Raphaël Boichot
email: raphael.boichot@simap.grenoble-inp.fr

Y Fan

A genetic algorithm for topology optimization of area-to-point heat conduction problem

Keywords: Genetic algorithm, topology optimization, heat conduction, area-to-point, optimal configuration

This paper presents a way of solving the classical area (volume)-to-point heat conduction problem by the means of a simple Genetic Algorithm (GA) in square configuration. After a short description of the numerical method, the optimal solutions proposed for minimizing the peak or mean temperature of a domain are presented. The effects of the conductivity ratio and the filling ratio on the configurations of the conductive tree are also analyzed and discussed.

A numerical benchmark is then established to assess the influence of mesh resolution and the reproducibility of the GA optimization. Results show that GA is capable of proposing solutions having almost the same cooling effectiveness for different mesh resolutions or random seed generators. GA is also relevant compared to other optimization techniques presented here. It can be considered as a simple, easy to adapt and robust but computation time consuming method for addressing the general area (volume)-to-point heat conduction problem.

Nomenclature :

A, R non-dimensional thermal resistances -

H

Introduction

The problem of cooling a continuous heat generating area (or volume) is widely recognized in electronic industries because the accumulation of generated heat, if not timely removed, will cause severely damage of the electronic components. One usual solution is to integrate a certain quantity of highly conductive material in the area (volume) to drain the generated heat to a heat sink (point) by pure heat conduction. The cooling effectiveness depends not only on the quantity (filling ratio, ϕ) and quality (conductivity ratio, kp/k0) of the available highly conductive material, but also on the topological configuration of the material to form the conducting path. How to determine the optimal configuration of the highly conductive material subjected to different objectives (e.g. minimum thermal resistance, minimum peak temperature, etc.) within the framework of the general "area (volume)-to-point" heat conduction problem has received a great attention in the literature. [START_REF] Bejan | Constructal-theory network of conducting paths for cooling a heat generating volume[END_REF] analytically studied this issue by proposing a "constructal approach" through first determining the shape of the optimal rectangular elemental areas and then assembling them scale by scale to pave the whole surface. The final optimal shape of the conducting path turns out to be tree-like networks, not inferred by assumptions but deduced by optimization.

The fundamental constructal theory [START_REF] Bejan | Shape and structure, from engineering to nature[END_REF][START_REF] Bejan | Design with constructal theory[END_REF] was followed and extended in various aspects through formulating different objective functions, releasing the constraints or adding degrees of freedom, as summarized in the review article of [START_REF] Chen | Progress in study on constructal theory and its applications[END_REF].

These extensive analytical efforts offer an elegant way of solving the basic area (volume)-topoint problem with reduced global thermal resistance and enhanced cooling effect.

Nevertheless, the analytical method becomes mathematically difficult, sometimes impossible, when dealing with irregular or unspecified geometries.

So emerges the idea that classic analytical approaches may benefit from modern numerical computing by adding more degrees of morphologic freedom. The numerical methods consist in freely paving the surface to be topologically optimized with the only constraint of the meshing, and so, the computational power available. The first attempt in this direction was made by Ererra and Bejan (1998) by analogy with fluid flow in river drainage basins. After, [START_REF] Cheng | Constructs of highly effective heat transport paths by bionic optimization[END_REF] and [START_REF] Xia | Bionic optimization of heat transport paths for heat conduction problems[END_REF] used the bionic optimization method based on a gradient attraction to enhance heat drain topology efficiency. Within the same class of local attraction algorithm, Mathieu-Potvin and [START_REF] Mathieu-Potvin | Optimal conduction pathways for cooling a heatgenerating. body: A comparison exercise[END_REF] and [START_REF] Boichot | Tree-network structure generation for heat conduction by cellular automaton[END_REF][START_REF] Boichot | A simple Cellular Automaton algorithm to optimize heat transfer in complex configurations[END_REF] successively proposed cellular automaton (CA) algorithms driven by thermal gradients, by temperature or by both. It was reported (Marck, 2012) that the CA algorithms (including bionic optimization) can offer a simple way (in terms of computational efforts) to obtain a sub-optimal but acceptable solution to the area-to-point problem with rational values of conductivity ratio (kp/k0). Nevertheless, the drawbacks of the CA algorithms are apparent: no clue implies that the final configuration is the optimum, i.e. no objective function to minimize is explicitly used but only local attraction based on intuitive hypotheses.

More recently, a variety of numerical algorithms were proposed to tackle this problem. To list some, Xu et al. (2006) used a simulated annealing (SA) method. [START_REF] Zhang | Design of conducting paths based on topology optimization[END_REF] proposed a Solid Isotropic Material with Penalization (SIMP) model for topology optimization. The method of moving asymptotes (MMA) algorithm was used by [START_REF] Dirker | Topology Optimization for an Internal Heat-Conduction Cooling Scheme in a Square Domain for High Heat Flux Applications[END_REF] in order to reduce the average internal temperature for square domains [START_REF] Burger | Three-dimensional conductive heat transfer topology optimisation in a cubic domain for the volume-to-surface problem[END_REF]. Marck et al. (2012) coupled the SIMP model with an aggregated objective function approach (AOF) to tackle this topology optimization problem through a multi-objective strategy. A systematic and quantitative robustness analysis of solutions of various optimization algorithms was provided by Song and Guo (2011).

Meanwhile, another class of numerical optimization algorithms that is extensively used in heat transfer problems [START_REF] Gosselin | Review of utilization of genetic algorithms in heat transfer problems[END_REF])-the Genetic algorithm (GA)-has rarely been applied for solving this heat conduction problem. Genetic algorithms, based on the Darwin's theory of evolution (Darwin, 1872), are characterized by a poor sensitivity to local minima so that the global best solutions can theoretically be reached. Moreover, GA is well-adapted to objective functions with a huge number of parameters (or dimensions) and non-differentiable (discrete) problems. The only effort in the literature in this regard was carried out by [START_REF] Xu | Optimization of heat conduction using combinatorial optimization algorithms[END_REF], who formulated a GA for the minimization of peak or mean temperature of a 2D domain. Their pioneer work [START_REF] Xu | Optimization of heat conduction using combinatorial optimization algorithms[END_REF] showed that GA proposed better solutions compared to those of bionic optimization, especially for high conductivity ratio conditions.

Due to the computational time consuming nature of GA however, only a few cases were studied, with limited meshing fineness, limited values of conductivity ratio (kp/k0=3; 10; 100) and filling ratio (ϕ=0.1). In fact, besides the fitness (the objective function), a number of more intrinsic parameters in the sense of numerical methods (e.g. the mutation probability, the crossing probability, the rate of selection of the best individuals, etc.) have to be assessed to optimize the convergence of the GA. In-depth investigations and systematic analyses are still in need in this area due to the strong influence of these parameters on convergence.

The goal of this study is to formulate a GA for efficiently solving the area-to-point heat conduction problem on one hand, and to analyze the effects of various parameters of practical use on the evolved topology on the other hand. We shall first present the GA method with a general 2D case, appropriate for introducing the notations and for describing in detail the basic principles of optimization. Then, the performances of the algorithm will be assessed for the minimization of two different objective functions: the peak temperature and the mean temperature across the domain. The effects of different conductivity ratios (kp/k0=2; 10; 50; 250) and filling ratio (ϕ=0.1; 0.3; 0.5) on the optimal configuration of conductive paths will be analyzed and discussed. After that, a dedicated numerical benchmark will be developed to investigate the meshing sensitivity and the reproducibility of GA method. Comparison with other methods addressing this problem is also proposed. Finally, technical remarks and main conclusions will be summarized.

Genetic algorithm

The GA is coded using software suite Matlab, assisted by the Parallel Computing toolbox (www.mathworks.com/products/parallel-computing/). The basic idea is to imitate the natural selection and survival of the fittest that exists in the genetics of the species. A synthesis of GA principles, applications and examples are given by Goldberg (1989).

Geometry and boundary conditions

The sketch of a typical area-to-point heat conduction problem to be solved by GA is given in Figure 1. The entire domain was discretized into small and homogeneous square elements, each element having determinate conductivity, uniform temperature and heating rate. In this study, a mesh of 100×50 square elements was used, considering a compromise between calculation time and precision. Other mesh resolutions (12×25, 25×50 and 100×200) were also assessed and the effects of mesh resolution on the complexity of optimal conductive trees will be discussed in the later section. Five kind of elements are defined to cover the entire calculation domain, as described below.

• Heat sink elements (blue): cells with a constant temperature (Tsink=293K) and a thermal conductivity equal to kp. The isothermal heat sink has an aperture width of 20% of one side of the domain.

• Symmetry elements (red): the right boundary of the domain shown in Figure 1 is defined as symmetry to represent a square domain. Numerically, these elements are coded similar to adiabatic elements (quasi-perfect insulators).

• Adiabatic elements (grey): cells defined with a constant temperature (Tsink=293K, in order to relax constraints on the finite difference formulation used) and a negligible conductivity (typically kad/k0=10 -9 ). The adiabatic elements and the heat sink elements enclose the entire domain. We have verified that the kad/k0 ratio used do not influence the estimation of temperature of the calculation domain (no thermal leaks at the borders).

• Heat generating elements (white): heat generating cells with heat generation rate p and relatively low conductivity k0.

• Conductive elements (black): cells with relatively high conductivity kp and zero heat generation rate (p=0). The filling ratio ϕ is defined as the ratio between the number of conductive elements and the sum of conductive and heat generating elements.

Calculation domain is enclosed by symmetry, adiabatic or heat sink elements. The heat equation for conduction is solved. The domain is considered as solid and no other thermal resistances than the conduction between elements is considered.

GA procedure

The general principle of GA is to assess the best configurations among a starting random population of configurations, keep the best (those meeting best the objective function or fitness), and then cross and mutate them to get a new child population of the same size, and so on. The main procedure of GA for the area-to-point heat conduction problem is described below.

•

Step 1: generate 1000 configurations with random paving of conductive and heatgenerating elements with fixed ϕ and kp/k0 ratio;

• Step 2: evaluate the fitness (objective function) of each configuration;

• Step 3: keep the best (in terms of fitness) 200 of 1000 evaluated configurations (future parents); eliminate the others from reproduction.

• Step 4: keep the best parent as the first individual of the children population (elitism);

• Step 5: cross two random parents among the best (with uniform probability) with a random crossing probability between 0 and 20% to give two children; see next paragraph for the chromosome format.

• Step 6: mutate the two children with a random mutation probability between 0 and a maximal value for each element (see next paragraph). Correct the children so that ϕ remains constant by random substitutions;

• Step 7: go to step 5 until 1000 children are made;

• Step 8: go to step 2 until the relative residuals in temperature reach 10 -6 or less for at least 1000 iterations or no better children are found for 1000 iterations.

We introduced only three constraints in our modeling. The first constraint used is keeping the ϕ parameter constant from one iteration to the next, which is ensured by a small sub-routine of random balancing of elements type into the mutation function -which is not conservative by essence -in step 6. This means that the fraction of each element is forced constant after mutation to restore the balance of element types. The second constraint use is the avoidance of single (non connected to similar elements in terms of property) elements: a mutated element must have a property similar to one of its neighboring element. This fastens the convergence of shapes and still allows non connected islands of more than one isolated element to appear and survive. Finally the heat sink, symmetry and adiabatic elements are not allowed to move or mutate to keep the boundary conditions constant.

The maximal mutation probability is first initialized at 5% (first iteration), then exponentially decreased each iteration so that it is divided by two every 700 iterations. Indeed, it was found by trial and error tests that this rate allows to maximize the probability of finding better children at each iteration and consequently to speed up the convergence without "freezing" the topology into local minima.

The algorithm is stopped when the relative tolerance on temperature between two consecutive best children stays below 10 -6 for 1000 iterations or if not better children are found for 1000 iterations (near convergence, each iteration or generation does not give "birth" to better configuration or children). It should be noted that the cross-over function is written to choose a random crossing direction (horizontal or vertical on each configuration) each time it is called, in order to avoid any crossing direction artifact. The "chromosome" used for crossing and mutation is just a vector of 5000 variables (50×100), each variable being an integer coding the type of elements for each element of the domain.

Calculation of the temperature field

One important step in the algorithm is the fast calculation of temperature field in the studied domain. This involves the solving of heat transfer equation numerically by a finite difference method. Details of the finite difference method for the calculation of temperature field may be referred to our earlier papers [START_REF] Boichot | Tree-network structure generation for heat conduction by cellular automaton[END_REF]; Boichot and Luo 2010) and will not be repeated here. The linear system obtained with finite difference method is solved with numerical methods optimized for sparse diagonal-dominant matrices and parallel computing and implemented with Matlab Software. Comparison with finite volume method will be given in the "Results and analysis section".

Example of topology convergence

Results and analysis

In this section, the results of GA for area-to-point heat conduction problem will be presented and analyzed. Two objective functions are identified for the study: the peak temperature and the mean temperature, with different values of ϕ and kp/k0 values. The minimization of both objective functions will lead to the generation of conductive tree configurations.

We first performed the GA to minimize the peak temperature, then the mean temperature of the domain. The performance of the evolved configuration was assessed by the nondimensional thermal resistances R [START_REF] Bejan | Constructal-theory network of conducting paths for cooling a heat generating volume[END_REF] or A (Marck 2012) in the case of peak temperature minimization and mean temperature minimization, respectively. The two parameters are expressed as:

max sink 0 / T T R pLH k - =
(1)

mean sink 0 / T T A pLH k - =
(2)

Figure 3 presents the optimal configuration of conductive path proposed by GA for peak and mean temperature minimization (ϕ=0.3; kp/k0 varying from 2 to 250). Figure 4 presents the optimization results for ϕ varying from 0.1 to 0.5 and kp/k0=10. From Figure 3, we may observe that similar optimal configurations (ramified trees) are obtained for two different objective functions, with the exception for kp/k0=2. For mean temperature minimization (kp/k0=2), the optimal conductive path consists in the agglomeration of high conductivity material around the heat sink: the heat generating elements are pushed far against the heat sink. Whereas for peak temperature minimization (kp/k0=2), a part of the high conductivity materials fill the farthest corner of the domain rather than making continuous cooling trees. This obviously counter-intuitive result is due to the fact that peak temperature usually appears in the farthest corners of the domain where the conductive trees try to grow toward. In case that the conductivity heat generating elements is close to that of conductive material (e.g. kp/k0=2), the conductive material will be optimally used for reshaping the heating area. This trend is also reported by [START_REF] Mathieu-Potvin | Optimal conduction pathways for cooling a heatgenerating. body: A comparison exercise[END_REF].

Figure 3 indicates the influences of conductivity ratio on the optimal configuration of conductive path, for both objective functions. Lower kp/k0 values imply thicker trunk for draining the heat current. In contrast, higher kp/k0 values lead to slenderer trunks and more branching levels. The high conductivity material tends to disperse through the calculation domain. Figure 4 illustrates the influence of filling ratio on the optimal configuration of conductive path. It can be observed that the higher the φ values, the thicker the trunks for a same kp/k0 ratio. It is worth noting that trees evolved based on local attraction parameters (CA algorithms) or evolutionary algorithms (GA method) present similar trends on branching properties with respect to the kp/k0 and ϕ parameters [START_REF] Boichot | Tree-network structure generation for heat conduction by cellular automaton[END_REF]) despite the absence of objective function in CA algorithms (local attraction).

By comparing the left column and right column on figures 3 and 4 for the same values of kp/k0 and φ, one may observe that the optimal conductive trees proposed by minimizing peak temperature are generally slenderer than those proposed by minimizing mean temperature.

The conductive trees tend to cover the farthest zones from the heat sink on the domain, which consequently leads to the thermal resistance equipartition from the adiabatic boundaries to the heat sink. This may be observed by examining the temperature isolines for converged solution shown in Figures 5 and6 (particularly figure 5). It appears that in case of peak temperature minimization, the GA proposes optimal conductive tree that tends to equalize as possible the temperature at the adiabatic borders, in particular for high kp/k0 ratios. This point is of particular interest: to decrease at most the peak temperature of a domain, an intuitive reaction is trying to equalize the temperature along the adiabatic borders with a conductive tree. The reductio ad absurdum is to consider that if it is not the case (temperature equipartition not reached), you will always be able to lower the maximal temperature point by modifying the conductive tree in exchange of an increase of the temperature in another area of the border (which is not the hottest). We can observe in figure 5 that the higher the kp to k0 ratio, the closer to this optimal property the converged shape tends to. There must be a particular condition in minimal kp/k0 or (kp/k0)φ that allows borders temperature equipartition by sufficient bending of thermal streamlines.

On other hand, the isolines of figure 6 for the case of mean temperature minimization clearly indicate that such an optimality criterion is not relevant for tree evaluation.

The performance of GA optimized conductive trees, measured by the non-dimensional thermal resistance R (peak temperature minimization) and A (mean temperature minimization)

for different values of (kp/k0)φ product can be compared with some other numerical methods to assess the efficiency of genetic algorithm to solve the area-to-point problem in square configuration. The A and R relationship with (kp/k0)φ for our genetic algorithm can be fitted by the following relationships:

Figure 7 and 8 present the comparison of A and R parameters between Cellular Automaton method (CA) proposed by [START_REF] Boichot | Tree-network structure generation for heat conduction by cellular automaton[END_REF][START_REF] Boichot | A simple Cellular Automaton algorithm to optimize heat transfer in complex configurations[END_REF] and improved by Marck (2012),

Constructal approach proposed by [START_REF] Bejan | Constructal-theory network of conducting paths for cooling a heat generating volume[END_REF] and modified by [START_REF] Ghodoossi | Exact solution for cooling of electronics using constructal theory[END_REF], Evolutionary Structural Optimization (ESO) by extension proposed by Marck (2012) and SIMP model with an aggregated objective function approach (AOF) developped by Marck et al. (2012), data compiled and proposed by Marck (2012). It appears that GA algorithm performance overcomes both constructal approach, CA and ESO algorithms. Results with SIMP algorithm are similar in terms of performance for A parameter (data not available for R parameter). It can be emphasized that GA algorithm and SIMP algorithm should be close to the optimal method to solve the area-to-point problem. Shapes obtained are interestingly very similar between GA and SIMP method also. It should be finally noted that Cellular Automaton algorithm, which is not an optimization technique per se, can provide interesting trees, close to the GA algorithm, with a much reduced computational effort (CA algorithm converges 3.10 4 faster than GA algorithm on the same computer, see Luo ( 2013)).

To assess whether the numerical method used in this study to calculate the temperature map (direct finite difference method) introduces a bias, we have performed cross validations with a finite volume method implemented into a commercial CFD software (CFD-Ace version 2014.0.0.11217 from ESI group), with the two converged trees obtained with the highest kp/k0 ratio (250).

Figure 9 present the comparison between Matlab temperature map calculated with direct finite difference method (5000 elements) and CFD-Ace temperature map obtained with 68772 elements (unstructured meshing), an external adiabatic frontier and nodes frontier coinciding with frontier between low and high conductivity materials for the shape converged that minimizes maximal temperature. The relative difference for R parameter is 6.3% (Matlab code slightly overestimates R compared to CFD-Ace), while the thermal isolines nearly coincide. The calculation time is 0.019 second for direct method with 5000 elements versus more than one hour for finite volume method with 68772 elements and the commercial software.

Figure 9: temperature map obtained with Matlab and CFD-Ace softwares for the shape obtained with ϕ=0.3 and kp/k0 = 250, converged for maximal (peak) temperature minimization.

Figure 10 present the comparison between Matlab temperature map calculated with direct finite difference method (5000 elements) and CFD-Ace temperature map obtained with 64137 elements (unstructured meshing), an external adiabatic frontier and nodes frontier coinciding with frontier between low and high conductivity materials for the shape converged that minimizes mean temperature of the domain. The relative difference for A parameter is 5.6%

(Matlab code slightly overestimates A compared to CFD-Ace), while the thermal isolines nearly coincide. The calculation time ratio is about the same than previously evocated. Cross validations with finite volume method implemented into a commercial software confirmed the performance of the evolved trees and indicated that the direct finite difference method used in this study tends to slightly underestimate the real physical performance of the trees converges.

Robustness analysis

In this section, a numerical benchmark is dedicated to the robustness of the GA for area-topoint heat conduction problem. More precisely, the sensitivity of mesh resolution and the reproducibility with a different random seed generator for GA are assessed, for constant values of ϕ=0.3 and kp/k0=10 and peak temperature minimization.

Mesh sensitivity

Figure 11 shows the optimal conductive trees proposed by GA, for different mesh resolutions varying from 25×25 to 200×200. The conductive tree obtained at convergence clearly depends of the mesh resolution of the domain. The coarser the mesh, the larger the smallest dimensions that will be reached by the conductive tree is. Since the configurations at convergence should barely suffer from any issue of local optimum, they can be considered as close to the best that can be expected for a given mesh resolution. By examining Figure 11, it is clear that the optimal solution complexity changes with the mesh resolution (and the freedom allows to algorithm), not excluding an intrinsic infinite complexity if element size tends to zero. By increasing the mesh resolution, at least at the scales performed, the converged configuration is allowed to develop slender branches and more branching levels. Each refinement of the mesh gives a refinement of the observed converged configuration. Then one question arises: is the complexity of optimal conductive trees infinite or not? More precisely, will the branching levels always increase with increasing mesh resolution? At the still very coarse mesh resolutions we tested, it does appear that way.

The solutions proposed by GA are surprisingly robust regarding different mesh resolutions, at least in the range tested. Indeed, the standard deviation of the four R values obtained fall below 0.5% for the trees presented in figure 9, and more surprisingly, the geometry 200x200 leads to the less performing tree (the best is the 100x100 elements geometry). To our point of view, this is mainly due to the poor convergence that deteriorates quality of tree for a too refined geometry: the stopping criterion of the GA algorithm is too coarse to allow the 200x200 elements geometry to converge adequately (i.e. the change of few elements should modify the objective function below the convergence criterion). The convergence for 200x200 elements geometry took 17 days, so a parametric study of stopping criterion in this particular case is currently out of reach. Additionally, the values of R parameters for trees of figure 11 are represented on figure 8: the mark size does not allow seeing any difference in value. Each tree converged is optimal at the mesh size used, which is a strong constraint by itself. It should be however noticed than with such coarse mesh as 25x25 geometry, a bias between mathematical and physical thermal performance may exist.

Reproducibility

Another aspect of the robustness study is whether the configuration obtained at convergence is reproducible. In fact, a diversified configuration may possibly be obtained each time by running the GA with a different seed generator (with a Mersenne twister algorithm generating random numbers), as shown in Figure 12. It can be clearly observed that the solution converges to globally similar but locally slightly different configurations. However, the three final conductive trees have almost the same cooling effectiveness, measured by a standard deviation on R values that falls below 0.6%. As a result, these solutions may be considered as very close to the global optimum. This observation may also lead us to retrospectively consider that the differences in R values obtained by refining the mesh must be considered with caution. If the converged configurations having slight differences in local details present very close cooling performances, it means that the objective function is very flat regarding to variation of its 5000 discrete parameters near the global optimal point. Then the open question is: is there any interest of pushing the convergence limit further for the purpose of designing real objects of practical use for thermal engineering?

From the engineering point of view, once the conductive trees of a given GA iteration "condensates" (see for example in Figure 2) to a configuration with sharp borders, the further small adjustment of local details, very time consuming due to GA nature, may not noticeably augment the performance. In that sense, relaxing the convergence criterion could be a way of decreasing time needed for acceptable convergence.

From an industrial point of view, making efficient conductive tree shapes for practical use as heat sink could be easily achieved by micro fabrication, printing, laser/water cutting or metal extrusion. The issue is not how to make these objects, but what to do with these and where they can be efficiently included to improve thermal performances of real objects. This kind of fins minimizing space occupied and conductive mass could be embedded into sorption cooler beds, high temperature fuel cells (as interconnect), nuclear fuel claddings or high power electronics to decrease their thermal resistance, for example. Where thermal gradients impede the lifetime/function of objects, topology optimization could play a key role. Preliminary applications of GA algorithm applied to topology optimization to circular area-to-point problem [START_REF] Luo | Heat and Mass Transfer Intensification and Shape Optimization: A Multi-scale Approach[END_REF] showed that this method is very easily expendable to any geometry without any particular code modification and even produce counter intuitive optimal solutions.

Conclusion

We proposed here a genetic algorithm to solve the general area-to-point heat conduction problem. Conductive trees may be generated by GA for minimizing the peak or mean temperature of the domain. Their configurations depend strongly on the values of conductivity ratio as well as the filling ratio. A robustness study on the mesh sensitivity and the reproducibility also implies that the converged solutions proposed by GA approach the close-to-optimal configuration because of the absence of constraints used (except the square meshing of the domain) and the relative tolerance to local minima.

The main advantage of GA numerical method is clearly neither its velocity of convergence nor the easiness to define a stopping criterion, but the noticeable simplicity and more morphologic freedom offered for addressing unspecified problems. Further works should focus on numerical methods which allow fast temperature map calculation without any bias to verify if the absolute accuracy of temperature map has an influence on the evolved trees.

p

  Heat generation rate per unit volume or surface W m -3

Figure 1 :

 1 Figure 1: geometry and mesh used for this study. Note the symmetry on the right border.

Figure 2

 2 Figure 2 shows an example of topology convergence with iteration number for ϕ=0.3 and

Figure 2 :

 2 Figure 2: example of topology convergence for ϕ=0.3 and kp/k0=10 vs. generation (iteration). The conductive matter is black, the non conductive matter white. Grey levels indicate the cohabitation of several types of elements at a given position in the population for a given iteration.

Figure 3 :

 3 Figure 3: Optimal configuration proposed by GA for peak (left) or mean (right) temperature minimization (ϕ=0.3; kp/k0 from 2 to 250).

Figure 4 :

 4 Figure 4: Optimal configuration proposed by GA for peak (left) or mean (right) temperature minimization (kp/k0=10; ϕ from 0.1 to 0.5).

Figure 5 :

 5 Figure 5: temperature isolines into the geometry converged for maximal (peak) temperature minimization with ϕ=0.3 and kp/k0 from 2 to 250.

Figure 6 :

 6 Figure 6: temperature isolines into the geometry converged for mean temperature minimization with ϕ=0.3 and kp/k0 from 2 to 250.

Figure 7 :

 7 Figure 7: Benchmark of different optimization methods for A parameter.

Figure 8 :

 8 Figure 8: Benchmark of different optimization methods for R parameter. Black arrow indicates performance of trees represented in figures 11 and 12.

Figure 10 :

 10 Figure 10: temperature map obtained with Matlab and CFD-Ace for the shape obtained with

Figure 11 :

 11 Figure 11: optimal conductive tree as a function of the mesh resolution with ϕ=0.3 and kp/k0=10.

Figure 12 :

 12 Figure 12: Conductive trees at convergence with different seed generators for the random number algorithm with ϕ=0.3 and kp/k0=10.

Acknowledgement

The authors want to thank Johan Marigny and Gaëtan Patry, students of Grenoble Institute of Technology (France), for their great contribution to the writing and debugging of the GA algorithm.


Gülnur Taban-Çalişkan

Daniel Mesquita Fernandes

Jean-Claude Daran

Dominique Agustin
email: dominique.agustin@iut-tlse3.fr

Funda Demirhan
email: funda.demirhan@bayar.edu.tr

Rinaldo Poli
email: rinaldo.poli@lcc-toulouse.fr

[ Cp

Gülnur Taban-Çalışkan

] Daniel

Mesquita Fernandes

Rational Synthesis and Characterization of the Mixed-Metal Organometallic Polyoxometalates [Cp*Mo x W 6-x O 18 ] -( x = 0, 1, 5, 6)

Keywords: Organometallic oxides, tungsten, molybdenum, polyoxometalates, DFT calculation, X-ray crystallography

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

Introduction

Materials based on mixed oxides are interesting for heterogeneous catalysis [1][2][3][4][5][6][7] as well as for chromogenic materials. 8 Their properties (reactivity, light absorption, …) may be fine tuned by subtle modification of the nature and relative proportion of the different metals. Fine control of the composition and homogeneity of these materials, however, can only be partially achieved by the current techniques (sputtering, CVD), since it is difficult to control a mixed and homogeneous metal distribution at the atomic level. 9 In order to facilitate this task, the use of single source precursors is being sought but this strategy, often making use of mixedmetal alkoxides, needs strictly controlled environments (inert atmosphere, reduced pressure), since the precursors suffer from hydrolytic sensitivity. [10][11][12][13][14][15][16][17][18] Polyoxometalates (POMs), in addition to being interesting molecular compounds in their own right, [START_REF] Pope | Heteropoly and isopolyoxometalates[END_REF] may also be considered as "molecular scale" models of metal oxides and are useful for the understanding the interaction of substrates and oxide surfaces in heterogeneous oxidation catalysis. [START_REF] Masure | [END_REF][21][22] POMs are interesting compounds due their numerous potential applications especially in materials science and catalysis. 23 Although the elemental composition and morphological constitution of POMs can be precisely controlled, 23,24 the synthesis of well defined heterometallic POMs often suffers from serendipity and results in a random distribution of the different metals in the structure. Rational strategies that have been employed to prepare heterometallic POMs are (i) the assembly of predefined polyatomic fragments sometimes performed in organic solvents and with air-and water-sensitive organometallic precursors and (ii) the grafting of simple organic fragments on lacunary oxoclusters. [25][26][27][28][29][30][31][32][33][34] The use of hydrothermal methods starting from elementary bricks is another alternative strategy, the outcome of which is however largely affected by serendipity. 31,35 POMs grafted with organometallic fragment have attracted interest [36][37][38][39][40][41][42] 42 Recently, we have reported the use of Cp*2M2O5 (M = Mo, W), compounds that are stable in air and in aqueous solution in the entire pH range, 43 in combination with the inorganic salts Na2M'O4 (M' = Mo, W) in a 1:4 ratio, as precursors of the hexanuclear organometallic polyoxometallic complexes Cp*2MoxW6-xO17 (x = 0, 2, 4, 6) in a selective, high-yielding, room temperature aqueous reaction. 44,45 The formula of these compounds may also be written as

[(Cp*M)2(M'O)4(2-O)12(6-O)].
The relative position of the M and M' atoms is perfectly defined by the nature of the starting materials, the M element from the organometallic reagent ending up selectively in the (Cp*M) positions and the M' element from the inorganic reagent occupying selectively the (M'O) positions. Compared to the typical Lindqvist-type octahedral species [M6O19] 2-, [46][47][48] two adjacent metallic atoms bear a Cp* ligand in place of a terminal oxido ligand, leading to a neutral compound.

In addition to being isoelectronic with the Linqvist-type [M6O19] 2-ion, these neutral organometallic polyoxometallates are also isoelectronic with the known [Cp*Mo6O18] - ion, 49,50 which was best obtained as a Bu4N + salt from (Bu4N)[MoCp*O3] and (Bu4N)2[Mo4O10(OMe)4Cl2] in methanol in up to 40% yield. 50 It seemed interesting to enlarge the synthetic spectrum of our aqueous Cp*2M2O5/M'O4 2-method to a stoichiometry of 1:10 for the rational synthesis of the

[(Cp*M)(M'O)5(2-O)12(6-O)] -(or [Cp*MM'5O18] -) ions,
none of which has yet been described except for the above mentioned all-Mo example. We report in this paper the application of this strategy leading to the synthesis and characterization of the entire series of [Cp*MM'5O18] -ions (M, M' = Mo, W), obtained in the presence of a variety of different cations.

Results and Discussion

(a) Syntheses and characterization

The reaction between Cp*2M2O5 and M'O4 2-(M, M' = Mo, W) in a 1:10 molar ratio according to equation 1 selectively yields the anionic [Cp*MM'5O18] -complexes in goods yields and purity. The reaction is carried out by mixing the stoichiometric amounts of Cp*2M2O5 (dissolved in MeOH) and Na2M'O4 (aqueous solution), yielding the products after adequate acidification. The ions quantitatively precipitate upon addition of an appropriate organic halide salt. The cations used in this study are tetrabutylammonium 1a-4a, tetrabutylphosphonium 1b-4b, tetraphenylphosphonium 1c-4c, or butyl-pyridinium in case of compounds 1d and 3d. The compounds colour varies from orange through pale green to yellow depending on the Mo/W ratio. Compound 1a has already been previously reported and characterized, including by a single crystal X-ray diffraction study, but was synthesized by a different method in a lower yield (see Introduction). 50 The selectivity of the synthetic strategy is worthy of note: the reaction could have resulted, for instance, in a mixture of the previously reported [Cp*2M2M'4O17] compounds and the parent Lindqvist anions [M'6O19] 2-. This indicates the thermodynamic stability of the [Cp*MM'5O18] -anions with respect to a hypothetical fragment redistribution process.

Cp*2M2O5 + 10 M'O4 The success of this synthesis, as well as that of the related neutral Cp*2MoxW6-xO17 compounds, 45 relies on the high stability of Cp*M bond in Cp*2M2O5 toward protonolysis at any pH. 43 Thus, it is possible to use the "Cp*M" moieties as an elementary building block, especially at low pH, due to the ionic splitting of Cp*2M2O5 into Cp*MO2(H2O) + and Cp*MO2 + . The occurrence of this process was clearly demonstrated for the Mo system 43,51 and 1 H NMR evidence shows that it also takes place for the W system. 52 Thus, the syntheses performed herein can be considered as the assembly of individual organometallic {Cp*MO2} + fragments and inorganic {M'O4} 2-species, using specific stoichiometry and pH conditions.

The products are stable in water at low pH. Three of the compounds were obtained in the form of single crystals; their molecular structures will be described in the next section. All anions were also investigated by mass spectrometry using electrospray method. The spectrum in negative mode showed the expected molecular ion with an isotopic pattern in good agreement with the simulation, see Figure 1. Notably, metal compositions different from MM'5 (for instance M2M'4) were absent from the spectra of the mixed-metal products 2 and 3, The fragmentation pattern is not identical for each type of anion, but as a general feature we can observe loss of the Cp* fragment to yield [MM'5O18] -, followed by subsequent loss of both MO3 and M'O3. Table 1 reports the relevant bond distances (angles are in the SI, Table S1). The numbering scheme used for the anions in compounds 1c and 2c is identical (with the metal atom M4 being Mo in 1c and W in 2c). The numbering scheme of the anion in 2a is the same as the other two anions for what concerns the first half of the molecule and only this half is described in Table 1 in comparison with the other two structures. The opposite half has metric parameters in close correspondence with the first one, in spite of the absence of crystallographic mirror symmetry in this case. A full table of bond distances and angles of 2a is available in the Supporting Information (Table S1) and all distances are also displayed later in the DFT section. The Cp*-M distance is slightly shorter when M = W. All structurally equivalent M-O bonds have quite similar distances in the two compounds. The ideal C4v symmetry of the ions, however, is broken by a distortion that renders the bridging Mo-O-Mo moieties asymmetric and this distortion is more pronounced in compound 1 than in compound 2. This distortion will be further discussed later in the DFT section. In addition, the central a The bond angles are provided in the Supporting Information (Table S1). Ct = Cp* ring centroid.

(c) DFT calculations

By analogy with the previously reported mixed-metal neutral compounds, 45 DFT geometry optimizations were carried out on the isoelectronic anionic [Cp*MM'5O18] - complexes in order to interpret the IR absorption spectrum in the M=O stretching region and to understand the effect on them of the metal nature. To save computational time, simplified models were used where the Cp* ligands were replaced by Cp rings. The models are numbered by roman numerals corresponding to the compounds numbering scheme:

[CpMoxW6-xO18] -with x = 6 (I), 5 (II), 1 (III) and 0 (IV). If we consider free rotation of the Cp ligand linked to M, the cluster geometry can be idealized to C4v. This is justified at least on the NMR timescale, because the five methyl groups of the Cp* ligands are equivalent in the 1 H NMR spectrum. In order to facilitate the discussion, the five (M=O) metals will be labelled according to their idealized symmetry equivalence: M' is used for the four equivalent equatorial metals and M" for the axial metal trans to the M atom that bears the Cp ring. The terminal O atoms will be labelled Ot, the doubly bridging atoms Ob and the central (µ6-O)

atom Oc (see Scheme 1). No symmetry constraint was imposed to the ions during the geometry optimizations. The full Cartesian coordinates of the optimized geometries are available in the Supporting Information (Table S2) and relevant metric parameters are summarized in Table 2. For comparison, Table 2 reports also all distances of the three structures described in this contribution, as well as the previously described structure of 1a. 50 Scheme 1. Atom labelling scheme used in Table 2.

Table 2. Selected bond distances for the geometry optimized models I-IV and comparison with the experimental structures of 1c and 2c. [a] System 1a [b] 50 .

The agreement between experimental and calculated geometries for the [MM'4M"O18] - core in the case of 1/I and 2/II is generally quite good. The calculated distances to the terminal and bridging ligands are generally only slightly longer than the experimentally observed ones (the maximum deviation is 0.05 Å for M-Ob(M") and M"-Ob(M) in I), whereas the distances to the central O atom are slightly underestimated for M-Oc (by 0.01 Å). As for neutral compounds, the atom Oc is much closer to M than to M' and M" and the calculations tend to place Oc even closer to M relative to the experimental structure. This discrepancy could of course be related to the use of the simplified model.

The experimentally observed asymmetry of the central MM'4M"O18 core is also shown in the optimized structure and decreases, in agreement with the experimental observations, upon replacing Mo with W atoms in the structure, tending toward the symmetric limiting structure for the all-W member of the series. A trans-alternation pattern of long and short bond lengths in {M4(Ob)4} rings yielding distorted octaedra, originally described for the parent Lindqvist anions, 53 is a common feature of poyoxometalates and is notoriously more pronounced for molybdates than for tungstates. However, formal replacement of a terminal oxo ligand by a Cp ring, 54 as in [CpTiM5O18] 3-, or an imido group, 55,56 as in [M6O18(NAr)] 2- (M = Mo, W) appears to somewhat attenuate this irregularity. The introduction of a single W atom renders the structures of 2a and 2c very close to the symmetric limit and the optimized anion structure (II) is also very little distorted. The reason for this trend is not quite clear and its understanding goes beyond the scope of the present work. This trend has also been noted for the isoelectronic Cp*2MoxW6-xO17 series. 45 However, a close analysis of this asymmetry for compounds 1/I reveals interesting features. The asymmetry is most clearly visible in the distances between the Ob atoms and the metal atoms. The two experimental structures and the optimized geometry, however, exhibit three different kinds of asymmetry. Focusing first on the M'-Ob-M" moieties, the structure of 1a, although of low quality, shows that the four independent M'-Ob(M") distances have a narrow spread around their average, 1.96(2) Å, and the same is true for the M"-Ob(M') distances, 1.86(2), while these two averages are very different from each other. For the structure of 1c, on the other hand, the asymmetry is manifested within each set of structurally equivalent distances. The optimized distances in I follow the same trend as 1c. For the M'-Ob-M' moieties, however, the situation is quite different. There is an insignificant distortion in both experimental structures (averages of 1.94(2) Å for 1a and 1.930(2) Å for 1c), whereas the calculated distances in I yield two very different sets of four distances each, with averages 1.841(4) and 2.049(7) Å. Finally, the M-Ob-M' moieties show the same type of distortion in all three geometries, namely two sets of two different distances for each structurally equivalent M-Ob(M') and M'-Ob(M) set, the difference between the short and long sets being smaller for 1a (0.04(2) and 0.03(2) Å), intermediate for 1c (0.057(2) and 0.038(2) Å), and greater for I (0.11(1) and 0.110 Å). This analysis clearly shows that there is a driving force for the POM structure to distort away from maximum symmetry, not solely related to crystal packing. However, the crystal packing (nature of the cation) has the effect of driving this distortion in different directions. The effect of packing forces is also revealed by the discrepancy in the experimental structures (Table 1) between the W4-Ct distances in compounds 2a and 2c.

We also wish to note that the effect of the metal nature on the distortion of the octahedral M6(Ob)12 scaffold is also quite evident for the parent [M6O19] 2-Linqvist anions (M = Mo, W). The statistical analysis of all the salts of these two dianions for which a structure is reported in the Cambridge Structural Database shows a greater average distortion for the Mo structure relative to the W structures. Of the 77 Mo structure, the difference between the maximum and minimum Mo-Ob distances (Δ) goes from zero (for the hydrated NEt4 + salt where the dianion sits on a crystallographic m m position) 57 to a maximum of 0.22 Å (for a crown ether-containing ammonium salt) 58 with an average Δ of 0.104 Å over the entire set. In the most distorted structure, two equal sets of short, 1.85(2) Å, and long, 2.01(3) Å, Mo-Ob distances can be identified like for the [Cp*Mo6O18] -structures analyzed here. For the 70 W structures, on the other hand, the average Δ is reduced to 0.060 Å. Like for the [Cp*MM'4M"O18] -ions analyzed in this contribution, the nature of the cation and the consequent packing forces seem to greatly influence the degree of distortion, since both undistorted Mo structure (see above) and very distorted W structures (such as the Et4N + salt with Δ = 0.21 Å) 59 exist. It can be concluded that this structural motif has a natural tendency to distort from the maximum symmetry but the potential energy surface along this distortion is rather flat, allowing a great degree of control to be exerted by the crystal packing. The greater average distortion for the Mo structures is probably caused by the weakness of the Mo-Ob bonds relative to the W-Ob bonds.

The trends of the bond distances, beyond the above discussed asymmetry, due to the change of M or M'/M" along the series of structures I-IV can be divided into primary (distances to the metal being changed) and secondary (distances to M due to a change of M'/M", or viceversa) effects. The calculations show similar trends to those already observed for the neutral [Cp2M2M'2M"2O17] analogues. 45 The primary effect is relative important in the M-Cp distance (Δ = 0.013 from I to II and 0.012 from III to IV), but a secondary effect in the same distance is also notable (Δ = -0.011 from I to III and -0.012 from II to IV). In other words, the M-Cp distance is lengthened on going from Mo to W, but is shortened when the M'/M" atoms are changed from Mo to W. On going from Mo to W, the terminal M'/M"=Ot distances show a slight primary lengthening and no significant secondary effects. Conversely, the M-Oc and M"-Oc show a primary shortening and a slight secondary lengthening, whereas no large effects are visible in the M'-Oc distances.

(d) IR characterization

Infrared spectroscopy is a good characterization tool for symmetrical polyanions. For the Lindqvist M6O19 2-anions (M = Mo, W), the number of observed vibrations is determined by their Oh symmetry 60,61 and a correlation between experimental and calculated vibrations could be established. 62 As already discussed above, the symmetry of the polyoxometallate cluster in compounds 1-4 is reduced at best to C4v (considering Cp* as a rapidly rotating ligand). The observed structures, backed up by the DFT calculations, show however a further symmetry reduction to Cs, which is more pronounced for the Mo-richer compounds.

The observed (for 1a-4a) and calculated (for I-IV) IR spectra in the metal-oxygen stretching region are shown in Figure 3 (see Experimental section for the list of frequencies) and the calculated frequencies, symmetry labels and assignment are listed in Table 3. Views of the normal modes are shown in the Supporting Information (Table S3). As found for all POM derivatives, the terminal M'=Ot and M"=Ot vibrations have higher frequency (observed, 950-1000 cm -1 ; calculated, 1020-1050 cm -1 ) than the M-Ob-M vibrations (observed 750-890 cm -1 , calculated 780-850 cm -1 ). Some contribution of Cp C-H bending modes is found to mix with the lowest frequency vibration (840-960 cm -1 ). There is a rather good match between calculated and observed spectra (Figure 3), the frequency shift certainly being related the computational method and/or to the model (Cp vs. Cp*). This gives us confidence in the reported band assignment. The trends of calculated frequencies as the metal atoms are changed are amenable to a detailed analysis. First of all, the structural distortion away from the idealized C4v symmetry, even for system I where this distortion is more pronounced (see previous section), is not significantly reflected in the shape of the normal modes. For instance, the pseudo-degenerate E-type modes, which split into A'+A" in Cs symmetry, remain practically degenerate (maximum difference: 1 cm -1 ). Therefore, the labels of the higher symmetry C4v point group are used in Table 3 and 3.

From the experimental spectra only two (M=O) bands can be unambiguously determined, sometimes with shouldering. Comparison with the calculated spectra suggest that the strongest one is the E-type vibration, whereas the second most intense band is most probably the highest frequency A1-type band Experimentally, the most significant effect on the spectrum in this region is seen for a change of the inorganic metal, whereas a change of the organometallic one produces hardly any difference.

Conclusions

We have presented here a new, rational and facile synthesis of new organometallic Group 6 Lindqvist-type polyanions of type [Cp*MM'5O18] -(M, M = Mo, W). This family was previously represented only by the homometallic Mo member, obtained by two different and less efficient synthetic strategies. The thermal decomposition of these compounds (at least those with N-containing cations) yields the mixed-metal oxides Mx/6M'1-x/6O3 with a homogeneous distribution of the two metals, which may be of interest for the study of the metal influence in various applications.

Experimental Section

All experiments were performed in air. Compounds Cp*2Mo2O5 and Cp*2W2O5 were synthesized according to the literature. 63 Water was deionized and methanol (Carlo Erba,

X-ray analyses.

A single crystal of each compound was mounted under inert perfluoropolyether at the tip of glass fibre and cooled in the cryostream of a Bruker APEXII CCD diffractometer for 2a or an Agilent Technologies XCALIBUR CCD diffractometer for 1c and 2c. Data were collected using the monochromatic MoK radiation (= 0.71073). The structures were solved by direct methods (SIR97) 64 and refined by least-squares procedures on F 2 using SHELXL-97. 65 All H atoms attached to carbon were introduced in calculation in idealised positions and treated as riding models. The drawing of the molecules was realised with the help of ORTEP32. [START_REF] Burnett | Report ORNL-6895[END_REF][START_REF] Farrugia | [END_REF] Crystal data and refinement parameters are shown in Table 4.

Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 866670 -866672.

Copies of the data can be obtained free of charge on application to the Director, CCDC, 12

Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk). From the resulting optimized geometries, starting geometries for III and IV were generated by changing the metal. All optimizations were carried out on the isolated ions using the Gaussian 03 suite of programs 68 with the B3LYP functional, which includes the threeparameter gradient-corrected exchange functional of Becke 69 and the correlation functional of Lee, Yang, and Parr. 70,71 The standard 6-31G** basis set was used for the C, H and O atoms, while the CEP-31G* basis set 72 was adopted for Mo and W. Analytical frequency calculations were also run on the optimized geometries, yielding positive frequencies for all normal modes. The calculated IR spectra shown in Figure 3 were generated from the DFT-generated frequencies and intensities by applying Lorentzian functions and adjusting the linewidth to best fit the experimental spectra.

  because the organometallic fragment may impart a different reactivity to the molecule with respect to the all-inorganic POMs, obtaining mixed-metal clusters under mild conditions (often using aqueous solutions). As illustrative examples, the reaction of Na2MO4 (M = Mo, W) with [Cp*RhCl2]2 or [( 6 -arene)RuCl2]2 in water or acetonitrile yielded the octanuclear compounds [(LM)(M'O)(-O)3]4 [LM = Cp*Rh and (p-MeC4H4iPr)Ru for M' = Mo; 36,37,40 LM = (C6Me6)Ru and (p-MeC4H4iPr) for M' = W].

The 1 H

 1 NMR spectra of the isolated salts in DMSO show the Cp* signal at δ 2.2 (when linked to Mo) or 2.4 (when linked to W), plus the resonances of the cation with suitable intensity for the 1:1 Cp*/cation stoichiometry. The 31 P NMR spectrum of the phosphonium salts shows the expected cation resonance at δ 23.4 for Ph4P + and 35.1 for Bu4P + . The compounds show characteristic M=O and M-O-M vibrations in the IR spectrum. These will be analyzed in detail later in section d on the basis of the DFT calculations. All compounds were also investigated in terms of their thermal behaviour by thermogravimetric analyses (TGA) in air. The salts with N-based cations (a, d) led to complete loss of the organic part,with formation of the mixed-metal trioxides Mx/6M'1-x/6O3 (x = 0, 1, 5, 6), with a relative good match between experimentally observed and theoretical mass losses upon warming up to 500°C. The TGA analysis of the salts with phosphonium cations (b, c) gave indication of phosphorus loss or not depending on the anion (P2O3 is volatile at the temperatures used in the experiments) but a precise stoichiometry could not be established.

Figure 1 .Figure 2 .

 12 Figure 1. Experimental (left) and simulated (right) isotopic patterns for the [Cp*MoxW6-xO18]-ions measured for compounds 1a, 2b, 3b and 4c by electrospray mass spectrometry (negative mode) in acetone/methanol solution.

Figure 2 .

 2 Figure 2. ORTEP views of the [Cp*MMo5O18] -ions. (a) M = Mo in compound 1c; (b) M = W in compound 2c; (c) M = W in compound 2a. The ellipsoids are drawn at the 30% probability level and the Cp* H atoms are not shown.

(

  µ6-O) atom is drawn closer to the Cp*-bearing axial metal (Mo(4) in 1c and W(4) in 2c) and farther away from the opposite axial Mo(1) atom, the distances to the equatorial Mo(2) and Mo(3) atoms being intermediate. This type of distortion is however less pronounced than in the Cp*2Mo6O17 structure.44

Figure 3 .Table 3 .

 33 Figure 3. Experimental (left) and DFT calculated (right) IR spectra in the Mo-O stretching region for compounds 1a/I (A), 2a/II (B), 3a/III (C), 4a/IV (D).The range 700-1100 cm -1

  in the discussion. Five terminal metal-oxido (M'=O and M"=O) vibrations are expected and indeed found by the calculations (2A1+B1+E). The B1 band, corresponding to as(M'=O), is very weak due to small overall dipole moment, whereas the Etype pair has the highest intensity. The two A1-type vibrations correspond to the in-phase (stronger) and out-of-phase (weaker) elongations of the M'=O and M"=O bonds. Thus, only three major bands are essentially observed in this region. The calculated frequencies do not show significant trend as a function of the metal nature. The analysis of the bridging Mt-O-Mt vibrations (Mt = M, M' or M") is more complex, because of more extensive vibrational coupling not only among structurally different bonds but also with vibration of other nature (notably Cp bending modes). The five most representative bands (2A1+B1+E) are listed in Table

  analytical grade) was used as received. Sodium molybdate and tungstate dihydrates (Na2MO4•2H2O, M = Mo, W), tetrabutylammonium bromide (Bu4NBr), tetrabutylphosphonium bromide (Bu4PBr), tetraphenylphosphonium bromide (Ph4PBr) and N-butylpyridinium bromide (BuPyrBr) were purchased from Aldrich and used as received. Elemental analyses (C, H and N) were performed by the LCC Analytical Service Laboratory. The IR spectra were recorded on KBr pellets at room temperature with a Mattson Genesis II FTIR spectrometer and the data were processed with WinFirst software. The TGA measurements were carried out on a SDT Q600 V20.9 thermal analyzer. A quantity of each sample was placed into a nickel/platinum alloy crucible and heated at 0.83 K•s -1 under reconstituted air flow up to 500 K. An empty crucible was used as a reference.1 H and 31 P NMR spectra were on a Bruker Avance DPX-200 spectrometer. Mass spectrometry analyses were performed at the "Service Commun de Spectrométrie de Masse » of Université Paul Sabatier on a API 365 Perkin Elmer Mass spectrometer, in electrospray ionization mode and negative polarity. General synthetic procedure of Cat[Cp*MM'5O18]. The same procedure was used for all compounds. One equivalent of Cp*M2O5 (M = Mo or W) was dissolved in the minimum amount of methanol. In a second flask, ten equivalents of Na2M'O4•2H2O (M' = Mo or W) were dissolved in the minimum amount of water. Both solutions were mixed without apparent change. Aqueous 1 M HNO3 (18 equivalents) was then added to the mixture, resulting in a color change (the color depends on the M/M' nature, see equation 1

Table 1 .

 1 Relevant bond lengths [Å] for all structures. a

		1c (M(4) = Mo) 2c (M(4) = W) 2a (M(4) = W)
	Distances
	Mo(1)-O(1)	2.504(3)	2.512(4)	2.502(7)
	Mo(1)-O(6)	1.923(2)	1.914(4)	1.894(8)
	Mo(1)-O(7)	1.878(2)	1.899(4)	1.911(8)
	Mo(1)-O(11)	1.673(3)	1.669(5)	1.688(8)
	Mo(2)-O(1)	2.3285(19)	2.335(3)	2.343(7)
	Mo(2)-O(3)	1.908(2)	1.898(3)	1.911(8)
	Mo(2)-O(4)	1.9287(15)	1.933(3)	1.939(7)
	Mo(2)-O(5)	1.928(2)	1.931(4)	1.950(8)
	Mo(2)-O(6)	1.890(2)	1.894(3)	1.918(8)
	Mo(2)-O(21)	1.679(2)	1.678(4)	1.667(7)
	Mo(3)-O(1)	2.3391(19)	2.346(3)	2.338(6)
	Mo(3)-O(2)	1.870(2)	1.883(3)	1.912(8)
	Mo(3)-O(5)	1.933(2)	1.941(4)	1.947(8)
	Mo(3)-O(7)	1.928(2)	1.908(3)	1.898(9)
	Mo(3)-O(8)	1.9317(15)	1.937(3)	1.957(7)
	Mo(3)-O(31)	1.674(2)	1.671(4)	1.683(7)
	M (4)-O(1)	2.109(3)	2.104(4)	2.136(7)
	M(4)-O(2)	1.974(2)	1.958(3)	1.925(7)
	M(4)-O(3)	1.917(2)	1.931(3)	1.934(8)
	M(4)-Ct(1)	2.092(2)	2.085(2)	2.105(4)

  For the definition of the symbols used, see Scheme 1. [b] From ref.

	M-Cp	2.08(3) 2.092(2) 2.150 2.105(4) 2.085(2) 2.163 2.139 2.151
		1.65(2)	1.674(2)	1.688	1.667(7)	1.678(4)	1.687	1.701	1.701
	M'=Ot	1.67(2) 1.68(2)	1.679(2)	1.688 1.688	1.675(7) 1.676(7)	1.671(4)	1.687 1.687	1.701 1.701	1.701 1.701
		1.68(2)		1.688	1.683(7)		1.687	1.701	1.701
	M"=Ot	1.62(2) 1.673(3) 1.681 1.688(8) 1.669(5) 1.681 1.696 1.696
	M-Oc	2.14(1) 2.109(3) 2.074 2.136(7) 2.104(4) 2.067 2.127 2.103
		2.33(1)	2.328(2)	2.378	2.338(6)	2.335(3)	2.369	2.365	2.369
	M'-Oc	2.33(1) 2.33(1)	2.339(2)	2.382 2.383	2.338(6) 2.343(7)	2.346(3)	2.372 2.379	2.366 2.367	2.370 2.370
		2.34(2)		2.386	2.353(6)		2.380	2.368	2.375
	M"-Oc	2.48(1) 2.504(3) 2.609 2.502(7) 2.512(4) 2.629 2.585 2.602
		1.87(2)	1.917 (2)	1.886	1.921(7)	1.931(3)	1.910	1.915	1.918
	M-Ob(M')	1.88(2) 1.91(2)	1.974 (2)	1.890 1.995	1.925(8) 1.934(7)	1.958(3)	1.921 1.946	1.923 1.947	1.927 1.940
		1.94(2)		2.009	1.952(8)		1.964	1.962	1.951
		1.90(2)	1. 870 (2)	1.881	1.906(8)	1.883(3)	1.910	1.914	1.921
	M'-Ob(M)	1.92(2) 1.93(2)	1. 908 (2)	1.881 1.990	1.911(8) 1.912(8)	1.898(3)	1.927 1.951	1.926 1.947	1.930 1.942
		1.95(2)		1.990	1.918(8)		1.962	1.954	1.950
		1.90(2)	1.928(2)	1.837	1.933(8)	1.931(3)	1.924	1.924	1.926
		1.92(2)		1.838	1.936(7)		1.924	1.925	1.926
		1.93(2)	1.929(2)	1.843	1.939(7)	1.933(4)	1.924	1.926	1.927
	M'-Ob(M')	1.94(2) 1.94(2)	1.932(2)	1.845 2.043	1.947(8) 1.949(7)	1.937(4)	1.925 1.935	1.929 1.929	1.927 1.930
		1.95(2)		2.044	1.950(8)		1.935	1.931	1.931
		1.96(2)	1.933(2)	2.051	1.957(8)	1.941(3)	1.936	1.932	1.932
		1.96(2)		2.059	1.957(7)		1.937	1.934	1.933
		1.94(2)	1.890(2)	1.875	1.898(9)	1.894(3)	1.902	1.906	1.910
	M'-Ob(M")	1.95(2) 1.97(2)	1.928(2)	1.883 1.968	1.913(8) 1.918(8)	1.908(3)	1.912 1.934	1.913 1.930	1.916 1.928
		1.97(2)		1.976	1.919(8)		1.950	1.941	1.935
		1.84(2)	1. 878(2)	1.875	1.894(8)	1.899(4)	1.895	1.903	1.907
	M"-Ob(M')	1.85(2) 1.86(2)	1. 923(2)	1.880 1.963	1.902(8) 1.911(8)	1.914(4)	1.909 1.932	1.913 1.929	1.913 1.926
		1.87(2)		1.975	1.915(8)		1.943	1.937	1.932
	[a]
			1c	I	2a	2c	II	III	IV
	M		Mo	Mo	W	W	W	Mo	W
	M'=M"		Mo	Mo	Mo	Mo	Mo	W	W

  ). The mixture was left stirring at room temperature for two hours. The bromide salt with the desired (> 3 equivalents) dissolved in water was then added to the solution leading to a precipitate of the expected compound. The product was filtered off, washed with small portions of water, methanol and diethylether and finally dried under vacuum at 70°C.

	(calcd): 26.5 (25.4). 1 H NMR (DMSO-d6): δ 2.4 (s, 15H, Cp*), 7.6-8.1 (m, 15H, Ar). 31 P
	NMR (DMSO-d6): δ 23.6. MS: m/z = 1525.3 (theor. 1524.7), [Cp*W6O18] -.
	Bu4N[Cp*Mo6O18], 1a. 92% yield. IR (ν, cm -1 ): 979sh, 957s, 796s. Anal. Calc. for
	C26H51O18NMo6: C, 25.1; H, 4.1; Found: C, 25.1; H, 4.1 TGA: formal loss Bu4NCp*; % exp.
	(calcd): 30.5 (30.4). 1 H NMR (DMSO-d6): δ 0.96 (q, 12H, Me), 1.35 (m, 8H, CH2), 1.59 (m,
	8H, CH2), 2.27 (s, 15H, Cp*), 3.18 (m, 8H, CH2). MS: m/z = 999.8 (theor. 999.5),
	[Cp*Mo6O18] -.
	Bu4P[Cp*Mo6O18], 1b. 92% yield. IR (ν, cm -1 ): 979sh, 967s, 916sh, 798s, 759sh. Anal.
	Calc. for C26H51O18PMo6: C, 24.8; H, 4.1. Found: C, 25.0; H, 3.8. TGA: loss of 4 Bu and
	Cp*; % exp. (calcd): 28.3 (28.9). 1 H NMR (DMSO-d6): δ 0.94 (q, 12H, Me), 1.46 (m, 8H,
	CH2), 2.20-2.30 (m, 16H, CH2), 2.27 (s, 15H, Cp*). 31 P NMR (DMSO-d6): δ 35.1.
	Ph4P[Cp*Mo6O18], 1c. 92% yield. IR (ν, cm -1 ): 967s, 878s, 796s, 760sh. Anal. Calc.
	for C34H35O18PMo6: C, 30.5, H 2.6. Found: C, 30.2; H, 2.8. TGA: formal loss of Ph4PCp*; %

exp. (calcd):

35.3 (35.4

).

1

H NMR (DMSO-d6): δ 2.27 (s, 15H, Cp*), 7.6-8.2 (m, 15H, Ar).

31

P NMR (DMSO-d6): δ 23.6. Single crystals of this compound could be grown from acetone.

Table 4 .

 4 Crystal data and structure refinement for all compounds.

	Compound	1c•(CH3)2CO	2a	2c•(CH3)2CO
	Empirical formula C37H41Mo6O19,P	C26 H51 Mo5 N O18 W	C37H41WMo5O18P
	Formula weight	1396.31	1329.23	1484.22
	Crystal system	Orthorhombic	Monoclinic	Orthorhombic
	Space group	Pnma	P21/n	Pnma
	a, Å	14.8994(7)	11.8997(5)	14.9089(5)
	b, Å	13.8507(7)	25.9172(13)	13.8674(5)
	c, Å	21.2967(8)	12.7160(6)	21.3702(8)
	=	90	90	90
	=	90	90.405(5)°.	90
	 =.	90	90°	90
	Volume, Å 3	4394.9(3)	3921.6(3)	4418.2(3)
	Z	4	4	4

Computational Details. The guess geometries for I and II were based on the crystallographically determined structures of 1a and 2a, replacing all Cp* CH3 groups by H atoms.

Acknowledgments.

We are grateful to the CNRS, to the IUF (Institut Universitaire de France) and to Celal Bayar University (FBE 2010/082) for financial support, and to CINES (Centre Informatique National de l'Enseignement Supérieur) and CICT (Centre Interuniversitaire de Calcul de Toulouse, project CALMIP) for granting free computational time. We also thank the French Embassy in Ankara and the Syndicat Mixte of the "Communauté D'agglomération Castres-Mazamet" for financial support of the Ph.D. thesis of G.T.C.


Nadja Gaudillière

Justin Dirrenberger

Olivier Baverel

Cyrille Sollogoub

Additive Μanufacturing

The invention of the computer and of the numerous digital technologies that followed started what is now called by some the Digital Turn [START_REF] Carpo | The Digital Turn in Architecture 1992-2012[END_REF] . Although the turn is in some ways already behind us (for example through the omnipresence of computers, cameras and various digital technologies in our lives), huge potentials still remains unexploited, and a large part of the history of digital still has to be written. In particular in architecture, the Digital Turn lies ahead of us. Whereas already deeply discussed and explored by a few architects, researchers and historians of architecture, digital technologies are not yet widely applied in most of the world's architectural production.

The Digital Turn in architecture does not only concern technical matters, but also a conceptual change. The emergence of new digital fabrication techniques has been changing the fabrication of architecture, but the emergence of computer programming has also an impact on the conception of architecture. Architecture must open to the potential of digital and bond with otherdisciplines in order to accompany the Digital Turn. The work presented in this paper has involved architects, structural engineers and material scientists, in order to explore the potentials of additive manufacturing in architecture, particularly in the production of structural assembling systems.

Additive Manufacturing

Rise of additive manufacturing

Since its development in the 1980s, the use of additive manufacturing has kept increasing, and it is now considered as one of the most promising production technologies. Over the past years, several additive manufacturing processes have been developed and improved: extrusion and deposition, laser sintering, photopolymerization, etc. The number of printable materials has also risen to encompass most engineering materials: metals and alloys, thermoplastic and thermosetting polymers, cement, ceramics, paper and wood, even sugar and lately carbon ber. Additive manufacturing is now used in many elds of application: aerospace and automotive industries, medical engineering, design, architecture, producing objects as diverse as motors, personalized smartphone cases, furniture, moulds, plaster casts, clothing, jewellery, and now buildings… To illustrate the current state of possibilities and applications of additive manufacturing, two examples can be detailed.

Everyday Additive Manufacturing

One of the most widespread additive manufacturing processes is the extrusion and deposition of a molten material, in a large majority of cases thermoplastic polymers. The material is deposed in layers, and the accumulation of solidi ed material eventually forms the designed object. This technique has been used to develop a ordable, easily usable 3D printers for everyday use, such as the famous Makerbot Replicator 2 machine. Given the price range of these printers (US$500 to US$2000 depending on the level of pre-assembling of the machine's parts), it is available to many people wishing to use additive manufacturing at home, in order to produce various everyday objects. Most of the time, les for these objects are found on the Internet, and therefore this kind of everyday additive manufacturing does not necessarily involve any form of creativity or innovation. It is used to produce mass customisation objects, such as smartphone cases, small items to repair other home objects, or toys, jewellery or other small everyday objects.

Industrial Additive manufacturing

The second example, that can be deepened further in order to highlight the current state of additive manufacturing, is the case of selective laser sintering and/or melting of granular materials, which has been thoroughly developed for aerospace applications with speci c powder metal alloys. Just like in the case of extrusion and deposition and other additive manufacturing techniques, the accumulation of solidi ed material eventually forms the object. The machines required by such process are much more advanced and expensive printers, and the applications are mostly reserved to industrial additive manufacturing. It is used for academic and industrial research and production in elds such as aerospace or automotive industries, mainly for prototyping, but more and more for production of items such as engine parts or moulds for foundry.

A potential intermediary

This duality in application leads on the one hand to mediocre materials and printers producing mostly basic and non-challenging objects, mainly dedicated to mass customization, and on the other hand, to top-of-the-range printers and tailored materials producing expensive items in response to particular industrial problems. An intermediary between those two ways of using additive manufacturing could exist, it might be possible to use the average printers and materials usually reserved to mass customization to produce viable solutions for some research or industrial problems. In particular, that might be potential architectural applications for this kind of additive manufacturing. Despite the massive research conducted in order to increase the size of 3D printers, resulting in objects being the size of a building, most available printers still produce small or object scale items. This paradox of small 3D printing versus large architectural scale might be overcome by considering structural assembly. As a matter of fact, many types of architecture rely on structural part assembly, and this is one of the applications where the intermediary, architectural potential of additive manufacturing resides. While making an inventory of existing structures such as nexorades [START_REF] Baverel | Nexorades Based on Regular Polyhedra[END_REF] , gridshells [START_REF] Hennicke | Gitterschalen Grid shells[END_REF] and geodesic domes [START_REF] Popko | Geodesics[END_REF] , it appeared to us that gridshells could be an ideal case study, given the issues still remaining when building such structures, especially regarding the particular case of structural parts connections.

Gridshells: De nition and Speci cations

Historical overview

The rst gridshell was built by Frei Otto, a german architect and structural engineer, in 1975 in Mannheim, Germany [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF] , with the help of the Arup engineering team (Fig. 1). Frei Otto was looking for a way to build a large span, light structure, and he conceived the double curved wooden grid structures, shown in Fig. 1. The rst built example is the Mannheim Multihalle gridshell, initially a pavilion for the horticultural show held that year in the city. Only a few other examples of built gridshells exist in the world, such as the Japan Pavilion at the Hannover Exhibition in 2000, by Shigeru Ban 7 , and the Weald and Downland Museum gridshell, in 2002, by Edward Cullinan Architects 8 , both with the help of Buro Happold structural engineering team. Except for a particular family of metallic grids which is not considered in this paper, the rst gridshells were built with a grid consisting of wooden laths, but recent gridshells built by the Navier laboratory experimented the possibility of using composite glass bre reinforced polymer (GFRP) tubes for the structure, such as the Creteil and the Solidays gridshells [START_REF] Douthe | Gridshell structures in glass bre reinforced polymers[END_REF][START_REF] Douthe | Gridshell in composite materials: Towards wide span shelters[END_REF] . Despite the progress made in the knowledge we have of gridshells and in our process of construction of these structures during the building of these few examples, several issues remain in the conception and construction of gridshells, particularly regarding their connections.

Characteristics

Gridshells' most spectacular characteristic lies in their shaping process. As it can be seen in Fig. 2, the grid is assembled at by joining two perpendicular layers of laths and then deformed step-by-step unto its de nitive position by raising the grid and bringing its borders to their place on the ground. The grid can either be assembled at on the ground and lifted, as it was made for most gridshells, or assembled at on top of a sca olding with its borders lowered, as it was made for the Weald and Downland Museum gridshell [START_REF] Miyake | Paper in Architecture[END_REF] . In some cases, the gridshell is made of four layers of laths instead of two, for structural improvement and for a better control of the nal shape: it is a quite di cult process to master the deformation in order to bring the grid to its de nitive form. Once in place, the structure is triangulated either by a third, or fth, direction of laths, or by cables. Sometimes the triangulation is provided by the covering of the structure, in this case made of plates, as for the Weald and Downland Museum gridshell. In other cases, the cover is made of a tensile membrane and the triangulation has to be provided by one of other aforementioned solutions, as for the Mannheim gridshell. Initially, gridshell shapes stem from dynamic relaxation [START_REF] Day | An introduction to dynamic relaxation[END_REF] , a method used to nd surfaces with a geometry such that all forces are at a state of equilibrium. Given this property, these are ideal shapes in term of stress and load transmission, but it connes gridshells to particular shapes depending on this form-nding method. In the last years, several form-nding methods have been developed, so that now almost every shape can be approximated and studied to be turned into a gridshell: these structures have now the same freedom in terms of shape than any other. Once the desired surface obtained through one of the now various existing form-nding methods, it has to be meshed, in order to obtain the grid that will form the structure. The usual and basic method is the so-called compass method, proposed by Frei Otto [START_REF] Hennicke | Gitterschalen Grid shells[END_REF] . As it can be seen in Fig. 3, the shape is made of two main curves along the surface, crossing each other in one point, and a series of circles at each intersection between the curves and the circles. A net of points, corresponding to the intersections of the gridshell laths grid is therefore obtained.

Speci cations

The strength of the structure, and therefore the possibility to have very large spans, comes from the doubly-curved shape. As their name indicates, gridshells are wooden grids behaving like shells, with stress distributed on the whole structure. Therefore, we have a good knowledge of the structural behaviour of the global structure of a gridshell. However, uncertainty remains regarding the mechanics of gridshells: the exact determination of the stress state exerted on each node remains challenging, rst during the deformation of the grid, and then during the life of the structure. Nevertheless, a conservative worst-case-scenario estimate of the stress state within the nodes can be given, thus yielding mechanical speci cations to be respected when design of an assembling system. We also established a list of required geometrical speci cations for a gridshell node to be fully functional. In order to achieve the peculiar installation of a gridshell structure, the nodes connecting the two layers of laths must allow them some movements; some degrees of freedom must be free, whereas some other must be prescribed to prevent unwanted displacements. First, the node position, along the two laths it connects, must not vary (Fig. 4). The deformation of the grid would not take place properly with such translations. Furthermore, the form of the grid being pre-de ned, it would change it if the laths were not well fastened together, and the nal shape of the structure would be impossible to obtain. Secondly, while the node cannot translate along the lath, it must be able to rotate around it, in order to allow the necessary movement freedom for the grid deformation. Thirdly, for the same reason, both laths must be able to rotate around the node axis.

Finally, as mentioned earlier, since gridshells can be made of 2, 3, 4 or 5 (or even more) layers of laths, various types of triangulations and covers can be thought of. The ideal connection would take these elements into account, and its geometry should allow an easier implementation and adaptation of the assembling system. The connection mechanism and function should also be reversible, so that gridshells could be assembled and disassembled.

Connections

The connections in various examples gridshells considered so far are all di erent (Fig. 5). The Mannheim gridshell node is made of a metal part piercing through the wood layers and maintaining them together. The main problem of this connection is that by removing matter in order to place the node, it weakens the structure at its most loaded locations: intersections. In Hannover, the connection is as peculiar as the gridshell: the structure is made of cardboard, temporary connections were used during the shaping process of the structure and then replaced by nodes made of adhesive rope. For the Weald and Downland Museum, the node is made of metal platforms placed between each lath layer and fastened together by bolt and screws. Although allowing the gridshell to be shaped, the geometrical speci cations are obtained in a very approximate way. The Creteil and Solidays gridshells build by the Navier Laboratory use a di erent material for the laths, which therefore changed shape: initially wooden rectangular struts, the laths are in this case GFRP round tubes. This change in strut section geometry allowed the use of a particular object as connection: steel sca olding elements. These are made of two openable rings with a circular junction enabling the rotation of the two tubes around the axis of the node. Although this connection is much closer to the geometrical requirements than the others, it still exhibits drawbacks that could be overcome with a 3D printed connection. The geometrical requirements are not perfectly met, and could therefore be improved by designing a new, tailored geometry rather than using an object initially designed for another purpose.

Additive manufacturing is making any geometry possible, even enabling to produce objects unbuildable with any other technology, such as a ball in a closed hollow sphere. The secondary requirements mentioned earlier (adding laths layers, xing the triangulation and the cover) are not taken into account by the sca olding element, and by designing a new node, one could anticipate these steps in order to ease the implementation and speed up the process. Optimized solutions for each aspect of the node can be conceived, prototyped and implemented with additive manufacturing in order to improve the gridshell assembling system. Last but not least, the sca olding elements are extremely heavy when compared to the weight of the lath structure. Each of these connections weights about 1 kg, which represents up to 2/3 of the total weight of the structure. A change of material and a new design could solve this issue and considerably relieve the structure. Gridshells seem like a challenging case study: speci c problems still remain unsolved regarding their connections, and yet these problems could be solved by considering the geometric possibilities o ered even by, even low-cost, additive manufacturing, possibilities o ered by no other technologies until now and the characteristics of the materials usually shaped by this kind of printers.

Material Choice and Manufacturing Process

Material selection approach

This leads us to the second part of this study: which material would be relevant for the gridshell assembling system? The previous connections, as explained, were made of steel. As a material, steel exhibits high performances, as shown by its very common application in every eld of production, especially when considering structural connections. But as said, a particular manufacturing process was chosen, dictating us a preselection of materials, and steel was not a part of this pre-selection. Steel was therefore not an option. Moreover, some of the problems raised by the steel connection could be resolved by changing the material, in particular with regards to its weight. The geometrical requirements of the structure could easily be solved by the use of additive manufacturing. Although steel is one of the most e cient materials available, it is relatively expensive to manufacture this way.

A rst selection had been made by the choice of the manufacturing process, but a large range of available materials still remained. By applying a materials selection approach [START_REF] Hennicke | Gitterschalen Grid shells[END_REF] to the materials which can be processed through 3D printing, the most adequate materials were short-listed. This rst materials performance map, shown on Fig. 6, corresponds to thermoplastic polymers spread according to their Young modulus (Y-axis, in GPa) versusspeci c price (X-axis, in US$.kg -1 ).

From this map, several materials are selected because of their good performance and are gathered in Table 1. Comparison is made for their supposed price, real market price, and their initial avaliable shape. Polylactic Acid (PLA) and Acrylonitrile Butadiene Styrene (ABS) both stand out, most notably because of their availability as strand, the shape needed for common 3D printers, meaning there is no need to reshape the feedstock before printing products: it saves money by reducing the production time.

Performance tests

The typical performance of both ABS and PLA suit our needs for the structure nodes. However, ABS and PLA test specimens are usually not manufactured by 3D printing. Hence, mechanical characterization of 3D printed samples was conducted in order to compare their performance to, typically injected, ABS and PLA specimens. The orientation of the layer construction is also of prime importance, as depicted on Fig. 7, showing the tensile behaviour of 3D printed samples for 3 di erent orientations: 0°, 45° and 90°.

These results show that the best orientation for the layers is 0°, meaning that a node produced with such a machine should be optimized to be printed with the layer construction direction oriented along the direction of maximum stress. Fig. 7 compares also the mechanical performance of the same sample (same orientation, same material), but printed with two di erent machines: a Makerbot Replicator and a Stratasys Mojo [START_REF] Ashby | Material selection in mechanical design[END_REF] . The most remarkable feature in these results is the presence of ductility for the Mojo specimen, i.e. irreversible plastic deformation, as expected from a thermoplastic polymer sample, whereas it vanishes for the Replicator-printed specimen. Those tests give an idea of the performances of PLA when shaped by a 3D printer as well as an insight on the inherent variability of material properties depending on the printer considered. When compared with the design requirements for the assembling system/ gridshell connection, both 3D printed ABS and PLA are appropriate, whether shaped by a mid-range or low-cost 3D printer, respectively the Mojo and the Replicator print- ers. This capacity, along with the geometrical possibilities o ered by additive manufacturing makes this technology very promising for solving problems posed by gridshell assembling.

Designing the Assembling System

A node divided in parts

In the second part of this paper, several geometrical speci cations for gridshells, that need to be respected while conceiving a node, were presented. In order to properly inspect each geometrical problem and nd a solution to it, they were separated, as well as the parts of the connection responsible for each speci cation. Several intuitive potential solutions were investigated. Although this work is still in progress, some of these solutions are currently being tested. Designing the node that way, in several parts, allowed us to solve, or partly solve several problems from the beginning. The idea was to conceive a half-node, locked around one lath and connectable to other half-nodes. One could therefore stack them easily, and use the assembling system for as many lath layers as wanted. The triangulation and cover xation could also be solved, by adding one last piece on top of the stacked half-nodes, a di erent part that would allow the necessary xations of the cover and triangulation in a better way that what was previously proposed, avoiding the wear and possible failure of the cover because of the nodes rubbing against it.

First requirement: No translation

The rst requirement, as explain in the second part of the paper, was to prevent the translation of the node along the lath, as well as to allow its rotation around it. In order to achieve this, two di erent systems were considered, either internal, or external. The external one would consist of two rings, xed on each side of the node around the lath and preventing it to glide. We explored several options for the material these stop rings could be made of, retaining two solutions: steel or heat-shrinkable sheath (the same as the one used for cables). The node itself, trapped between the stop rings, would be free to rotate and to translate, but kept from gliding by the rings. This solution has already been tested on the Navier laboratory gridshells, with metallic rings and has proved itself e cient. But it is an additional step in the implementation of the node that could be avoided by the invention of an internal system. This system could for example consist of a double ring, with a bearing system in-between. The rst ring is xed to the lath in a way preventing it from moving, whether by rotating or translating. The second ring is free of rotating around the rst, but is kept from translating by the bearing system. This solution would be quicker to implement, but more complex and longer to design and using more material, therefore more expensive; and a question still remains: how to guarantee the adhesion of the rst ring to the lath.

Second requirement: Junction of the parts

The second issue in the designing of the node was the junction of the node parts (two or more, as explained). Three di erent solutions were considered, all based on the same plug system: each half-node would possess a male and a female part, and could be plugged to two other half-nodes, making the stacking of elements and the accumulation of lath layers possible. For a good answer to the stress, the junction had to be as short and large as possible. It also had to allow the rotation of both laths. The rst plug was a plug to clip, with a gap allowing it to shrink and then to expand. The other part of the plug was rst slim and then larger. When implemented, the shrinking and then expanding of the plug would allow it to be clipped inside the other part and then locked in it. Despite a very easy implementation, this plug was the longest junction of the three and the chosen material's sti ness made the allowance to shrink more di cult to obtain. The second plug was made of several side wings and the host to this plug was designed to match the side wings, organised in a particular way: it could only be plugged or separated in one position, and in case of a su cient rotation to place each wing in front of an exit, the organisation of the wings would keep it from falling apart. But despite this technique, if the plug rotated enough to be in a dangerous position, only the hooks would be supporting the stress, and the plug could break much more easily. In order to know which angle between the wings and therefore which number of wings would be the less dangerous and the most e cient, the angles formed by the laths of two of the Navier laboratory gridshell were studied. As shown in Fig. 8, three wings seemed to be the safest solution for this kind of plug, but it is still a fragile option. The third junction considered to link the di erent parts of the node was made of a bolt and screw system. To make the rotation of the laths possible, the two parts would not entirely be screwed together: the pressure of the laths would make the junction a little further screwed or a little unscrewed, therefore authorising this necessary movement. Although tests are necessary to make sure it would work even in torsion, and although it is a little longer to implement, this system is still easy to put in place and the variants possible in the screw thread would allow several improvements: a solid enough screw thread to support the stress without breaking, a shorter junction.

Third requirement: Implementation and fastening of the nodes

The last question addressed in this work is about how the half-node would be placed on the laths, and then maintained. In the previous connection, it was necessary to use a bolt and screw system, to x the node to a given position on the lath and force it to stay in place. It had the disadvantage of damaging the laths if not protected, and to be rather empiric: a compromise between xing the node in position well enough to keep it from sliding and avoid crashing the lath by doing so had to be found for each junction. In this study, one goal of the node was to guarantee the set position of the node without having to squeeze it on the lath, so that is would not need a bolt and screw system. The node could be placed on the lath by clipping it on it, and then fastened with a simple closing system, inspired by a bike saddle xing system for example. We could either design a connection with a closing system already part of it, a simple one that would only need to be clipped during the implementation of the node, or design a connection with a host for an external closing element, such as a bike saddle xing or a zip-tie.

Further research

The multiple solutions considered in this work for the assembling system are gathered in Table 2 for comparison, in order to determine if one of them really stand out and if some of them tted better together than others. The next step will consist in producing several node prototypes resulting of several combinations, testing and comparing some of the solutions proposed here. Finally, the assembling system will have to be implemented on an actual 1:1 scale gridshell structure.

Conclusion

This project results from the collaboration of architects, structural and material scientists. It consists in a multidisciplinary, collective design method, based on the deep relations between material selection, process selection, as well as geometrical and mechanical requirements. Our contribution illustrates the very promising possibilities of 3D printing as a means to solve various issues and innovate in architecture, structural engineering and materials science.

Fig. 1

 1 Fig. 1 Mannheim Gridshell.

Fig. 2

 2 Fig. 2 Shaping process.

Fig. 3

 3 Fig. 3 Compass method.

Fig. 4

 4 Fig. 4 Geometrical speci cations.

Fig. 5

 5 Fig. 5 Previous connections.

Fig. 6

 6 Fig. 6 Material selection diagram.

Fig. 7

 7 Fig. 7 Orientation of the layers and tensile test characterization.

Fig. 8

 8 Fig. 8 Junction of the parts: side wings solutions.

Table 1

 1 Materials comparison.

Table 2

 2 Assembling system solutions.

This potential raises the question of the role additive manufacturing already has today, and of the role it will have in the next years. Since its appearance, this technology has been presented as a key item to a new industrial revolution, and the many breakthroughs in the materials or processes used and in the rst applications found for additive manufacturing seem to guarantee, even if not an industrial revolution, a leading role for additive manufacturing in prototyping and production processes at every scale of fabrication. Along with the potentials of this manufacturing process left to explore and the applications left to invent are also issues to be solved, such as nding a cleaner, more respectful to the environment, way of printing.