Jamel Saadaoui
email: jamelsaadaoui@gmail.com.

Global Imbalances: Should We Use Fundamental Equilibrium Exchange Rates?

Keywords: JEL Classification: F31, F32, C23 Global Imbalances, Fundamental Equilibrium Exchange Rates, International Monetary Cooperation, Global Recovery JEL Classification: F31, F32, C23 Global Imbalances, Fundamental Equilibrium Exchange Rates, International Monetary Cooperation, Global Recovery

. This approach is often labelled as normative mainly because the return to the equilibrium is not described in the model. If the FEER is not related neither in the short nor in the long to the real exchange rates, we see no clear justification to intervene in foreign exchange markets based on these equilibrium rates. In this case, the FEER is a normative approach and should not be used to reduce global imbalances. This paper provides empirical evidences robust to cross-sectional dependence that the FEER is related to real exchange rate in the long run and thus could be a useful tool to prevent the resurgence of large global imbalances and associated risks.

Introduction

As witnessed by the evolution of current account balances and net foreign assets, the reduction of global imbalances observed during the climax of crisis is incomplete. Indeed, current account imbalances in flow have been reduced with the global slowdown and the collapse of the world trade in 2009. However, these evolutions of current account imbalances have not been sufficient to reduce net foreign assets positions in stock. After the climax of the crisis, global imbalances in stock (i.e. the net foreign assets positions) represent more than 15% of world GDP in absolute value as we can see in Fig. 1. Note: Data are preliminary for 2011. EUR surplus: Austria, Belgium, Denmark, Finland, Germany, Luxembourg, Netherlands, Sweden, Switzerland. EUR deficit: Greece, Ireland, Italy, Portugal, Spain, United Kingdom, Bulgaria, Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovak Republic, Turkey, Ukraine. Emerging Asia: Hong Kong S.A.R. of China, Indonesia, Korea, Malaysia, Philippines, Singapore, Taiwan province of China, Thailand. Oil exporters: Algeria, Angola, Azerbaijan, Bahrain, Republic of Congo, Ecuador, Equatorial Guinea, Gabon, Iran, Kazakhstan, Kuwait, Libya, Nigeria, Norway, Oman, Qatar, Russia, Saudi Arabia, Sudan, Syria, Trinidad and Tobago, United Arab Emirates, Venezuela, Yemen. Rest of the world: remaining countries.

As pointed out by [START_REF] Blanchard | Why) Should current account balances be reduced?[END_REF], the persistence of large current account imbalances and large net foreign assets positions is a threat for the world economy. Firstly, large current account imbalances increase the systemic risks as countries with large deficits can be subject to sudden stops and their macroeconomic consequences. Secondly, they increase political tensions as a number of countries, which are suspected of unfair competition with undervalued exchange rates, could be threatened by retaliatory measures. Thirdly, in the current context of weak growth in advanced countries, the perpetuation of export-led growth strategies in some emerging countries could be a menace for the global recovery.

Currencies realignments are still proposed to ensure global macroeconomic stability. These realignments are based on equilibrium rates derived from equilibrium exchange rate models. Among these models, we have the fundamental equilibrium exchange rate (FEER) model introduced by [START_REF] Williamson | Estimating Equilibrium Exchange Rates[END_REF]. This approach is often labelled as normative mainly because the return to the equilibrium is not described in the model. If the FEER is not related neither in the short nor in the long to the real exchange rates, we see no clear justification to intervene in foreign exchange markets based on these equilibrium rates. In this case, the FEER is a normative approach and should not be used to reduce global imbalances. This paper provides empirical evidences robust to cross-sectional dependence that the FEER is related to real exchange rate in the long run and thus could be a useful tool to prevent the resurgence of large global imbalances and associated risks. This paper is organized as follow. Section 2 presents a general framework of the FEER approach. Sections 3 focusses on the empirical results robust to crosssectional dependence. Section 4 concludes on the usefulness of the FEER approach to reduce global imbalances.

FEER Methodology

In the literature on equilibrium exchange rates, the FEER approach have several variants. We can quote [START_REF] Cline | Estimating consistent fundamental equilibrium exchange rates[END_REF], [START_REF] Jeong | Exchange rate misalignments at world and European levels: a FEER approach[END_REF] and [START_REF] Carton | Estimation of consistent multi-country FEERs[END_REF] for example. These variants differs on the type and size of modelling (general equilibrium, partial equilibrium, reduced form relationship), on the determination of the sustainable current account in the medium term (econometric estimates, judgemental assessment, arithmetic average) and on the trade elasticities (calibration to balance the trade model in volume and value, econometric estimates in a panel setting to ensure consistency of the world trade model).

In spite of all these differences, we present a general framework adapted to describe every FEER approach. We start with a simple current account model based on [START_REF] Clark | Exchange rates and economic fundamentals -a methodological comparison of BEERs and FEERs[END_REF]:

CA = -KA (1) CA = ntb + nf ar (2) ntb = b 0 + b 1 q + b 2 ydpot + b 3 yf pot (3) nf ar = f (q) ( 4
)
Where CA is the current account balance, KA is the capital account, ntb is the net trade balance, nf ar represents returns of net foreign assets, q is the real effective exchange rate (when q increases, we observe a real effective depreciation), ydpot is the domestic full employment output and yf pot represents full employments output of foreign economies.

A real effective depreciation and an increase of full employments output of foreign economies improve the net trade balance (b 1 > 0, b 3 > 0), an increase of the domestic full employment output deteriorates the net trade balance (b 2 < 0).

Combining Equations 1 to 4 gives:

CA * = f (q reer , ydpot, yf pot) = -KA * (5)
Where CA * is the sustainable current account in the medium term.

To determine the FEER, every approach have to solve the following equation:

q f eer = f (KA * , ydpot, yf pot) (6)
We obtain the fundamental equilibrium exchange rate (q f eer ), which realizes simultaneously the external and internal equilibrium for all trading partners.

In our approach, we use a two-step procedure to obtain the fundamental equilibrium exchange rate for each trading partners [START_REF] Jeong | Exchange rate misalignments at world and European levels: a FEER approach[END_REF]. Firstly, we use a partial equilibrium model of world trade for the main countries at the world level (US, China, Japan, Euro area, UK and the Rest of the World). We solve Equation 6 to obtain fundamental equilibrium exchange rates for these countries in a partial equilibrium model of 35 equations. Secondly, we use simple national model in which world demand and world price are exogenous for smaller economies. National estimates are linked with the estimates of the main countries at the world level 1 . In that case the misalignments (i.e. the difference between observed rates and equilibrium rates), written in differential logarithmic (r = dLogR = (Ri -R e )/R e ), are computed as2 :

r = 1 sx . b mx + ηm.di -ηx.d * (7)
Where b is the difference between the observed current account and the equilibrium one, as percentage of GDP, d and d * stand for internal and world demand in volume, also written in differential logarithmic, ηm and ηx are import and export volume elasticities, sx and mx are coefficients derived from the foreign trade model in which mark-up behaviours are allowed.

Concerning the determination of the sustainable current account in the medium term, following [START_REF] Chinn | Medium term determinants of current accounts in industrial and developing countries: an empirical exploration[END_REF], we regress the current account on several medium-term determinants of investment and saving behaviours. The consistency of current account targets is ensured by using the Rest of the World as a residual.

At the world level, the sum of current account targets expressed in the same currency is equal to zero. The trade elasticities of the world trade model comes from econometric estimates. These estimates are generally made in a panel setting to ensure that elasticities are mutually consistent3 . Although, there are several variants of the FEER approach in the literature on equilibrium exchange rates. This simplified framework contains the essential principles which are included in all FEER approaches.

Empirical Results

The purpose of this section is twofold. First, we estimate FEERs for seventeen industrialized and emerging countries (the United States, the United-Kingdom, the Euro area, Japan, Korea, China, Brazil, India, Mexico, Argentina, Chile, Colombia, Indonesia, Malaysia, Philippines, Thailand and Uruguay) over the period 1982 to 2007 with the methodology described above4 . Secondly, we test empirically the usefulness of the FEER approach to reduce global imbalances.

After the estimation of FEERs for these seventeen countries over the period 1982-2007, we test the following long-run relationships:

reer i,t = α i + βf eer i,t + µ i,t (8) f eer i,t = δ i + θreer i,t + ε i,t (9)
Where f eer is the fundamental equilibrium exchange rate and reer is the real effective exchange rate5 . Variables in minuscule represents natural logarithms.

When the time dimension (T = 26 in our sample) is superior to the cross-section dimension (N = 17 in our sample), we can test the existence of cross-sectional dependencies with a Lagrange multiplier test as pointed out by De [START_REF] De Hoyos | Testing for cross-sectional dependence in panel-data models[END_REF]. Consequently, we apply an LM test on an ARDL(1,1) specification with fixed effects as in [START_REF] Persyn | Error-correction based cointegration tests for panel data[END_REF]. To test cointegration, we use the panel and the "mean group" statistics suggested by [START_REF] Westerlund | Testing for error correction in panel data[END_REF]. The existence of negative error-correction term is taken as proof for cointegration. To take into account cross-sectional dependence, critical values need to be obtain through bootstrapping. As we can see in Table 3, variables are cointegrated. [START_REF] Persyn | Error-correction based cointegration tests for panel data[END_REF] for the details.

The previous results have established that the variables are integrated and cointegrated in presence of cross-sectional dependence. Now, we use a Cross-Sectionally augmented pooled Mean Group (CPMG) estimator introduced by Pesaran ( 2006) and implemented recently by [START_REF] Mohaddes | Commodity price volatility and the sources of growth[END_REF] to estimate the long-run relationships in presence of cross-section dependence. In this approach, we augment the PMG estimator [START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF] with cross sectional average of independent and dependent variables in order to capture the common factors or the heterogeneous time effects.

More precisely, we start with the ARDL(1, 1) model as specified in Equation 10:

reer i,t = δ 0i + δ 1i f eer i,t + δ 2i f eer i,t-1 + λ i reer i,t-1 + u i,t (10)
The error correction equation yield:

∆reer i,t = φ i (reer i,t-1 -θ 0i -θ 1i f eer i,t ) -δ 2i ∆f eer i,t + u i,t (11)
Now, we assume that the error term u i,t follow multi-factor error structure:

u i,t = γ i f t + ε i,t (12)
where f t is a factor of unobserved common shocks. The error terms dependencies across individuals are captured by f , whereas the impacts of these factors on each country are governed by the idiosyncratic loadings in γ i .

By using Equations 10 and 12 and by averaging across i, we obtain:

reer t = δ0 + δ1 f eer t + δ2 f eer t-1 + λreer t-1 + γf t + εt (13)
Where the variables with a bar denote the simple cross section averages of the corresponding variables in year t. The common factors can be captured through a linear combination of the cross-sectional averages of the dependent variable and of the regressors:

γ i f t = -c i δ0 -c i δ1 + δ2 f eer t + c i 1 -λ reer t-1 + c i ∆reer t + c i δ2 ∆f eer t (14)
where c i = γ i γ . Replacing Equations 12 and 14 in Equation 11 yields the error correction equation:

∆reer i,t = φ i reer i,t-1 -θ 0i -θ 1i f eer i,t -a * i reer t-1 + b * i f eer t -δ 2i ∆f eer i,t + c i ∆reer t + c * i ∆f eer t + ε i,t (15)
Where

φ i = -(1 -λ i ); θ 0i = δ 0i -c i δ0 /(1 -λ i ); θ 1i = (δ 1i + δ 2i )/(1 -λ i ); a * i = c i 1 -λ /(1 -λ i ); b * i = c i δ1 + δ2 /(1 -λ i ); c * i = c i δ2 .
Since the CPMG estimator imposes long-run coefficients to be constant for all individuals, while it allows short run heterogeneity, the error correction models6 are written:

∆reer i,t = φ reer i,t-1 -θ 0 -θ 1 f eer i,t -a * reer t-1 + b * f eer t -δ 2i ∆f eer i,t + c i ∆reer t + c * i ∆f eer t + ε i,t (16) ∆f eer i,t = φ f eer i,t-1 -θ 0 -θ 1 reer i,t -a * f eer t-1 + b * reer t -δ 2i ∆reer i,t + c i ∆f eer t + c * i ∆reer t + ε i,t (17)
The results are presented in Tables 4 and6. The estimations give clear cut results. They clearly show a positive and significant long-run relationship between fundamental rates and observed rates in presence of cross-sectional dependence.

The results are robust to different groups of countries since the results in Tables 5 and7 are very similar to those for the entire sample.

As pointed out by [START_REF] Saadaoui | Exchange rate dynamics and fundamental equilibrium exchange rates[END_REF], in case of cyclical evolution of competitiveness (Equation 8), the half-life7 is equal to 3.8 years (3 years for emerging countries only). For structural evolution of competitiveness (Equation 9), the half-life is equal to 2.31 years (2.15 years for emerging countries only). When a country experienced a cyclical evolution of its competitiveness, it can slow the return to equilibrium in case of unfavourable evolutions hence a longer half-life8 . We provide robust empirical evidences that the FEER approach is related in the long run with observed rates even if the dynamic of real exchange rates is not explicitly described in the model. These results confirm the usefulness of the FEER approach to reduce global imbalances. The FEER approach should be used as a tool to prevent the return of large imbalances and associated risks.

Conclusion

The reduction of global imbalances observed during the climax of crisis is incomplete as witnessed by the evolution of net foreign assets positions. In this context, currencies realignments are still proposed to ensure global macroeconomic stability. These currencies realignments are based on equilibrium (or reference) rates derived from equilibrium exchange rate models. Among these models, we have the FEER approach introduced by [START_REF] Williamson | Estimating Equilibrium Exchange Rates[END_REF]. This approach is often labelled as normative as the exchange rate dynamic is not explicitly described in the model. We provide robust empirical evidences that fundamental rates are related in the long run with observed rates in presence of cross-section dependence. These empirical results are supportive of the usefulness of the FEER approach to reduce global imbalances and associated risks. A return of large imbalances could dampen the global recovery [START_REF] Blanchard | Why) Should current account balances be reduced?[END_REF].

In July 12, the IMF has adopted the FEER concept to strengthen its surveillance activities on bilateral and multilateral levels (International Monetary Fund, 2012).

In its Pilot External Sector Report, the IMF produce a set of deviations between real effective exchange rates and those consistent with fundamental and desirable policies for 28 economies. Even if this new decision does not create new formal obligations, it could be considered as a step in the recognition that members must have mutually consistent objectives to ensure global macroeconomic and macrofinancial stability.

Our empirical results are consistent with the IMF's decision as they support the usefulness of the FEER approach to reduce global imbalances. This decision could be preliminary step towards a larger discussion on the future of the international monetary system.

Fig. 1 :

 1 Fig. 1: Net foreign assets (in percent of world GDP)

Table 1 :

 1 Breusch-Pagan LM test of independence

	p-value

Source: author's calculations.

As we can see in Table

1

, we strongly reject the null hypothesis of cross-sectional independence. In order to take into account cross-sectional dependence, we implement panel unit root tests, panel cointegration tests and a new estimator which allow cross-sectional dependence. We use the CADF test introduced by

[START_REF] Pesaran | A simple panel unit root test in the presence of cross-section dependence[END_REF]

to test the unit root properties of the variables in presence of cross-sectional dependence. This test is robust to cross section dependencies by subtracting cross section averages of lagged levels in addition to the standard ADF equation. As shown by Table

2

, series are nonstationary I(1) series as a I(1) series achieves stationarity after first differencing.

Table 2 :

 2 Integration of the variables

		Level	First Difference
	feer	0.223	-4.271
		(0.588)	(0.000)
	reer	0.375	-3.083
		(0.646)	(0.001)

Source: author's calculations. Note: p-values in parentheses.

Table 3 :

 3 Cointegration of the variables

		Gτ	Gα	P τ	P α
	Equation (8)	-2.162	-6.414	-6.661	-4.005
		(0.001)	(0.013)	(0.070)	(0.031)
	Equation (9)	-2.481	-7.460	-9.441	-6.548
		(0.000)	(0.000)	(0.001)	(0.001)

Source: author's calculations. Notes: p-values in parentheses. p-values for cointegration tests are based on bootstrap methods, where 800 replications are used. See

Persyn and

Table 4 :

 4  Source: author's calculations. Notes: p-value for the Hausman test of homogeneity of long run coefficients. The symbol *** indicates statistical significance at the 1% level.

		Long-run coefficient (β)	z-stat / p-value
	CPMG	0.53***	7.38
	Error-correction term (φ)	-0.20***	-4.27
	Hausman test	1.13	0.77
	Number of cross-section	17
	Number of periods	26
	Number of observations	442

Table 5 :

 5 Long-relationship (Equation8) for emerging countries only

		Long-run coefficient (β)	z-stat / p-value
	CPMG	0.63***	7.77
	Error-correction term (φ)	-0.26***	-6.06
	Hausman test	3.79	0.28
	Number of cross-section	13
	Number of periods	26
	Number of observations	338
	Source: author's calculations. Notes: p-value for the Hausman test of homogeneity of long
	run coefficients. The symbol *** indicates statistical significance at the 1% level.

Notice that the FEER estimates are not obtained country-by-country but in a consistent framework by relying on a world trade model for the main economic areas.

R e is the equilibrium exchange rate.

See Jeong et al. (2010) for more details and complete description of the model and the methodology.

Estimates for emerging countries are presented and discussed in[START_REF] Aflouk | Exchange rate misalignments and international imbalances: a FEER approach for emerging countries[END_REF].

Source: Bank for International Settlements.

Causality tests have been conducted thanks to the Pooled Mean Group estimator. They clearly show that the causal relationship is bi-directional[START_REF] Saadaoui | Exchange rate dynamics and fundamental equilibrium exchange rates[END_REF].

The half-lives are computed by using the following formula: h = -ln (0.5)/ln (1 + |φ|) . They correspond to the number of periods for a deviation (from the long run equilibrium) to decay by 50%. Here, deviations correspond to misalignments.

See Saadaoui (2011) for a distinction between cyclical and structural evolutions of competitiveness.


Pierre Frankhauser
email: pierre.frankhauser@univ-fcomte.fr

The Fractalopolis model -a sustainable approach for a central place system

Introduction

Urban sprawl generates often diffuse settlement patterns where residential areas are localized far away from jobs as well as from retail centers and services. Hence periurbanization tends to increase the number and the length of trips and hence energy consumption and pollution. This negative impact has essentially been made evident by Newman and Kenworthy ([NR89]). Sustainable development requires better managing this dynamics what risks, moreover, undermining natural and agricultural areas. Numerous authors recommend coming back to compact cities in order to limit urban sprawl. However policies favoring the compact city turned out to be less efficient as expected. They induce an increase of the costs of housing, traffic congestion and reduce the accessibility of leisure areas ( [START_REF] Breheny | Urban compaction: feasible and acceptable?[END_REF]). A large number of households who choose these areas flee urban density since they prefer living in individual houses surrounded by a garden and enjoy a green and calm environment. Even if this lifestyle contributes to urban sprawl ( [START_REF] Hoffman | The historical origins and causes of urban decentralization in the united states. joint center for housing studies[END_REF]), such households will reject densifying ( [START_REF] Breheny | Urban compaction: feasible and acceptable?[END_REF], [START_REF] Gordon | Are compact cities a desirable planning goal?[END_REF], [START_REF] Fouchier | La densification: une comparaison internationale entre politiques contrastées[END_REF]). As pointed out Schwanen et al. ( [START_REF] Schwanen | Policies for urban form and their impact on travel: The netherlands experience[END_REF]) households tend to minimize the distance or the travel time for acceeding to their jobs ( [START_REF] Lm Mc Dowell | The new service class: housing, consumption and lifestyle among london bankers in the 1990s[END_REF]), but also to retail service centers ( [START_REF] S R Lerman | Location, housing, automobile ownership, and mode to work: A joint choice mode[END_REF]) or even leisure areas ( [START_REF] Guo | Residential location modeling: Accommodating sociodemographic, school quality and accessibility effects[END_REF]). Hence, in order to manage mobility, to reduce energy consumption and to prevent from fragmentation of build-up areas as well as of open landscape, we develop in the following a new planning concept. By improving accessibility to the different kinds of sites residents frequent more or less regularly, this concept aims to reduce the negative impacts of periurbanisation without rejecting it. Leisure areas of in the neighborhood of urbanized areas provide the desired quality of live for residents.

Several authors emphasize the importance of developing secondary centers or polycentric urban networks ( [START_REF] Fouchier | La densification: une comparaison internationale entre politiques contrastées[END_REF])what incites us to reflect about an planning apporoach which starts by reflcting on central place theory and which introduces a hierarchical system of cities according to the catchment areas of their public and private services. A critical reflection about the underlying spatial system leads us to introduce a multi-scale approach of planning which is inspired from fractal geometry [START_REF] Frankhauser | Fractal geometry for measuring and modeling urban patterns[END_REF]).

Towards a new central place model

The classic central place theory of Christaller ([Chr80]) introduces a system of cities consisting of different levels. The town belonging to a particluar level provide a certain type of public and private services. The system follows a hierarchical logic where few big cities offer high level services with large catchment areas including an important number of small and mean-sized cities. The lower level services are present in a huger number of cities whose catchment areas are smaller, finally very small cities ensure only supply in daily needs.

Globally the Christaller systems follows a "multiplicative logic" (geometrical series) for both the systems, the number of cities belonging to a certain supply level, and the mean population numbers of these cities. It is well-known that linking two geometrical series allows obtaining a hyperbolical distribution law relating in the present case the size of cities to their numbers of occurrence. This is in concordance with the rank-size distribution of cities, the so-called Zipf law ([D82], [F95]), but such relationships are often observed in economics, in nature and correspond also to fractal geometry [START_REF] Mandelbrot | Die fraktale Geometrie der Natur[END_REF]).

Figure 1: The Christaller system of cities (Christaller, 1933) A particular feature of Christallers' system is the fact that cities are uniformly dsitributed in space (cf. figure 1). Arlinghaus ([Arl85]) introduces the same logic in her paper aiming to introduce a fractal approach of central place theory. Let us however remind that this papers generates just the borderlines of the catchment areas of the towns belonging to the different hierarchical levels by means of generateor resembling to that of teragons or Koch curves.

The development of a settlement system is closely linked to the transportation network ensuring accessibility. Covering territories uniformly by a transportation network is rather expensive and urban sprawl is just one of the consequence of providing good accessibility all over space. Hence we may ask the question if in the context of regional and urban planning it is really the good solution referring to such a uniform distribution principle for central places or if it would be more useful to concentrate activities in the vicinity of some transportation axes. Such ideas are not new. In Northern European metropoles such concept have been proposed for managing urban sparwl as shows the famous finger plan for Copenhagen, he palm-plan for Hamburg or the development strategies developped for Berlin favouring an extension along the suburban railway axes (fig. 2). More recently the idea of a "transport oriented development" (TOD) refers to similar reflexions [START_REF] Calthorpe | The next American Metropole[END_REF]).

Figure 2: The urban growth of Berlin along suburban railway axes and the first steps for generating a multifractal Sierpinski carpet However we should be aware that only the TOD and the Berlin concept introduce really subcenters in the hinterland of the main center and thus refer to a central place hierarchy. By distiguishing facilities according to their attendance rate such a hierarchy seems indeed of interest. Indeed we may accept longer distances for acceeding to rarely used facilities, but daily needs should benefit from a good accessibility in order to reduce taffic flows and hence pollution, but also consumption of space, since the construction of transportation axes on a regional level consumes much space. E.g. it has been shown that in the fringe of the greater Paris region between 1987 and 1997, 1,4% of space has been consumed for housing or mixed use buildings, wheras more then 50% has been used for constructing roads [START_REF] Fp Tourneux | L'évolution de l'occupation du sol dans les franges franciliennes : des artificialisations concentrées plus qu'un étalement urbain ?[END_REF]).

This incited us to reflect about a planning concept which aims combining a central place hierarchy with a transport oriented logic. Hence the spatial organization of the proposed central place system follows strongly a hierarchical principle for the offer of facilities, similar to that of Christaller, but concentrates city development close to transportation axes which should rather be interpreted as public transportation network axes. In order to develop a spatial model the concept makes use of the scaling properties of fractal geoometry and refers more particularly to the logic of Sierpinski carpets 1 . The concept should be considered as descriptive and normative, serving as reference for planning purposes. We focus here on the theoretical framework, but the model can be applied to real world situation without loss of the basic principles. These principles have been implemented in a planning support system which provides GIS-facilities for developping and evaluating planning scenarios 2 . The basic principles based on a unifractal approach as well as concrete applications for planning have been presented up to now in several, mostly French or German written papers ( [START_REF] Frankhauser | Une approche multi-échelle pour le développement résidentiel des nouveaux espaces urbains[END_REF], [START_REF] Frankhauser | Maîtriser l'étalement urbain en recourant à la géométrie fractale -propositions méthodologiques[END_REF], [START_REF] Frankhauser | Von der strukturanalyse zur Raumplanung : Der Beitrag der fraktalen Geometrie zur Stadtgeographie[END_REF], [START_REF] P (dir | Vers des déplacements péri-urbains plus durables : proposition de modèles fractals opérationnels d'urbanisation[END_REF], [START_REF] Tannier | Simulation fractale d'urbanisation -mup-city, un modèle multi-échelle pour localiser de nouvelles implantations résidentielles[END_REF]), [START_REF] Czerkauer-Yamu | MUP CITY c : Multi-Skalare Planung als nachhaltiges Verflechtungsprinzip von bebauten Zonen und Freiraum[END_REF]).

The goal of the present working paper is to enlarge the model by using a multifractal logic (cf. also [START_REF] Czerkauer | A planning concept for a sustainable development of metropolitan areas based on a multifractal approach[END_REF]) and by focussing of the population distribution in the considered city system. But first we remind the the underlying logic of the concept which takes into account simultaneously different kinds of objectives:

• reducing of the travel length for acceeding to higher order facilities,

• respecting the diversity of social demand i.e. taking into account that certain types of households prefer living in a calm, low dense environment which allows good access fo green amenities,

• avoiding leapfrogging that lengthens the distances to acced to centers,

• avoiding the fragmentation of natural or agricultural areas.

For this aim the proposed spatial model uses iterative mapping procedures similar to those used for generating multifractal Sierpinski carpets. We assume that there exists a hierarchically structured system of central places according to different levels of services and commercial offer they provide. However, contrarily to Christaller, we will see later that towns belonging to the same hierarchical level have no longer the same population. Towns of a given level but lying close to a rather high ranked center are assumed to concentrate more population than those lying close to lower ranked centers. Figure 3 shows a system which corresponds to our logic. This system reminds by his hexagonal shape the Christaller scheme. However we see that the towns are concentrated in the vicinity of axes which may be interpreted as public transportation axes. Between the axes exists another connected spatial system which corresponds to undeveloped areas and which we interpret as natural and Figure 4: Generating stepwise a multifractal Sierpinski carpet agricultural reserves including rural settlements. By its shape this system avoids obviously a fragmentation of these rural and natural zones. On the other hand the peripheric branches of the transportation network are not linked. This helps reducing traffic flows between supply centers belonging to the same hierarchical level.

In order to introduce a quantitative modeling approach in a convenient way, we consider now a simpler model version which follows the same logic (cf. figure 4). Let us start by drawing a large sized square of a certain base length which we normalize to one (figure 4(a). We assume that the most important center of our system is localized in the centroid of the square. The surface (in grey) of the square represents in some sense the catchment (or attraction) area of our central place. We now introduce a generator, represented in figure 4(b) consisting of one central square of base length r 1 < 1 centered on the previously introduced first order center dotted in the figure. This square is surrounded by N = 4 smaller ones with base length r 0 < r 1 . Let us emphasize that the generator lies just within the initial square so that the outer corners of peripheral squares are identical to that of the initial square. Moreover no overlapping of squares is allowed. We assume now that this first step corresponds to the implementation of N = 4 second order centers localized in the centroids of the smaller squares. The surface of the squares correspond now to the direct catchment areas of these centers, or more precisely, they define the areas wherein we favour future development. This means that we assume that distances to the center from settlements lying within this area are acceptable. According to its level the central square has a bigger second level catchment area than the peripheral centers. In the next step we reiterate the procedure shown in figure 4 b. Each of the existing squares is then replaced by a smaller replication of the generator. According to our logic, we conserve of course the already generated first order and second order central places and we add third order central places lying within the direct catchment areas of the second order centers. Again these centers are localized in the centroids od the generated smaller squares. By the iteration process the reduction factors r 1 and r 0 are combined according to all possible combinations what yields e.g. for the second step:

r 1 • r 1 , r 1 • r 0 , r 0 • r 1 , r 0 • r 0
Of course since permutations are allowed we have

r 0 • r 1 = r 1 • r 0
This is the reason why the catchment area of the second order centers is the same then that of the third order centers belonging to the highest ranked center. This corresponds to a particularity of multifractal structures and we will come back to this topic when considering population distribution.

Another consequence of this feature of multifractals is that the direct catchment areas belonging to the third order centers have no longer the same size. We have small squares of base length r 0 • r 0 and larger ones with base length r 0 • r 1 .

The next step adds another hierarchical level and we discover again that the size of the catchment areas of centers issued from different iteration steps and thus corresponding to different hierarchical levels is the same, and on the other hand that the catchment areas of centers belonging to the same level are different. Two logics can thus be distinguished:

• the first one generates the central place hierarchy by adding a lower level at each iteration step. Hence the iteration step where the centroid has been generated, determines its service level in the central place hierarchy.

• the second one is linked to the mentioned "degeneration" effect. Since permutations are allowed we have direct catchment areas which have the same size but belong to different service levels.

We should however emphasize that the logic of the spatial configuration of the centers corresponds to the logic of the central place theory. The fact that the area affected to the centers are of different size according to their localization seems an interesting feature since we can assume that cities lying closer to important high level centers are usually bigger than those lying close to low level centers. This logic will be reconsidered when defining the theoretical population numbers.

By going on with iteration, it is of course possible to generate a more hierarchical spatial system. Let us just remind that Christaller, e.g., distinguishes 7 different service levels. However in order to conserve a certain legibility, we restrict ourselves here to the four levels already introduced. These levels may be associated to the following kinds of attendance rates:

• level 1: rarely frequented services or shops

• level 2: monthly frequented services or shops

• level 3: weekly frequented services or shops

• level 4: daily frequented services or shops

The coding system

We now introduce a coding system which allows distinguishing the different centers according to their service level. Hence for the first iteration step we distinguish the large central square which we denote by the digit 1 and the four smaller peripheral squares denoted by 0. In each following step we add now on the right of each digit another one, according the same logic. This makes evident that the hierarchy is created just by combining two factors. Hence in the next step the highest order central square is now called 11, the four adjacent generated smaller ones 10. The four peripheral squares generated in the previous step are replaced, too, by the generator. The occurring central place are called 01 and the four peripheral ones 00 (figure 5(a). This procedure is reiterated in the third step (cf.figure 5(b)). We obtain then a set of 8 different codes, each one consisting of three digits. The first level center with the highest facility level m = 1 has the code 111. The four directly adjacent squares of level m = 2 have the codes 110. They correspond to suburban areas of the main center. The four centers 011 correspond to the four centers of level m = 2 generated at the first iteration step. The peripheral centers centers 101 and 001 are issued from the second iteration step and correspond to centers of the facility level m = 3. Of course the 101 centers belong to the catchment area of 111 for higher level facilities, whereas the centers 001 belong to the catchment area of the second level centers 011. The small elements 100 and 000, adjacent to these third level centers, are all low level centers m = 4 (cf. figure 5(b)).

The stepwise generation of the elements can hence be represented as follows: 111 where we have set the basic surfaces as (r 1 ) 2 = S 1 and (r 0 ) 2 = S 0 .

The codes inform us directly about the facility levels. Using a generalized code ijk, who obtain

k = 0 ⇒ m = 4 jk = 01 ⇒ m = 3 ijk = 011 ⇒ m = 2 ijk = 111 ⇒ m = 1
By introducing these codes we have given up the previously discussed commutativity. Indeed in the introduced system the codes 101 and 110 or 011 are not equivalent, even if the surface of their direct catchment area is the same. Hence the code introduces a non-commutative operation.

This has the consequence that the system shows some properties which correspond rather to properties of unifractals than to mulifractals, what is interesting in the given context. Hence, making abstraction of their size, we verify that the total number of centers belonging to the different levels follows a geometrical series, excepted the passage from the highest to the next level: This corresponds to the usual hierarchical logic observed in fractal structures.

The population model

We now focus on the population numbers what needs some preliminary reflections. We assume that the population model affects to the introduced catchment areas a certain amount of population. But first we should remind some fundamental features of multifractal geometry. Two basically different iteration concept must be distinguished :

• the first one used before resembles to the usual iteration procedure used e.g. for generating Sierpinski carpets and corresponds to the procedure we used before for generating our central place hierarchy. The procedure reduces an initially given figure subsequently by using several -in the present case two -reduction factors. As pointed out before, this corresponds to generate progressively a set of points corresponding to the centroids of the smaller copies of the initial figure. Hence the total areas belonging to the prefractal set -in our example the direct catchment areas of the centers of different order -is reduced at each iteration step.

• another way of proceeding is to give a certain mass, e.g. a population, and to dispatch this mass progressively according to an iterative mapping procedure on different parts of space. E. g. we may cover a given area by a grid consisting of large meshes and distribute the mass according to some weighting factors p 0 , p 1 . . . among these different meshes. Then the iteration procedure generates within each mesh smaller ones and distributes the mass within this meshes according to the previously introduced distribution procedure. By this procedure the weighting factors are again combined in the same way as the previously introduced reduction factors, i.e. we obtain combined factors like p 2 0 , p 0 • p 1 , p 2 1 . This procedure is reiterated. However since we are reasoning in parts, the sum of mass over all grid elements remains constant over iteration, whereas the mass is more and more concentrated in the meshes which combines the high values of weighting factors.

In our case the situation is peculiar, since we combine a surface model with a weighting logic rather reminding the second approach. Let us remind that the first one serves generating a subset of areas which we consider as suitable for further urbanization. The second corresponds rather to o the distribution of population in the areas. However this does not really hold since, at each iteration step, we reduce the total amount of surface for which we assume future development as possible. This means that we subsequently put out of the system settlements, and thus population. This must be taken into account in the model.

Hence we propose the following model. We assume that when introducing the generator in the first step we split the population p living in the square-like initially selected area in two parts:

p = α • p + (1 -α) • p (1) = α • p + p rur(1) (2)
The population p rur(1) is the amount of the population living in the zones cut away by the generator and (1 -α) is the corresponding part of the population. This makes evident that α is directly given by empirical data by the relation:

α = p -p rur(1) p
The amount α • p of the population lives in the "urban system" as it defined at this first iteration step. Hence we are reasoning in parts of population what becomes obvious by dividing (1) by p:

1 = α + (1 -α) (3)
The urban system consists of the generator, i.e. of the center with code 1 and the four subcenters coded by "0". We assume now that this urban population is distributed among these 5 elements so that the main center concentrates the part a 1 of the urban population and each of the subcenters a 0 . Hence we obtain:

p = α(a 1 + 4a 0 )p + p rur(1) = α(a 1 + 4 1 -a 1 4 )p + p rur(1) (4)
The second equation holds since we are reasoning in parts, i.e. (a 1 +4a 0 ) = 1. Hence the model affects the following population numbers to the different elements of the prefractal: code level population surface density 1 1 αpa 1 (S 1 )

αpa1 (S1) 0 2 αpa 0 (S 0 ) αpa0 (S0)
Since α is already determined we have to compute a 1 and a 0 respecting the normalization requirement. Again we compute both the parameters directly by referring to empirical data distinguishing the center "1" from the four subcenters "0". If we call p1 the population living in the area "1" and p(i) 0 with i = 1 . . . 4 the empirical populations living in the surrounding subcenters, we obtain

a 1 = p1 αp a 0 = 1 4αp i p(i) 0
Of course the normalization requirement is strictly fulfilled. We now go on with iteration. In the next step we assume again that an amount of population p rur(2) lives in the parts of space now cut away by iteration (this ares is represnetd in grey in figure 5(a). We obtain the following relation:

p = αβ • p + α(1 -β) • p + p rur(1) (5) = αβ • p + p rur(2) + p rur(1) (6)
Again β is easy to compute according to the relationship:

β = p -p rur(2) -p rur(1) α • p
This means that we estimate the parameters -here α, β etc. stepwise. Hence each iteration is consistent with itself -sums over the ratios of population affected to the different areas are by definition identical to the initally given total population. This seems strictly coherent with the iteration logic which can be stoppped at an arbitray iteration step what should not affect parameter values determined for previous steps. We vapply the same logic for all further introduced parameters.

We now must reflect about how distributing the population living within the new "urban system" among the elements belonging to our multifractal. If we would, according to the iterative logic of fractal geometry, use the logic we applied for constructing the multifractal Sierpinski carpet, we would obtain the following relation corresponding to (4):

p = αβ(a 1 (a 1 + 4 • a 0 ) + 4a 0 (a 1 + 4 • a 0 ))p + p rur(2) + p rur(1) (7) = αβ(a 2 1 + 4a 1 • a 0 + 4a 0 • a 1 + 4 2 a 2 0 )p + p rur(2) + p rur(1) (8)
Again we would have dispatched the population according to the parts a 1 and a 0 among the different elements of the multifractal Sierpinski carpet. Since a 1 • a 0 = a 0 • a 1 , the population would be the same for the cities with code 10, correponding to peripheric zones of the most important center and 01 what is the second order main center. By going on with iteration at the next step a center 110 with facility level 4 would have the same population as the level 3 center 101 or the level 2 center 011. Such a logic seems unrealistic since we should expect that e.g. the density in the level 2 center 011 is superior to that of the level 3 center center 101.

It is evident that this is a consequence of the commutative logic of multifractal iteration3 .

Hence we modify the strict iteration logic by allowing other ratios for the distribution of population in the subsequent iteration steps. Thus by introducing parts b 1 and b 0 for the second iteration step, we may rewrite the relation (7):

p = αβ(a 1 (b 1 + 4 • b 0 ) + 4a 0 (b 1 + 4 • b 0 ))p + p rur(2) + p rur(1) = αβ(a 1 • b 1 + 4a 1 • b 0 + 4a 0 • b 1 + 4 2 a 0 • b 0 )p + p rur(2) + p rur(1)
Of course this assumption destroys the commutative logic and we clearly may distiguish centers of type 10 from those of type 01 with respect to there population numbers what holds, too, for the next step if we introduce additional factors c 1 and c 0 .

It is evident that b 1 and b 0 are again linked by the requirement of normalisation, so that:

b 0 = 1 -b 1 4 what yields for (9) p = αβ a 1 b 1 + 4 • 1 -b 1 4 + 4 • 1 -a 1 4 b 1 + 4 • 1 -b 1 4 p+p rur(2) +p rur(1) (9)
Hence we obtain again the model values of the population affected to the different elements generated at this iteration step: code level population surface density 11

1 αβpa 1 b 1 (S 1 ) 2 αβpa1b1 (S1) 2 10 3 αβpa 1 b 0 (S 1 ) • (S 0 ) αβpa1b0 (S1)•(S0) 01 2 αβpa 0 b 1 (S 1 ) • (S 0 ) αβpa0b1 (S1)•(S0) 00 3 αβpa 0 b 0 (S 0 ) 2 αβpa0b0 (S0) 2
For determining the parameters b 1 and b 0 we proceed according to the previously defined logic which respects the different iteration steps. Hence α, β, a 1 ,a 0 are known. If we introduce the empirical data according to the same logic as previously, we obtain the empirical equation which is the equivalent to to (9):

p -p rur(2) -p rur(1) = p11 + 4 i=1 p(i) 10 + 4 i=1 p(i) 01 + 16 i=1 p(i) 00 (10)
where we introduced empirical population numbers by a "hat" and we just restricted ourself to consider the only "urban population". We rewrite this equation and ( 9

1 + 4a 0 • b 1 + 4a 1 • b 0 + 4 2 a 0 • b 0 )p
We now require that the part of the urban system which contains the parameter b 1 correponds to the total real population of this part what yields:

p11 + 4 i=1 p(i) 01 = αβ(a 1 • b 1 + 4a 0 • b 1 )p = αβp • b 1 (11)
where we took into account the normalisation a 1 +4a 0 . The same is required for b 0 what yields:

4 i=1 p(i) 10 + 16 i=1 p(i) 00 = αβ(4a 1 • b 0 + 4 2 a 0 • b 0 )p = αβp • 4b 0 (12)
Both the relations are of course coherent, there sum is by definition the empirical population and the normalization b 1 + 4b 0 is verified. Relationships So the fractal rules just forbids that already generated lacunae i.e. conserved "free" space are not "occupied" by squares in further iteration steps4 .

The software allows identifying the population living in the introduced squarelike elements. These empirical data can be used to estimate the above introduced model parameter, what means that we have for each scale (iteration step) the amounts α, β . . . of population living in the rural hinterland i.e. the zones making no longer part of the zones to be developed in future. Moreover are computed the means of the centers belonging to the same code according to the previously introduced logic, i.e. by a stepwise estimation of a 1 , a 0 , b 1 , b 0 etc.

Moreover, the user can define himself values for these parameters in order to increase or reduce population concentration in certain types of centers which are, of course again distinguished according to their code. The software tool provides the information about these "theoretical" population numbers to the user, too and allows representing again the deficits or surpluses with respect to reality. Hence different kinds of scenarios can be constructed and compared. Similar simulation actually developed for the metropolitan area of Vienna (cf.

[CYF11], [START_REF] Czerkauer | Strategic planning for the development of sustainable metropolitan areas using a multi-scale decision support systemthe vienna case[END_REF]). Indeed the first railway line in France linked both these cities. Even the decline of the old industrial activities did not slow down the economic success and Lyon still remains a highly attractive region. Urbanization is constrained in the West by a mountain region, which is however attractive for high standard residential areas. In the East and the South of Lyon, the flat Rhone valley was propitious for setting up industrial areas and constructing social housing and lower level residential areas, whereas a swampy nowadays protected area prevented the North of strong urbanization. Several axes with subcenters can be identified :

• in the North the Saône Valley with Villefranche-sur-Saône and farther away Macon,

• in the North-East the well-developped axe in direction of Bourg-en-Bresse,

• in the South the Rhone valley with Vienne,

• in the South-West the Gier valley with Saint Etienne,

• in the South-East an axe going in direction Chambéry.

These axes are served by railway lines and are well accessible by road axes. In our simulation we considerd the four firstly enounced axes as propitious for further development and we localized the subcenters on existing important cities and areas favoured for further urbanization follows mainly the local railway lines5 . Hence enlarging settlement will be restricted to already settled-up areas with good accessibility to the main center. Of course other axes could have been chosen or even the fifth axis could have been included by chosing another genertor consisting of 5 subcenters. In the example we have generated three iteration steps (figures (b) to (d)). The figure represents the situation where we have chosen some parameter values and computed the differences between real world situation and the population numbers computed by means of the model parameters. In the figures 6 (b) -(d) dark blue corresponds to an important surplus of population whith respect to the chosen parameter values, whereas red would indicate a high lack of population. Pale yellow corresponds to a rather good concordance between the real world situation and the model. Here we have illustrated a potential "redistribution" of population and we see that according to the iteration steps and the chosen parameters some surplus of population may occur at a certain step and disappear in the next one. This is due to the fact that population is really distributed among all the areas according to the model parameters and this may, in a next step, generate a spatial dsitribution being in better concordance with real world situation. We could also have added population and estimated how this surplus would have to been dispatched among the different centers.

The chosen parameters concentrate the population by using the following parameters (cf. figure 6(d) at the bottom at right): αa 1 = 0.6 αa 0 = 0.05 (1 -α) = 0.2 βb 1 = 0.55 βb 0 = 0.04 (1 -α) = 0.29 γb 1 = 0.5 γb 0 = 0.05 (1 -γ) = 0.30

This corresponds to a scenario where the concentration in the center is the highest on the global level and periurbanization is the more accepted the more the level is local. The rural hinterland is favored on the second and third level with respect to the first one, since (1 -γ) > (1 -β) > (1 -α), what corresponds to the idea that a slight concentration close to development zones seems more reasonable than completely uniform distribution in rural areas.

Outlook

For concrete applications it is useful to combine the population model with data sets about the localization of the different kinds of services and facilities according to the defined levels. Then it is possible to define rules which allow evaluating for a given square-like element the quality of accessibility to the existing services, but also to the leisure areas of different attendance rate. For the basic model referring to a unifractal approach and a fixed grid covering the study area such rule sets are presented in detail in [START_REF] Tannier | Spatial accessibility to amenities in fractal and nonfractal urban patterns[END_REF]). For the here discussed fractalopolis approach similar rule sets have been defined for the, adapted to the case of the Vienna agglomeration, which are presented in a companion paper which will be soon available ( [START_REF] Frankhauser | Accessibility rules for the multifractal Fractalopolis planning concept -the Vienna example[END_REF]).

Of course "virtual" services can be introduced if it is e.g. intended to establish somewhere a shopping mole or other types of facilities. By varying the position of the squares their best position for future urbanization can be tested.

Acknowledgement

This research has been supported by the French Ministry of Ecology, Sustainable Development and Energy in the frame of the project Vilmodes maling part of the research program PREDIT 4.

Figure 3 :

 3 Figure 3: A modified Christaller scheme

  Figure 5: Illsutration of the coding sytem. The figure (a) shows what ares are cut off from the urban system when going fom the generator to the nexrt iteration step. Figure (b) shows the generated central place hierarchy (cf. text)

p

  ) in order to group the data referring to b 0 and those referring to b 1 : p -p rur(2) -p rur(1) = p11 + -p rur(2) -p rur(1) = αβ(a 1 • b

Figure 6 :

 6 Figure 6: Application of the model on the urban area of Lyon: (a) shows the chosen area with the train network, (b) represents the first iteration step, (c) the second one and (d) the third one -on the bottem at the right the waiting factors for poppulation distribution ribtion used at the different steps Figure 6 illustrates how we may apply the concept in a real world case, the area of the Lyon agglomeration. Lyon the second important metropolitan area in France. Lying at the western border of the Rhone valley, it benefits from a good accessibility to Mediterranean area as well as to Italy, but also to the Northern and Eastern part of France and to central Europe. This strategic position favored urban development since the antiquity. In the 19th century textile and silk industry boosted the urban development. Moreover heavy industry rose up at the same time in the South-West in the Gier valley around Sainte Etienne.

There use for describing urban paterns has been shown e.g. in[START_REF] Frankhauser | La fractalité des structures urbaines. Villes. Anthropos[END_REF] or more recently in[START_REF] Thomas | The morphology of built-up landscapes in wallonia (belgium): A classification using fractal indices[END_REF]

This software paquage has been developped by Gilles Vuidel, software engeneer at the

TheMA Research Institute, Université de Franche-Comté and CNRS, Besançon, France

This is why the distribution function of the surfaces follows not a Pareto-distribution as unifractals do, but a binomial distribution, cf.[START_REF] Feder | Fractals[END_REF]

Let us remind that in our case the lacunae may contain settlements, but this areas are not opened to future urbanization

The crossing high speed railway lines are not of interest in the given context.

(11) and (12), respectively the normalization, allow determining the parameters b 1 and b 0 :

In the same sense we now can go on with iteration. We then introduce a ratio γ of the population now affected to the elements belonging now to the urban system. Again normalized ratios c 1 and c 0 are introduced according to c 1 + 4 • c 0 . We restrict the discussion just by giving the populaton numbers as they are generated by the model:

The parameters are estimated according to the previously introduced logic by determining first γ and then c 1 and c 0 .

Applying the concept

Simulating population distribution scenarios

The previously introduced concept looks of course rather theoretical. Let us first emphasize that according to fractal logic the localizaton of the square-like elements can be modified within a certain range without affecting the fractal properties. Indeed we may chose the localization of the squares for the first step as we like if respecting the already enounced two rules:

• the squares must lie within the initial square;

• the squares are not allowed to overlap.

Hence we can e.g. put the big square in a corner of the initial square and place the four smaller ones around. In the next step the same logic holds. This means that within each square generated at the previous step, we can again place the now generated smaller squares by respecting the two introduced rules.


V Cazajus
email: vcazajus@enit.fr

B Lorrain
email: blorrain@enit.fr

H Welemane
email: chweleman@enit.fr

M Karama

Residual stresses in ceramic metal assembly after brazing process

Keywords: Ceramic-metal assembly, tensile strength, residual stresses, brazing process, thermo-mechanical analysis

The framework of this study is the thermo-mechanical analysis of the brazing process of ceramic metal assemblies. The thermal expansion gradient between ceramic and metallic materials leads to the development of residual stresses during the cooling phase of the brazing process which induce consequently an important reduction of the strength of these composite structures. In the present work, numerical simulations are performed in order first to predict the residual stresses distribution after the brazing process and in a second step, to study their influence on the tensile strength of metallized ceramic seals. Results obtained are compared with experimental tests.

Introduction

Brazing is a welding process currently used to produce ceramic to metal assemblies that are interesting because ceramic and metallic materials have dissimilar thermal, mechanical, electronic and chemical behaviour. Such assemblies are integrated in complex systems designed for high technology applications (medical, aircraft, spatial, electronic or nuclear). This process allows then to link two base materials (ceramic and metal) which exhibit quite different thermo-mechanical behaviour. The difference between the thermal expansion coefficient (CTE) of these materials leads to the development of residual stresses during the cooling phase of the brazing process. Such residual stresses clearly reduce the strength of the brazed joint and can lead to catastrophic failure at the interfaces, even during the brazing process itself [1;2]. Many analytical and numerical approaches have already been carried out to estimate the residual stresses within ceramic metal assemblies [START_REF] Charles | Identification d'un mode de vieillissement dans un assemblage céramique-métal[END_REF][START_REF] Hsuech | Residual stresses in metal-ceramic bonded strips[END_REF][START_REF] Yu | Residual stresses in ceramic-interlayer-metal joints[END_REF][START_REF] Iancu | Residual stress state of brazed ceramic/metal compounds[END_REF][START_REF] Santacreu | Sur la mécanique des assemblages: calcul des contraintes résiduelles dans les assemblages céramique-métal obtenus par brasage réactif et caractérisation mécanique des interfaces[END_REF][START_REF] Munz | Thermal stresses in ceramic-metal joints with an interlayer[END_REF][START_REF] Levy | Thermal residual stresses in ceramic to metal brazed joints[END_REF]. In order to define the mechanical strength of such assemblies, it is important to estimate these residual stresses. This paper aims first at determining through numerical simulations the residual stresses induced by the cooling phase of the brazing process within standard tensile specimens of ceramic metal assemblies. Then, these additional stresses are taken into account to determine the consecutive bond strength of such composite structures for standard tensile tests [START_REF] Astm) | Standard test method for tension and vacuum testing metallized ceramic seals[END_REF]. Numerical and experimental results are compared and discussed.

Brazing process

Brazing is a welding process which produces the coalescence of two or more like or unlike base materials by means of a filler metal alloy with a lower melting point. The assembly is heated to a suitable temperature upper the liquidus temperature of the filler metal which is drawn into the joint by capillary attraction. During the cooling phase, the filler metal solidification produces the joint of the base materials. In our case, after this point (cooling from filler metal solidus temperature to room temperature), the base materials and the filler metal are not submitted to any metallurgical transformation. Besides, the two base materials (ceramic and metal) are very dissimilar. A low thickness metallization at the ceramic surface is used to increase wettability by the filler metal at the liquid state and allow capillary attraction. The capillary attraction of the filler metal by the metallic part occurs then naturally.

Figure 1. Thermal cycle of the brazing process

Heating process is performed in two phases into a controlled atmosphere furnace. The assembly is heated in vacuum atmosphere to protect the metallic part from oxidation. The heating and cooling rate is defined to protect the ceramic part from thermal shock. The figure 1 describes the thermal cycle of the brazing process. Note that there is only thermal loading and no mechanical loading on the ceramic metal assembly during the brazing process. The figure 2 shows the microstructure of a ceramic to metal assembly after brazing. We can observe three domains: metal, joint and ceramic. The joint is composed of the filler metal, the metallized surface of the ceramic and the interfaces generated at high temperature by diffusion between the base materials surfaces and the filler metal. The materials (ceramic, metal and filler metal) considered in this study exhibit different thermo-mechanical behaviour. It is important to define precisely such response in order to estimate the residual stresses induced by the brazing process and the tensile strength of the ceramic metal assembly.

Ceramic. The ceramic material (alumina with purity of 97%) is known to be linear elastic and its properties do not depend on temperature for the temperature range of the brazing process (20°C to 850°C). Elastic parameters have been identified through ultrasonic tests at room temperature (Table 1). Besides, alumina is a brittle material which has different strength in tension and compression. Tensile tests and compression tests have provided the corresponding strengths of this material: respectively 175 MPa in tension, and 1750 MPa in compression. Finally, the Coefficient of Thermal Expansion (CTE) according to temperature has been identified through dilatation test (Table 1).

Metal and filler metal. The metal (stainless steel) and the filler metal (silver-copper eutectic alloy) exhibit elastoplastic behaviour depending on temperature: filler metal is perfectly elasto-plastic and metal exhibits linear hardening (Table 1). The evolution of the yield stress with temperature presented in Table 1 is given by the bibliography [START_REF] Levy | Thermal residual stresses in ceramic to metal brazed joints[END_REF]. As before, dilatation tests have provided the coefficient of thermal expansion (CTE) according to temperature (Table 1). Note that the ductile damage within the metallic materials is neglected as the failure of the specimen occurs preferentially within the ceramic material.

Tensile test

Test method. The tensile test method of ceramic metal seals is based on the standard test method for tension and vacuum testing of metallized ceramic seals (ASTM F 19-64) [START_REF] Astm) | Standard test method for tension and vacuum testing metallized ceramic seals[END_REF]. In this paper, this test method is used to determine tensile strengths of ceramic metal specimens. The specimen is composed of two identical washers of filler metal (or braze alloy) which join a metal washer located at the centre of the specimen with two identical ceramic components on each side. The geometrical dimensions of the ceramic component are detailed in Figure 3 (the metal washer thickness is equal to 0.3 mm and the filler metal washer thickness is equal to 0.06 mm). The gripping device used to apply the load is presented on Figure 4. Such design allows a self aligning of the axis of the test specimen with the centreline of the test machine heads. The results dispersion for the identification of the tensile strength can then be significantly reduced.

Filler metal washer

Metal washer

Ceramic component Ceramic component

Experimental results. Five specimens have been prepared and tested for this study. For each sample, Figure 5 shows the evolution of the load on the specimen according to the applied displacement of the grip. At the beginning of the test, we note a low non linear increase of the load, which corresponds to the aligning of the grip. Then, the response increases linearly up to the brittle failure of the sample which occurs within the ceramic material near the bond interface. Although the linear coefficient is quite the same for each specimen, the applied load that leads to the failure is quite different between the five samples (9.5 kN to 12 kN). The average of these five values is equal to 10.7 kN with a standard deviation of 1 kN. Note that the variability of the alumina tensile strength could explain such deviation of the results.

Numerical model

Numerical simulations have been performed with the finite element ABAQUS code in accordance with the standard specimen geometry (Fig. 3) and the processes involved. The numerical simulation is composed of two steps:

1. First, the cooling phase of the brazing process which provides the induced residual stresses in the materials, 2. In a second step, the tensile test itself which gives the tensile strength of the ceramic metal assembly. For both stages, the following assumptions have been taken :

• the structure is 2D axisymmetric and symmetric with the median plane;

• the mesh of the assembly is constituted with 8-nodes quadratic axisymmetric quadrilateral elements. The element size is lowest near the interfaces and the free side surface because the residual stresses are essentially localised in theses zones and the failure occurs near theses zones during the tensile test (Fig. 6). A converging study on maximum principal stresses has fixed the nodes number at 5210 in our case to estimate residual stresses after brazing process and also stresses during the tensile test with a good precision; • the interfaces between the filler metal and the base materials (ceramic and stainless steel) are supposed to be perfect. Concerning the brazing process model:

• the starting point of the simulation is the solidification of the filler metal during the cooling phase of the brazing process (Fig. 1) and the simulation ends when the ceramic metal assembly temperature is equal to the room temperature;

Sample number:

• there is no mechanical loading on the ceramic metal assembly;

• the thermal loading is the cooling cycle (Fig. 1) from the filler metal solidus temperature to room temperature with the uniform cooling hypothesis: the temperature of all the nodes is assimilate to the furnace temperature according to time. Finally, for the tensile test simulation:

• the grip, made of hard steel, has been introduced in the finite element model (Fig. 6-b) : a linear elastic behaviour has been considered for this material (Young modulus 200000 MPa, Poisson ratio 0.3) and the contact between the grip and the specimen is simulated with a frictionless model; • two initial stress states have been considered :

1. first, free state of stress : the stresses equal to zero in all the specimen, 2. then, pre-stressed state : the stresses are the residual stresses given by the previous simulation stage of the brazing process; • a negative vertical displacement is applied on the lower surface of the grip (Fig. 6-b);

• the Rankine criterion is used to detect the crack initiation within the ceramic component : a crack is supposed to be initiated when the maximum principal tensile stress exceeds the tensile strength of the alumina. For the brazing process model:

• Residual strains within the metallic materials are smaller than the failure strain limit. So the metallic materials are not damaged after the brazing process; • The figure 7-a) gives the evolution of the maximum value within the ceramic part of the maximum principal stress according to the furnace temperature. The residual stresses are quite small within the assembly at high temperature because they are limited to the low yield stress value of the filler metal and associated to low strains values in metal and ceramic. The filler metal yield stress given by Levy [START_REF] Levy | Thermal residual stresses in ceramic to metal brazed joints[END_REF] increases faster for the range of low temperature and then induces bigger stresses. • Concerning the ceramic material, the maximum value of the maximum principal stress which controls the crack initiation (Rankine criterion) is located on the radial free edge of the specimen under the ceramic filler metal interface (Fig. 7-b). This value is equal to 85 MPa after brazing which represents the half value of the ceramic strength. If ceramic is not damaged at the end of the cooling process, residual stresses induced are yet not negligible. For the tensile test, we have studied the evolution of the load on the specimen according to the stress evolution for the two initial stress states described before (Fig. 8):

• For the free state of stress, the model shows that the failure load corresponding to the initiation of a crack within the ceramic part is around 20 kN; • If the residual stresses are taken into account, this value comes to 10.5 kN, which is very close to the experimental results (mean value of 10.7 kN). This result is consistent with the fact that the residual effects reduce the amplitude of stress that can be supported by the ceramic. Consequently, it is essential to account for the residual stresses in order to estimate precisely the strength of the ceramic metal brazed assemblies. The brazing process model presented in this paper allows an estimation of the residual stresses within ceramic metal assemblies induced by the cooling phase. The introduction of such stresses in the tensile test model leads to a better estimation of the tensile strength of standard specimens, consistent with experimental results. In view of the design of joints in engineering structures, the numerical model developed in this work represents then an essential tool to study the mechanical behaviour and the reliability of ceramic metal brazed assemblies. Note finally that it may be interesting to complete this work by simulating the damage growth within the ceramic material and to compare to the crack pattern generated during experimental tests.

Figure 2 .

 2 Figure 2. Microstructure of the ceramic to metal brazed assembly

Figure 3 .

 3 Figure 3. Tensile specimen geometry of ceramic metal seal (distances in mm)

Figure 4 .

 4 Figure4. Self aligning grip for tension test[START_REF] Astm) | Standard test method for tension and vacuum testing metallized ceramic seals[END_REF]

Figure 5 .

 5 Figure 5. Tensile tests on ceramic metal assemblies.

Figure 6 .

 6 Figure 6. Mesh of the specimen: a) brazing process -b) tensile test -c) detail of the link zone

Figure 7 .

 7 Figure 7. Maximum principal stress a) Evolution of the maximum value during the brazing process -b) final distribution within the specimen

Figure 8 .

 8 Figure 8. Tensile loading on the specimen according to the maximum principal stress evolution within the ceramic part

Table 1

 1

			Young modulus (GPa)
	T(°C) 20 100 200 300 400 500 600
	A	330 330 330 330 330 330 330
	FM	79	79	79	79	79	79	79
	SS	193 191 183	-	168	-	148
				Poisson ratio
	T(°C) 20 100 200 300 400 500 600
	A	0.26 0.26 0.26 0.26 0.26 0.26 0.26
	FM 0.33 0.33 0.33 0.33 0.33 0.33 0.33
	SS	0.22 0.27 0.27	-	0.31	-	0.37
	Coefficient of Thermal Expansion (× × × ×10 -6 °C-1 )
	T(°C) 20 100 200 300 400 500 600
	A	-	6.2 7.4 7.8	8	8.1 8.2
	FM	-	16.7 19.2 20.2 20.4 20.7 20.5
	SS	-	18.4 20.3 20.9 21.3 21.4 21.7
			Yield stress (MPa)
	T(°C) 20 100 200 300 400 500 600
	FM	247 243 230 185 40	15	8
	SS	247 243 169 148 136 133 125
			Tensile strength (MPa)
	SS	973 774 667 635 633 589 517
			Tensile failure strain (%)
	SS	48	44	38	34	35	34	34

. Mechanical and thermal properties of each constituent material according to temperature: Alumina (A), Filler Metal (FM), Stainless Steel (SS).