total variation diminishing runge-kutta schemes[END_REF] in relation [START_REF] Mckinzie | The US nuclear war plan: A time for change[END_REF].

The simplified model designed for blast waves propagation in presence of obstacles is finally composed of geometrical equations ( 13)-( 14)-( 15) and the ad-hoc A-M relation [START_REF] Mckinzie | The US nuclear war plan: A time for change[END_REF]. This model is referred as Geometrical Blast Dynamics (GBD) in the remainder of this article.

As discussed in section 3.1 for GSD model, GBD model naturally makes arise Mach stems for compressive flows. Consequently, GBD model does not taken into account the Regular Reflection. This point will be fully studied in section 5.

Analysis of the blast closure for TNT

In this section, the blast closure is analyzed for a spherical diverging blast wave issue from an explosion of a TNT charge. A similar analysis may be possible for a gaseous charge. The blast term, E r (r)/M a (r), is modeled with Kinney's law [START_REF] Kinney | Explosive shocks in air[END_REF]. The dimensionless blast contribution [START_REF] Huynh | Accurate monotone cubic interpolation[END_REF] then depends only on the scaled distance

Z = r/W 1/3 : E r (r) M a (r) W 1/3 Z c = - 2Z c Z - λ 1 + γ+1 2γ ζ K (Z) 2 ζ K (Z) ζ K (Z) Z c , (37)
where Z c ≈ 0.053 m/kg 1/3 is the scaled charge radius (36). From (1), one obtains:

ζ K (Z) ζ K (Z) = 2 Z 1 + (4.5/Z) 2 - 1 Z 1 + (0.048/Z) 2 - 1 Z 1 + (0.32/Z) 2 - 1 Z 1 + (1.35/Z) 2 . (38)
In free field, the blast wave propagation results from a competition between the geometrical expansion,

A (r) A(r) W 1/3 Z c = 2Z c Z ,
written as dimensionless, and the blast effects (37). This observation results from the significant size of the explosive in Kinney's law (1), as

ζ K (Z)/ζ K (Z) ∝ Z.
Indeed, for the ideal case of a point source in the free field, the non-dimensional overpressure theoretically decreases as r -3 in the very near field [START_REF] Taylor | The formation of a blast wave by a very intense explosion. i. theorical discussion[END_REF]. This leads to ζ a (r)/ζ a (r) ∝ r -1 and blast effects are proportional to Z -1 . Consequently, contributions of geometrical expansion and blast effects may be of the same order. Taylor expansion of (38) gives:

ζ K (Z) ζ K (Z) ∼ - 1 Z - C Z 3 , with C = 4.5 2 -0.048 2 -0.32 2 -1.35 2 ≈ 18.32 m 2 /kg 2/3 .
As λ ∼ 4 for M ∼ 1, the blast closure is expressed as:

E r (r) M a (r) W 1/3 Z c ∼ 2CZ c Z 3 > 0.
Consequently, blast effects contribution in the model keeps slowing down the shock front far field.

Numerical scheme

From the numerical point of view, a two-dimensional axisymmetrical conservative Lagrangian scheme has been developed and validated for GSDT model in [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF]. The mean space step is denoted ∆s. In the vicinity of a shock-shock, the transverse closure in (30) is deactivate for points at a distance d c from the discontinuity for the sake of stability. The reader is referred to [START_REF] Ridoux | Beyond the limitation of geometrical shock dynamics for diffraction over wedges[END_REF] for technical details.

The extension of the Lagrangian scheme to GBD model is direct. As proof of the validation for GBD closed with Kinney's law (1), the numerical solution in free field is compared with the analytical solution [START_REF] Kinney | Explosive shocks in air[END_REF] in figure 11. in [START_REF] Ridoux | Comparison of Geometrical Shock Dynamics and Kinematic models for shock wave propagation[END_REF]. The shock-shock trajectory is compared with the TP path from experimental laws from literature.

These last ones are taken from UFC [START_REF]Structures to resist the effect of accidental explosions[END_REF], defined up to a distance of 7 meters, and NRDC [START_REF] Mckinzie | The US nuclear war plan: A time for change[END_REF]. In order to show Figure 13 shows a wider comparison of Triple Point trajectories between GBD model and experimental data from [START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF]. These data are issue from experiments for C4 and they are adapted here for TNT with TNT equivalency E q = 1.28. Let us recall that SHoB, SD and SHT P stand for the scaled HoB, the scaled distance at the ground and the scaled height of burst respectively: Let us consider the reflection of a spherical blast wave over a concave corner of deflection angle θ w > 0 as sketched in figure 14. The blast wave is issue from the explosion of 1 kg of TNT at the ground. GBD model is closed with the empirical Kinney's law (1) for spherical charges. Consequently, the amplification of the blast wave over the ground is taking into account at twice the mass of TNT [START_REF] Kinney | Explosive shocks in air[END_REF], that is 2 kg of TNT here. As dis- From error analysis at 2 m, three trends take shape depending on the kind of reflection:

SHoB = HoB W 1/3 , SD = D W 1/
-The interaction of the blast wave with the concave corner directly results in a Mach Reflection (MR). m from the corner. Accordingly, some additional mechanisms should be considered in GBD in order to remove Mach stem for RR configurations. Let us consider the diffraction of a spherical blast wave over a convex corner of deflection angle θ w < 0 as sketched in figure 18. The blast wave is issue from the explosion of 1 kg of TNT at the ground. GBD model is closed with the empirical Kinney's law (1) for spherical charges. Consequently, the amplification of the blast wave over the ground is taking into account at twice the mass of TNT [START_REF] Kinney | Explosive shocks in air[END_REF], that is 2 kg of TNT here.

Diffraction of a spherical blast wave over a convex corner

The overpressure along the wall estimated with GBD model is compared with reference solutions for distances Table 3 Incident Mach number at the corner according to (1) and ( 6) for W = 1.0 kg of TNT at the ground, i.e. W = 2.0 kg.

The gap between GBD results and reference values may come from the discrepancy between GSDT and experiments as observed in figure 9. Indeed, for θ w = -75

•
for example, the gap between GSDT model and experiments is more important for M 0 = 4 than for M 0 = 2.

Incident Mach number, M 0 , decreases when the charge is farther from the structure as seen in table 3. For GBD model is closed with the stoichiometric propaneoxygen gas law [START_REF]Structures to resist the effect of accidental explosions[END_REF]. For Lagrangian simulations, we sug- gest to discretize the initial hemispherical shock wave with one point per degree, that is a space step, ∆s, equals to 5.0 × 10 -4 m for charge C3 and 10.0 × 10 -4 m for charge C6. This point will be discussed hereafter. tion over the top side and the back side: 18.9% at the captor Gi4 and 18.3% at the captor Gi4. Behind the barrier, the overpressure extracted above the shockshock is in good agreement with data, with errors from -8.5% to 33.3%. Accordingly, some additional mechanisms should be considered in GBD in order to account for the regular reflection, especially for weak shocks.

In the light of the results and the computational time

  The objective of this paper is to present a simple extension of the GSD model to blast waves, much simpler than the high-order extension of the post-shock flow. The new model, called GBD model (Geometrical Blast Dynamics), is based on a decoupling between the shock front geometrical changes and the blast effects in the A -M relation. It consists in modeling the post-shock flow term with an empirical law for spherical charges.

Fig. 2

 2 Fig. 2 Evolution of the peak overpressure in free field for the explosion of 1 kg of TNT from Kinney's law (1). GSD model is drawn as an indication. Results for GSD are obtained by solving (29) with a RK4 algorithm. The initial data are r 0 = 0.25 m and M 0 ≈ 10.124, value calculated with (1)

Fig. 3

 3 Fig. 3 Schematic representation of the interaction of an expanding spherical shock wave over the ground. Figure comes from [6].

Fig. 5

 5 Fig. 5 Configuration for blast wave diffraction over a convex corner (a) and diffracted pressure coefficients (b) from [3].

  γ the specific heat ratio of the gas, supposed constant and equals to 1.4 for air. These variables are in-dexed by 0 for the initial state of the gas at rest. The standard conditions for air are p 0 = 101, 325 Pa, that is p 0 = 1.01325 bar, ρ 0 = 1.225 kg/m 3 and c 0 = 340.3 m/s.

3. 1

 1 Geometrical Shock Dynamics model The Whitham's GSD model is a simplified model for shock wave propagation based on a geometrical vision of the evolution of the shock front. The key idea behind GSD model consists in splitting the shock front into elementary areas, A, propagating along ray tubes in which the cross-flow is neglected. The model of shock propagation is thus reduced to the 1D problem of a planar shock moving into a channel with varying cross section.

Fig. 6

 6 Fig. 6 Shock position at times t 1 and t 2 . Quantities indexed by 0 are fluid parameters of the initial state of the gas, and quantities indexed by -are fluid parameters disturbed by the shock.

with M 0

 0 a reference Mach number. It is straightforward to prove the hyperbolicity of GSD model provided that A (M ) < 0, which is verified as λ(M ) > 0 and M ≥ 1. Some waves may thus develop on the shock front. They are responsible for the modification of the intensity, shape, and orientation along with front evolution. In particular, discontinuities, called shock-shocks, appear. They correspond to the triple point position on the shock when a Mach stem arises. It is worth noting that Mach reflections systematically occur for compressive flows due to the geometrical vision of the shock wave. Nevertheless, for regular reflection conditions, it

Fig. 7

 7 Fig. 7 Scheme of a planar shock diffraction over a convex corner in GSD theory. The solution is a rarefaction wave linking the initial state (M 0 , θ 0 = 0) to the wall state (M w , θ w ).

Fig. 8

 8 Fig.8Restriction curve[START_REF] Oshima | Diffraction of a plane shock wave around a corner[END_REF] for a solution to exist for the diffraction of a planar shock over a convex wall. There is no solution up to the wall if (M 0 , θ w ) is strictly below the curve.Two particular cases discussed in the text are indicated as examples by red and blue dotted lines.

Fig. 9

 9 Fig. 9 Diffraction of a planar shock over a convex corner. Wall Mach number, M w , with respect to deflection angle, θ w in degrees, for different incident Mach numbers M 0 . Comparison between Skew's experimental data [18] (black dots), GSD model (23) (black solid line), and GSDT model (25) (red dashed line).

3. 4

 4 Radial solutions for GSD and GSDT For a planar (d = 1), cylindrical (d = 2) or spherical (d = 3) shock wave, the evolution of the shock position (7) verifies the Ordinary Differential Equation (ODE): dr dα = M (r).

  Figure 10 draws each contribution versus the scaled distance. Near field the source, the blast closure values are negative for Z c ≤ Z ≤ 0.079 m/kg 1/3 and the geometrical expansion dominates up to Z = 0.357 m/kg 1/3 .

Fig. 10

 10 Fig. 10 Competition between the geometrical expansion, A (r) A(r) W 1/3 Z c , and the blast effects, E r (r) M a (r) W 1/3 Z c , in GBD model (31) with Kinney's closure (1) for free field propagation up to 1.6 m (a) and 100 m (b) from the source.

Fig. 11 Fig. 12

 1112 Fig. 11 Evolution of the peak overpressure in free field for the explosion of 1 kg of TNT from Kinney's law (1). The numerical solution is performed with ∆s = 0.01. Initial data: r 0 = 0.2 and M 0 ≈ 11.70.

Figure 12 presents

 12 Figure 12 presents results for the explosion of 1 kg

  the contribution of the blast closure, results are compared with GSD model. As expected, GSD model overestimates the shock positions as well as the TP path due to the truncation of post-shock flow. This discrepancy is drastically reduced with GBD model. One notes the correct prediction of the shock wave positions in comparison with the Eulerian simulation as well as the good restitution of the TP path with experimental laws.

Fig. 13 5 . 2 Fig. 14

 135214 Fig. 13 Blast wave reflection over the ground generated by the explosion of TNT at a scaled height SHoB = HoB/W 1/3 . Comparison of the TP paths between GBD with Kinney's closure and experimental data from [6]. The Lagrangian simulations have been performed with ∆s = 0.005 m and d c = 0.05 m. Dashed lines correspond to FIT (4)-(5)from[START_REF] Boutillier | Evaluation of the existing triple point path models with new experimental data: proposal of an original empirical formulation[END_REF].

Fig. 15

 15 Fig. 15 Reflection of a spherical blast wave over a concave corner of deflection angle θ w as sketched in figure 14. The structure is placed at dist = 2 m from TNT charge. Wall overpressure versus distance from the corner. Errors are calculated at 2 m and 5 m from the corner with relation (39).

  cussed in section 4.1, GBD model systematically provides a Mach Reflection in this configuration (compressive flows). The overpressure along the wall estimated with GBD model is compared with reference solutions for distances from the charge to the structure, dist, ranging from 1 m to 4 m and for deflection angles, θ w , ranging from 15 • to 90 • . The length of the ramp is equals to 5 meters. Results are extracted from simulations at 5 mm height. Parameters ∆s, d c and the height of extraction have little influence over results from a distance of 0.2 m from the corner. Reference solutions (wall peak overpressures) are obtained from Kinney's law (1) and the reflection table in figure 4. Results for dist = 2m. Results for dist = 2 m are presented in figure 15. Errors at 2 m and 5 m from the corner are evaluated with relation (39). A good agreement between GBD model and reference solutions can be observed. For θ w ranging from 15 • to 45 • , the interaction of the blast wave with the concave corner theoretically results in a Mach Reflection (MR) according to (3). One observes a relative error lower than 40% for theses cases. For θ w ≥ 60 • , the reflection is first regular (RR) and then leads to a Mach reflection. The discrepancy between GBD model and reference solutions increases with the later Mach stem arising. Indeed, for θ w = 60 • , a MR arises around 0.57 m from the corner. The error at 5 m from the corner is evaluated at 53.7%. For θ w = 90 • , the MR occurs farther from the corner, around 1.86 m, and the error is 96.2%. As underlined in the previous subsection, this discrepancy is a consequence of GBD to systematically make arise a Mach stem for compressive flows. It should be noted that GBD model correctly describes the decay of peak overpressure with an increasing error along the wall. Moreover, one notes an overestimation of reflected overpressure in comparison with reference solutions. These observations are similar for dist = 1 m, dist = 3 m and dist = 4 m. All results are gathered in table 10 in appendix A. Error analysis. The relative errors between GBD model and reference solutions at 2 m and 5 m from the corner are presented in figure 16. Although GBD model makes arise a Mach stem, errors do not increase with the later transition to MR. Indeed, for θ w = 75 • for example, the Mach stem arises around 0.57 m at dist = 1 m and around 1.27 m at dist = 2 from (3). Nevertheless, error at 5 m is higher at dist = 1 m than at dist = 2m: 132% versus 70%. It suggests that the gap between GBD model and reference solutions for compressive flows do not completely depends on the kind of reflection. Differences may come from the decoupling between local shock curvature changes and blast effects in the model. This point should be addressed in further investigations.

Fig. 16 -

 16 Fig.[START_REF] Catherasoo | Shock dynamics in non-uniform media[END_REF] Reflection of a spherical blast wave over a concave corner as sketched in figure14. Relative errors (39) for wall overpressure between GBD model and reference solutions (Kinney (1)+reflection coefficients, figure4) at 2 m and 5 m from the corner. Circle points and triangular points correspond to MR and RR respectively, according to relation[START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF]. Square points are configurations with a transition between RR and MR. Results are gathered in table 10 in appendix A.

Fig. 17

 17 Fig. 17Reflection of a spherical blast wave over a concave corner for dist = 4 m from the charge and with deflection angles θ w = 75 • (a) and θ w = 90 • (b). Comparison between wall overpressure and overpressure around 0.01 m above the shock-shock. Free field estimation (1) is drawn as an indication.

Fig. 18 Fig. 19

 1819 Fig. 18 Scheme a spherical blast wave diffraction over a convex corner.

Fig. 20

 20 Fig. 20 Diffraction of a spherical blast wave over a convex corner as sketched in figure 18. Relative errors (39) for wall overpressure between GBD model and reference solutions (Kinney (1)+diffraction coefficients, figure 5(b)) at 1 m and 2 m from the corner. Results are gathered in table 11 in appendix A.

θ 5 . 4

 54 w = -75 • , the gap between GBD model and reference solutions are larger for dist = 1 m where M 0 = 3.86 than for dist = 2 m where M 0 = 2, as underlined for GSDT. Interaction of a spherical blast wave with a mound As a last example, let us consider the interaction of an hemispherical blast wave generated by a gaseous charge (stoichiometric propane-oxygen mixture) with a protective barrier, also called a mound. This study comes from[START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF] 38] in which small-scale experiments are carried out (scale 1/15). Figure21shows the schematic representation of the experimental bench. The r 0 radius charge is placed at the distance d from the mound and it releases an energy E. The mound is character-ized by its height, h, the top thickness, e, the front side inclination angle, α 1 , and the back side inclination angle, α 2 . In[38], captors are located on each face of the barrier (numbered from Gi1 to Gi5 or Gi6) and behind the mound (GH1 to GH7) which record the peak overpressure. This experimental results are compared with 2D-axisymmetrical Eulerian simulations. These latter ones required 256 processors on the TERA100 supercomputer during 12 hours[START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF].We propose here to compare the history of the wall peak overpressure between GBD model, the Eulerian results and the experimental data. The features for each case treated in this section are gathered in table 4. Two charges are considered, referred as C3 and C6 with a radius of 3 cm and 6 cm respectively. Two kinds of mound are treated so-called 1A and 1B as shown in figure 21. Both mounds differ in the top thickness, e, and the back side inclination angle, α 2 . The different faces of the mound and the ground behind it are numbered from 1 to 4 as shown in figure 23(a). The detonation of the charge generates a blast wave in the air which interacts with the mound. The shock interaction with the obstacle results in four successive reflection or diffraction : reflection on the faces 1 and 4 and diffraction on the faces 2 and 3.

Fig. 21

 21 Fig. 21 Schematic representation of test bench of small-scale experiments from [38] for mound 1A (a) and mound 1B (b).

Fig. 22

 22 Fig. 22 Case 1AC3d8,5. Wall peak overpressure on the front side (a), the top side (b) and the back side (c) of the barrier. Eulerian and experimental data are taken from [38]. The free field estimation comes from (2). The Lagrangian simulation has been performed with ∆s = 2.5 × 10 -4 m.

Fig. 23

 23 Fig. 23 Case 1AC6d7. Successive shock front positions (a) (red dashed lines correspond to TP paths) and wall peak overpressure on each face of the barrier (b)-(e). Eulerian and experimental data are taken from [38]. The free field estimation comes from (2). The Lagrangian simulation has been performed with ∆s = 5.0 × 10 -4 m.

(around 2

 2 figurations. Moreover, GBD model only gives the peak

  )

		14									∆P i (bar)
		12									344.7 206.8
											137.9
		10									68.95 34.47
											27.58
	i ∆P	8									20.68 13.79
	r / ∆P	6									10.34 4.826 3.447
											2.068
		4									1.379 0.345
											0.138
		2									0.069 0.034
											0.014
		0
		0	10	20	30	40	50	60	70	80	90
				Angle of incidence β (degree)

Table 1

 1 Scaled distances of transition between RR and MR,

	3 , SHT P =	HT P W 1/3 ,

SR 0 (in cm/kg 1/3 ), for several SHoB (in cm/kg 1/3 ) according to (4).

Table 2

 2 Results at 5 m from the corner Results at 8 m from the corner θ w ( • ) ∆P w ref (bar) ∆P w GBD (bar) ∆P c GBD (bar) ∆P w ref (bar) ∆P w GBD (bar) ∆P c GBD (bar) Reflection of a spherical blast wave over a concave corner for dist = 4 m from the charge and with deflection angles of 75 • and 90 • . Comparison between wall overpressures, ∆P w , and overpressures around 0.01 m above the shock-shock, ∆P c .

	75	0.611	1.162	0.714	0.384	0.638	0.407
			error: 90.2%	error: 16.9%		error: 60.1%	error: 5.99%
	90	0.787	1.602	0.980	0.486	0.816	0.513
			error: 104%	error: 24.5%		error: 67.9%	error: 5.56%

• . Results at 5 m and 8 m from the corner are presented in table 2. The estimation of overpressure is significantly improved, with errors lower than 6% at 8

Table 4

 4 Features for cases from[38] as seen in figure21. Cases are called as the original ones for consistency.

	Case	r 0 (m)	E (MJ)	d (m) α 1 ( • ) α 2 ( • ) e (m) h (m)
	1AC3d8,5	0.03	1.72 × 10 -3	0.085	45	45	0.03	0.19
	1AC6d7	0.06	13.75 × 10 -3	0.070	45	45	0.03	0.19
	1BC6d7	0.06	13.75 × 10 -3	0.070	45	90	0.19	0.19
	1BC6d14	0.06	13.75 × 10 -3	0.140	45	90	0.19	0.19

Table 5

 5 Computational time to solve GBD model with the Lagrangian algorithm. Simulation made on Intel(R)

	Core(TM) i5-8300H CPU @ 2.30GHz. t f is the simulated
	time.

Table 6

 6 Case 1AC3d8,5. Peak overpressure at the captors (seen figure 21(a) for captors positions). Errors are calculated with relation (39). ∆P w /∆P c GBD (bar) Error for Face Captors ∆P w exp. (bar) ∆s = 10.0 × 10 -4 m ∆s = 5.0 × 10 -4 m ∆s = 10.0 × 10 -4

	1	Gi1	1.520	1.860	1.870	22.4%
		Gi2	0.703	0.952	0.957	35.4%
		Gi3	0.433	0.617	0.614	42.5%
	2	Gi4	0.170	0.202	0.193	18.8%
	3	Gi5	0.031	0.061	0.061	96.8%
	1	Gi1	6.360	6.85	6.99	7.70%
		Gi2	2.280	3.07	3.10	34.6%
		Gi3	1.340	1.81	1.81	35.1%
	2	Gi4	0.508	0.604	0.549	18.9%
	3	Gi5	0.120	0.142	0.145	18.3%
	4	GH1	0.102	0.096	/	-5.9%
		GH2	0.094	0.086	/	-8.5%
		GH3	0.077	0.079	/	-2.5%
		GH4	0.061	0.070	/	14.8%
		GH5	0.054	0.061	/	13.0%
		GH6	0.045	0.056	/	24.4%
		GH7	0.039	0.052	/	33.3%

Table 7

 7 Case 1AC6d7. Peak overpressure at the captors (seen figure 21(a) for captors positions). Results at the face 4 correspond to overpressure around 2∆s above the shock-shock and they are similar for ∆s = 5.0 × 10 -4 m. Errors are calculated with relation (39).

	flows, while the reflection is regular in this configura-	The table 7 gathered peak overpressure at the cap-
	tion. However, the overpressure extracted around 2∆s	tors. In light of the results, ∆s = 10.0 × 10 -4 m is
	above the shock-shock is in excellent agreement with	enough to achieve nearly-converged results. Errors for
	data.

GBD increase as the blast wave travels on the face 1, from 7.70% to 35.1%. The error is low after the diffrac-

Table 8

 8 Cases 1BC6d7 and 1BC6d14. Peak overpressure at the captors (seen figure 21(b) for captors positions). The Lagrangian simulation has been performed with ∆s = 10.0 × 10 -4 m. Errors are calculated with relation (39).

Table 9

 9 Incident overpressures and incident Mach numbers are given in table 9 as an indication. Interaction of a spherical shock wave over a convex or a concave corner. Incident overpressures (1), ∆P i , and incident Mach numbers (6), M i , at the corner for W = 1.0 kg of TNT at the ground, i.e. W = 2.0 kg.Results at 1 m from the cornerResults at 2 m from the corner dist θ w ∆P w GBD ∆P w ref Error ∆P w GBD ∆P w ref Error

	dist (m)	1	2	3	4
	∆P i (bar) 16.43 3.562 1.387 0.727
	M i	3.86	2.00	1.47	1.27

Table 11

 11 Spherical blast wave diffraction over a convex corner as sketched in figure18. Wall overpressure values, ∆P w , are extracted from simulations at 5 mm height. Reference solutions are calculated from Kinney's law (1) and diffraction coefficients from[START_REF] Éveillard | Towards a fast-running method for blast-wave mitigation by a prismatic blast wall[END_REF] (see figure5(b)). Errors are calculated with (39).

Acknowledgements Part of this work has been possible thanks to the LETMA collaboration: a contractual research laboratory between CEA, CNRS, École Centrale Lyon, C-Innov and Université Pierre et Marie Curie.


Sébastien Gauthier
email: sebastien.gauthier@univ-rennes1.fr

Françoise Robin-Le Guen

Laurianne Wojcik

Nicolas Le Poul

Aurélien Planchat

Yann Pellegrin
email: yann.pellegrin@univ-nantes.fr

Patricia Guevara Level

Nadine Szuwarski

Mohammed Boujtita

Denis Jacquemin
email: denis.jacquemin@univ-nantes.fr

Fabrice Odobel
email: fabrice.odobel@univ-nantes.fr

Comparative Studies of New Pyranylidene-Based Sensitizers Bearing Single or Double Anchoring Groups for Dye-Sensitized Solar Cells

Mono-and di-anchoring γ-pyranylidene-based organic dyes featuring D-π-A and D-(-A) 2 architectures have been engineered as sensitizers for applications in Dye-Sensitized Solar Cells (DSSCs). Their photophysical, electrochemical and photovoltaic properties were further investigated. TD-DFT calculations were performed to rationalize the trends observed in the optical and electrochemical properties of the dyes. The investigation of the photovoltaic performances of this series of new dyes provided structure-property relationships where their Power Conversion Efficiencies (PCE) could be correlated to structural features, such as the length of the -conjugated spacer, the nature of the substituents on the pyranylidene electron donor moiety and the number of anchoring groups. Dye-Sensitized Solar Cells based on mono-anchoring dyes were more efficient than the corresponding cells based on di-anchoring analogues due to high dye loading. The highest Power Conversion Efficiency of 5.23% was achieved with the mono-anchoring 17a dye containing t-butyl substituent groups on the pyranylidene fragment and with one thienyl -conjugated spacer.

Introduction

The development and implementation of renewable energy sources with low carbon emissions have become a paramount environmental requirement to reduce the massive use of fossil fuels. Harvesting the solar power is an attractive, environmentally friendly and renewable energy source, since the sun provides the earth with an energy power about 4000 times the current forecast energy needs of the whole humanity in 2050 [START_REF] Bandara | Dyesensitized, nano-porous TiO2 solar cell with poly(acrylonitrile): MgI2 plasticized electrolyte[END_REF][START_REF] Hagfeldt | Dye-Sensitized Solar Cells[END_REF][START_REF] Hsu | Femtosecond Excitonic Relaxation Dynamics of Perovskite on Mesoporous Films of Al 2 O 3 and NiO Nanoparticles[END_REF][START_REF] Lewis | Powering the planet: Chemical challenges in solar energy utilization[END_REF]. Photovoltaic devices, such as solar cells, are certainly the most valuable artificial systems able to directly convert light energy into electricity, and have been the subject of increased research efforts for their development and optimization over the past few years [START_REF] Imahori | Large π-Aromatic Molecules as Potential Sensitizers for Highly Efficient Dye-Sensitized Solar Cells[END_REF][START_REF] Wu | Molecular engineering of cocktail co-sensitization for efficient panchromatic porphyrin-sensitized solar cells[END_REF][START_REF] Wu | Organic sensitizers from D-π-A to D-A-π-A: effect of the internal electron-withdrawing units on molecular absorption, energy levels and photovoltaic performances[END_REF].

Among them, Dye-Sensitized Solar Cells (DSSCs) have emerged as a promising and competitive technology with sizeable Power Conversion Efficiency (PCE) and facile device fabrication, as demonstrated in the seminal work of O'R ga a d G ätzel in 1991 [START_REF] Higashino | Porphyrins as excellent dyes for dye-sensitized solar cells: recent developments and insights[END_REF]O'Regan and Grätzel, 1991). DSSCs consisting of a dyed wide-band gap n-type metal oxide semiconductor such as TiO 2 as photoanode, a Pt counter-electrode and a redox electrolyte, operate through a process whereby the energy of the absorbed sunlight is transferred from the anchored dye to the acceptor metal oxide semiconductor [START_REF] Grätzel | Recent Advances in Sensitized Mesoscopic Solar Cells[END_REF].

In their or g al wo k, O'R ga a d G ätzel used ruthenium-based photosensitizers as dyes in their DSSCs. In recent years, research efforts have been devoted to eliminate the need for expensive metals and have focused on the development of metal-free organic dye sensitizers, usually based on push-pull type donor-(spacer)-acceptor (D-π-A) molecular structures [START_REF] Mishra | Metal-Free Organic Dyes for Dye-Sensitized Solar Cells: From Structure: Property Relationships to Design Rules[END_REF][START_REF] Ooyama | Photophysical and Electrochemical Properties, and Molecular Structures of Organic Dyes for Dye-Sensitized Solar Cells[END_REF][START_REF] Qu | New D-π-A dyes for efficient dye-sensitized solar cells[END_REF]Y. Wu et al., 2012). In addition to their affordability, these D-π-A molecular systems present several advantages, such as their simple preparation process, their easily adjustable structures, their high molar extinction coefficients and their excellent intra-molecular charge transfer (ICT) photoexcitation properties [START_REF] Bessho | Highly Efficient Mesoscopic Dye-Sensitized Solar Cells Based on Donor-Acceptor-Substituted Porphyrins[END_REF][START_REF] Mao | Stable Dyes Containing Double Acceptors without COOH as Anchors for Highly Efficient Dye-Sensitized Solar Cells[END_REF][START_REF] Marszalek | Structural variations of D-π-A dyes influence on the photovoltaic performance of dye-sensitized solar cells[END_REF]. In order to enhance the Power Conversion Efficiency (PCE) of D-π-A dyes, multiple modifications have been performed on these molecular systems that consisted of varying the donor, spacer, and acceptor moieties [START_REF] Cai | Organic dye-sensitized solar cells with a cobalt redox couple: influences of π-linker rigidification and dye-bath solvent selection[END_REF][START_REF] Choi | Highly Efficient and Thermally Stable Organic Sensitizers for Solvent-Free Dye-Sensitized Solar Cells[END_REF][START_REF] Ji | Rational design criteria for D-π-A structured organic and porphyrin sensitizers for highly efficient dye-sensitized solar cells[END_REF][START_REF] Yao | Donor/Acceptor Indenoperylene Dye for Highly Efficient Organic Dye-Sensitized Solar Cells[END_REF]. Numerous metal-free organic dyes containing conjugated electron-donor groups, such as triphenylamine [START_REF] Wang | Triarylamine: Versatile Platform for Organic, Dye-Sensitized, and Perovskite Solar Cells[END_REF], dialkylaminophenyl [START_REF] Liang | Arylamine organic dyes for dye-sensitized solar cells[END_REF], indoline [START_REF] Alagumalai | Effect of Out-of-Plane Alkyl Group' Position in Dye-Sensitized Solar Cell Efficiency: A Structure-Property Relationship Utilizing Indoline-Based Unsymmetrical Squaraine Dyes[END_REF][START_REF] Yang | Influence of the Donor S z D-π-A Organic Dyes for Dye-Sensitized Solar Cells[END_REF], carbazole [START_REF] Sathiyan | Review of carbazole based conjugated molecules for highly efficient organic solar cell application[END_REF][START_REF] Soni | Improved molecular architecture of D-π-A carbazole dyes: 9% PCE with a cobalt redox shuttle in dye sensitized solar cells[END_REF], dithiafulvene [START_REF] Luo | Co-sensitization of Dithiafulvenyl-Phenothiazine Based Organic Dyes with N719 for Efficient Dye-Sensitized Solar Cells[END_REF][START_REF] Wan | Novel organic sensitizers containing dithiafulvenyl units as additional donors for efficient dye-sensitized solar cells[END_REF] and tetrathiafulvalene (TTF) [START_REF] Brunetti | π-Extended TTF: a versatile molecule for organic electronics[END_REF][START_REF] Wenger | Efficient Electron Transfer and Sensitizer Regeneration in Stable π-Extended Tetrathiafulvalene-Sensitized Solar Cells[END_REF] groups have been successfully used as sensitizers in DSSCs, the most efficient exhibiting PCE reaching up to 12%. It has been reported that the PCE performances of DSSCs can be enhanced by extending the conjugation of the dyes, increasing the strength of the donor/acceptor units and improving the adsorption of the dyes onto the TiO 2 surface [START_REF] Yeh-Yung Lin | Y-shaped metal-free D-π-(A)2 sensitizers for high-performance dye-sensitized solar cells[END_REF]. Due to the importance of the anchoring group on the binding energy of the dye onto TiO 2 and the electron injection rate, selecting an appropriate anchoring moiety is particularly crucial for enhancing the device' performance.

In the majority of organic dyes, carboxylic or cyanoacrylic acids are used as they play the dual role of electron acceptor and anchoring site to tether the dye molecules onto the TiO 2 surface [START_REF] Zhang | Anchoring Groups for Dye-Sensitized Solar Cells[END_REF]. The strong bonding of the carboxylic acid group onto the TiO 2 surface allows for a good electronic coupling between the dye and TiO 2 , thus providing a high electron injection rate [START_REF] Ambrosio | What Is the Best Anchoring Group for a Dye in a Dye-Sensitized Solar Cell?[END_REF][START_REF] Srinivas | A Combined Experimental and Computational Investigation of Anthracene Based Sensitizers for DSSC: Comparison of Cyanoacrylic and Malonic Acid Electron Withdrawing Groups Binding onto the TiO 2 Anatase (101) Surface[END_REF][START_REF] Wiberg | Effect of Anchoring Group on Electron Injection and Recombination Dynamics in Organic Dye-Sensitized Solar Cells[END_REF].

Improving light-absorbing capability, suppressing charge recombination, increasing electron extraction channels and enhancing the binding strength with the semiconductor surface are important factors for improving the overall performance of DSSCs. These requirements can be, at least partially, met by incorporating two or more anchoring groups in the dye molecular structures [START_REF] Abbotto | Di-branched di-anchoring organic dyes for dye-sensitized solar cells[END_REF][START_REF] Grisorio | Anchoring stability and photovoltaic properties of new D(-π-A)2 dyes for dye-sensitized solar cell applications[END_REF][START_REF] Gupta | Carbazole based A-π-D-π-A dyes with double electron acceptor for dye-sensitized solar cell[END_REF][START_REF] Zhang | Novel dithieno[3,2-b:2′,3′-d]pyrrole-based organic dyes with high molar extinction coefficient for dye-sensitized solar cells[END_REF]. Indeed, DSSCs organic dyes synthesized with several anchoring groups were recently reported to display enhanced photovoltaic properties compared to their counterparts containing only one binding group [START_REF] Berhe Desta | Reversed Y-shape di-anchoring sensitizers for dye sensitized solar cells based on benzimidazole core[END_REF][START_REF] Jia | Enhanced performance of dyesensitized solar cells with Y-shaped organic dyes containing di-anchoring groups[END_REF][START_REF] Li | Double-anchoring organic dyes for dye-sensitized solar cells: the opto-electronic property and performance[END_REF]. γ-Pyranylidene fragments have been incorporated as electron-donating groups in organic molecules and in various platinum-and ruthenium-based organometallic complexes for the preparation of push-pull chromophores for second-order Non-Linear Optics (NLO) [START_REF] Durand | Push-pull D-π-Ru-π-A chromophores: synthesis and electrochemical, photophysical and second-order nonlinear optical properties[END_REF][START_REF] Durand | Incorporation of a platinum center in the piconjugated core of push-pull chromophores for nonlinear optics (NLO)[END_REF][START_REF] Gauthier | Mono-and Diplatinum Polyynediyl Complexes as Potential Push-Pull Chromophores: Synthesis, Characterization, TD-DFT Modeling, and Photophysical and NLO Properties[END_REF][START_REF] Gauthier | Methylenepyran based dipolar and quadrupolar dyes: synthesis, electrochemical and photochemical properties[END_REF][START_REF] Marco | Polarization, second-order nonlinear optical properties and electrochromism in 4H-pyranylidene chromophores with a quinoid/aromatic thiophene ring bridge[END_REF][START_REF] Marco | Push-pull systems bearing a quinoid/aromatic thieno[3,2-b]thiophene moiety: synthesis, ground state polarization and second-order nonlinear properties[END_REF] and/or DSSCs [START_REF] Andreu | 4 H -Pyran-4-ylidenes: Strong Proaromatic Donors for Organic Nonlinear Optical Chromophores[END_REF][START_REF] Bolag | Enhanced performance of dyesensitized solar cells with novel 2,6-diphenyl-4H-pyranylidene dyes[END_REF][START_REF] Ferreira | New D-π-A-conjugated organic sensitizers based on α-pyranylidene donors for dye-sensitized solar cells[END_REF][START_REF] Gauthier | Synthesis, photovoltaic performances and TD-DFT modeling of push-pull diacetylide platinum complexes in TiO 2 based dye-sensitized solar cells[END_REF]. Various γpyranylidene fragments have been found to be effective donors in D-π-A sensitizers for DSSCs yielding promising PCE of up to 6% [START_REF] Marco | Pyranylidene/thienothiophene-based organic sensitizers for dye-sensitized solar cells[END_REF]. Recently, one di-anchoring D-(-A) 2 type organic dye based on γ-pyranylidene core with two anchoring acceptor groups was designed and synthesized by Franco and co-workers [START_REF] Andrés-Castán | 4Hpyranylidene organic dyes for dye-sensitized solar cells: Twisted structures towards enhanced power conversion efficiencies[END_REF]. They found that the Photo Conversion Efficiency of this di-anchoring γ-pyranylidene-based D-(-A) 2 dye was higher than that of the mono-anchoring D--A analogue due to improved photocurrent and stability.

Despite a demonstrated improvement of the PCE in DSSC systems containing di-anchoring dyes, the development of dyes with multiple anchoring sites has been relatively unexplored so far. To address this shortcoming, we have designed and synthesized a new series of dianchoring D-(-A) 2 push-pull dyes 10a-c and 12c, based on the pyranylidene group (Fig. 1).

The photophysical properties of these di-anchoring D-(-A) 2 push-pull dyes have been compared to their mono-anchoring analogues, such as compound 17a presented in this paper and compounds D1, D2 and D3 previously investigated in our research group (Fig. 1.) [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF] Our approach was to investigate the effects of a variety of structural modifications, such as donors with different substituents on the pyranylidene ring (i.e. phenyl (10b), t-butyl (10a), thienyl (10c)), the nature of the linker (10c versus 12c), the number of anchoring groups (10a versus 17a, 10b versus D1, 10c versus D2 and 12c versus D3), and the length of the linker for the dyes with a single anchoring group (i.e. one thienyl (17a), two thienyl (18a), and three thienyl units (19a)), on the optical, electrochemical and photovoltaic properties of the dyes.

The organic compounds, prepared in this study, were fully characterized by various techniques and their efficiency was evaluated in TiO 2 -based DSSCs. Theoretical studies were also conducted to support the experimental data and to correlate the dye structures to their electronic properties. [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]

Results and Discussion

Synthesis of the Dyes

The key intermediate in the preparation of di-anchoring D-(-A) 2 push-pull dyes is the dialkynes 5a-c and 7c (Scheme 1). First, the di-alkyne 5b is prepared according to previously described synthetic routes [START_REF] Gauthier | Mono-and Diplatinum Polyynediyl Complexes as Potential Push-Pull Chromophores: Synthesis, Characterization, TD-DFT Modeling, and Photophysical and NLO Properties[END_REF]. Identical synthetic pathways are used for 5a, 5b, and 7c, which are obtained in two steps from bis [4-(2trimethylsilylethynyl)]benzophenone 2 or bis [5-(2-trimethylsilylethynyl)]thienone 3 and the desired phosphonium tetrafluoroborate salt 1a-c [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF][START_REF] Gauthier | Synthesis, photovoltaic performances and TD-DFT modeling of push-pull diacetylide platinum complexes in TiO 2 based dye-sensitized solar cells[END_REF] (Scheme 1). The first step is a Wittig reaction leading to the intermediates 4a-c and 6c obtained in good yield, whereas the second step consists of the deprotection of the trimethylsilyl group with quasiquantitative yield. The mono-and di-carboxaldehyde intermediates 9a-c, 11c, 14a, 15a, and 16a, are then obtained from a copper/palladium catalyzed Sonogashira cross-coupling reaction of 5a-c, 7c

and 13 with the corresponding aldehydes 8 (Scheme 2 and 3). Finally, the dyes 10a-c, 12c, 17a, 18a, and 19a are obtained in good yields (84-95%) through Knoevenagel reactions of the corresponding precursors with cyanoacetic acid in the presence of piperidine and subsequent acid wash (Scheme 2 and 3). All targeted sensitizer dyes have been characterized by 1 H and 13 C NMR, and high-resolution mass spectrometry. All the NMR spectra of the new compounds are presented in the Supplementary Information (SI). These new dyes are also perfectly stable in their solid state and do not require any specific storage conditions.

Electronic UV-visible absorption

The electronic spectra of compounds 10a-c, 12c, 17a, 18a, and 19a obtained in dichloromethane (1.1 × 10 -5 M) in the 300-800 nm wavelength range are shown in Fig. 2 and the key spectroscopic features are summarized in Table 1. The absorption bands of the dyes can be separated into two groups falling in the regions 300-450 nm and 450-600 nm. For all compounds, the absorption bands in the high-energy region correspond to localized * transitions originating from the pyranylidene-oligothienyl segment, whereas the least energetic absorption bands in the visible region can be assigned to an ICT process between the pyranylidene donor unit to the cyanoacrylic acid acceptor (see theoretical calculations below).

Molar extinction coefficients (ε) obtained for compounds 10a-c, 12c, 17a, 18a, and 19a are larger than 23,900 M -1 cm -1 with a maximum at 50,000 M -1 cm -1 for compound 10a. These coefficients exceed those of the standard ruthenium dyes N3 [START_REF] Horiuchi | Highly-efficient metal-free organic dyes for dyesensitized solar cells[END_REF] and N719 [START_REF] Wang | A High Molar Extinction Coefficient Sensitizer for Stable Dye-Sensitized Solar Cells[END_REF] (13,900 and 14,000 M -1 cm -1 , respectively), indicating that most of the dyes of this series should be good light harvesting compounds.

The groups placed in the 2-and the 6-positions of the pyranylidene fragment (i.e., phenyl, thienyl, t-butyl) have a limited influence on the absorption properties, which is consistent with the topology of the involved excited states (see below). Finally, the potential effects on the absorption properties of the nature of the substituents placed on the exocyclic carbon of the pyranylidene moiety of 10c and 12c dyes were examined. While the replacement of a phenyl by a thienyl substituent (10c versus 12c) has a limited influence on the absorption band wavelength, it leads to a significantly lower molar extinction coefficient (the same observation can be made when comparing D2 to D3) [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]. In addition, the main visible absorption of the di-anchoring D-(-A) 2 10a, 10b and 10c dyes appears at a wavelength similar to that of the mono-anchoring D--A 17a, D1 and D2 analogues.

However, a significant red-shift ( = 15 nm) is observed when comparing the absorption spectra of the mono-anchoring D--A D3 and its di-anchoring D-(-A) 2 12c analogue.

The increase of the maximum absorption and the red-shift of the maximum absorption of 19a compared to that of 18a can be explained by the enhanced electron delocalization over oligothienyl spacers of increasing lengths. The same phenomenon has been observed for other organic dyes when the -conjugation of the spacers is extended [START_REF] Bolag | Enhanced performance of dyesensitized solar cells with novel 2,6-diphenyl-4H-pyranylidene dyes[END_REF][START_REF] Fischer | D-π-A Sensitizers for Dye-Sensitized Solar Cells: Linear vs Branched Oligothiophenes[END_REF]. a Estimated from the wavelength with the intersections tangent method ( onset ).

Electrochemical study and electron transfer driving forces

The redox properties of the dyes were investigated by using cyclic voltammetry (CV). All pyranylidenes display similar voltammetric signatures, i.e., two successive oxidation processes (excluded compound 17a which shows only one), and one (or two) broad reduction peaks starting from the open-circuit potential. This redox behavior is reminiscent of those observed with analogous compounds such as D1-D3 [START_REF] Durand | Incorporation of a platinum center in the piconjugated core of push-pull chromophores for nonlinear optics (NLO)[END_REF][START_REF] Gauthier | Synthesis, photovoltaic performances and TD-DFT modeling of push-pull diacetylide platinum complexes in TiO 2 based dye-sensitized solar cells[END_REF][START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]. Fig. 3A shows CVs for both compounds 10a and 17a in dichloromethane. The first oxidation process at E 0 (1) is more positive by 60 mV for the V-shaped compound 10a

than for the linear analogue 17a, whereas the second peak at E pa (2) remains at the same value for both compounds. This suggests that the incorporation of a second withdrawing group decreases the net electron density on the pyranylidene moiety, making its oxidation at E pa (1) slightly more difficult than with only one withdrawing group. The same tendency (50-70 mV potential difference) is observed for the di-anchoring compounds 10b, 10c and 12c compared to their mono-anchoring analogues D1, D2 and D3, respectively, although measurements have not been carried out in the same conditions (solvent). Interestingly, the E 0 (1) value is almost independent of the R substituting group (R=phenyl, thienyl, t-butyl) for V-shaped compounds 10a-c (see values in CH 2 Cl 2 /THF in Table 2), as for similar linear pyranylidenes [START_REF] Durand | Incorporation of a platinum center in the piconjugated core of push-pull chromophores for nonlinear optics (NLO)[END_REF]. Scanning towards negative potentials leads to the appearance of broad and irreversible peaks (3) and (4) in the 0.2 V to -0.8 V potential range (vs. NHE), corresponding to the reduction of the cyanoacetic anchoring groups, consistently with literature [START_REF] Gauthier | Synthesis, photovoltaic performances and TD-DFT modeling of push-pull diacetylide platinum complexes in TiO 2 based dye-sensitized solar cells[END_REF]. The appearance of one or two peaks is not clearly understood at this stage.

Increasing of the number of thienyl linkers induces a negative shift of the reduction, as clearly emphasized on Fig. 3B with compounds 17a, 18a and 19a, whereas oxidation peaks remain almost unaffected.

Quantum chemical calculations

To describe the electronic transitions, and especially to quantify the ICT strength of all molecular structures presented in this paper, we performed (Time-Dependent) Density Functional Theory Calculations using a range-separated hybrid (see the SI for computational details). The optimal ground-state geometries of all compounds did not reveal any unexpected aspect. For instance, in 17a, the cyanoacrylic, thienyl, and ethynyl-phenyl moieties are almost perfectly coplanar (dihedral < 1°), whereas the pyranylidene moiety and the two vicinal sixmember rings are arranged in a propeller-shape with respect to each other with dihedral angle in the 42-52° range. The frontier orbitals HOMO and LUMO are displayed in Fig. 4 for 10a and in Fig. S43 (SI) for the other dyes. As can be seen, the HOMO is centered on the electrondonor pyranylidene moiety, while the LUMO is split between the two accepting branches of the dye, with strong contributions on both the cyanoacrylic and the thienyl moieties. This TD-DFT was used to obtain the vertical transition energies as well as the ICT parameters following L Bah ' d CT metric [START_REF] Jacquemin | Wha h "b " a om c cha g model to describe through-space charge-transfer excitations?[END_REF][START_REF] Bahers | A Qualitative Index of Spatial Extent in Charge-Transfer Excitations[END_REF]. The results are shown in Table 3. First, the systems with two anchoring groups display two nearly degenerated states, corresponding to transitions from the HOMO to the LUMO and LUMO+1, the latter two combining orbitals localized on the two anchoring branches (see the SI). While it is important to mention that the computed TD-DFT transition energies in the vertical approximation cannot be strictly compared to experimental wavelength of maximal absorption, we can still notice that the computed transition wavelengths are in good qualitative agreement with experimental trends with i) almost no spectral change in 10a, 10b, and 10c, which is consistent with the lack of contribution of the substituting groups (t-butyl, phenyl and thienyl) on the density difference plots (Fig. 5); ii) a significant redshift when going from 10c to 12c; iii) red and hyperchromic shifts in the 17a, 18a and 19a series.

Additionally, the energies of the frontier MOs are not strongly affected by substitution (see the SI), which confirms the electrochemical study described above. As can be seen in Table 3, all the lowest transitions involve strong ICT with a charge transfer of ca. 0.6-0.8 electron over several Å. The values given in Table 3 are large compared to those reported in the literature obtained with a very similar approach [START_REF] Ciofini | Through-Space Charge Transfer in Rod-Like Molecules: Lessons from Theory[END_REF]. When going from 17a, to 18a and then 19a, the bands are redshifted, but this is not accompanied with an increase of the ICT character. On the contrary, the ICT character decreases, which is understandable from the density difference plots shown on Fig. 5. While 17a displays a net ICT from the donor group to the accepting moiety, in 19a, the donor group becomes a more passive element, and the ICT starts from the oligothiophene bridge to the cyanoacrylic moiety. This confirms that longer linkers do not necessarily provide an improved ICT character [START_REF] Jacquemin | Wha h "b " a om c cha g model to describe through-space charge-transfer excitations?[END_REF].

We also noted that the ICT properties of 17a were similar to these of the previously described D1 and D2 [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]. Our theoretical study shows that all di-anchoring dyes display a strong ICT from the electron donor pyranylidene core to the accepting groups on the two branches (see also the SI for density difference plots for the second state), indicating that these dyes could be attractive candidates for DSSC applications.

Table 3

Theoretical vertical excitation wavelength [PCM(THF)-CAM-B3LYP/6-311+G(2d,p)] in nm, oscillator strength (between brackets) a d L Bah ' ICT parameters of all dyes: d CT and q CT are given in Å and in e, respectively.

Dye

 vert /nm d CT /Å q CT /e

Photovoltaic measurements

All photosensitizers were used to dye TiO 2 mesoporous electrodes (12 µm thick) using the conditions optimized in our previous work [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]. Briefly, the TiO 2 photoelectrodes were immersed in a mixture of tert-butanol and acetonitrile which was used to dissolve the organic molecules (0.25 mM) in presence of 1 mM chenodeoxycholic acid whose role is to limit the aggregation of the dyes on the semi-conductor. Aggregation is known to promote the premature deactivation of the excited state of the chemisorbed dye and dramatically affects the performances of the solar devices [START_REF] Zhang | Dye aggregation in dye-sensitized solar cells[END_REF]. After one night, the obtained photo-electrodes were rinsed, quickly assembled with a platinum counterelectrode and impregnated with a triiodide/iodide electrolyte. The detailed fabrication of the devices is given in the Supporting Information and the current/voltage characteristic of the cells are gathered in Figure S45. The Incident Photon-to-Electron Conversion Efficiency (IPCE) for all dyes are given in Fig. 6, and indicate that the pyranylidene dyes exhibit good sensitizing properties with light absorption capacity extending as far as about 650 nm. a [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF].

The devices were then tested under AM 1.5 calibrated irradiation. Their performances are gathered in Table 4. At first glance, it appears clearly that the di-anchoring dyes are less performant than their analogues mono-anchoring dyes because of lesser Jsc and Voc. This is particularly obvious when comparing 10a with 17a, whose structures only differ by the number of anchoring sites (2 vs. 1 respectively). However, this observation can be extended to the sub-series 10a, 10b, 10c and 12c vs. sub-series 17a, 18a and 19a where substituents in positions 2 and 6 of the pyranylidene core are different, but known to only marginally affect the photovoltaic parameters [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]. Within the di-anchoring dye series, Jsc increases in the following order: 12c<<10a<10b<<10c. The dye 12c exhibits the lowest injection driving force (Table 4), combined with the lowest dye loading (Table 4) and the lowest d CT (Table 3) in the series, thus justifying its ranking as the least performant dye. 10a, 10b and 10c display comparable injection driving forces. Their charge transfers, quantitatively estimated by d CT and q CT (Table 3) are also similar and, therefore, the observed increase in Jsc more likely stems from the increasing dyes loading which follows a similar ranking: 12c<<10a<10b<<10c. Moreover, the absorption spectrum of 10c is the most intense between 400-550 nm implying a higher light harvesting efficiency.

The open circuit potential Voc decreases in the following order: 10a>10b≈10c>12c. The Voc is the difference of the Nernst potential of the counter-electrode and the Fermi level energy of the dyed TiO 2 photo-electrode. The latter is strongly affected by electrostatic and dipolar interactions between the TiO 2 surface and the dye (thermodynamic parameters) as well as by the charge recombination processes occurring at the dye/TiO 2 interface, e.g. between triiodide or oxidized dye (after electron injection) and the electrons diffusing in the semi-conductor conduction band (kinetic parameters). Since the same conditions were used to prepare the TiO 2 photo-l c od ( l c oly , ub a …) and the ICT characters of di-anchoring dyes 10a, 10b and 10c are comparable, thermodynamic effects are unlikely to account for this observation. Dye 10a is, however, considerably bulkier than dyes 10b, 10c and 12c due to the bulky t-butyl groups: this implies that the triiodide ions will experience more difficulties to penetrate the dye monolayer, and consequently to reach the TiO 2 surface. Thus, the surface is more passivated, hence limiting interfacial charge recombination and leading to the highest Voc in the di-anchoring dyes series. The average electron lifetime in TiO 2 was measured by Intensity-Modulated photovoltage Spectroscopy (IMVS) experiments. The electron lifetimes measured by IMVS are ranked in the following order: 10a>>10b>10c>>12c which fully supports the results described above for these dyes (Table 4 and Figure S46, SI), meaning that the TiO 2 surface shielding by the dye is maximum with t-butyl groups (dye 10a) and minimal with thienyl substituents (10c and 12c).

The DSSC features of the mono-anchoring dyes 17a, D1, D2 and D3 display significantly better performances than their di-anchoring dye counterparts because of higher Jsc and higher

Voc. This is due to several factors. First, the mono-anchoring dyes are substantially less bulky than the di-anchoring ones, and consequently display a much higher dye surface loading capacity, thus leading to a more favorable light harvesting efficiency and consequently higher Jsc. In addition, injection driving forces and calculated d CT and q CT (Table 3) are more favorable for mono-anchoring dyes than for di-anchoring dyes, thus favoring a high Jsc.

Finally, mono-anchoring dyes 18a and 19a exhibit a more intense absorption band in the 450-550 nm region, (therefore a higher light harvesting efficiency) than di-anchoring systems (Fig. 2), which favors the creation of a large photocurrent density since the photon flux of the solar spectrum is very intense in this region.

The better performances of the mono-anchoring dyes also originate from larger Voc. This can be assigned to the higher quality of the dye monolayer formed by the mono-anchoring dyes.

As mentioned above, the higher dye loading capacity of the mono-anchoring dyes and the presence of t-butyl groups lead to a dense organic passivating layer, which prevent interfacial charge recombination with triiodide. IMVS data confirm this assertion since the electron lifetime is shorter for di-anchoring dyes than for all mono-anchoring ones. Interestingly, the Voc decrease within the mono-anchoring series in the following order: 17a>18a>19a.

Namely, the output potential diminishes when the size of the oligothienyl spacers increases (from one to three thienyl units) and follows the same trend as the ICT character (Table 3) which can also induce a higher upward band bending of the conduction band of TiO 2 [START_REF] Rühle | Molecular Adjustment of the Electronic Properties of Nanoporous Electrodes in Dye-Sensitized Solar Cells[END_REF][START_REF] Zhang | Band Bending in Semiconductors: Chemical and Physical Consequences at Surfaces and Interfaces[END_REF]. A similar observation was made in previous works [START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF] where we also attributed the output potential decrease to a plausible bending of the dye backbone due to oligothienyl structure, to an increased aggregation or to a closer distance with the TiO 2 surface.

Conclusions

In this work, seven new push-pull sensitizers were synthesized and investigated for their photovoltaic performances in DSSC. The originality of this work stems from the synthesis of V-shaped di-anchoring D-(-A) 2 dyes containing the pyranylidene group as electron donor.

Such structures have been much less investigated than similar molecular systems based on trisarylamine moieties for DSSCs applications.

The major goal of this work was to investigate the impact of: i) the substituents on the pyranylidene group and ii) the number of anchoring groups of the dyes on the photovoltaic performances of the sensitizers. First, we found that the t-butyl substituents on the pyranylidene group resulted in better photovoltaic performances than the phenyl or thienyl ones, as the t-butyl moiety impose specific dye loading and conformation on the TiO 2 surface, which prevents interfacial charge recombination. Interestingly, the substituents in 2,6 positions on the pyranylidene group barely affect the electronic properties of the dyes, but have a significant impact on the structure and the organization of the dye on the TiO 2 surface.

The second major finding of this study is that the binding of these types of dyes to the TiO 2 surface through two anchoring sites does not seem to enhance photovoltaic performances, and the di-anchoring dyes were not found to exhibit systematically superior photovoltaic properties compared to their mono-anchoring counterparts. The higher photovoltaic performances displayed by the mono-anchoring dyes of this series stem from their lower molecular footprint which allows for a higher dye loading resulting in a better surface shielding from the electrolyte. Moreover, their higher ICT character increases the upward band bending. This investigation has provided valuable pieces of information for the future development of better performing organic push-pull dyes based on pyranylidene as electrondonor group.

Fig. 1 .

 1 Fig. 1. Structures of the dyes investigated in this work (10a-c, 12c, 17a, 18a, 19a) and previously studied (D1-3)[START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]

Scheme 1 .

 1 Scheme 1. Synthesis of alkyne compounds 5a-c and 7c. Reagents and conditions: a) THF, n-BuLi, -78°C to r.t., 2h. b) K 2 CO 3 , CH 2 Cl 2 /MeOH, r.t., overnight.

Scheme 2 .

 2 Scheme 2. Synthesis of dyes 10a-c and 12c. Reagents and conditions: a) Pd(PPh 3 ) 4 , CuI, THF, NHiPr 2 , 60° C, overnight. b) Cyanoacetic acid, piperidine, CHCl 3 , reflux, overnight.

Fig. 2 .

 2 Fig. 2. UV-Vis absorption spectra of 10a-c, 12c, 17a, 18a, and 19a in CH 2 Cl 2 solution (1.1 × 10 -5 M) at room temperature.

Fig. 3 .

 3 Fig. 3. Cyclic voltammetry at a Pt electrode of A) 10a (black) and 17a (blue) in positive initial scanning; B) 17a (blue), 18a (black) and 19a (red) in negative initial scanning. Experimental curves were measured in CH 2 Cl 2 / NBu 4 PF 6 0.1 M (v = 0.1 V s -1 , C = 0.5 mM). Redox potential values were recalibrated by taking E 0 (Fc + /Fc) = 0.71 V vs. NHE. Peak (*): not observed when scanning in initial negative direction.

  d G in eV, G reg = E 0 (I -35 V vs. NHE; G inj = E(S*/S + ) -E CB (TiO 2 ) with E CB (TiO 2 ) = -0.5 V vs. NHE. e Peak not detected. f For comparison with compounds 10b and 10c, the values found in CH 2 Cl 2 /THF (50:50 v:v) / NBu 4 PF 6 0.1 M are as following: E 0 (1) = 0.98 V vs. NHE; E pa (2) = 1.34 V vs. NHE; E pc (4) = -0.77 V vs. NHE; E(S*/S + ) = -1.28 V vs. NHE; G reg = -0.63 V vs. NHE ; G inj = -0.78 V vs. NHE.

  Fig. 4. Contour plots of the CAM-B3LYP HOMO and LUMO orbitals of dye 10a.

Fig. 5 .

 5 Fig. 5. EDD plots calculated for the lowest transition of all dyes. The blue (red) lobes represent regions of density increase (decrease) upon excitation. See also the SI. Contour threshold: 0.0008 au.

Fig. 6 .

 6 Fig. 6. IPCE of the DSSC using dyes 10a-c, 12c, 17a, 18a, and 19a as photosensitizers.

Table 1

 1 Absorption properties of the dyes 10a

-c, 12c, 17a, 18a, 19a, D1, D2 and

  D3 recorded in CH 2 Cl 2 (1.1 × 10 -5 M) at room temperature.

	Dye	 max /nm (10 4 M -1 .cm -1 )	 onset /nm a
	10a	396 (5,0), 485 (sh)	569
	10b	393 (4,8), 480 (sh)	556
	10c	393 (4,2), 470 (sh)	564
	12c	391 (2,4), 500 (sh)	630
	17a	379 (2.7), 472 (1.5)	566
	18a	442 (4.1)	543
	19a	366 (sh), 462 (4.8)	540
	D1	387 (4.8), 480 (sh)	568
	D2	389 (5.3), 472 (sh)	568
	D3	406 (3.4), 520 (sh)	655

Table 2

 2 in CH 2 Cl 2 /THF or THF) because of their low solubility in dichloromethane at the millimolar concentration. Electrochemical data obtained for compound 10a in both CH 2 Cl 2 and CH 2 Cl 2 /THF indicate that the solvent has a very limited impact on the voltammetry response (see Table2).

presents the electrochemical data obtained from experiments conducted with a platinum working electrode. Redox potential values are measured against ferrocene and recalibrated versus NHE. All experiments were carried out in dichloromethane except for compounds 10b, 10c and 12c (

Table 2

 2 Electrochemical NHE from the experimental curves obtained vs. Fc + /Fc, by taking E 0 (Fc + /Fc) = 0.47 V vs. SCE in CH 2 Cl 2 /NBu 4 PF 6 (measured experimentally) and E 0 (SCE) = 0.24 V vs. NHE.

	d	G inj d

data for 10a-c, 12c, 17a, 18a, 19a, D1, D2 and D3 (0.5 mM) at a Pt working electrode in CH 2 Cl 2 /NBu 4 PF 6 0.1 M (E / V vs. NHE a , v = 0.1 V s -1 ). G inj and G reg values for 10a-c, 12c, 17a, 18a, 19a, D1, D2 and D3. Dye E 0 (1) a E pa (2) a E pc (3) a,b E pc (4) a,b E(S*/S + ) c G reg a Values calibrated vs.

  Compound poorly soluble in CH 2 Cl 2 . Experiment performed in CH 2 Cl 2 / THF (50:50 v:v) / NBu 4 PF 6 0.1 M. h Compound poorly soluble in CH 2 Cl 2 and CH 2 Cl 2 / THF (50:50 v:v). Experiment performed in THF / NBu 4 PF 6 0.1 M.

g i

[START_REF] Gauthier | Synthesis and properties of novel pyranylidene-based organic sensitizers for dye-sensitized solar cells[END_REF]

.

Table 4

 4 Metrics of the solar cells fabricated illuminated with a calibrated sunlight AM1.5 (100 mW/cm 2 ). Electron lifetime recorded by IMVS at 150 W/m² irradiance.

	Dye	Jsc (mA/cm²)	Voc (mV)	FF (%)	PCE (%)	Dye loading (nmol/cm²)	Electron lifetime (ms)
	10a	8.26 (±0.40)	565 (±4)	72 (±1)	3.35 (±0.16)	79.2	40.1
	10b	8.52 (±0.35)	532 (±1)	72 (±1)	3.25 (±0.16)	95.8	31.8
	10c	9.74 (±0.28)	538 (±5)	68 (±2)	3.57 (±0.18)	110.4	25.3
	12c	4.75 (±0.60)	492 (±4)	70 (±1)	1.64 (±0.22)	67.5	14.2
	17a	11.43 (±0.10)	652 (±4)	70 (±1)	5.23 (±0.11)	110.3	52.6
	18a	9.84 (±0.80)	612 (±1)	72 (±1)	4.32 (±0.32)	138.0	60.8
	19a	9.73 (±0.06)	574 (±1)	70 (±1)	3.90 (±0.09)	137.4	46.9
	D1 a	13.1 (±0.1)	603 (±1)	70 (±1)	5.52 (±0.01)	174	43.67
	D2 a	10.8 (±0.1)	588 (±4)	70 (±1)	4.44 (±0.03)	173	24.19
	D3 a	6.7 (±0.1)	505 (±1)	68 (±1)	2.31 (±0.05)	161	13.70

Acknowledgments

We thank the Région des Pays de la Loire, for financial support via LUMOMAT program. This work used the computational resources of the CCIPL installed in Nantes.

Supplementary material

Supplementary data to this article (Synthesis and characterization details, Theoretical

Calculations and Photovoltaic Measurements) can be found online at …