Jesus Crespo-Cuaresma
email: jesus.crespo-cuaresma@uibk.ac.at

Neil Foster
email: foster@wiiw.ac.at

Robert Stehrer
email: stehrer@wiiw.ac.at

< C1 -Econometric and Statistical Methods: General < C -Mathematical and Quantitative Methods, C21 -Cross-Sectional Models|Spatial Models < C2 -Econometric Methods: Single Equation Models < C -Mathematical and Quantitative Methods, R11 -Regional Economic Activity: Growth, Development, and Changes < R1 -General Regional

Keywords: C11 -Bayesian Analysis Economic growth, Bayesian Model Averaging, Quantile Regression Regional Growth, Bayesian Model Averaging, Quantile Regression JEL Classification: C11, C21, R11 Quantilsregressionen analysiert. Dabei wird ein Bayesian Model Averaging (BMA) für Quantilsregressionen verwendet, um die relevanten Kovariaten, die unterschiedliche Effekte in den jeweiligen Wachstumsquantilen aufweisen können, zu ermitteln. Die Resultate zeigen, dass die robusten Wachstumsdeterminanten in den jeweiligen Quantilen tatsächlich unterschiedlich sind. Unter Berücksichtigung von länderspezifischen Effekten ist insbesondere die Variable Anlageinvestitionen ein robuster Erklärungsfaktor regionalen Wachstums; ohne Berücksichtigen dieser Effekte sind Humankapitalausstattung und das Pro-Kopf Einkommen robuste Determinanten. Auch wenn eine bestimmte Variable robust in mehreren oder allen Quantilen ist, sind die ermittelten Effekte auf das Wachstum der Regionen in den jeweiligen Quantilen oftmals unterschiedlich Regionales Wachstum, Bayesian Model Averaging, Quantilsregressionen

We analyse the robustness of growth determinants across EU regions using quantile regression (QR). We propose using Bayesian Model Averaging (BMA) on the class of QR models to assess the set of relevant covariates allowing for different effects across growth quantiles. The results indicate that the robust growth determinants differ across quantiles. The set of robust variables includes physical investment when taking country fixed effects into account and skill endowment and initial GDP per capita when not. Even when a variable is found to be robust across quantiles its estimated impact on growth is often found to vary across quantiles.

A great deal of effort has been expended in to the question of what are the most important determinants of differences in income growth rates across countries. The empirical literature on this subject tends to follow a common approach, regressing a usually small number of variables on output growth rates using a cross-section, or more recently a panel, of countries. The seminal contribution adopting this approach was [START_REF] Barro | Economic growth in a cross-section of countries[END_REF] which has now been copied and adapted in numerous papers. i This literature has included a large number of variables purporting to explain growth. [START_REF] Durlauf | Growth econometrics[END_REF] for example report more than 40 "general growth theories" and over 130 growth determinants in various crosscountry regressions. This has lead researchers to try and find a set of 'robust' variables that are important determinants of growth in a number of different models.

An early attempt at identifying the set of robust growth determinants was [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] who used the Extreme Bounds Analysis (EBA) of [START_REF] Leamer | Specification Searches[END_REF][START_REF] Leamer | Let's Take the Con Out of Econometrics[END_REF]. In this type of analysis the dependent variable is regressed on the explanatory variable of interest, , including different sets of other explanatory variables. If the maximum and minimum of the resulting coefficients on this variable all have the same sign (and are significant) the relationship is classified as 'robust', in the other case as 'fragile'. [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] report two variables only, initial income and gross fixed capital formation, as robust variables in this particular sense ii . Such a criterion has been criticised as being too strong however. [START_REF]I just ran two million regressions[END_REF] for example, moves away from looking at the maximum and minimum of the coefficients and concentrates instead on the entire distribution of the coefficients from the estimated models. He considers as an evaluation criterion the percentage of times a variable appears significant and of the same sign. Using this definition of robustness and a 95 percent cut-off level, Sala-i-Martin finds a larger set of growth determinants could be considered robust.

A further approach to seeking robust determinants has been to follow some model selection criteria. One such approach is the general to specific methodology often associated with David Hendry, with the paper by [START_REF] Hendry | We ran one regression[END_REF] being one example using this methodology to address the robust determinants of growth. Another approach (see [START_REF] Schneider | Catching Growth Determinants with the Adaptive LASSO, Background paper on the European Commission Directorate General Regional Policy Project[END_REF] uses consistent parameter estimation and model selection procedures based on the Least Absolute Shrinkage and Selection Operator (LASSO) estimator as proposed by [START_REF] Zou | The adaptive LASSO and its oracle properties[END_REF]. Bayesian Model Averaging (BMA) methods have also become a popular means of identifying the robust set of growth determinants. Examples where BMA has been applied to cross-country growth data include [START_REF] Brock | Growth empirics and reality[END_REF], [START_REF] Brock | Policy evaluation in uncertain economic environments (with comments and discussion)[END_REF], Sala-i-Martin, [START_REF] Doppelhofer | Determinants of long-run growth: A Bayesian averaging of classical estimates (BACE) approach[END_REF], [START_REF] Fernandez | Model uncertainty in cross-country growth regressions[END_REF] and Masanjala andPapageorgiou (2007 and[START_REF] Foster | The impact of trade liberalisation on economic growth: Evidence from a quantile regression analysis[END_REF].

The vast majority of the existing empirical growth literature concentrates on cross-country growth rates. There are however a smaller number of papers considering regional growth rates. A number of papers have examined the issue of convergence at the regional level. [START_REF] Barro | Economic Growth[END_REF] for example present results at the regional level for the US, Japan and the EU. They find evidence in favour of convergence. [START_REF] Boldrin | Inequality and convergence in Europe's regions: Reconsidering European regional policy[END_REF] and [START_REF] Egger | Spatial Convergence[END_REF] find evidence of only slow income convergence. Other studies employ spatial techniques: [START_REF] Baumont | The European regional convergence process, 1980-1995: Do spatial regimes and spatial dependence matter?[END_REF] and [START_REF] Le | A Spatial Econometric Analysis of Convergence across European Regions[END_REF] for 6 | P a g e example, examine the importance of convergence after allowing for spatial dependence. [START_REF] Egger | Spatial Convergence[END_REF] also show that spatial effects exert a non-negligible effect on regional convergence. A smaller number of papers consider the various potential determinants of growth at the regional level. For example, [START_REF] Cheshire | Endogenous processes in European regional growth: Convergence and policy[END_REF] consider growth in 122 Functional Urban Regions and find that measures of human capital and economic potential have the strongest impact on growth. [START_REF] Badinger | Trade, human capital and innovation: The engines of European regional growth in the 1990s[END_REF] consider data from 128 EU regions and find that capital accumulation and educational attainment are robust determinants of regional growth. Puigcerver-Peñalver (2007) estimates a hybrid growth model which allows for endogenous and exogenous determinants of growth over the period 1989-2000 for 41 Objective 1 regions. Apart from finding convergence, she also finds a significant and positive impact of structural funds. [START_REF] Egger | Spatial Convergence[END_REF] provide some evidence indicating that the sectoral structure has an impact on regional growth. [START_REF] Fingleton | Theoretical economic geography and spatial econometrics: dynamic perspectives[END_REF] provides support for one of the main tenets of new economic geography, namely that urbanisation, peripherality, the initial level of technology and across-region spillovers are determinants of regional productivity growth variations, operating via the rate of technical progress and labour efficiency variations.

Crespo [START_REF] Crespo-Cuaresma | Growth, convergence and EU membership[END_REF] estimate convergence for the EU-15 countries over the period 1960-1998 and find economic integration beneficial for poorer countries, though there are a number of potential factors for this, such as technological spillovers, the stabilisation of the exchange rate, financial transfers (structural funds) etc.

Thus there is some uncertainty where these benefits come from. More closely related to this paper however are contributions searching for robust determinants of growth. [START_REF] Lesage | Bayesian model averaging for spatial econometric models[END_REF], [START_REF] Lesage | Spatial growth regressions: Model specification, estimation and interpretation[END_REF] and Crespo Cuaresma, [START_REF] Crespo-Cuaresma | The determinants of economic growth in European regions, Background paper on the European Commission Directorate General Regional Policy Project[END_REF] for example all use BMA methods to identify the set of robust growth determinants at the regional level. Crespo Cuaresma et al. (2008b) show that human capital accumulation and convergence forces appear as the most relevant variables in explaining economic growth at the regional level in Europe when model uncertainty is explicitly accounted for in the estimation method.

In this paper we seek to identify the set of robust growth determinants using a dataset of EU regions. The paper builds upon previous work in a number of ways. Firstly, as opposed to the majority of the existing literature we identify the robust growth determinants for a sample of 255 NUTS2 European regions using BMA. Secondly, and most importantly, we combine BMA with Quantile Regressions (QR) by concentrating on a space of econometric models where the effect of growth determinants is allowed to differ across quantiles. Our paper proposes therefore a methodological generalization of BMA which allows us to obtain model averaged estimates based on QR and thus considers alternative sets of robust growth determinants for under-and over-achieving regions.

To date, the vast majority of empirical growth research has relied on the least squares methodology, which models the mean of the growth rate conditional on a set of explanatory variables. Quantile regressions on the other hand model the conditional quantile of the growth rate at any quantile on the conditional growth distribution. In recent years studies have begun to emerge that use QR methods to address the determinants of economic iii There are a number of reasons for employing QR in the context of growth regressions. One major advantage of QR over standard OLS is that the estimator is robust to outlying observations on the dependent variable. This is a particular advantage in the growth setting where growth rates have been found to be characterised by long right tails and where outlying countries or regions can have a marked effect on OLS results (see [START_REF] Temple | The new growth evidence[END_REF]. A further major advantage is that the QR estimator provides one method of capturing parameter heterogeneity across regions. As indicated by [START_REF] Durlauf | Econometric Analysis and the Study of Economic Growth: A Skeptical Perspective[END_REF], amongst others, the assumption of parameter homogeneity is neither an empirical nor a theoretical result. From a theoretical point of view, the fact that economic units which are hit by negative growth shocks may present different economic dynamics which would require the specification of a different data generating process has received a lot of interest in the economic growth literature. Poverty trap models, such as the one put forward originally by [START_REF] Azariadis | Threshold externalities in economic development[END_REF] emphasizing threshold models (see the recent survey by [START_REF] Azariadis | Poverty traps[END_REF] present a theoretical framework which justifies the need for empirical models with parameter heterogeneity. [START_REF] Barreto | Under performers and over achievers: A quantile regression analysis of growth[END_REF] argue that by using QR they are addressing the behaviour of countries in which the factors that are not included in the estimated model create an environment that is conducive to high (or low)

growth relative to conditions suggested by the variables that are included in the model. As an example, they argue that while investment is often found to be the most important tool to foster improved growth in studies based on OLS, if determinants outside the model are unfavourable, it is conceivable that increased investment will be wasted, resulting in a negligible impact on growth. QR, by potentially providing one solution for each quantile, allows one to assess how policy variables affect regions according to their position on the in the framework of regional datasets, where unmodelled spatial dependence in the form of geographical polarization of economic growth processes renders standard OLS estimates biased (see for example [START_REF] Lesage | Bayesian model averaging for spatial econometric models[END_REF]. Geographical polarization may lead to subsamples of observations being poorly modelled by standard linear regression models and leading to a better fit using QR methods. iv A further advantage of QR is that by considering the entire conditional growth distribution it allows one to consider the magnitude of the effects of the explanatory variables at the tails of the conditional distribution, which may be more interesting and useful than finding the magnitude of such effects at the conditional mean.

The paper closest in spirit to ours is the paper of [START_REF] Barreto | Under performers and over achievers: A quantile regression analysis of growth[END_REF] who combine QR with a variant of both [START_REF] Leamer | Let's Take the Con Out of Econometrics[END_REF] EBA and Sala-i-Martin's (1997) method of determining robustness to consider whether the set of robust growth determinants differ across quantiles. Using cross-country data they find that for under-achieving countries the most significant determinants of growth are latitude, social infrastructure, civil liberties and liquid liabilities, while for over-achieving countries trade, social infrastructure, the share of government expenditure, investment share and prices are the most significant determinants.

To highlight the importance of considering QR in the context of regional growth determinants, the following four figures show five estimated quantile regression lines (i.e.

the dotted lines) and the OLS regression line (i.e. the solid line) when considering the relationship between four standard growth determinants and the growth of income per capita across regions. v From these figures we can observe that for some of the variables, in particular the share of gross fixed capital formation in value-added and the share of high skilled labour we find a great deal of dispersion in the estimated regression lines, indicating that the response of growth to changes in these variables is sensitive to the quantile considered. In addition, we find that in a number of cases there is quite a difference between the mean (i.e. OLS) and median (i.e. 50 th percentile) regression lines, as well as regression lines for other quantiles. These figures are therefore suggestive of parameter heterogeneity and of the importance of considering alternatives to OLS.

<<< Figures 1-4 around here >>>

Combining the BMA approach with QR allows us to simultaneously address the issues of model uncertainty in growth regressions and the presence of heterogeneous effects across different quantiles of the conditional growth distribution. Our results indicate that while some variables appear to be robustly related to growth at all quantiles, examples being initial GDP per capita and a capital city dummy when excluding country effects, others are only found to be relevant at specific quantiles only, in particular human and physical capital variables. Moreover, even when variables are found to be robust across quantiles it is often found to be the case that the coefficients on such variables differ across quantiles. For example, we find that human capital tends to play a more important role for underperforming regions when including country fixed effects, while the opposite is true for physical capital accumulation. The results therefore indicate the problems of trying to draw policy conclusions from OLS regressions, with the impact of particular variables found to depend upon a number of (often unmodelled) characteristics.

The paper is set out as follows. Section 2 discusses the concepts of QR and BMA in further detail and describes how we combine these two approaches. Section 3 discusses the data and Section 4 presents our initial results. Section 5 presents the main results of the paper and Section 6 concludes.

2. Bayesian Averaging of Quantile Regression Models 2.1. Quantile Regressions Quantile regressions were introduced by Koenker and Bassett (1978), though the history of the Least Absolute Deviations (LAD) model from which quantile methods are derived predates OLS. vi Quantile regression analysis has recently received a great deal of attention with extensions to the existing literature that deal with the practical problem of estimating the covariance matrix, that consider the performance of the various estimators in small samples, as well as methods to deal with endogeneity, panel data and heteroscedasticity.

Moreover, a growing literature applies such methods to a wide range of economic issues.

Quantile regression models seek to model the conditional quantile functions, in which the quantiles of the conditional distribution of the dependent variable are expressed as functions of observed covariates. The main advantage of QR is that potentially different solutions at distinct quantiles may be interpreted as differences in the response of the dependent variable to changes in the regressors at various points in the conditional distribution of the dependent variable. In the cross-section growth literature therefore it is possible to interpret changing coefficients across the conditional distribution as the result of systematic differences between countries or regions [START_REF] Canarella | Parameter heterogeneity in the neoclassical growth model: A quantile regression approach[END_REF].

The quantile regression model, as described by [START_REF] Buchinsky | Recent advances in quantile regression methods: A practical guideline for empirical research[END_REF] is

, ' i i i x y θ θ ε β + = n i ,..., 1 =
where β θ is the parameter vector associated with the θ th quantile and ε θi is an unknown error term. It is assumed that ε θi satisfies the constraint

( ) , 0 = i i x Quant θ θ ε
such that the errors have zero conditional mean though no other distributional assumptions are required.

From a frequentist point of view, the quantile regression estimator of can be obtained by minimising a weighted sum of absolute errors, where the weights are symmetric for the median regression case ( ) and asymmetric otherwise. In general therefore, the linear model for the θ th quantile ( )

1 0 < < θ solves the following minimisation problem, ( )       - - + - ' : ' : ' 1 ' 1 min i i i i x y i x y i i i i i x y x y n
As one keeps increasing θ from zero to one, one can trace the entire conditional distribution of , conditional on the set of regressors. In terms of this paper therefore QR allow us to trace the entire distribution of the growth rate of income per capita, conditional on the regressors included. The resulting minimisation problem can be solved using linear programming methods. The coefficient for a regressor can be interpreted as the marginal change in the θ th conditional quantile of due to a marginal change in . vii The asymptotic theory of QR is provided by Koenker and Bassett (1978). One can use procedures to estimate the asymptotic standard error of the estimators, or alternatively one can use a bootstrap procedure.

The use of QR has a number of benefits. The major benefit being that the entire conditional distribution of the dependent variable can be characterised by using different values of θ. A further benefit relates to the fact that median regression methods can be more efficient than mean regression estimators in the presence of heteroscedasticity (though this problem is also addressed by robust estimation). QR are also robust with regard to outlying observations in the dependent variable. The quantile regression objective function is a weighted sum of absolute deviations, which gives a robust measure of location, so that the estimated coefficient vector is not sensitive to outlier observations on the dependent variable. Finally, when the error term is non-normal, quantile regression estimators may be more efficient than least squares estimators.

Bayesian Model Averaging

Bayesian Model Averaging (BMA) is a standard Bayesian solution to model uncertainty, and consists of basing prediction and inference on a weighted average of all the models considered, rather than on one single regression model. viii Model averaging in general and BMA in particular, are becoming more and more popular, and there are now numerous (2004), Sala-i-Martin et al ( 2004), [START_REF] Durlauf | Is God in the details: A re-examination of the role of religion in economic growth?[END_REF][START_REF] Durlauf | Are any growth theories robust?[END_REF], Crespo-Cuaresma and [START_REF] Doppelhofer | Model Averaging[END_REF], [START_REF] Eicher | Determining growth determinants: Default priors and predictive performance in Bayesian model averaging[END_REF], Masanjala andPapageorgiou (2007, 2008), Ley andSteel (2007, 2009).

Given where are the posterior model probabilities given by equation ( 2).

The likelihood function is thus of central importance when implementing the BMA approach, which creates a problem when implementing BMA on QR. Following [START_REF] Koenker | Goodness of fit and related inference processes for quantile regression[END_REF] and [START_REF] Yu | Bayesian quantile regression[END_REF] the marginal likelihood for a quantile regression model can be computed however by assuming that is distributed according to an asymmetric Laplace distribution, so that,

where . The use of the asymmetric Laplace distribution for implies that under the assumption of an improper uniform prior distribution on the parameter vector, β can be estimated by maximising, , which is just the minimisation problem proposed by [START_REF] Koenker | Regression quantiles[END_REF] for estimating quantile regression models in a frequentist framework. [START_REF] Yu | Bayesian quantile regression[END_REF] show that this likelihood function and an improper uniform prior on β lead to a proper posterior distribution of the parameter vector.

Consider the case of two competing models, and , the posterior odds for model 2 against model 1 can be readily written as the product of the Bayes factor and the prior odds. Further assuming equal priors across models, the posterior odds are equal to the Bayes factor, , which in turn can be approximated using the Laplace method as, where is the dimension of , is the inverse Hessian of the likelihood and is the maximum likelihood estimator of . Equation ( 2) can be further operationalised using [START_REF] Schwarz | Estimating the dimension of a model[END_REF] approximation (see [START_REF] Raftery | Bayesian model selection for social research[END_REF] as where is the standard likelihood ratio test statistic for the choice between model 1 and 2 based on the likelihood function given in equation ( 6). We use this approximation in order to calculate the posterior model probabilities. In our setting, the approximation based on the Schwarz criterion has the advantage that it does not require the explicit specification of priors over the parameter space (see also [START_REF] Kass | Bayes factors[END_REF] and thus can be easily implemented using frequentist estimation methods.

Data

The data used in the analysis covers 255 NUTS-2 regions in the 27 EU countries. For eight countries the NUTS-2 region is also the country (these countries being Cyprus, Denmark, Estonia, Latvia, Lithuania, Luxembourg, Malta and Slovenia). The maximum number of regions in a country is 39 (Germany). The period of coverage is from 1995-2005, though for some variables a shorter time-period is used due to data availability. The starting point in the dataset ensures that the post-transitional recession in the Eastern European countries had capturing the functioning of labour markets and factor input conditions. The initial unemployment rate captures the sound operation of labour markets and is also related to factor accumulation, regional flexibility and social cohesion. One should note that there is not necessarily a clear link between these sets of variables and a particular growth theory:

the same variable can have an important role in different growth theories, while a particular growth theory might emphasize more than one variable. For example, initial conditionsand in particular the initial level of GDP per capita -are particularly emphasised in the neoclassical growth theory where the convergence process is driven by capital accumulation.

However, the initial level of GDP per capita (as a proxy for productivity) is also important in theories emphasizing learning capabilities (for example, models emphasising the 'advantage of backwardness' or the 'technology gap').

In since the averaged estimates would be composed of some estimates referring to elasticities based on within-country relationships and others referring to differences across regions of different countries.

BMA results

As an initial step we implement the BMA approach described above using classical least squares estimates. The BMA approach requires a prior probability of each model and a prior probability distribution over the parameters of each model to compute the weights when averaging over models. We follow the usual approach in the literature and assume a flat prior (i.e. all models are equally likely) in the model space, which implies a prior inclusion probability of 0.5 for each variable. We employ a Markov Chain Monte Carlo Model Composition (MC 3 ) algorithm using random walk steps as described in [START_REF] Fernandez | Model uncertainty in cross-country growth regressions[END_REF] to deal with the very large model space, which allows us to only visit models that have a non-negligible posterior probability. All reported results are based on 2 million draws of the Markov Chain, after 1 million discarded burn-in draws. xiii Tables 1 and2 Despite the very large number of models entertained, a large part of the posterior model probability appears concentrated in a relatively small number of models. The relatively larger number of models visited by the Markov chain in the case of the setting with country fixed effects indicates that uncertainty across models is larger when we consider withincountry data. As expected, the results we obtain are found to differ depending on whether country effects are included or not, which implies that the determinants of regional growth between countries are of a different nature as those within countries. The variables with the highest inclusion probability when country dummies are excluded (Table 1) are whether the region hosts the capital city (CAPITAL), the initial GDP per capita (GDPCAP0), the initial share of high educated persons in working age population (SHSH) and the initial unemployment rate (URT0). Once country effects are allowed for (Table 2) however the inclusion probability of a number of the variables, in particular GDPCAP0 and URT0, falls dramatically. In this case there are three variables with an inclusion probability above 0.5, indicating that we consider them to be robust growth determinants, namely the share of gross fixed capital formation in gross value added (SHGFCF), CAPITAL and SHSH. xiv The results indicate that an indicator of human capital and a variable capturing whether the region houses the capital city are the most important determinants of regional growth, with physical capital investment (SHGFCF) becoming relevant when country effects are included. That human capital and investment are found to be relevant growth determinants is suggestive of the importance of factor accumulation for regional growth. The importance of these variables is also consistent with more recent endogenous growth models that emphasise the importance of learning-bydoing and schooling [START_REF] Lucas | On the mechanics of economic development[END_REF][START_REF] Stokey | Human Capital, Product Quality and Growth[END_REF] and capital accumulation [START_REF] Romer | Increasing Returns and Long-Run Growth[END_REF].

The capital city variable can be interpreted as summarizing several different effects from the European countries in our sample, and its effect is less clear-cut if the sample is reduced to old member states xv , which is in line with the fact that growth in Eastern European countries was concentrated in and around capital cities. The Williamson hypothesis [START_REF] Williamson | Regional inequality and the process of national development: a description of the patterns[END_REF] proposes that there exists a trade-off between economic growth and regional disparities for countries at lower levels of development, and the growth bonus of regions which contain the capital city in Eastern Europe may be capturing this effect. xvi

Interestingly, the importance of initial GDP per capita (GDPCAP0) is not found to be strong once we include country effects. The result that initial income is not relevant when country effects are included but is when they are excluded suggests that while countries across Europe appear to be converging, regions within countries do not show robust evidence of income convergence. This finding is again consistent with the above mentioned fact that economic growth has been concentrated in the capital city regions in Eastern European countries. This result is further consistent with the results of De la Fuente and Vives (1995) who show that while convergence has taken place in Europe, regions within countries have either failed to converge or have diverged.

In terms of the posterior means and standard deviations reported in these two tables we see that for the robust variables in each table the posterior mean of the coefficients are of the expected sign. As expected, in this setting we find a positive posterior mean for the parameters attached to SHSH, CAPITAL and SHGFCF, and a negative one for GDPCAP0. The posterior standard deviations indicate that the coefficients are well estimated when not including country fixed effects, but obtaining precise estimates of the quantitative effects of variables for regions within countries appears more difficult.

Results from the Bayesian Averaging of Quantile Regressions

In this section we report the results from implementing BMA on QR. We implement the BMA approach at each decile from the 10 th to the 90 th percentile again both including and excluding country effects. Table 3 (4) reports the inclusion probabilities at every decile along the conditional growth distribution when country effects are excluded (included). The variables are ranked according to the mean of the PIP across the quantiles (with variables showing a PIP greater than 0.50 considered robust and highlighted).

Considering the results in Table 3 where country effects are excluded we find that the initial GDP per capita (GDPCAP0) and the capital city dummy (CAPITAL) have a high inclusion probability across quantiles (with the exception of CAPITAL in the first decile). The share of high skilled workers (SHSH) tends to become robust at the highest quantiles (though not uniformly), while the variable indicating learning activities (SHLLL) is found to be robust at lower quantiles and internet access of firms (INTF) at the lowest quantile. Consistent with the least squares results therefore we find that GDPCAP0 and CAPITAL are robust growth determinants, and this appears to be true across the conditional growth distribution.

Different to the least squares results however we find additional variables (SHLLL and INTF)

to be robust growth determinants at particular quantiles. Such a result emphasises the relevance of moving beyond considering least squares results only, with potentially different drivers of growth and different policy recommendations needed for under-and overachievers. In addition, while SHSH is again found to be robust, this is only the case for certain quantiles, and the higher quantiles in particular This effect is partly driven by Eastern

European regions showing a high share of skilled workers and relatively high rates of economic growth. Such results leads to the nuanced policy conclusions that policies such as promoting higher skills, learning activities and communication facilities are expected to have a differential impact on growth across regions, and are only likely to be beneficial for some regions -namely over-performers.

In Table 4, i.e. when including country fixed effects, we also find that the set of robust determinants differs across quantiles. In particular, we find that the capital city dummy (CAPITAL) and the share of gross fixed capital formation (SHGFCF) are only found to be robust growth determinants at the higher quantiles (though the latter also at the lowest quantile), while the share of high educated workers (SHSH) tends to be robust at lower quantiles. This latter result is compatible with those reported above: when not including country fixed effects the share of highly educated workers is important as this was one of the driving forces behind the high growth rates in Eastern European countries. When including country fixed effects the result implies that human capital is an important factor of growth by enhancing technology adoption. Patenting activities (TP_0) are also found to be robust at the lowest quantiles. In this case, no general policy prescriptions can be made as there is no variable found to be robust across quantiles. Investment in physical capital is likely to benefit over-achievers, while investment in human capital is likely to benefit underachievers. results depending upon whether we include or exclude country effects. Secondly, we find that there are a number of variables that have a high inclusion probability across quantiles.

In the case when country effects are excluded these include whether the region is home to the country's capital and the initial per capita GDP. Thirdly, there are also variables that are only found to be robust for certain quantiles. Examples of such variables when country effects are excluded include the indicator of human capital, which is found to be relevant mainly for over-performers, while when country effects are included we find that the variable CAPITAL and the investment rate are only relevant for higher quantiles, while the share of high-skilled workers is more relevant at lower quantiles.

<<<Tables 3-6 around here >>>

The final two tables (5 and 6) report the posterior means and standard deviations of the estimated coefficients for the 10 th , 30 th , 50 th , 70 th and 90 th percentiles of the conditional growth distribution for those variables with a relatively high inclusion probability. xvii In terms of the posterior means of the model-averaged parameter, there are no surprises in terms of the signs of the variables. In Table 5 we find a negative mean on GDPCAP0 and a positive one on the remaining robust variables. There is some variation in the size of the posterior means across quantiles however. For GDPCAP0 the mean of the coefficient follows a U-shape being slightly larger (in absolute terms) at the lowest and highest quantiles indicating nonlinearities in the convergence process. For CAPITAL we find that the posterior mean of the parameter increases as we move to higher quantiles, while for the share of high educated workers (SHSH) the mean coefficient is highest at the middle and highest quantiles. This is also the case in Table 6 which reports the posterior means and standard deviations when including country fixed effects. We find positive means on all of the robust determinants as expected, but some differences in the size of the mean across quantiles. The mean on CAPITAL is again found to be increasing as we move to higher quantiles, as does that on the share of gross fixed capital formation (SHGFCF). For the share of high educated workers (SHSH) however the mean is found to be largest at the low and medium quantiles. For under performers the role of human capital endowment seems relatively important having positive effects on technology adoption and learning-by-doing. For high performers however other variables such as investment (i.e. embodied technical progress) become more relevant. From a policy perspective the effects of increasing the human capital stock is therefore expected to be larger for under-performers, whereas for over-performers policy measures geared toward efficient use and complementarities to the existing human capital stock would yield higher returns in terms of growth rates.

Conclusions

Growth within European regions in the recent past has been quite uneven. While many of these differences in regional growth can be accounted for by country performance and the convergence process of the Eastern European countries there remain significant differences in regional growth performance even after controlling for country effects. In this paper we seek to understand and identify the set of variables that robustly determine regional growth.

The paper differs from the previous literature to understand the robust growth determinants by allowing the set of robust determinants to differ across regions. In particular, we identify the set of robust determinants for both under-and over-achievers defined in terms of their growth performance. To do this we combine quantile regression analysis, which allows us to model regional growth at different points on the conditional growth distribution, and Bayesian Model Averaging (BMA) to select a small number of robust variables from a longer list of potential explanatory variables.

We obtain a number of interesting results from our analysis. Firstly, country specific factors are found to play an important role. The sign, size and significance of many variables differs depending upon whether we account for country effects or not. The list of robust variables we obtain using the BMA analysis (using both least squares and quantile regression models) is also found to differ depending upon whether country effects are accounted for or not.

Secondly, we find that there is considerable parameter heterogeneity across quantiles. This is reflected in two sets of results; those showing that the size of the parameters on a specific set of variables varies across quantiles and those showing that the set of robust variables differs across quantiles.

In terms of the robust set of variables we often find that measures of skill endowment (or human capital) are robust determinants, with a higher level of high skilled labour being associated with higher growth. When we account for country effects, investment in physical capital is also found to be a robust determinant of growth with the expected sign. In terms of the quantile results we tend to find that physical capital has a stronger association in overachievers, with the results on human capital depending upon whether we include country 29 | P a g e effects or not. While the policy relevance of these variables is clear, other robust variables lead to less clear-cut policy conclusions, in particular geography variables. The dummy for if a region is home to the country's capital city for example is often found to be robust across quantiles, with its association with growth being positive. This is likely to reflect a number of characteristics of capital cities, such as infrastructure, agglomeration economies and so on, but it is not clear how such effects can be replicated. Interestingly, initial GDP per capita which is often found to be relevant in existing studies is not found to be a robust variable when country effects are accounted for. i For a review of the empirical growth literature, see [START_REF] Temple | The new growth evidence[END_REF] and [START_REF] Durlauf | The new empirics of economic growth[END_REF].

ii [START_REF] Kalaitzidakis | A non-linear sensitivity analysis of cross-country growth regressions[END_REF] employ the same approach as [START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF] but allow for potential nonlinearities. They find more variables to be robustly related to growth, emphasising the importance of nonlinearities in the growth process.

iii Examples using cross-country data include Mello and Perrelli (2003), [START_REF] Osborne | The sources of growth at different levels of development[END_REF], [START_REF] Canarella | Parameter heterogeneity in the neoclassical growth model: A quantile regression approach[END_REF] and [START_REF] Foster | The impact of trade liberalisation on economic growth: Evidence from a quantile regression analysis[END_REF]. All of these papers find evidence of heterogeneous effects of some growth determinants across quantiles.

iv BMA using QR may be also embedded in classes of models which assess spatial correlation across variables or errors explicitly, but this falls outside the scope of this study.

v The figures are based on simple bivariate regressions of per capita GDP growth on each of the growth determinants.

vi Useful surveys of quantile regression methods include [START_REF] Buchinsky | Recent advances in quantile regression methods: A practical guideline for empirical research[END_REF] and [START_REF] Koenker | Quantile regression[END_REF]. A book length treatment of the subject is provided by [START_REF] Koenker | Quantile Regression[END_REF].

vii Quantile regression coefficients measure the marginal effect of changes in the independent variables on the dependent variable for representative under-and over-achieving countries in terms of growth and not slow and fast growing countries per se. This can be contrasted with OLS which considers the average behaviour of representative countries.

viii Overviews of BMA are provided by [START_REF] Raftery | Bayesian model averaging for linear regression models[END_REF], [START_REF] Hoeting | Bayesian model averaging: A tutorial[END_REF], [START_REF] George | Model uncertainty[END_REF] and [START_REF] Doppelhofer | Model Averaging[END_REF].

ix This section follows closely the description of [START_REF] Raftery | Bayesian model selection for social research[END_REF] and [START_REF] Raftery | Bayesian model averaging for linear regression models[END_REF] who provide a fuller description of BMA techniques.

x Originally we started with a slightly larger set of variables. Some of these were dropped however because of issues of multicollinearity.

xi Admittedly, endogeneity may still be present in some models despite the (Granger-) causal structure that we have imposed in our specifications by measuring the regressors at the beginning of the period. A more systematic account of the issue of endogeneity in the setting of quantile-BMA falls outside the scope of this piece of research and is proposed as a potentially fruitful avenue for further research. In particular, recent results by [START_REF] Moral-Benito | Determinants of Economic Growth: A Bayesian Panel Data Approach[END_REF] and [START_REF] Chernozhukov | Inference on Instrumental Quantile Regression Processes[END_REF] may prove helpful in this respect.

xii When country effects are controlled for this is done using the within transformation (i.e. subtracting from each observation the country mean of the relevant variable).

xiii We checked the convergence of the MC3 algorithm by computing the correlation between posterior model probabilities based on the Markov chain frequencies and the exact marginal likelihoods (as proposed by Fernández et al. 2001). In all reported results this correlation was above 0.95.

xiv We take the prior inclusion probability as the threshold to define robust variables. The intuition for this choice is that it helps us identify variables for which the probability of inclusion in the true model increases after observing the data.

xv These results are available from the authors upon request. The robustness of the other variables as growth determinants is not affected by the use of these subsamples. xvii The full set of results is available upon request.

  a g e conditional growth distribution. Parameter heterogeneity is potentially even more relevant

  techniques being applied in economics. Applications of BMA to economic growth include[START_REF] Zellner | Bayesian and non-Bayesian methods for combining model and forecasts with applications to forecasting international growth rates[END_REF] Zellner (1993), Fernandez et al (2001), Leon-Gonzalez and Montolio

  a rapid catching-up process beginning from 1995 onwards for most, though not all, of these countries. In addition, we are only able to obtain data on most of the explanatory variables we include from 1995 onwards in a comparable and consistent manner. The dataset thus covers the period of strong European integration, beginning with the expansion to 15 members in 1995 and to 25 in 2004, when ten of the twelve Eastern European countries joined the EU (Bulgaria and Romania becoming members in 2007).The dependent variable in our analysis is the average yearly growth rate of real GDP per capita (gGDPCAP) over the period 1995-2005. We use information on 35 potential determinants of growth. x Where possible the first year for which data are available is used when measuring the explanatory variables in order to minimise problems of endogeneity.xi The variables are listed and described in the appendix. The set of variables included is on the one hand motivated by the various growth theories but also by the availability of comparable data across the 255 regions. It should be noted here that we have to use a balanced dataset in that there are no missing values. In the appendix we have grouped the data into six groups comprising various explanatory variables. For example, one group includes initial conditions and factor accumulation which is particularly emphasised in neoclassical growth theories but also in models emphasising technology gaps and catchingup. The second group includes variables capturing human capital which plays a central role in endogenous growth models by supporting regional innovation and the dissemination of knowledge. Infrastructure and socio-geographic variables are particularly emphasised in economic geography and spatial growth models and capture the effects of proximity to labour and product markets. Variables related to innovation are again related to a g e endogenous growth theories. Finally, a set of employment related variables is included

  each econometric setting (BMA based on OLS and QR) we present the results corresponding to both models with and without country fixed effects.xii The use of country fixed effects has an important effect on the interpretation of the resulting parameters. The speed of income convergence, for instance, refers to the convergence process towards a unique, European steady state (after controlling for the other variables in the model) in terms of income per capita in the case without country fixed effects. On the other hand, the income convergence process (and its speed) refers to a country-specific income level for the setting with fixed effects. In principle, we could have included the individual country dummies as regular regressors in the BMA framework. While this is unproblematic from a statistical point of view, it makes the interpretation of results unnecessarily complicated,

  , infrastructure and the polarization of, for instance, administrative services. The importance of this dummy is however also related to the inclusion of Eastern

  Figure 1: Initial GDP per capita Figure 2: Share of gross fixed capital formation in

  xvi A deeper analysis of the Williamson hypothesis falls outside the scope of this paper. Crespo Cuaresma et al. (2009) investigate this issue further.

  The posterior variance reflects both the weighted average of the within-model posterior variances, and the between-model variation of the model-specific posterior means. In addition to the posterior means and standard deviations, BMA provides the posterior

				,			(2)
	where,
	inclusion probability of a candidate regressor,	(3) , by summing the posterior
	is the integrated likelihood of model model probabilities across those models that include the regressor. , is the prior density of	under
	F o r is the likelihood, and the true model (assuming that one of the models considered is true). The posterior model model , is the prior probability that is F o r If all possible subsets are considered as potential models then the cardinality of the set is F o r
	P e probabilities can thus be obtained as the normalised product of the marginal likelihood for each model and the prior probability of the model P P . Notice that for the e e
			e r e r e r
			R R R
			e e e
	v i e The posterior mean and variance of a regression coefficient, , are then given by, v i e v i e
					w w w
	conditional quantile of conditional on	as,	O n l O n l model O n l	for the	(4)
					y y y
	where	is the	quantile of	and	is a set of parameters at the	(5)
	Here quantile to be estimated. Bayesian inference about the parameter attached to denotes the posterior mean of under model , and is equal to zero if at the	is
	quantile is given by rewriting equation (1) as,
	not included in
						15 | P a g e 16 | P a g e

data on a dependent variable, , a number of observations, , and a set of candidate regressors the variable selection problem is to find the best model, or the most appropriate subset of regressors out of the total set of candidate regressors. In what follows we sketch out the basic intuition behind BMA methods. ix We begin by denoting the set of all models considered, where each model represents a subset of the candidate regressors, . Model has the form, where is a subset of , is a vector of regression coefficients to be estimated and is the standard iid error term. We denote by the vector of parameters in . Taking into account model uncertainty, Bayesian inference about the parameter attached to , a variable in , is, (1) i.e. the average of the posterior distributions under each model weighted by the corresponding posterior model probabilities. This is what is termed Bayesian Model Averaging (BMA). The posterior probability of model is, simple case the posterior odds for a model against the other can be readily written as the product of the Bayes factor and the prior odds. Further assuming equal priors across models, the posterior odds are equal to the Bayes factor. . The posterior mean is therefore the weighted average of the modelspecific posterior means, where the weights are equal to the model's posterior probabilities. . As such, even with a moderate number of regressors we have an extremely large number of models and estimating all is typically not feasible (e.g. with 30 regressors we have around one billion models and with 40 about two trillion). A number of approaches have been developed to help deal with this problem, examples including a Markov Chain Monte Carlo Model Composition algorithm

[START_REF] Madigan | Bayesian graphical models for discrete data[END_REF]

and a branch-and-bound algorithm developed by

[START_REF] Raftery | Bayesian model selection for social research[END_REF]

.

2.3. Combining Quantile Regression with BMA

In order to consider whether the set of robust growth determinants differs across quantiles we need to combine QR with BMA. To do this we can write

Table 3 :

 3 Inclusion Probabilities across Quantiles (no country effects)

						Regional Studies			Page 44 of 57
	Variable	10th	20th	30th	40th	50th	60th	70th	80th	90th
	GDPCAP0	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
	CAPITAL	0.220	0.810	0.895	0.988	0.997	0.999	1.00	1.000	1.000
	SHSH	0.120	0.158	0.654	0.394	0.378	0.511	0.608	0.293	0.916
	SHLLL	0.057	0.751	0.293	0.543	0.578	0.391	0.104	0.041	0.033
	INTF	0.798	0.123	0.084	0.036	0.007	0.007	0.005	0.014	0.011
	ERET0	0.015	0.036	0.036	0.068	0.095	0.111	0.055	0.038	0.283
	ART0	0.011	0.018	0.029	0.063	0.035	0.032	0.032	0.043	0.093
	URT0	0.011	0.014	0.031	0.046	0.057	0.054	0.029	0.017	0.068
	AIRPORTDENS	0.087	0.021	0.027	0.037	0.030	0.030	0.012	0.007	0.005
	PATENTHT	0.028	0.061	0.037	0.029	0.016	0.019	0.012	0.018	0.004
	PATENTICT	0.032	0.052	0.026	0.028	0.016	0.010	0.010	0.012	0.003
	TELH	0.003	0.007	0.013	0.017	0.009	0.010	0.024	0.091	0.007

http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl Page 49 of 57 http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl

Table 5 :

 5 Posterior Mean of Regressors across Quantiles (no country effects)

						Regional Studies				Page 50 of 57
		10th		30th		50th		70th		90th
	Variable
		Mean	S.D	Mean	S.D	Mean	S.D	Mean	S.D	Mean	S.D
	GDPCAP0	-0.02279	0.00392 -0.01759 0.00252 -0.01893	0.00180	-0.01810	0.00251	-0.02005	0.00346
	CAPITAL	0.00281	0.00549 0.01120 0.00514 0.01678	0.00494	0.02881	0.00388	0.03135	0.00548
	SHSH	0.00446	0.01316 0.03208 0.02517 0.01834	0.02491	0.03136	0.02703	0.03478	0.01478
	SHLLL	0.00215	0.00959 0.01335 0.02171 0.02444	0.02206	0.00364	0.01126	0.00135	0.00872
	INTF	0.03249	0.02074 0.00265 0.00947 0.00011	0.00186	0.00000	0.00122	0.00019	0.00300
	ERET0	0.00018	0.00196 0.00087 0.00516 0.00305	0.01060	0.00167	0.00759	0.00748	0.01303
	ART0	0.00014	0.00192 0.00089 0.00638 0.00118	0.00719	0.00107	0.00665	0.00280	0.01003
	URT0	-0.00016	0.00233 -0.00089 0.00571 -0.00221	0.00986	-0.00092	0.00613	-0.00201	0.00841
	AIRPORTDENS	0.43450	1.47956 0.13747 0.89176 0.13390	0.83511	0.02449	0.28517	-0.00039	0.13825

http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl

Table 6 :

 6 Posterior Mean of Regressors across Quantiles (country effects)

		10th		30th		50th		70th		90th
	Variable
		Mean	S.D	Mean	S.D	Mean	S.D	Mean	S.D	Mean	S.D
	CAPITAL	0.00001 0.00029 0.00002 0.00037 0.00126 0.00358 0.01315 0.00423 0.02001 0.01535
	SHSH	0.01692 0.03759 0.06295 0.02856 0.04245 0.03309 0.00328 0.01541 0.00894 0.02138
	SHGFCF	0.03027 0.02455 0.00057 0.00407 0.00089 0.00547 0.02597 0.02046 0.04381 0.02778
	TP_0	0.00189 0.00128 0.00003 0.00021 0.00000 0.00007 0.00000 0.00003 0.00000 0.00001
	PATENTBIO -0.00185 0.02763 0.00040 0.01889 0.00210 0.02418 0.01076 0.05178 0.11853 0.17419
	INTF	0.00073 0.00540 -0.00004 0.00120 0.00000 0.00095 0.00000 0.00085 0.02743 0.03364
	GDPCAP0	-0.00031 0.00216 -0.00006 0.00061 -0.00003 0.00045 -0.00002 0.00045 -0.00623 0.00894
	SHCE0	-0.00114 0.00602 -0.00142 0.00783 -0.00272 0.01019 -0.00119 0.00633 -0.00030 0.00353
	LEVSH	0.00062 0.00145 0.00005 0.00032 0.00001 0.00012 0.00000 0.00005 -0.00001 0.00015

Acknowledgement:

This paper is based on a background study written for the European Commission's DG Regional Policy within the project "Analysis of the Main Factors of Regional Growth: An indepth study of the best and worst performing European regions" (contract no. 2007.CE.16.0.AT.209). We would like to thank the participants of the WIIW Workshop on Regional Growth, as well as three anonymous referees for helpful comments on earlier drafts of this paper. The usual disclaimer applies.

Appendix: Data Description

Data used in this study are collected from various sources, in particular: the Eurostat Regio database, Eurostat LFS database, ESPON (for details on these variables see http://www.espon.eu/), and Cambridge Econometrics. The period covered is 1995-2005. Variables capturing initial conditions are taken for 1995 or the first year for which data are available.

<<< Table A1


L Champaney
email: laurent.champaney@lmt.ens-cachan.fr

A domain decomposition method for studying the effects of missing fasteners on the behavior of structural assemblies with contact and friction

Keywords: Missing Fasteners, Contact, Friction, Uncertainties, Assemblies, Optimization

The aim of the present work is to present an efficient numerical tool for the simulation of the behavior of 3D assemblies of structures with contact and friction when strong structural modifications occur. The paper focuses on the effects of missing connection elements (screws, bolts, rivets, . . . ) on the behavior of the assembly. A fast and efficient domain decomposition strategy for predicting the responses of an assembly for different configurations of missing elements is proposed. The efficiency is provided by the use of a pre-calculated solution as initialization for the calculation of a new design configuration. Applications are concerned with bolted joints under quasi-static loads.

Introduction

Many mechanical connections are made of multiple fastener assemblies so that the load is distributed on the different connecting elements. Transmission of forces in such assemblies is allowed by both friction forces due to pre-strain of the connectors and contact forces on the body of the fasteners (screws, bolts or rivets). The load capacity of the connection strongly depends on the way forces are distributed on each one of the fasteners. It is extremely important to be able to establish this distribution at the design stage. Moreover, being able to predict modifications of force distributions due to a change of state of the connection (loss of adherence, loosening of bolts, . . . ) or imperfections with respect to a nominal configuration (imperfect hole clearance, imperfection of friction properties, missing fasteners, . . . ) is also of great importance.

There is an extensive literature on the behavior of multi-fastener assemblies for nominal perfect geometries and given state of the connections. Analytical works [START_REF] Barrois | Stresses and displacements due to load transfer by fasteners in structural assemblies[END_REF][START_REF] Yen | Multirow joint fastener load investigation[END_REF][START_REF] Wei-Xun | Load distribution of multi-fastener laminated composite joints[END_REF] offer some quick evaluation techniques for establishing the load distribution in the assemblies. In more recent works [START_REF] Ekh | Finite element modeling and optimization of load transfer in multi-fastener joints using structural elements[END_REF], finite element calculations are used to optimize the load transfer in such connections. In order to precisely establish the load transfers, many simulations use 3D finite element models. They include a fine description of frictional contact zones. Simulations using such models under quasi-static loadings are obviously not straightforward and can be time consuming. Being able to carry out such simulations is still a challenge.

The effects of friction on the load capacities and the behavior of bolted joints have been intensely studied and a review is presented in [START_REF] Gaul | The Role of Friction in Mechanical Joints[END_REF]. Loosening of the fasteners can have critical effects on the distribution of friction forces and so on the distribution of loads on the fasteners. The effects of such phenomena have been studied by many authors in the last decade using either experimental or numerical approaches [START_REF] Pai | Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load[END_REF][START_REF] Yang | Detection of bolt loosening in CC composite thermal protection panels: I. Diagnostic principle[END_REF][START_REF] Jaglinski | Study of Bolt Load Loss in Bolted Aluminum Joints[END_REF][START_REF] Zhang | Finite Element Modeling of Self-Loosening of Bolted Joints[END_REF].

For assembling purposes, gaps (hole clearances) must be introduced between the assembled parts and the fasteners. Due to machining, the diameters and positions of the holes can be imperfect and so the gaps can be strongly uncertain.

The effect of uncertainties of the geometry (hole clearance for example) on the behavior of the assembly has been studied in very few experimental works [START_REF] Stanley | Measurement of load distribution in multibolt composite joints in presence of varying clearance[END_REF].

The simulation of the effects of all the types of imperfections and changes of state is still a challenge. The approach that is presented in this paper has been successfully used for the prediction of the effects of imperfections of pre-strain of bolts and of friction effects [START_REF] Boucard | A suitable computational strategy for the parametric study of problems with multiple contact[END_REF][START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF]. It has recently been extended to the study of imperfections of hole clearances [START_REF] Champaney | Computational strategy for taking geometrical uncertainties into account in structural assemblies[END_REF].

This paper deals with the study of the effects of missing fasteners on the behavior of connections. When the number of fasteners is high, it can frequently happen that a fastener is forgotten when the connection is assembled. Moreover, loosening of a bolt can end by a complete disappearance of the bolt. Some experimental studies [START_REF] Shimada | Detection of missing fastener based on vibration mode analysis using fiber Bragg Grating (FBG) sensors[END_REF] offer some procedures for the detection of missing connectors but no specific work on numerical simulations is available. Predicting the effect of a missing fastener might be trivial as one has just to simulate the behavior of the connection without modeling the missing component. When the number of fasteners is elevated, this can lead to a large number of different simulations. This paper details a computational strategy for an accelerated determination of the solutions of such simulations.

The proposed strategy relies on a specific approach that has been developed at Laboratory of Mechanics and Technology (LMT Cachan, France) during the last twenty years and which is dedicated to the simulation of the behavior of 3D assemblies of structures. This calculation technique is based on a decomposition of the assembly that emphasize the connections by modeling them by an interface behavior. The resolution scheme is the LArge Time INcrement (LATIN) Method [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. The approach builds, at each iteration, an approximate solution of the problem on the whole studied time interval. Its efficiency as already been proved on the calculation of solutions for quasi-static analysis of 3D assemblies of structures including a large number of frictional contact zones [START_REF] Champaney | Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions[END_REF]. This approach is highly parallel and its extension to dynamics on parallel computers is presented in [START_REF] Odièvre | A parallel, multiscale domain decomposition method for the transient dynamic analysis of assemblies with friction[END_REF]. This approach has successfully been developed for taking into account the influence of uncertainties on connection parameters such as friction coefficients and pre-strains [START_REF] Boucard | A suitable computational strategy for the parametric study of problems with multiple contact[END_REF]. The main idea is the reusing of one solution that was calculated for a set of design parameters to initialize the calculation for a new set of parameters. When the responses that are obtained for both sets of parameters are close, this initialization allows a great reduction of the computational costs for the second calculation. When, due to the nonlinear behavior of the assembly, the two solutions are not so close, the reduction of the computational costs is still significant. To achieve the initialization, the design parameters are simply changed along the iterations in order to efficiently build solutions by re-using the best pre-calculated solutions for different sets of parameters [START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF].

In this paper, the proposed approach is extended to the quick calculation of a series of design configuration of assemblies where fasteners are successively missing. The purpose of the strategy that is proposed here is to get fine informations on the effects of the lack of a connecting element. In the situations that are studied here, one only seeks for the force distributions in the connecting elements before any degradation occurs in the assembly. So, only small displacements and deformations are considered with no plasticity or damage. An accurate description of the contact and friction interactions is needed in order to get the load distribution in the connectors. Obtaining this level of accuracy is only possible if conformal meshes are used on the contact zones. However, taking non-conformal meshes into account is possible ; this procedure is not presented here but is detailed in [START_REF] Champaney | Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions[END_REF].

The resolution scheme and the way Coulomb's friction contact condition are taken into account are first presented. The principles of reusing solutions for initialization is detailed. Then the way the lack of fasteners is taken into account is shown. Two examples show the efficiency of the approach and some possible improvements that concern the selection of solution for initialization.

Resolution scheme: LATIN method

A brief presentation of the main aspects of the LATIN method is proposed here after. Details of the method are available in [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF] and the application to the treatment of frictional contact problems are proposed in [START_REF] Champaney | Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions[END_REF]. In this paper, we address structural mechanics problems including unilateral contact and friction.

That is to say that we need to solve systems of nonlinear partially differential equations in time and space. Here, a discrete formulation of the LATIN method is presented only. It is obtained using a classical finite element discretization and time discretization of the continuous formulation that can be found in [START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF], which also gives the details of the discretization issues.

Decomposition of an Assembly

The model of structural assembly that is used here is composed of substruc-

tures Ω E that communicate with each other through interfaces (see figure 1).

Substructures are the components of the assembly and interfaces model the connections between the components. In some cases, substructures model sub-parts of one component. In this paper, we only consider friction unilateral contact interfaces with Coulomb's friction law. The two substructures that are connected by interface Γ EE ′ are denoted Ω E and Ω E ′ .

On one interface are defined four fields. Two force fields, represented by vectors F EE ′ and F E ′ E , model the action of the interface on each substructure. Two dual velocity fields, represented by vectors U EE ′ and U E ′ E , model the motion of each substructure seen from the interface (figure 1).

Problems on the substructures

Classical finite element meshes are used on each substructure, and the different meshes match on the interfaces (figure 2). In the following, vectors U refers to nodal displacements values, vectors U to the nodal velocity values and vectors F to nodal forces.

A substructure Ω E must cope with the velocity field UEΓ of its neighboring interfaces at each time step:

UEΓ = C EΓ UE , (1)
where subscript EΓ collects all the neighboring interfaces of a substructure E.

U E is the displacement vector of the whole substructure. C EΓ is a boolean matrix mapping the substructure degrees of freedom on the boundary.

The same substructure must also satisfy to the equilibrium equation when subject to forces due to the neighboring interfaces F EΓ , and to the given body forces F d E , at each time step:

K E U E = F d E + C T EΓ F EΓ , (2)
where superscript T denotes transposition.

Problems on interfaces

At each time step, the equilibrium of the interface Γ EE ′ between substructures E and E ′ is enforced by the equation:

F EE ′ + F E ′ E = 0 ( 3
)
and a constitutive equation in which the forces depend on the displacement discontinuity:

F EE ′ = f (U E ′ E -U EE ′ ), (4)
where the function f can be nonlinear. This function is used to describe the nonlinear behavior of the connection that is modeled by the interface.

Subscript EE ′ refers to the side of substructure E along the interface Γ EE ′ ;

especially

F EE ′ (respectively F E ′ E
) is the action of the same interface on substructure E (resp. on the substructure E ′ ). The examples that are proposed in this paper only use compatible meshes on the interfaces (figure 2). The interface is composed of pairs of nodes each of them belonging to one of the opposite substructures. The nodal values in the velocities and forces vectors are ordered such that the interface equations can be written under a vectorial form.

Note that subscript EΓ refers to a data structure that is related to each substructure independently, while subscripts EE ′ and E ′ E refer to the same quantities, but related to an interface-oriented data structure. Obviously, data transfers are required between the two descriptions. Each of these two descriptions will be used in the following, depending if the expressions are related to a substructure or an interface.

The iterative scheme

The iterative scheme that is used here is derived from the LATIN method proposed by Pierre Ladevèze [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF]. The main principle is to separate the equations in order to avoid the solving of both a global and a nonlinear problem.

As the behaviour of the material is linear, a solution is completely given by the boundary displacement and force vectors on each substructure or by the interface displacement and forces on each interface:

s = E ( UEΓ , F EΓ ) = Γ EE ′ ( UEE ′ , F EE ′ , UE ′ E , F E ′ E ) .
The solution procedure is to search solutions s that will alternatively satisfy the global linear equations (kinematic admissibility and equilibrium on the substructure) and then to the local equations (interface equations). This leads to a decoupling of the problem. This procedure can be summarized on Figure 3 for which s n is the solution that satisfies to the global equations at iteration n and s n+ 1 2 is the one that satisfies to the interface equations between iterations n and n+1 (subscript n + 1 2 is given away in the following). The solution s n lies in a subspace represented by a straight line because it corresponds to linear equations. The solution s lies in a subspace represented by a curved line because the interface behavior can be nonlinear for more complex connection conditions.

Searching for s n is called the linear stage, and searching for s, the local stage. At each stage, the solutions are found thanks to local and linear search directions E -and E + , represented by straight lines on Figure 3.

Local stage

At this stage, a solution s is searched from the known solution of the previous linear stage s n and with the upward search direction E + . This search direction is defined interface by interface as for a augmented Lagrangian procedure:

F EE ′ = F EE ′ + k( U EE ′ -UEE ′ ) (5)
and

F E ′ E = F E ′ E + k( U E ′ E -UE ′ E ), ( 6
)
where k is a scalar parameter. As for augmented Lagrangian formulations, the rate of convergence may depend on this parameter, but the final solution does not.

The constitutive relations of the interface for a perfect connection behavior are:

U EE ′ = U E ′ E ; F EE ′ + F E ′ E = 0. ( 7
)
Using the search direction, the interface vectors can thus be computed as:

F EE ′ = -F E ′ E = 1 2 (F EE ′ -F E ′ E ) - 1 2 k( UEE ′ -UE ′ E ) (8)
and

U EE ′ = U E ′ E = 1 2 ( UEE ′ + F EE ′ ) - 1 2k (F EE ′ + F E ′ E ). ( 9
)
The resolution of the local stage for frictional contact behavior is presented later in the paper.

Discretization

Due to the form of the search directions ( 5) and ( 6), a discretization of the interface forces is also adopted. The same finite element interpolation functions are used for both the velocities and the forces on each side of one interface. Thus, the force vectors F EE ′ and F E ′ E contain the nodal values of the components of the forces. On substructure E, F EΓ is a vector of finite element generalized forces that is an assembly of the forces coming from each interface. So, F EΓ is linked to the boundary force vectors F EE ′ by a unit interface mass matrix.

Linear stage

At this stage, a solution s n is searched from the known solution of the previous local stage s and with the downward search direction E -. On each interface, this search direction is conjugate to the previous one:

F EE ′ = F EE ′ -k( UEE ′ -U EE ′ ). ( 10
)
Assembled on the neighboring interfaces on substructure E, it leads to:

F EΓ = F EΓ -k( UEΓ -U EΓ ). ( 11
)
The global linear system to solve at this stage is obtained with this search direction [START_REF] Boucard | A suitable computational strategy for the parametric study of problems with multiple contact[END_REF], the boundary conditions (1) and the equilibrium of substructure E (2):

K E U E + C T EΓ kC EΓ UE = F d E + C T EΓ ( F EΓ + k U EΓ ). ( 12
)
This problem is of first order in time. It is solved at each time step starting from initial conditions by the use of an implicit integration scheme. Finally, this

gives the boundary velocity UEΓ = C T EΓ UE and the boundary forces using the downward search direction at each time step.

The frictional contact interface

Contact problems are characterized by constraints such as non-penetration conditions, and an active area of contact -that is, an area where contact effectively occurs -that is unknown a priori. For these reasons, the problems lead to stiff non-linear systems of equations. Several approaches exist for solving static contact problems [START_REF] Zhong | Static contact problems. A review[END_REF][START_REF] Lebon | Contact problems with friction: models and simulations[END_REF][START_REF] Wriggers | Computational contact mechanics[END_REF]. In most of them, the numerical methods that are employed for enforcing the contact constraints can be grouped into Lagrange multiplier and penalty methods. The penalty methods [START_REF] Kikuchi | Penalty/finite element approximations of a class of unilateral contact problems[END_REF][START_REF] Armero | Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems[END_REF] are closely related to the regularization of the contact constraints. They are usually formulated in terms of the displacement variables, and therefore are primal methods. They allow treating contact as a material behavior, as exemplified by the method of joint finite elements [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution methods[END_REF]. Penalty methods can experience various numerical difficulties, especially ill-conditioning, when a too large or too small penalty parameter is introduced. Lagrange multiplier methods are dual methods where the multipliers, which represent the contact reaction forces, are introduced in order to enforce exactly the non-penetration conditions. Augmented Lagrange multiplier methods [START_REF] Arora | Multiplier methods for engineering optimization[END_REF][START_REF] Klarbring | Mathematical programming and augmented Lagrangian methods for frictional contact problems[END_REF][START_REF] Radi | Treatment of the frictional contact via a Lagrangian formulation[END_REF][START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF][START_REF] Koko | Uzawa block relaxation domain decomposition method for the two-body contact problem with Tresca friction[END_REF] result in mixed formulations involving both displacement and force unknowns. The numerical solution schemes underlying both the Lagrange multiplier and augmented Lagrange multiplier methods are often related to the Uzawa algorithm [START_REF] Arrow | Studies in linear and non-linear programming[END_REF][START_REF] Simo | An augmented lagrangian treatment of contact problems involving friction[END_REF].

Considering the strategy that was presented in the previous sections, the approach that is proposed here is a mixed domain decomposition method for a contact and friction problem of Augmented Lagrangian type. A dual domain decomposition method for the same type of problems was recently proposed in [START_REF] Avery | The FETI family of domain decomposition methods for inequality-constrained quadratic programming: Application to contact problems with conforming and nonconforming interfaces[END_REF]. Moreover, some recent examples of domain decomposition strategies for contact problems are proposed in the following papers [START_REF] Danek | Domain decomposition for generalized unilateral semi-coercive contact problem with given friction in elasticity[END_REF][START_REF] Dostal | Feti based algorithms for contact problems: scalability, large displacements and 3d coulomb friction[END_REF][START_REF] Krause | A non-smooth multiscale method for solving frictional twobody contact problems in 2d and 3d with multigrid efficiency[END_REF].

The following sections describe the form of function f in the case of classical Coulomb's frictional contact conditions. Considering that the contact status (closed, open, sticking, sliding) can be different on each point of the interface, the local stage is solved node by node. As compatible meshes are used on the interfaces (figure 2), one point corresponds to two opposite nodes. W EE ′ is the three-component displacement vector that is the restriction of the boundary displacements to the considered node on substructure E side and W EE ′ on substructure E ′ side. T EE ′ and T E ′ E are the three-component force vectors that is the restriction of the boundary forces to the same nodes.

In order to separate the normal behavior and the tangential one, local coordinates are used. The outward local normal at each point of an interface is defined by the three-component vector N . N is oriented from substructure E towards substructure E ′ . The 3 × 3 projection operator on the local tangential plane P is defined by :

P W = W -(N T W )N. ( 13
)
For the definition of contact and sliding conditions, the jump in displacement on the interface is:

D EE ′ = (W E ′ E -W EE ′ ). ( 14
)

The unilateral contact problem

At each time step, the non-penetrating contact conditions are:

• Open: if N T D EE ′ > 0 then N T T EE ′ = N T T E ′ E = 0, • Contact: if N T D EE ′ = 0 then N T T EE ′ = -N T T E ′ E ≤ 0.
These conditions can be simply written as:

• Open: c n > 0,

• Contact:

c n ≤ 0,
where the contact indicator c n is:

c n = 1 2 N T D EE ′ - 1 2k N T ( T E ′ E -T EE ′ ). ( 15
)

The friction problem

At each time step, Coulomb's friction conditions can be written as:

• Stick : if P T EE ′ < µ|N T T EE ′ | then P ḊEE ′ = 0, • Slip: if P T EE ′ = µ|N T T EE ′ | then ∃λ > 0 such that P ḊEE ′ = -λ T EE ′ .
These conditions can be simply written as:

• Stick : G t ≥ µ|N T T EE ′ |, • Slip: G t < µ|N T T EE ′ |,
where the slip indicator G t is the vector:

G t = k 2 P ḊEE ′ - 1 2 P ( T E ′ E -T EE ′ ). ( 16
)

Resolution

The resolution is carried out by projecting the solution of the previous linear stage onto the contact and friction conditions following the search directions:

( T EE ′ -T EE ′ ) = k( Ẇ EE ′ -Ẇ EE ′ ), (17)
(

T E ′ E -T E ′ E ) = k( Ẇ E ′ E -Ẇ E ′ E ). ( 18
)
The status (Open, Contact, Stick or Slip) of each point of the interface is obtained explicitly since the indicators c n and G t can be derived from the previous solution using the search directions. An implicit time integration scheme is used to calculate the contact indicator c n using an expression in terms of the displacements (see Eq. 15):

D (t+1) EE ′ = D (t) EE ′ + ∆t Ḋ(t+1) EE ′ . (19)
The initial condition of the time integration is used to take into account the initial gap. Thus, the contact indicator is written as:

c n = 1 2∆t N T D EE ′ - 1 2k N T ( T E ′ E -T EE ′ ), ( 20
)
in order to obtain, using the time integration (Eq. 19) and the search directions (Eq. 17 and 18):

c n(t+1) = 1 2∆t N T D (t+1) EE ′ - 1 2k N T ( T (t+1) E ′ E -T (t+1) EE ′ ) = 1 2 N T Ḋ(t+1) EE ′ - 1 2k N T ( T (t+1) E ′ E -T (t+1) EE ′ ) + 1 2∆t N T D (t) EE ′ = 1 2 N T Ḋ(t+1) EE ′ - 1 2k N T (T (t+1) E ′ E -T (t+1) EE ′ ) + 1 2∆t N T D (t) EE ′ . So c n(t+1) = 1 2 N T Ḋ(t+1) EE ′ - 1 2k N T (T (t+1)
E ′ E -T (t+1) EE ′ ) + 1 2∆t N T D (t) EE ′ . (21)
Thus, the contact indicator is calculated incrementally from the known solution of the previous global stage. The slip indicator G t (Eq. 16) is calculated explicitly using the search directions (Eq. 17 and 18):

G t = k 2 P ḊEE ′ - 1 2 P (T E ′ E -T EE ′ ). ( 22
)

Properties of the iterative scheme

As a conclusion, this iterative scheme presents the following properties:

• the strategy gives an approximation of the solution, over the complete time interval, at each iteration,

• the problems that must be solved on each substructure, at each time step and each iteration [START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF], can have a large size, depending on the way the assembly is decomposed and on the mesh size. However, It is important to notice that the involved operators do not depend on the interface parameters and on the contact statuses. Thus they can be factorized once during the initialization. This reduces the numerical cost of each iteration.

• when frictional contact is involved, the resolution of the local stage is completely explicit and does not require any iterative process,

• as any Augmented Lagrangian type scheme, the convergence is not rapid but the algorithm is very robust and the iterations are not very costly.

The solution that is obtained when convergence is reached is not affected by the parameter k, which affects only the rate of convergence

• The quality of the solution is measured using an error indicator that gives an energy norm of the distance between two successive solutions s and s n .

The expression of this error indicator is given in [START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF] • For contact and friction problems where discontinuities can occur, the resolution strategies can be very sensitive to the time and space discretization.

The LATIN method acts as a regularization method and present a smooth convergence. This property has already been shown for buckling problems and is presented in [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics -New Approaches and Non-Incremental Methods of Calculation[END_REF].

Taking into account structural changes

Principles of multiple resolution

In order to calculate the influence of the design parameters, an multiple resolution strategy using the LATIN method has been implemented [START_REF] Boucard | A suitable computational strategy for the parametric study of problems with multiple contact[END_REF]. During the design process, several configurations, corresponding to different values of the sets of design parameters, must be simulated. Thus, some very similar calculations must be carried out in order to get the solutions corresponding to the different sets of design parameters. Multiple resolution means that, the similarity of the problem is used in order to reduce the computational costs.

The numerical cost is expected to be less than the cost of one computation multiplied by the number of studied configurations.

Our strategy consists in accelerating the calculation of one solution by using another solution as an initialization. Using the LATIN method, the iterative resolution for one set of parameters is initialized by the solution calculated for another set. This is done very simply by changing the parameters between two successive iterations. The strategy is described in figure 4. As each iteration of the LATIN method generates a solution on the whole time interval, the initialization overacts on the whole time interval. Due to the strong mechanical content of this initialization, the number of iterations needed to reach convergence for this new calculation is reduced. Indeed, if the two solutions are close, the number of iterations can be very small.

So far, the parameters whose influence have been taken into account are associated to the connection behavior: gaps, pre-strains, friction coefficient.

For such design parameters, this multiple resolution strategy leads to a great reduction of computational costs as shown in [START_REF] Champaney | Adaptive multi-analysis strategy for contact problems with friction. Application to aerospace bolted joints[END_REF]. In the example presented in this last paper, the initialization solution was the one computed just before. An optimization of the choice of the initialization is presented in [START_REF] Boucard | A multiscale strategy for structural optimization[END_REF].

In the present work, the same multiple resolution strategy is used to study cases where the design parameters are the presence of the different fasteners (bolts, screws, . . . ). In order to study the influence of missing fasteners on the behavior, a first iterative calculation, including all the fasteners, is performed until convergence is reached. Then, the substructures that correspond to missing fasteners (bodies, heads, nuts, . . . ) are de-activated. De-activation means that the substructures stay in the model but do not interact with the neighboring interfaces. Then, the iterative process is continued until convergence to the solution of this new configuration. As the iterative process is continued, the calculation of this last solution is then initialized by the solution of the previous situation where the fastener is present.

De-activating a substructure is a very simple action. The linear stage of the LATIN method is not performed on such structures. It only returns a null force field

F EΓ = 0, (23)
and the velocity field associated by the search direction:

UEΓ = U EΓ - 1 k (F EΓ -F EΓ ) = U EΓ + 1 k F EΓ . (24)
The next iterations, until convergence, calculate the new solution for the assembly without the missing fasteners, as if the interfaces between the missing substructures and the rest of the assembly were prescribed null force interfaces.

At the end of this second calculation, the concerned substructures are reactivated and some others are de-activated in order to study the influence of another missing fastener in a third calculation.

Improvements

As presented in the examples at the end of this paper, it is often observed that the solution for one missing fastener is closer to the solution with no missing fasteners than the solution for any other missing fastener. In our activation deactivation process during the iterations, the calculation for one missing fastener is naturally initialized by the solution of the previous calculated solution that often corresponds to another missing fastener.

In order to initialize each new calculation with the solution obtained for the assembly featuring all its fasteners, this last solution is computed first and then saved. A solution consists only in the velocity and force vectors on each

interface s = Γ EE ′ ( UEE ′ , F EE ′ , UE ′ E , F E ′ E ),

Example

In the following examples, only linear material behavior is considered. This means that the only non-linearities are unilateral contact and friction. One only seeks for the effects of missing connecting elements on the distribution of forces and the loading conditions on the most loaded element. These effects are studied under the assumption of small deformations and displacements. From these results a local re-analysis can be conducted in order to simulate the behavior of the most loaded connector. This last simulation can include non-linear behavior such as plasticity, damage and failure. Such simulations are not presented here.

As announced in the introduction, the proposed examples present conformal meshes on the contact zones. The meshes are obtained from CAD models using an automatic procedure. The CAD model does not include the connecting elements (bolts or screws) but only their localization and their type. A geometrical patch is automatically introduced where an element is located and conformal meshes are automatically built on this patch.

First example : bolted connection

This example deals with the assembly of three aluminium plates fastened by two steel bolts (figure 5). The assembly is submitted to traction. One seeks for the solution for the case where the two bolts are present in the assembly and the solutions in which each one of the bolts is successively missing. For symmetry reasons, only one fourth of the assembly is modeled (figure 6). A coarse mesh (7,819 dof) and a fine one (169,380 dof) are used for his example. Figure 8 shows the evolution of the LATIN method convergence indicator along iterations. One can easily notice the strong discontinuity of the evolution whenever substructures are activated or dis-activated. The number of iterations needed to reach convergence for the second calculation is only 40% of the number needed for the first calculation. The strong effect of the initialization is clearly shown here.

On the contrary, the number of iterations of the third calculation is the same as for the first one, when no solution selection technique is used. On such a two bolt structure, it is obvious that the solution of the second configuration can not be a useful initialization for the third one. When this last calculation is initialized with the solution of the first calculation (see figure 9), acceleration is observed. The efficiency of the acceleration is the same as the one observed for the second calculation.

Table 1 presents the number of iterations for each calculated configuration and for three different used strategies:

• without multi-resolution acceleration: one different calculation for each configuration

• multi-resolution without initial solution selection: each calculation is initialized by the calculation of the previous one,

• multi-resolution with initial solution selection: calculations #2 and #3 are initialized by the solution of calculation #1 (two bolts present).

Second example : wheel-shaft assembly

This example deals with the assembly of a gear wheel on a hollow shaft using eight pre-strained screws (figure 10). The hollow shaft is cantilevered on one end and the wheel is submitted to a force that corresponds to the force applied on the tooth of an helical gear.

Figure 11 presents the distribution of the von Mises equivalent stress for each of the nine calculated configurations. Configuration #1 is the assembly with its eight screws. For each configurations #2-9, one of the screws is missing.

Figure 8 shows the evolution of the LATIN method convergence indicator along iterations. One can easily notice the strong discontinuity of the evolution whenever substructures corresponding to screws are activated or deactivated.

The number of iterations needed to reach convergence for configuration #1 is nearly the same as the total number of iterations needed to get the solutions for configurations #2-9 (see figure 12).

Conclusions

An accelerated strategy for the quick calculation of the influence of missing fasteners on the behavior of connections was presented. It is based on the use of a decomposition of the assembly into substructures and interfaces and on an iterative resolution scheme based on the LATIN Method. The reusing of stored solutions for initialization allows a strong reduction of the number of calculated iterations and so of the computational costs. The model that is stored in the computer stays the same during the whole process. Structures corresponding to missing fasteners are simply deactivated. This limits the data exchanges. Moreover, the main operators remain constants during the process and are factorized only once at the beginning. The proposed examples of assemblies showed the efficiency of the approach and some possible improvements. tables

Figures

! E ! E' " EE'

  at each time step. Indeed, as the behavior of the substructures is linear, the internal solutions can easily be reconstructed from the boundary fields. Such a solution corresponds to a small amount of stored data. Each time a new calculation is started, i.e. each time fastener substructures are deactivated or reactivated, this solution is restored: forces and velocity vectors of the linear stage are set back to the stored values before continuing the iterations. Thus, the new calculation is initialized by the stored solution. The efficiency of this improvement based on the selection of the initial solution is shown on the next example.

Figure 7

 7 Figure 7 presents the fine mesh and the distribution of von Mises equivalent stress for the three studied configurations. Configuration #1 includes both the bolts. Left bolt is deactivated in configuration #2 and right bolt is deactivated in configuration #3.

Figure 1 :

 1 Figure 1: Decomposition of the structure

Figure 2 :

 2 Figure 2: Compatible meshes on the interfaces

Figure 3

 3 Figure 3: LATIN scheme

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Studied model, dimensions in millimeters

Figure 9 :

 9 Figure 9: Error Indicator along the iterations when the second and third calculations are initialized by the solution of the first one (coarse mesh).

Figure 12: Error indicator for the complete simulation (nine calculations).