Janet Gardner

Mark Clayton

Richard Allen

John Stein

Timothée Bonnet

The effects of temperature extremes on survival in two semi-arid Australian bird communities over three decades, with predictions to 2104

Keywords: semi-arid eastern Australia arid zone, Australian birds, avian guilds, climate change, energy and water budgets, population projections, species persistence, survival, temperature extremes, thermoregulation

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The effects of temperature extremes on survival in two semi-arid Australian bird communities over three decades, with predictions to 2104

INTRODUCTION

A signature of contemporary climate change is the changing frequency, intensity and duration of extreme weather events [START_REF] Ummenhofer | Extreme weather and climate events with ecological relevance: a review[END_REF][START_REF] Stillman | Heat waves, the new normal: summertime temperature extremes will impact animals, ecosystems, and human communities[END_REF]. Birds in the arid and semi-arid regions of the world are particularly vulnerable to temperature extremes, in part, because their high body temperatures are close to the lethal limit for endotherms, and small changes in temperature can cause thermal stress at both the hot and cold ends of the temperature range. Exposure to weather extremes can disrupt energy and water budgets with consequences for fitness that affect both survival and reproduction [START_REF] Mitchell | Revisiting concepts of thermal physiology: predicting responses of mammals to climate change[END_REF][START_REF] Huey | Predicting organismal vulnerability to climate warming: roles of behaviour, physiology, and adaptation[END_REF][START_REF] Boyles | Adaptive thermoregulation in endotherms may alter responses to climate change[END_REF].

Mortality associated with exposure to summer extremes can be a direct result of succumbing to acute heat stress via hyperthermia or lethal dehydration, exemplified by mass die-offs of budgerigars (Melopsittacus undulatus), zebra finches (Taeniopygia guttata) and Carnaby's cockatoos (Calyptorhynchus latirostris) during heatwave events in Australia [START_REF] Mckechnie | Feeling the heat: Australian land birds and climate change[END_REF][START_REF] Saunders | The impact of two extreme weather events and other causes of death on Carnaby's Black Cockatoo: a promise of things to come for a threatened species[END_REF]. In such cases, avian body temperature exceeding about 46°C is lethal [START_REF] Boyles | Adaptive thermoregulation in endotherms may alter responses to climate change[END_REF].

Fitness in terrestrial animals may also be affected by high temperatures well below those causing die-offs events. Growing evidence suggests that air temperatures in the mid to high 30 o Cs are harmful for endotherms [START_REF] Cunningham | Missed-opportunity costs and the response of birds and mammals to climate change[END_REF]. For example, prolonged exposure to temperatures >38 o C has been shown to increase the cost of avian thermoregulation by increasing the need for evaporative cooling to maintain body temperatures below lethal limits [START_REF] Mckechnie | Avian thermoregulation in the heat: evaporative cooling in five Australian passerines reveals within-order biogeographic variation in heat tolerance[END_REF]. Increasing exposure to these sub-lethal conditions increases the risk of mortality, when exposure is prolonged over hours, days and weeks [START_REF] Sharpe | Weighing the cost: the impact of serial heatwaves on body mass in a small Australian passerine[END_REF][START_REF] Sharpe | Too hot to handle? Behavioural plasticity during incubation in a small, Australian passerine[END_REF]. Indeed, recent work on Australian desert birds in the laboratory showed that chronic heat stress can cause organ damage (e.g. liver or kidneys) which can contribute to avian mortality over time [START_REF] Xie | Organ histopathology and hematological changes associated with heat exposure in Australian desert birds[END_REF].

High temperatures are also associated with changes to foraging patterns and microhabitat use (du Plessis et al., 2012;[START_REF] Edwards | The impact of high temperatures on foraging behaviour and body condition in the western Australian Magpie Cracticus tibicen dorsalis[END_REF][START_REF] Cooper | The field metabolic rate, water turnover, and feeding and drinking behaviour of a small avian desert granivore during a summer heatwave[END_REF]Fungi et al., 2019) which can lead to a reduction in food intake and loss of body condition. Southern pied babblers, Turdoides bicolor and southern yellow-billed hornbills (Tockus leucomelas), for example, were unable to maintain body mass on days when temperatures exceeded thresholds in the mid 30s°C (van de [START_REF] Van De Ven | The costs of keeping cool: behavioural trade-offs between foraging and thermoregulation are associated with significant mass losses in an arid-zone bird[END_REF]du Plessis et al., 2012) and Jacky Winters Microeca fascinans lost 2% of their mass on days ≥ 42 o C [START_REF] Sharpe | Weighing the cost: the impact of serial heatwaves on body mass in a small Australian passerine[END_REF]. Given that body condition is closely linked to survival and reproductive success [START_REF] Paniw | Life history responses of meerkats to seasonal changes in extreme environments[END_REF], increasing exposure to high summer temperatures is likely to have major fitness consequences as the climate warms.

Exposure to cold extremes can also impose thermoregulatory and energetic challenges for endotherms, although we have much less understanding of how birds respond to reductions in cold extremes that are expected to be beneficial for fitness as the climate warms. Energetic constraints and cold stress can drive reductions in body condition and impose carry-over costs on subsequent survival and reproduction [START_REF] Williams | Cold truths: how winter drives responses of terrestrial organisms to climate change[END_REF]. In Australia, changes in survival were associated with temperature extremes for two coexisting bird species monitored over 37 years, red-winged fairy-wrens Malurus elegans and white-browed scrubwrens Sericornis frontalis [START_REF] Gardner | Effects of extreme weather on two sympatric Australian passerine bird species[END_REF]. In scrubwrens, winter temperatures <5 o C were associated with lower survival within the same season while in both species, survival was associated with body size, and there was evidence that size-dependent mortality was mediated by carry-over effects of climate in the previous season [START_REF] Gardner | Effects of extreme weather on two sympatric Australian passerine bird species[END_REF]. Moreover, energetic costs of thermoregulation in the cold during periods of low food availability can be greater than those in warmer conditions during the breeding season [START_REF] Harrison | Carry-over effects as drivers of fitness differences in animals[END_REF][START_REF] Williams | Cold truths: how winter drives responses of terrestrial organisms to climate change[END_REF]. For example, energetic stress on cool days was greater than that associated with exposure to high temperatures between 40 and 45 o C in an arid zone population of zebra finch [START_REF] Cooper | The field metabolic rate, water turnover, and feeding and drinking behaviour of a small avian desert granivore during a summer heatwave[END_REF].

Warming winters can, or are predicted to, reduce winter mortality and lead to improvement in body condition [START_REF] Robinson | Weather-dependent survival: implications of climate change for passerine population processes[END_REF][START_REF] Ozgul | Coupled dynamics of body mass and population growth in response to environmental change[END_REF]. For example, warming winters led to longer growing seasons with concomitant improvement in the body condition of yellow-bellied marmots, Marmota flaviventris and this was associated with changing patterns of survival and reproduction [START_REF] Ozgul | Coupled dynamics of body mass and population growth in response to environmental change[END_REF]. Benefits of warming conditions in winter may therefore offset negative effects of warming summers on survival.

Studies of how changes in seasonal weather influence vital rates are rare in arid zone species, largely because of a paucity of long-term demographic data. This limits our ability to project population trends in coming decades. Here we investigate variation in survival associated with exposure to high summer and low winter temperatures for 37 avian species captured over a 30year period at two sites in a semi-arid landscape in western NSW, Australia. Using mark recapture time-dependent Cormack-Jolly-Seber models we first test for the effects on 6 monthly survival of exposure to: (1) summer temperatures above 38 o C, and (2) winter temperatures below 0 o C. We classify these temperatures as extreme because chronic effects which re-duce survival appear to begin at these thresholds in Australian bird species [START_REF] Bailey | Using different body size measures can lead to different conclusions about the effects of climate change[END_REF][START_REF] Gardner | Individual and demographic consequences of reduced body condition following repeated exposure to high temperatures[END_REF]2017;[START_REF] Mckechnie | Avian thermoregulation in the heat: evaporative cooling in five Australian passerines reveals within-order biogeographic variation in heat tolerance[END_REF]; obviously more extreme temperatures can have acute effects via immediate mortality. At our study sites these thresholds are currently reached on around 5% of days: cold extreme in winter 8% at Charcoal Tank Nature Reserve, 9% at Weddin Mt National Park; hot extreme in summer 5% at Charcoal Tank, 3% at Weddin Mt.

We then test whether the changes in 6 monthly survival associated with warming winters will be sufficient to offset the negative effects on survival of rising summer temperatures based on climate projections for the region to the year 2104. Current climate projections for Australia indicate more hot summer and fewer cold winter temperature extremes (BOM & CSIRO, 2020).

METHODS AND MATERIALS

Study sites and species

The study was carried out at two sites: The Charcoal Tank Nature Reserve (hereafter Charcoal; -33.9831 o S, 147.1575 o E) and Weddin Mountains National Park (hereafter Weddin; -33.9386 o S, 147.9872 o E) in western New South Wales in south-east Australia (Fig. S1). Both sites have been the subject of long-term (>30 years) ringing programs, with data collected in each year since 1986. The study sites are located 75 km apart within a fragmented agricultural landscape (see Appendix S1 in Supporting Information for site description).

Birds were captured in mist nets and sampled 2-7 times annually [START_REF] Gardner | Individual and demographic consequences of reduced body condition following repeated exposure to high temperatures[END_REF]. At each capture, we recorded body mass using a Pesola balance (± 0.5 g). Wing length was measured as the length of the flattened wing chord from the carpal joint to the tip of the longest primary feather using a butt-ended ruler (± 1.0 mm), and primaries were scored for moult. Data collection was overseen by Mark Clayton (Charcoal) and Richard Allen (Weddin) ensuring consistency in methods over time and between sites.

We obtained weather data from the Australian Government Bureau of Meteorology, from the Grenfell (Manganese Rd) station for the Weddin site, and from the West Wyalong Airport station for the Charcoal site. We defined 6-month periods from 1 st September to 29 th February as "Summer" or from 1 st March to 31 st August as "Winter". For each 6-month period between 1970 to 2016 we extracted (1) mean daily extreme temperatures for each season (mean daily maximum in summer, mean daily minimum in winter), (2) the sum of precipitation for each 6-month period, (3) the number of days with temperatures beyond an extreme threshold for each season (days above 38°C in summer, days below 0°C in winter).

Future climate predictions

To predict the number of temperature extremes until the end of the century under difference scenarios of greenhouse gas concentration we chose the model CESM1-CAM5 (Community Earth System Model version 1 that includes the Community Atmospheric Model version 5, [START_REF] Neale | Description of the NCAR Community Atmosphere Model (CAM5.0)[END_REF]. The model CESM1-CAM5 performs very well for short-term predictions of climate in Eastern and Southern Australia (Moise et al., 2015). We extracted daily predicted minimum and maximum temperatures for the years 2016 to 2104 from the West Wyalong Airport weather station, located 10.5km from Charcoal, under three Representative Concentration Pathway (RCP) scenarios: RCP 2.6 corresponding to a very stringent scenario of reductions in greenhouse gases emissions, RCP 4.5 an intermediate scenario and RCP 8.5 the worst-case climate change scenario [START_REF] Meinshausen | The RCP greenhouse gas concentrations and their extensions from 1765 to 2300[END_REF]. We combined climate projections to observed data (1986 to 2015) and for each 6-month period calculated the number of days below 0°C in winter and above 38°C in summer. For both seasons and for each of the three RCP scenarios we fitted generalised additive models of the number of extreme temperature days as a function of year. Year was modeled as a thin-plate regression spline [START_REF] Wood | Thin plate regression splines[END_REF]. We used the predictions from each of these generalised additive models to project survival in the future.

Mark-recapture data

The full data set contains captures for 39 species for which > 100 captures were recorded between 1986 and 2016 (Table S1) with a total of 33550 captures for 21621 individuals. Those data included only adults which were identified on the basis of plumage. We classified birds as belonging to one of four feeding guilds: nectarivore, insectivore, omnivore or granivore. Most individuals are captured only once, presumably due to movements of non-resident birds or postfledging dispersal. Our study populations, typical for much of the Australian arid zone, contain both resident and migratory species, including partial migrants and local nomads, making it difficult to classify movement patterns for individual species (see [START_REF] Chan | Partial migration in Australian landbirds: a review[END_REF]. This means that apparent local mortality represents a mixture of true mortality and permanent emigration. To reduce the influence of emigration we removed the first capture of each individual to focus our analysis on resident birds, increase the proportion of apparent survival (or mortality) related to local survival (or mortality) rather than emigration and decrease computation time. In two species, not a single individual was recaptured so we were left with only 37 species totalling 4136 individuals. We present results derived from the full data set in Fig. S8. We used capture data to calculate a temporal proxy of bird density within each guild (see Appendix S2).

To create regular time intervals corresponding to seasons and to decrease the number of variables in models we aggregated capture sessions by 6-month periods from 1 st September to 29 th February ("Summer") or from 1 st March to 31 st August ("Winter"). Thus, we obtained 62 virtual capture occasions instead of the initial 748. This procedure violates the assumption of mark-recapture models that survival intervals are long relative to the marking intervals but was essential to obtain a manageable number of capture occasions and reasonably high recapture probabilities.

The noise introduced by this violation is expected to only blur the statistical signal and not bias it (see Appendix S3, Fig. S2).

Statistical analyses

Mark-recapture model

We modelled apparent survival probability across species and time periods, as a function of summer and winter weather. We used time-dependent Cormack-Jolly-Seber models which describe capture histories as the product of a survival probability and a recapture probability. We modeled variation among species, among time periods and among locations for survival and recapture probabilities (see Appendix S4).

For the linear predictor corresponding to recapture probability, the fixed effects were a global intercept, the effect of site (Weddin vs. Charcoal), the effect of season (the 6-month capture period being the non-reproductive or the reproductive season), and the effect of a bird being caught on a previous occasion to control for trap-dependence.

For the linear predictor corresponding to survival probability, the fixed effects were a global intercept for the effect of site (Weddin vs. Charcoal), a linear effect of time (i.e. 6-months periods fitted as a continuous covariate) to de-trend the response and avoid over-estimating the effect of climate (Grobois et al., 2008), the effect of the 6-month survival period being in winter (as opposed to summer), a dummy variable indicating whether an individual was captured for the first time to account for transience effects, a proxy for within-guild-site bird density fitted as a continuous covariate (see Appendix S2, Fig. S3, S4), and the effect of weather variables, for which we tried different combinations involving our three weather variables, measured for each site and each 6-month period: (1) the number of days with temperature reaching a threshold (either above 38°C for summer survival periods or below 0°C for winter survival periods); (2) the sum of precipitation because extreme temperatures are correlated with the sum of precipitation, and precipitation is known to affect bird survival [START_REF] Kennedy | Direct effects of rain on birds: a review[END_REF][START_REF] Robinson | Weather-dependent survival: implications of climate change for passerine population processes[END_REF][START_REF] Gardner | Individual and demographic consequences of reduced body condition following repeated exposure to high temperatures[END_REF]; (3) the average daily temperature (average winter minimum, average summer maximum). Weather variables were standardized to a mean of zero and a standard deviation of one.

In a first model (model 1 hereafter), we fitted the linear effects of the number of days with temperatures beyond thresholds in each season as these were the main predictors of interest, and a linear effect of the sum of precipitation in each season. That model is at the center of our work because it estimates the independent effect of extreme temperatures when accounting for the confounding effect of precipitation. For both seasons, separately, we calculated the proportion of temporal variance in mean survival among years explained by extreme temperatures (see Appendix S5 for details). To predict the net effect of changes in the balance of deleterious winter versus deleterious summer temperatures, we predicted seasonal-survival and year-survival based on model 1 and on predictions of temperature extremes based on the CESM1-CAM5 model for the three RCP scenarios, from 1986 to 2104. Survival was predicted while integrating the random effects of year and species.

Beyond model 1, we fitted four extra models to address secondary questions regarding the effect of weather on survival probability. In a second model (model 2 hereafter) we added the interaction between sum of precipitation and the numbers of days >38 o C or <0 o C in each season. In wetter summers the negative effect of extreme high temperatures may be reduced because water availability helps birds cope with heat, while in wetter winters the negative effect of extreme cold may be exacerbated because the insulating capacity of wet feathers is poor [START_REF] Kennedy | Direct effects of rain on birds: a review[END_REF][START_REF] Robinson | Weather-dependent survival: implications of climate change for passerine population processes[END_REF]. These expectations, however, assume that extreme temperatures co-occur with increased precipitation or water availability, within each time period, which may not be the case. Next we fitted model 3, an extension of model 2, in which we also included a linear effect of mean maximum or minimum daily temperature in each season. With the model we test whether it is possible to disentangle the effects of the number of days of extreme temperatures from the effect of the average season temperature.

We first fitted models 1 to 3 for all species together. However, dietary guilds are expected to respond differently to weather conditions because food availability varies with temperature and rainfall, and because water content varies between diets, so we fitted models 1 to 3 separately for each of the 4 guilds.

Model 4 included random slopes of species for the effects of the number of days of extreme temperatures to test for differences among species in their responses (see Appendix S4, results in main text). Model 5 included species average size and its interaction with the number of days of extreme temperatures (see Appendix S4, results Fig. S7). With model 6 we tested whether our results were robust to the exclusion of the first capture: model 6 is identical to model 1 but is fitted on the full data set, which includes all captures of all individuals, and includes a second transience fixed effect (see Appendix S4, results Fig. S8).

We fitted all models in the Bayesian probabilistic programming language Stan [START_REF] Carpenter | Stan: A probabilistic programming language[END_REF], using No-U-Turn Hamiltonian Monte Carlo sampling with Rstan (Stan Development Team, 2020). For fixed effects we used normalising Gaussian priors with a mean of 0 and a standard deviation of 3. For random effect standard deviations we used half-Cauchy priors of location 0 and scale 0.1. We used 4 chains each with 2000 warm-up iterations and 2000 samples.

We assessed model convergence using visual diagnostics (trace plots, pairwise posterior plots), in particular using Bayesplot [START_REF] Gabry | Visualization in Bayesian workflow[END_REF] and checked that all R-hat values were below 1.05 and effective sample sizes were above 200. We report posterior means and 95% highest posterior density credible intervals. All derived parameters and model predictions were calculated by integrating over the full posterior distribution. The code to run all models is provided (see Appendix Code S1).

RESULTS

Unless mentioned otherwise, parameter estimates are from model 1, fitted to all species together. Survival probability was slightly higher at Weddin compared to Charcoal (β=0.25, 95% highest posterior density credible interval [0.02; 0.50]) and did not clearly differ between summer and winter (β=-0.04 [-0.68; 0.59]) (Table S2). The effect of transience was negative (β=-0.78 [-0.998;-0.55]), meaning that birds have a lower probability of apparent survival just after being captured for the first time. This effect may correspond to non-resident birds that permanently leave the study site after first capture. There was heterogeneity in survival probabilities both among occasions, even after accounting for weather variables (random standard deviation, σ= 1.09 [0.77;1.54]) and among species (σ= 0.28 [0.14; 0.49], Fig. 1). Species mean 6-month survival probabilities varied from 0.88 [0.83; 0.92] in the white-throated treecreeper, to 0.76 [0.67; 0.83] in the superb fairy-wren (Fig. 1).

Recapture probability was consistently lower at Weddin compared with Charcoal and did not clearly differ between seasons (Table S2). Trap dependence was positive (β=0.33 [0.23;0.43]), meaning that a living bird is more likely to be recaptured just after having been captured. There was heterogeneity in recapture probabilities both among occasions (σ= 0.55 [0.46;0.66]) and among species (σ= 0.68 [0.44; 1.05]). Species recapture probabilities varied from 0.34 [0.27; 0.42] in the white-throated treecreeper, to 0.074 [0.038; 0.12] in the spiny-cheeked honeyeater (Fig. 1).

Effects of number of days of extreme temperatures

The effect of the number of extreme summer days was strongly negative (β=-0.57 [-1.01; -0.08]), corresponding to a decline in survival probability from 0.86 [0.78;0.95] when there are no days above 38°C to 0.59 [0.45; 0.77] for 30 days above 38°C (Fig. 2). The effect of the number of extreme winter days was marginally negative (β=-0.39 [-1.01; 0.04]), corresponding to a decline in survival probability from 0.84 [0.76;0.90] for 0 days below 0°C to 0.53 [0.28;0.78] for 44 days below 0°C (Fig. 2; Table S2). The effect of extreme temperatures explained 43% [16%; 70%] of temporal variation in survival among years in summer and 13% [0%; 36%] in winter. The effects of the number of extreme summer and winter days were also apparent when including transients, in Model 6 (summer β=-0.33 [-0.57; -0.10], winter β=-0.37 [-0.62; -0.15]), with the effect in summer being a bit weaker and the effect in winter being of similar strength but more precisely estimated (Fig. S8).

Model 4 revealed that species differed significantly in their responses to extreme summer temperatures (σ= 0.14 [0.02;0.32]) and to extreme winter temperatures (σ= 0.12 [0.02;0.27]), but those differences were biologically small so that the pattern of survival was qualitatively consistent across species (Fig. 2, Fig. S5, Table S8). Moreover, as is often the case in mixed models [START_REF] Houslay | Avoiding the misuse of BLUP in behavioural ecology[END_REF], despite significant variance among species the uncertainty in species-specific slopes was large so that it is not possible to identify which species responded more or less strongly (Fig. S5). In particular, those differences were not significantly explained by differences in the mean body size of species (Model 5, Table S9).

Guild-specific analyses confirm the consistency of the responses among species. The negative effect of extreme temperatures in summer was independently found in insectivore, nectarivore and omnivore species (Table S3, Fig. S6). On the other hand, the effect of extreme temperatures in winter was present only in nectarivore species, where it was significant and strong (β=-0.66 [-1.56;-0.43]), with a predicted decline in survival probability from 0.89 [0.78;0.97] with 0 days below 0°C to 0.41 [0.05;0.81] with 44 days below 0°C. For the granivore guild, a small sample size prevented a precise estimation of any parameter for weather effects.

Winter precipitation had a negative effect on survival (β=-0.52 [-0.94; -0.17]). The effect was statistically supported in the nectarivore guild model (β=-0.92 [-1.56;-0.43]) but not found in other guilds (Table S3). Summer precipitation had no clear effect on survival across species (β=-0.24 [-0.57; -0.57]), or in any of the guilds (Table S3).

When fitting model 2 to all species together (Table S4), there was no evidence of interactions between number of days beyond extreme temperatures and precipitation, in summer (β=0.04 [-0.4; 0.52]) or in winter (β=0.13 [-0.44; 0.13]). In guild models, there was support for an interaction only in omnivorous species, for summer, with increased survival to extreme temperatures at higher precipitation(β=0.67 [0.03; 1.42]) (Table S5).

Fitting model 3 to all species together allowed us to disentangle the effects of mean temperatures from the number of days with extreme temperatures. For summer, there was sufficient evidence that the number of days above 38°C had a negative effect on survival probability after accounting for effects of mean temperature (β=-0.32 [-0.55; -0.03]) (Table S6). In winter the trend for the effect of the number of days beyond 0°C was marginally negative but the uncertainty was large (β=-0.28 [-0.68; 0.04]).

In summary, winter extremes had a weak effect on 6-monthly survival, except in the case of nectarivores which are predicted to have low survival (ca 41%) in the coldest winters with > 40 days below freezing. In contrast, the negative effect of summer extremes was strong and consistent across species (Model 4), showing little variation among species (Fig. S5) and all three guild-models showing similar trends (Fig. S6). Our results are robust to the inclusion of the correlated effects of mean extreme temperatures (Model 3, Table S6), to the inclusion or exclusion of transient individuals (Model 6, Fig. S8, Table S10), and in accounting for species (Model 4, Fig. S5) and guild differences (Fig. S6).

Climate predictions:

At Charcoal the number of winter days below 0°C is projected to decrease from 12.0 in the year 2016 to almost 0 in 2104 (2.4 RCP 2.6, 1.4 RCP 4.5, 0.17 RCP 8.5), while the number of summer days above 38°C is projected to increase from 13.8 in 2016 to 28.7 (RCP 2.6), 36.4 (RCP 4.5) and 67.4 (RCP 8.5) in 2104 (Fig. 3). Based on these climate projections, model predictions for survival to 2104 showed broad-scale declines in annual survival across species (Fig. 4). Assuming median precipitation, in winter, 6-month survival is projected to increase slightly as winters warm,whereas summer survival and annual survival were predicted to decline substantially, with the severity of the decline depending on the RCP scenario (Fig. 4).

DISCUSSION

We quantified patterns of adult survival for two bird communities in semi-arid south-eastern Australia as a function of exposure to temperature extremes in winter and summer, using data from 37 species that have been the subject of continuous ringing programs over 30 years. Survival probability did not differ between summer and winter, but survival declined at a higher rate with increasing exposure to summer days with thermal maxima >38 o C, than with increasing exposure to winter days with minima <0 o C. The effect of hot days in summer was consistent across species and guilds whereas the effect of cold days in winter was consistent across species but only statistically clear in nectarivores. Model projections to 2104 suggest that survival may be reduced by a third from current levels; the increase in survival associated with milder, warming winters will be insufficient to offset the negative effects on survival of more frequent extreme days above >38 o C in summer.

Survival

Our study does not elucidate the causes of temperature-related summer mortality or the specific timeframe over which such mortality occurs, but several mechanisms might be at play. Mortality might be a direct result of hyperthermia or lethal dehydration during heatwave events as previously reported in arid Australia [START_REF] Mckechnie | Feeling the heat: Australian land birds and climate change[END_REF][START_REF] Saunders | The impact of two extreme weather events and other causes of death on Carnaby's Black Cockatoo: a promise of things to come for a threatened species[END_REF]. However, increasing exposure to high temperatures >38 o C is also likely to increase mortality over broader timeframes via increasing the need for evaporative cooling to maintain body temperatures below lethal limits or reducing food intake with consequences for water and energy budgets [START_REF] Mitchell | Revisiting concepts of thermal physiology: predicting responses of mammals to climate change[END_REF]. Recent work by [START_REF] Mckechnie | Avian thermoregulation in the heat: evaporative cooling in five Australian passerines reveals within-order biogeographic variation in heat tolerance[END_REF] examined the evaporative cooling capacity and heat tolerance of five insectivorous and nectarivorous passerine species in Australia's arid zone, among them the spiny-cheeked honeyeater (Acanthagenys rufogularis) included in this study. They confirmed the reliance of these species on evaporative heat loss via panting at high air temperatures. When air temperatures exceeded 38°C, rates of evaporative water loss increased rapidly and linearly to at least sevenfold (670-860%) above basal rates under field laboratory conditions [START_REF] Mckechnie | Avian thermoregulation in the heat: evaporative cooling in five Australian passerines reveals within-order biogeographic variation in heat tolerance[END_REF]. Previous work at both Charcoal and Weddin showed that prolonged exposure to days with air temperature >38°C was associated with reductions in body condition of white-plumed honeyeaters (Ptilotula penicillatus) [START_REF] Bailey | Using different body size measures can lead to different conclusions about the effects of climate change[END_REF], and at Charcoal, heat-exposed individuals in poor condition were less likely to be recaptured the following spring [START_REF] Gardner | Individual and demographic consequences of reduced body condition following repeated exposure to high temperatures[END_REF]. This apparent mortality may have been a result of thermoregulatory constraints but could also result from heat-related changes in foraging patterns and microhabitat use (du Plessis et al., 2012;[START_REF] Edwards | The impact of high temperatures on foraging behaviour and body condition in the western Australian Magpie Cracticus tibicen dorsalis[END_REF]Fungi et al., 2019) that lead to a reduction in food intake and loss of body condition, although we have no direct evidence in our study species.

Diet is an important factor influencing the way birds respond to high temperatures [START_REF] Albright | Mapping evaporative water loss in desert passerines reveals an expanding threat of lethal dehydration[END_REF][START_REF] Smit | Differences in the use of surface water resources by desert birds are revealed using isotopic tracers[END_REF]. Diets vary in water content, potentially making some dietary guilds more vulnerable to rising temperatures than others. For example, the dry seeds eaten by granivores have the lowest water content of any food, leading to a high dependency on surface water [START_REF] Fisher | Drinking patterns and behavior of Australian desert birds in relation to their ecology and abundance[END_REF]. Nectarivores are similarly dependent on surface water because they need water to process diets high in sugar [START_REF] Fleming | Nectar concentration affects sugar preferences in two Australian honeyeaters and a lorikeet[END_REF]. Both groups drink more frequently as ambient temperature increases [START_REF] Fisher | Drinking patterns and behavior of Australian desert birds in relation to their ecology and abundance[END_REF]. By contrast, omnivores and insectivores have animal-based diets or a combination of animal and plant-based (omnivores) diets, drink infrequently from freestanding water, and rely on water from their prey during the hottest, driest times of year [START_REF] Smit | Differences in the use of surface water resources by desert birds are revealed using isotopic tracers[END_REF]. This has led to the suggestion that drinkers would be more vulnerable to mortality in hot conditions when surface water is unavailable [START_REF] Smit | Differences in the use of surface water resources by desert birds are revealed using isotopic tracers[END_REF]. [START_REF] Czenze | The costs of keeping cool in a warming world: implications of high temperatures for foraging, thermoregulation and body condition of an arid-zone bird[END_REF] found that drinkers had greater heat tolerance than non-drinkers and could withstand air temperatures of ≥52°C based on metabolic chamber measurements for wild-caught individuals of 12 passerine species in arid South Africa that had ad libitum access to water prior to testing. However, in the absence of surface water or during long periods of inactivity in hot conditions, they concluded that drinking species are likely to exceed their limits of dehydration tolerance at lower temperatures than non-drinkers, and thus may be more vulnerable to mortality in dry, hot conditions.

In an attempt to understand the causes of collapse of bird communities in the Mojave Desert USA over the last century [START_REF] Iknayan | Collapse of a desert bird community over the past century driven by climate change[END_REF], [START_REF] Riddell | Cooling requirements fueled the collapse of a desert bird community from climate change[END_REF] modelled changing water requirements for thermoregulation using biophysical models and found that species declines were associated with increasing water requirements for body cooling in hot conditions. They found that birds inhabiting the hottest, driest sites without surface water had the lowest probability of persistence. Increasing water requirements were correlated with declines in insectivores and carnivores (non-drinkers with animal-based diets) but not herbivores and omnivores (drinkers with plant-based diets), in conditions where surface water was present.

In contrast to these studies, we found that dietary guilds displayed similar patterns of summer survival with increasing exposure to daily maxima >38°C. We found no evidence for an interaction between rainfall and increasing exposure to extremes >38°C across species, although the negative effect of summer extremes was reduced in higher rainfall conditions in the omnivorous guild. Despite this result, our study design is unlikely to detect the effects of rainfall-related water availability on summer survival because we calculated rainfall as the sum of rain for each 6month period and is not temporally related to the occurrence of extreme days >38°C. The structure of our analyses does not permit finer-scale resolution of the co-occurrence of rainfall and temperature >38°C, which seems critical for understanding responses to heat. Indeed, previous study of mass loss in white-plumed honeyeaters at Charcoal found that apparent mortality was associated with high summer temperatures but only in low rainfall conditions, characterized as rainfall in the 4 weeks prior to capture [START_REF] Gardner | Individual and demographic consequences of reduced body condition following repeated exposure to high temperatures[END_REF]. This suggests that water availability in hot conditions may be critical for drinking species, but we have no comparable data for non-drinking species. Although these studies suggest that drinkers might fare worse in the absence of surface water in hot conditions, the broad-scale ecological significance of such findings are unclear. Behavioural field studies are needed to understand the trade-offs individuals make while foraging in the heat in conditions of varying water availability, and the influence dietary guild has on mortality risk, to complement lab-based studies of physiology.

Climate Projections and survival

Under all RCP scenarios the number of hot days >38°C is predicted to increase and the number of cold days <0°C is predicted to decrease until the end of the century. Given that increasing exposure to cold winter days is associated with decreased survival, warmer winters are projected to increase 6-monthly survival. Conversely, hot summer days are deleterious to survival and projected to become more common reducing 6-monthly survival by 2104. The increase in winter survival will be more than offset by decreasing survival in summer. Indeed, annual survival is projected to decrease from 0.631 to 0.430 at best, 0.11 at worst, by 2104. We note that these calculations come with assumptions and should not be taken as a prophecy. First, we predicted survival based only on the predicted changes in extreme temperatures, while ignoring other changes that may affect bird survival in the next 80 years. Second, in the case of hot days the calculation is based on an extrapolation since a summer with more than 25 days above 38°C has never been observed at our study sites, whereas climate models predict yearly averages of 30 to 70 days above 38°C. It is possible that the effect of days above 38°C is not consistent for the range of values observed at our study sites and for the larger values in the range predicted by climate models. Third, our calculations assume no significant adaptation of the bird populations to the changing climate (see below).

Nevertheless, the prediction of a substantial decrease in annual survival for the bird community currently present in our study sites is likely robust. Indeed, the deleterious effects of temperature extremes we identified statistically are consistent with physiological mechanisms (see next paragraph). Even if the shape of the relationship between survival and number of hot days is not accurately represented by our extrapolation (for instance, being optimistic there may be a plateau beyond which survival does not decline), we know of no plausible mechanism that would reverse the trend of decreasing survival across a broader range of hot days. Therefore, more hot days on average will certainly decrease mean survival compared to the current situation. In addition, cold days are already quite rare in our study sites and are predicted to almost disappear within a few decades under all RCP scenarios, whereas hot days will become more and more common until the end of the century (Fig. 3). Trading very few cold days for a dozen or more hot days logically decreases overall survival.

Our results add to concerns about the persistence of avian populations in arid and semi-arid regions of Australia and on other continents. Modelling avian water requirements for temperatures predicted for the 2080s, [START_REF] Mckechnie | Climate change increases the likelihood of catastrophic avian mortality events during extreme heat waves[END_REF] concluded that desert birds in Australia will experience reduced survival times more frequently during summer due to large increases in thermoregulatory water requirements, regardless of water availability. Similarly, [START_REF] Conradie | Avian mortality risk during heat waves will increase greatly in arid Australia during the 21st century[END_REF] modelled the risks of lethal hyperthermia and dehydration for 10 Australian arid zone bird species to 2104 based on thermal physiology data. They concluded that increased water demands for evaporative cooling will greatly increase mortality risk and several species will also be vulnerable to lethal hyperthermia, particularly smaller species. In combination, increasing temperatures and unpredictable water availability pose a significant risk to the avifauna of the Australian interior, particularly in the far western parts of the continent [START_REF] Conradie | Avian mortality risk during heat waves will increase greatly in arid Australia during the 21st century[END_REF].

Our analyses, based on demographic data, suggest that climate change also poses a threat to birds in semi-arid regions of eastern Australia.

Elsewhere, Albright et al. (2019) modelled the consequences of 4°C warming to the year 2104 for desert passerines in the United States, and concluded warming would greatly increase the ex-tent, frequency and intensity of dehydration risk over large parts of the south-western United States, affecting water balance, daily activity and geographic distribution of desert birds in these regions. Building on that work, simulation models of heat flux showed that observed declines over the last century in the Mojave desert were positively associated with climate-driven increases in water requirements for evaporative cooling: predictions suggest an increase in water requirements of 50 to 78% under future climate scenarios, all else held equal [START_REF] Riddell | Cooling requirements fueled the collapse of a desert bird community from climate change[END_REF]. Finally, Conradie et al. (2019) synthesized physiological and behavioural data to evaluate the risks of lethal extreme heat events versus sublethal costs of chronic exposure to hot weather for birds in the Kalahari Desert. They concluded that the risk of acute heat exposure will remain relatively low to the end of the century compared with Australia and south-western USA due to lower rates of warming. By contrast, sublethal costs of chronic heat exposure are likely to drive large declines in avian diversity in the southern African arid zone by the end of the century.

These laboratory-based studies provide a physiological basis for climate-driven local extinctions that are likely to apply to the current study. Birds in our study will inevitably be exposed to greater risk of acute hyperthermia and dehydration as summers warm in these semi-arid environments. Model projections for the region indicate a 50% increase in the frequency of days with air temperatures >38°C, a temperature threshold known to affect white-plumed honeyeaters at both sites [START_REF] Gardner | Dynamic size responses to climate change: prevailing effects of rising temperature drive long-term body size increases in a semi-arid passerine[END_REF], 2016[START_REF] Bailey | Using different body size measures can lead to different conclusions about the effects of climate change[END_REF]. Moreover, recent heatwaves in semiarid South Australia, provide some insights into the consequences of a 4°C rise for the survival of small passerines. In January 2019 the mean daily maximum temperature was 37.5°C, 4.2°C above average, identical to the temperature increase predicted for that region in 2090. Birds at the site were exposed to up to 10 h per day of air temperatures above body temperature (40 o C) [START_REF] Sharpe | Weighing the cost: the impact of serial heatwaves on body mass in a small Australian passerine[END_REF], the threshold at which evaporative cooling is required to maintain body temperatures below lethal limits [START_REF] Boyles | Adaptive thermoregulation in endotherms may alter responses to climate change[END_REF][START_REF] Albright | Mapping evaporative water loss in desert passerines reveals an expanding threat of lethal dehydration[END_REF]. Apparent mortality among the colour-ringed population of insectivorous jacky winters, Microeca fascinans, increased almost three-fold during the two-month heatwave period (20% of the population died) and all breeding attempts were abandoned. Birds were never observed to forage, provision nestlings or incubate at air temperatures > 38°C [START_REF] Sharpe | Too hot to handle? Behavioural plasticity during incubation in a small, Australian passerine[END_REF]. More recently, in December 2019, 30% of the ringed population disappeared, and presumably died, within 24hr of air temperatures reaching 49°C [START_REF] Sharpe | Too hot to handle? Behavioural plasticity during incubation in a small, Australian passerine[END_REF]. Body size will inevitably affect avian responses to high temperatures, with smaller birds at particular risk of mortality. Balancing energy and water budgets against the risk of fatal hyperther-mia is a challenge for smaller birds because they have smaller energy stores and body water pools but higher rates of heat gain and higher mass-specific rates of evaporative water and energy use [START_REF] Smit | Can behaviour provide a basis for rapid assessment of relative vulnerability of desert birds to climate change?[END_REF][START_REF] Albright | Mapping evaporative water loss in desert passerines reveals an expanding threat of lethal dehydration[END_REF]. Consequently, smaller birds have shorter survival windows during extreme heat events than larger species. [START_REF] Albright | Mapping evaporative water loss in desert passerines reveals an expanding threat of lethal dehydration[END_REF] estimate that a 9.7g Lesser Goldfinch (Spinus psaltria) will lose body mass at about twice the rate of 71g Curve-billed Thrasher (Toxostoma curvirostre) (3.4% versus 1.8% body mass per hour). An increase in air temperature from 40 to 44°C will result in a 150% increase in rates of evaporative water loss, but time to death through dehydration differs greatly because of differences in massspecific rates of evaporative water loss [START_REF] Albright | Mapping evaporative water loss in desert passerines reveals an expanding threat of lethal dehydration[END_REF]. However, we found no effect of body size on responses to temperature extremes, although the response to temperatures >38°C was consistent with poorer survival of smaller individuals. Our data may be too limited to detect strong effects, as we analysed only 37 species with a limited range of sizes (from 48 to 133mm).

Our study looks at just one component of fitness: longitudinal adult survival. Clearly, estimates of reproduction are necessary to model population viability per se, but such data are not available for our study species. Nevertheless, our estimates of survival are likely to correlate with population decline, regardless of recruitment. Even if populations maintain recruitment at current levels by advancing the start of seasonal breeding as winters warm, increasing exposure to extreme heat represents a clear threat to survival, affecting all individuals, suggesting a downward trend in population size is likely. Assuming the decrease in annual survival of around 30% (RCP 2.6), recruitment would have to increase by ca. 40% to offset the decline in survival (1/(1-0.3) = 1.43), which seems unrealistic. In general, adaptive micro-evolution may be substantial over a few generations [START_REF] Bonnet | Genetic variance in fitness indicates rapid contemporary adaptive evolution in wild animals[END_REF], but current evidence suggests that adaptive responses are most likely insufficient to offset the current rate of climate change [START_REF] Radchuk | Adaptive responses of animals to climate change are most likely insufficient[END_REF]. Moreover, the potential for genetic adaptation to higher temperatures that could reduce mortality is unclear given the lack of evidence for such effects in birds (Gienapp et al., 2008;[START_REF] Chevin | Evolution of phenotypic plasticityin extreme environments[END_REF]. In Australia, there is evidence for microevolution in genes that are associated with heat-related dessication in Drosphila melanogaster [START_REF] Umina | A rapid shift in a classic clinal pattern in Drosophila reflecting climate change[END_REF], but we are unaware of similar examples in birds. This reflects the rarity of examples of microevolution in response to climate change in general, and heat tolerance in particular (Gienapp et al., 2008;[START_REF] Chevin | Evolution of phenotypic plasticityin extreme environments[END_REF]. In addition to microevolutionary changes in the mean response to extreme temperatures, a genetic response to natural selection may mitigate the negative impact of climate change on survival by reducing the temporal variation of vital rates, via demographic buffering [START_REF] Hilde | The Demographic Buffering Hypothesis: Evidence and Challenges[END_REF].

CONCLUSIONS

Our study highlights the significance of temperature extremes for species' persistence in arid and semi-arid regions, comprising some 70% of the terrestrial land mass of Australia, and about 40% globally. We used long-term ringing data and a mark-recapture framework to estimate survival as a function of temperature extremes, rather than make projections based on the physiological costs of heat exposure. Despite differences in methodology and approach, our conclusions are similar to physiological-based projections, and suggest that rising summer temperatures pose a risk to population persistence for birds in arid and semi-arid regions.

Figure 1. Species-specific 6-monthly mean survival and mean recapture probabilities as estimated from a random intercept model (Model 1). Species are arranged by mean-survival on the y-axis and color-coded by guild. Survival and recapture probabilities are calculated conditional on a median number of days with temperature extremes, median rainfall, and integrating sites, seasons and temporal variance. Points represent posterior means and horizontal bars represent 95% highest posterior density credible intervals.

Figure 2. Predicted probability of 6-monthly survival as a function of the number of days of extreme summer temperatures recording daily maxima >38° C and extreme winter temperatures recording minima <0° C. Average predictions, over all species, are shown in thick lines with shaded areas representing 95% credible intervals, and were derived from model 1. Thin lines represent species-specific predictions derived from model 4. Predictions were estimated for a median rainfall and population density, but integrated over temporal variation, and in the case of the average predictions, over species variation.

Figure 3 .

 3 Figure 3. Predictions of daily temperature extremes under three emission scenarios from the model CESM1-CAM5. (A) Number of days on which temperatures are below 0°C in winter; (B) Number of days on which temperatures are above 38°C in summer. Numbers are observed from 1986 to 2015 (black line) and simulated according to the model from 2016 to 2104. Smoothed lines and 95% confidence intervals are produced with negative-binomial generalised additive models using thin-plate smooths of year from 1986 to 2104.

Figure 4 .

 4 Figure 4. Projected 6-monthly survival as a function of predicted change in the number of (a) winter and (b) summer days with temperature extremes >38° C or <0° C, respectively; (c) annual survival to 2104 (averaged over all species, sites and time periods), for three scenarios of emissions and associated climate change. For winter the predictions from the three scenarios largely overlap. Dashed lines indicate the initial predicted survival probability to provide context for trends in predicted survival. Shaded areas represent 95% credible intervals.


N Bur
email: n.bur@estia.fr

P Joyot
email: p.joyot@estia.fr

On the Use of PGD for Optimal Control Applied to Automated Fibre Placement

Automated Fibre Placement (AFP) is an incipient manufacturing process for composite structures. Despite its conceptual simplicity it involves many complexities related to the necessity of melting the thermoplastic at the interface tape-substrate, ensuring the consolidation that needs the diffusion of molecules and control the residual stresses installation responsible of the residual deformations of the formed parts.

The optimisation of the process and the determination of the process window cannot be achieved in a traditional way since it requires a plethora of trials/errors or numerical simulations, because there are many parameters involved in the characterisation of the material and the process.

Using reduced order modelling such as the so called Proper Generalised Decomposition method, allows the construction of multi-parametric solution taking into account many parameters. This leads to virtual charts that can be explored on-line in real time in order to perform process optimisation or on-line simulation-based control. Thus, for a given set of parameters, determining the power leading to an optimal temperature becomes easy.

However, instead of controlling the power knowing the temperature field by particularizing an abacus, we propose here an approach based on optimal control: we solve by PGD a dual problem from heat equation and optimality criteria. To circumvent numerical issue due to ill-conditioned system, we propose an algorithm based on Uzawa's method. That way, we are able to solve the dual problem, setting the desired state as an extra-coordinate in the PGD framework. In a single computation, we get both the temperature field and the required heat flux to reach a parametric optimal temperature on a given zone.

INTRODUCTION

Automated Fibre Placement (AFP) is one the main technologies employed today to manufacture advanced composite laminates from unidirectional prepregs. This technique consists in laying and welding tapes of prepregs, building a laminate having the geometry and the desired mechanical characteristics.

This process has been widely studied [START_REF] Sonmez | [END_REF]2,3,4] with numerical models becoming more accurate, robust and complex. New techniques of computation [5,6], based on variables' separation, have enabled models' enrichment by adding parameters as extra-coordinates [7]. Thus, the Proper Generalised Decomposition (PGD) method leads to multi-parametric virtual charts providing a whole set of solutions for each combination of the considered parameters [8,9,10,11,12].

Then, the computational vademecum can be exploited on-line for process control or process optimisation purposes. Indeed, within the AFP we want to efficiently control the heating power: tapes have to be heated enough to ensure the melting of the matrix coating the fibres and the cohesion with the previously laid tapes, while not exceeding a threshold from which material burns.

Therefore, we took advantage of the PGD to build off-line virtual charts in order to determine the best power associated to a draping velocity profile [13].

However, in those simulations, solutions were computed from equations provided by underlying physics of the studied phenomena, the optimisation being carried out as post-process.

We propose here to compute directly the solution of an optimisation problem in order to get the separated representations of both the field and the control to obtain it. That is to say the optimisation is made directly off-line, reducing the cost of the post-process and improving the real-time control of the AFP.

Within the next section, we present the equations governing the phenomenon under consideration. In order to have a reference solution the system is solved by standard finite element method (FEM). Thereafter, section b) focuses on the writing and solving of the optimal system. We improve the obtained results by applying Uzawa's method in section b). Lastly section b) addresses few conclusions and perspectives.

PROCESS MODELLING

AFP process can be modelled with an heat equation associated with the next boundary conditions. Heat source applies on boundary Γ P (see Figure 1); considering a wide enough domain Ω, we set a Dirichlet's condition taking into account a fixed temperature on Γ R ; then, for the sake of simplicity, an homogeneous Neumann's condition is applied on others boundaries.

Γ B Γ L Γ T Γ P Φ Ω FIGURE 1. Domain of study.
This leads to the following advection-diffusion equation 1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -div (K∇u) + ρC p V∇u = 0i n Ω; u = 0o n Γ R ; -K∂ n u = 0o n Γ B ∪ Γ L ∪ Γ T ; -K∂ n u = -Φ on Γ P ; (1)
with

K = k 0 0 k ⊥ et V = v 0 T .
This system can obviously be solved in a classical way with standard FEM. However, since we want a reference solution to latter compare with results from other methods, we have to implement comparable systems. Consequently, in order to compare solution in separated representation, we take advantage of the tensor product method introduced by R. E. Lynche, J. R. Rice and D. H. Thomas [14] [15]. The writing of discrete form of Equation ( 1) separating the two space coordinates and using tensor forms of operators allows the handling of shape functions in the separated representation, the system remaining solved with a global, non separated FEM.

We can thus obtain the temperature field from the input Φ modelling the laser heat flux. For some couples power/speed we compute the corresponding reference solutions.

From these results we extract the temperature profiles on boundary Γ P where the source is applied. Indeed, we want to control the temperature on this same boundary during the process, to reach an optimum, providing the ideal heat flux. We compute also P S ray , the power seen by Γ P , as the integrand of the flux on this boundary multiplied by the width of the shone surface S ray .

We gather in Table 1 some key values for these four reference solutions, for the purpose of latter comparison. Due to velocity values, we increase the number of nodes to contain the Péclet number, avoiding the requirement of a stabilisation technique

v = 0.001 m • s -1 v = 0.01 m • s -1 v = 0.1m• s -1 v = 1m• s -1 Pw = 600 W Pw = 1897 W Pw = 6000 W Pw =

SETTING UP THE OPTIMAL SYSTEM

As announced in the Introduction, The PGD method provides multi-parametric virtual charts that can be used to control the AFP process [13].

Another way to go on is to use the optimal control theory. Within the AFP process, a heat flux provided by a laser melt the thermoplastic. The difficulty consists in determining the best power of the laser to reach an optimal temperature to melt the thermoplastic enough but not too much.

Thus we consider the following cost-function to be minimised, since the flux is applied only on Γ P , part of the boundary

J(u, Φ) = 1 2 Γ P (u -u d ) 2 + αΦ 2 dγ, (2)
subject to the state advection-diffusion equation ( 1), with α cost parameter of the command. That way Φ is used as control and we want to reach u d on boundary Γ P . The domain of study is depicted on Figure 1.

The corresponding Lagrangian writes

L (u, Φ, p) = J(u, Φ) + Ω p -div (K∇u) + ρC p V∇u dx. ( 3
)
To find a stationary point of the Lagrangian we set

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ u L (u, Φ, p) = 0; ∇ Φ L (u, Φ, p) = 0; ∇ p L (u, Φ, p) = 0. ( 4
)
Expanding these equations leads to the non-linear optimality system (see [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF]), whom weak form writes, with test functions

u ⋆ et p ⋆ , ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ω K∇u ⋆ •∇u + Ω ρC p V∇uu ⋆ -Γ P 1 α pu ⋆ dγ = 0 ∀u ⋆ ; Ω K∇p ⋆ •∇p + Ω ρC p Vp∇p ⋆ + Γ P up ⋆ dγ = Γ P u d p ⋆ dγ ∀p ⋆ . (5)
In Equation ( 5), fields u and p are coupled. The computation of such a problem can be achieved using mixed formulation, within a standard FEM as well as in the PGD framework as described thereafter.

The discrete form of the state variable u (x, z) and the adjoint parameter p (x, z) are expressed in tensor product form

U = ∞ i U i x ⊗ U i z and P = ∞ i P i x ⊗ P i z . (6)
The vector Ψ which brings together nodal values of u and p takes the form

Ψ= U P = ∞ i U i x ⊗ U i z P i x ⊗ P i z . (7)
The discretised weak form of Equation ( 5) is written Ψ ⋆T AΨ=Ψ ⋆T B were A = 8 i=1 A j

x ⊗ A j z and B = B x ⊗ B z will be expressed in a tensor form, and with the test function defined by

Ψ ⋆ = U ⋆ x ⊗ U z + U x ⊗ U ⋆ z P ⋆ x ⊗ P z + P x ⊗ P ⋆ z . (8)
This discretised problem can then be solved with the PGD method, for different values of the velocity, the desired state u d coming from the corresponding reference solution.

Since the goal of optimal control is to minimise the distance between the unknown u and the desired state u d on the boundary Γ P , we retrieve the wanted temperature profile on this boundary. However, to reach it, the flux, computed by PGD, to be applied take substantially away compared to the FEM input, as speed increases (see Figure 2).

v = 0.001 ms -1 v = 0.1ms -1 v = 0.01 ms -1 v = 1ms -1

UN-MIXING THE OPTIMALITY SYSTEM

Applying the PGD framework on the optimality system (5) can produce, in our case, weird results. Instead of searching a workaround within numerical stabilisation techniques, we propose to solve this system without mixed elements. For this purpose, we use Uzawa technique [17] [18].

Given an initial guess p (0) for p, Uzawa's method consists in our case of the following coupled iteration:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ A u u (k+1) = A pu p (k) p (k+1) = p (k) + ω A pu u (k+1) -u d + A p p (k) ( 9
)
where ω>0 is a relaxation parameter. Tensors are defined by

A u = k K u x ⊗ M u z + M u x ⊗ k ⊥ K u z + ρC p vH u x ⊗ M u z A pu = - 1 α M pu x ⊗ M pu z A p = k K p x ⊗ M p z + M p x ⊗ k ⊥ K p z -ρC p vH p x ⊗ M p z A up = M up x ⊗ M up z
This coupled iteration is computed within a fixed-point loop, each field u and p being solved by PGD. As previously, Figure 3 shows the computed flux to reach the desired temperature u d on the boundary Γ P .

v = 0.001 ms -1 v = 0.1ms -1 v = 0.01 ms -1 v = 1ms -1 FIGURE 3. Flux on Γ P with Uzawa's method TABLE 2. Summary of key values v = 0.001 m • s -1 v = 0.01 m • s -1 v = 0.1m• s -1 v = 1m•

FIGURE 2 .

 2 FIGURE 2. Flux on Γ P for different speeds.

TABLE 1 .

 1 Key values for reference solutions

x z l 1 l 2 l 3 h Γ R

CONCLUSION

In order to control the AFP process, we proposed a method based on the PGD technique allowing the inclusion of parameters as extra-coordinates. Due to numerical instabilities, we transformed a mixed formulation to a coupled problem, improving significantly the results.

Next step consists in considering the desired state u d as a new coordinate. Thus we will be able to build directly the process control as a multi-parametric virtual chart.

Table 2 summarises the previous simulations, collecting some key-values. Thus, solving the optimality system with the PGD algorithm, but without mixed elements using Uzawa method leads to consistent results, since both temperature and heat flux remain similar to those expected (using standard FEM).


domain: phys spi

text:
Jeanne Bernard Dphy

Malo Cadoret
email: malo.cadoret@lecnam.net

Yannick Bidel
email: yannick.bidel@onera.fr

Clément Salducci
email: clement.salducci@onera.fr

Nassim Zahzam
email: nassim.zahzam@onera.fr

Sylvain Schwartz
email: sylvain.schwartz@onera.fr

Alexis Bonnin
email: alexis.bonnin@onera.fr

Cédric Blanchard
email: cédric.blanchard@onera.fr

Alexandre Bresson
email: alexandre.bresson@onera.fr

Progress towards the development of a cold-atom inertial measurement unit for onboard applications

with performances at the level of the best classical inertial sensors [START_REF] Liu | Sensitive quantum tiltmeter with nanoradian resolution[END_REF]- [START_REF] Mcguirk | Sensitive absolute-gravity gradiometry using atom interferometry[END_REF]. There was only one demonstration of a complete atom interferometer IMU [START_REF] Canuel | Six-Axis Inertial Sensor Using Cold-Atom Interferometry[END_REF]. But it was a laboratory experiment which can work only in static condition and with rotation measurements not at the state of art level. There are still several blocking points to address in order to develop a compact atom IMU working in onboard environment.

Here, we will give our progress toward a cold atom inertial measurement unit. First, we will present the hybridization strategy that we choose for the IMU. Then, we will present our work on the development of an onboard horizontal accelerometer. Finally, we will conclude and present the remaining developments.

II. PRINCIPLE OF THE HYBRIDIZED ATOM INTERFEROMETER INERTIAL MEASUREMENT UNIT

The hybridized cold-atom inertial measurement unit relies on the periodic calibration of 3 classical accelerometers and 3 classical gyroscopes by a multi-axis atom interferometer sensor (see Fig. 1). This is an extension of the method developed for atom accelerometer [START_REF] Bidel | Absolute marine gravimetry with matter-wave interferometry[END_REF], [START_REF] Lautier | Hybridizing matter-wave and classical accelerometers[END_REF]. At each measurement cycle, the atom interferometer is configured to measure one of the six inertial components. This absolute measurement is then used to estimate the bias and the scale factor of the corresponding classical inertial sensor. On the other hand, the measurements Abstract-In this paper we present our progress towards the development of a complete cold-atom Inertial Measurement Unit (IMU). Our goal is to have a single atomic sensor that will alternatively measure each inertial component (3 accelerations and 3 rotations). Every atomic measurement will be hybridized with a set of classical inertial sensors. Hybridization enables to combine the advantages of both technologies and to provide accurate continuous measurements with high dynamic range, in the scope of onboard inertial navigation compatibility.

Index Terms-inertial sensor, atom interferometry, cold atoms, hybridization

I. INTRODUCTION

Atom interferometer sensor is a promising technology for inertial measurement. It has the main advantage compared to classical technology to provide absolute measurements (no calibration needed) and high long term stability (no drift). An Inertial Measurement Unit (IMU) i.e 3 accelerometers and 3 gyroscopes using this technology could thus provide a breakthrough in navigation application in which the biases of the sensors lead to position errors. Vertical accelerometers i.e gravimeters based on atom interferometry have already demonstrated better performances than classical sensors for static [START_REF] Karcher | Improving the accuracy of atom interferometers with ultracold sources[END_REF] [2] and onboard measurements [START_REF] Bidel | Absolute marine gravimetry with matter-wave interferometry[END_REF] and start to be commercialized [START_REF] Ménoret | Gravity measurements below 10-9 g with a transportable absolute quantum gravimeter[END_REF]. Concerning other inertial sensors (horizontal accelerometer, gyroscope, gradiometer), the level of maturity is much lower with only few laboratory instruments of the classical sensors are necessary to process the atom interferometer measurements in onboard environments. The output signal of a Mach Zehnder type atom interferometer consisting of three laser pulses is given by the following simplified expression:

P = P m - C 2 cos k eff • ( a + 2 v × Ω) T 2 (1)
where P m and C are the offset and the contrast of the fringe, a is the acceleration, v is the atom velocity, Ω is the rotation vector, k eff is the effective wave vector of the laser interrogating the atom and T is the time interval between the laser pulses.

The measured inertial component is selected with the direction of laser k eff and the atom velocity. By imposing a zero velocity for the atom, we measure only the acceleration in the direction of the laser k eff . By imposing a non-zero velocity, we measure a combination of a and Ω. To retrieve only the rotation, the acceleration is estimated by the calibrated classical accelerometer. Due to the periodic aspect of the signal of the atom interferometer, many values of the inertial quantities are possible for a given value of the atom signal. This ambiguity is lifted by using an estimated value of the atom phase provided by the classical inertial sensor. Finally by using this strategy, the six classical inertial sensors are regularly calibrated providing thus an absolute and continuous measurement of 3 accelerations and 3 rotations. A multi-axes atom interferometer sensor has thus to be developed. The vertical acceleration measurement is mature with the demonstration of airborne [START_REF] Bidel | Absolute airborne gravimetry with a cold atom sensor[END_REF] and shipborne measurement [START_REF] Bidel | Absolute marine gravimetry with matter-wave interferometry[END_REF]. The next steps of development are the horizontal accelerometer and the gyroscopes.

III. HORIZONTAL ACCELERATION MEASUREMENTS

A. Zero-velocity atom interferometry

In a light-pulse atom interferometer, laser pulses are used to coherently split, deflect, and recombine matter waves in order to create atom interference. Inertial sensors usually use stimulated two-photon (Raman or Bragg) transitions to create the beamsplitters and mirrors required for the interferometer. Stimulated two-photon transitions consist of an atom coherently absorbing and then emitting a photon from a pair of counterpropagating laser beams with different frequencies. This process results in a transfer between two internal ground states of the atom, accompanied with a momentum change of ± k eff , where k eff is the effective wave vector. For high precision measurements, the two counterpropagating Raman lasers are usually implemented in a retroreflected geometry, where a single laser beam with two laser frequencies is retroreflected off a mirror.

In an atom interferometer, the direction of the acceleration measurement is given by the Raman (or Bragg) lasers direction. Therefore, in order to measure horizontal accelerations, the Raman lasers must be horizontal. This configuration is more challenging than a vertical accelerometer where the Raman beams are vertical. First, the atoms fall under gravity perpendicular to the laser beam, therefore limiting the falling distance to the beam diameter for an instrument using a single laser beam. The use of several laser beams is possible, but complicates the setup. Second, the atoms are interrogated with zero velocity in the Raman beam direction. In a retroreflected configuration, two pairs of Raman beams with opposite effective wave vector ±k eff can perform the twophoton transitions (see Fig. 2). When the atoms have a non zero velocity in the Raman beam direction, the Doppler effect naturally lifts the degeneracy between the two transitions ± k eff , and one can easily address a single transition. This is not the case in a horizontal accelerometer : lifting the degeneracy requires to give a velocity to the atoms thanks to a moving molasses technique for instance. The best horizontal accelerometers [START_REF] Liu | Sensitive quantum tiltmeter with nanoradian resolution[END_REF], [START_REF] Canuel | Six-Axis Inertial Sensor Using Cold-Atom Interferometry[END_REF] based on atom interferometry both use this technique. However, it leads to more complex and larger instruments. Another solution is to exploit the double diffraction scheme, but this only works if the two Raman transitions are perfectly degenerated. For onboard applications, where the atoms acquire an uncontrolled velocity under the effect of the vehicle accelerations, it is impossible to address the double diffraction scheme with high efficiency. This method is thus not appropriate to build an embedded horizontal accelerometer. Finally, a last method consists in selecting atoms with nonzero velocity within the atomic velocity distribution [START_REF] Geiger | Detecting inertial effects with airborne matter-wave interferometry[END_REF]. This technique leads to an important loss of atoms which is not satifying. In this paper we present two new techniques using single-diffraction two-photon Raman transitions despite a zero Doppler shift in a retroreflected geometry.

B. Retroreflected frequency-chirped laser

The first technique demonstrated in 2019 [START_REF] Perrin | Zero-velocity atom interferometry using a retroreflected frequency-chirped laser[END_REF] consists in applying a frequency chirp β = 1 2π dω1 dt = dω2 dt to the Raman laser frequencies ω 1 and ω 2 (see Fig. 3). Since the reflected beams are delayed with respect to the incoming ones by a time duration t d = 2L c (with L the distance between the atoms and the mirror), the incoming laser frequencies are shifted by δω = 2πβt d at the position of the atoms. Therefore, one transition is detuned from the other by 2δω : such a frequency-chirped laser method lifts the degeneracy between the two transitions ± k eff , allowing one to selectively address Using this method, we demonstrated a Mach-Zehnder type atom interferometer hybridized with a force balanced accelerometer providing horizontal acceleration measurements with a short-term sensitivity of 3.2 × 10 -5 m.s -2 / √ Hz (see Fig. 4). This is comparable to state of the art for horizontal configurations [START_REF] Xu | Quantum tiltmeter with atom interferometry[END_REF], despite shorter interrogation times (32 versus 226 ms). We cannot draw conclusions regarding the longterm stability of the hybridized acelerometer because of the fluctuations of the Raman mirror angle. Future improvements of the experimental setup could include the implementation of an auxiliary tilt sensor monitoring the angle between the Raman beam and the horizontal plane during the measurement.

C. σ + -σ -Raman transitions between magnetically sensitive states

For the second technique, we implemented a horizontal accelerometer based on atom interferometry using counterpropagative Raman transitions between the hyperfine states |F = 1, m F = ∓1 and |F = 2, m F = ±1 (see Fig. 5) [START_REF] Bernard | Atom interferometry using σ + -σ -Raman transitions between |F = 1, m F = ∓1 and |F = 2, m F = ±1[END_REF]. Atom interferometry usually exploits the |F = 1, m F = 0 ↔ |F = 2, m F = 0 transition, but our scheme presents several advantages. First, due to selection rules, only a single counterpropagating transition is allowed even if the Raman laser beams are retroreflected (see Fig. 5). Secondly, it uses Raman beams with the same polarization configuration as the magneto-optical trap. Finally, it allows the control of the atom trajectory with magnetic forces.

With this method, we performed the measurement of the horizontal component of acceleration with a hybridized accelerometer and reached a short-term sensitivity of 25 × 10 -5 m.s -2 / √ Hz (see Fig. 6). Specific features of our technique, in comparison with usual magnetically-insensitive Raman-based atom interferometers, are the following : higher spontaneous emission rate (as the Raman lasers require to be tuned closer to atomic transitions), additional sensitivity to magnetic field inhomogeneity and light shifts (the differential light shift can not be cancelled out by tuning the intensity ratio between the Raman pair). The bias due to the magnetic field gradient is the main limitation to this hybridized atom accelerometer accuracy (2×10 -4 m.s -2 ).

Using the D 1 line (instead of the D 2 line) for the Raman excitation scheme could enhance the short-term sensitivity of this sensor, because it would drastically reduce spontaneous emission. In addition, the bias could be reduced by using a high-performance magnetic shielding. Moreover, that acceleration bias could be canceled out by implementing a fountain geometry.

IV. CONCLUSION AND PERSPECTIVES

We have demonstrated two new methods for measuring horizontal accelerations in a close-to-zero velocity regime. Both methods can be implemented in an embedded IMU. Next step of development are the gyroscopes. Here the challenge is be to perform interferometric measurements of rotations in a compact device but with state-of-the-art performance. Large momentum transfer beamsplitters will be necessary since the sensitivity of the interferometric sensor is proportional to its enclosed space-time area.

Fig. 1 .

 1 Fig. 1. Possible implementation of the hybridized cold atom IMU (SF: Scale Factor).

Fig. 2 .

 2 Fig. 2. Left : Schematic setup of two-photon Raman transitions in the commonly used retroreflecting geometry. Right: Scheme of the Raman transition between the two hyperfine ground states of an alkaline atom in the absence of Doppler shift. Both pairs ± k eff are simultaneously resonant.

  Fig. 3. Left : The frequency chirp β applied to the Raman lasers frequency shifts the incident laser frequency by δω = 2πβt d where t d = 2L/c. Right : Energy diagram showing that the frequency chirp β allows to lift the degeneracy between the two resonant transitions by an amount 2δω.

Fig. 4 .

 4 Fig. 4. The blue line represents the Allan standard deviation (ADEV) of the hybridized atomic accelerometer. τ -1/2 scaling is represented by the dashed line, whereas the τ scaling appears in the green line.

Fig. 5 .Fig. 6 .

 56 Fig. 5. (a) Energy diagram of the two hyperfine ground states of 87 Rb when a magnetic fied is applied. The degeneracy between the Zeeman sublevels is lifted and one can perform σ + -σ -Raman transitions. The circularly polarized beams which perform the |F = 1, m F = -1 ↔ |F = 2, m F = 1 transition appear in solid lines, whereas the beams performing the |F = 1, m F = 1 ↔ |F = 2, m F = -1 transition are represented in dashed lines. (b) Usual implementation of two-photon Raman transitions in a retroreflected configuration. The atom is a two-level system which interacts with two pairs of counterpropagating light fields with σ + -σ - polarizations. Such a polarization scheme allows only one pair of couterpropagating light beams to drive the Raman coupling. This results in a single diffraction process even if there is no Doppler shift.


Jean-Yves Buffiere

Emilie Ferrié

Wolfgang Ludwig

Anthony Gravouil

Jean-Yves Buffière

Characterisation and modelling of the three dimensional propagation of short fatigue cracks

Keywords: fatigue, micro-tomography, cast alloy, fine grain, 3D

Introduction

In the last twenty years, a lot of experimental and theoretical efforts have been devoted to the characterisation and modelling of small fatigue crack propagation [START_REF] Newman | The merging of fatigue and fracture mechanics concepts : a historical perspective[END_REF], Despite of this intense activity, the growth rate predictions for this kind of defects are far from being completely satisfactory for the moment.

Small fatigue cracks have a complex 3D shape and their propagation at the surface of a fatigue sample, where they are readily observed, is not necessarily representative of the bulk behaviour unlike long cracks (typically more than several millimetres in length). Although this problem has been recognised for at least twenty years [START_REF] Miller | The short crack problem[END_REF] it has hardly been taken into consideration in practice. From a characterisation point of view, the resolution of experimental techniques such as beach marking or heat tinting conventionally used to visualise the 3D shape of long cracks is not sufficient for small cracks. Thus, up to now, experimental characterisation has remained quite limited.

For such a characterisation, X-ray micro-tomography is a very attractive technique which enables the visualisation of internal features in opaque samples. Being a non destructive technique, it also enables, in principle, in situ visualisation of damage during loading and hence the chronology of damage initiation and growth. In the last ten years, significant progress has been made in terms of resolution with both the availability of new third generation synchrotron X-ray sources and new detectors [START_REF] Baruchel | Tomography in Material Science[END_REF]. A spatial resolution close to that of an optical microscope can now be achieved in 3D which opens (or re-opens) wide areas of research.

This paper presents some recent experiments which show the 3D development of fatigue cracks inside two aluminium alloys with very different grain sizes. The crack morphologies are studied in relation to the microstructure along the crack front. Examples of the use of 3D growth data for modelling the propagation of fatigue cracks are given.

Experiments and methods

Materials Two aluminium alloys have been studied. One is a model AS7GO3 cast Al alloy (Al-7Si-0.3Mg, in wt%) containing artificial pores introduced during the casting process. This was used in the T6 conditions (yield stress 200 MPa). The pores located at the surface or just below it act as stress raisers in this material and promote local cyclic plasticity which leads to crack initiation. Because of the casting process, the size of the grains is large (∼ 300 µm) and their 3D shape is irregular. This can be visualised by X ray tomography via a Gallium infiltration technique. Although this method is destructive (Gallium renders the material extremely brittle) it can be used at the end of a fatigue test to reveal the shape of the grains surrounding a fatigue crack. The 3D shape of a crack can therefore be studied as a function of the grain boundaries encountered locally by the crack front [START_REF] Ferrié | 3D characterisation of the nucleation of a short fatigue crack at a pore in a cast Al alloy using high resolution synchrotron microtomography[END_REF]. More detailed descriptions of the microstructure and fatigue deformation mechanisms of this model material and of the infiltration technique can be found elsewhere [START_REF] Buffière | Experimental study of porosity and its relation to fatigue mechanisms of model Al-Si7-Mg0.3 cast Al alloys[END_REF], [START_REF] Ludwig | Study of the interaction of a short fatigue crack with grain boundaries in a cast Al alloy using X-ray microtomography[END_REF].

The second material studied is a powder metallurgy Al-Li alloy based on AA5091 alloy (Al-1.2Li-4.0Mg-1.0C-0.5O in wt%). The material was selected because of its grain size which is of the order of 1 µm. The ultrafine grains promote homogeneous deformation and prevent crystallographic cracking so that the crack shape appears very regular with the spatial resolution of the 3D imaging technique employed in this study (∼ 1 µm). The material was produced from powders using a mechanical alloying process, followed by hot isostatic pressing (hipping) and forging. It was used in the as-forged (T1) condition (Yield stress 450 MPa) [START_REF] Ferrié | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF].

Micro-tomography All the experiments described in this paper have been performed at the European Radiation Synchrotron Facility (ESRF) in Grenoble (France) on beam line ID19. The details of the tomography experimental set up of ID19 can be found in reference [START_REF] Ludwig | Study of the interaction of a short fatigue crack with grain boundaries in a cast Al alloy using X-ray microtomography[END_REF], In the following paragraphs, only the main experimental features are presented. Obtaining a tomographic image consists, firstly, of recording a series of radiographs (called scans) of a sample which is rotated around one axis. These radiographs are further used by a reconstruction algorithm to obtain a 3D numerical image of the sample which is, in its classical form, a 3D map of the attenuation coefficient in the sample. The two dimensional (2D) radiographs are recorded on a Charge Coupled Device (CCD) camera with a square array of 2048*2048 elements. This detector is coupled with a fluorescent screen via optical lenses. The white beam coming from the synchrotron ring is rendered monochromatic by a multilayer monochromator. The energy of the beam is set to 20 keV. The monochromatic beam is parallel so no geometric enlargement is possible, instead, the voxel size is a result of the camera optics used. The size of the isotropic voxels in the reconstructed images is 0.7µm, a value which gave a good compromise between the specimen size and the spatial resolution. The specimen section to be imaged is hence restricted to 1 × 1mm 2 .

In situ fatigue tests A dedicated fatigue machine with a reduced size and a low vibration level has been designed in order to monitor crack development in situ. The machine was directly installed on the rotation stage of the micro-tomography setup. The samples have been cycled in air with a constant stress amplitude (R=0.1). Crack initiation was first monitored by radiography. Once a crack was initiated, full 3D tomographic scans of the crack were recorded at regular intervals during the (interrupted) fatigue test which was stopped before the final fracture of the sample. During image acquisition the sample was maintained under maximum load to improve crack detection. The cyclic conditions were different for the two materials.

For the cast alloy a maximum cyclic frequency of 5Hz was attainable. Therefore a maximum stress of 180 MPa was used in order to make the fatigue life "fit" within a 24 hour experiment. This rather high value somewhat restricts the probability of observing crack arrests, which are typical of the short crack regime below the macroscopic crack propagation threshold.

For the Al-Li alloy a cyclic frequency of 40 Hz could be achieved by changing the loading system from pneumatic to mechanical. The maximum load value was set at 220 MPa.

Similar hour glass specimens were used for both materials (section 1*1 mm 2 ) but in the case of Al-Li a thin (2 µm) rectangular notch, 100 µm wide and 20 µm deep, was produced in the sample using Focused Ion Beam (FIB) machining. This notch was located at the centre of one of the specimen faces and acted as a crack initiation site [START_REF] Ferrié | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF]

Results and analysis Cast alloy

The development of a fatigue crack initiated on a sub-surface pore in the corner of a sample of the cast material is illustrated in Fig. 1. This figure shows that, for the experimental conditions investigated, the front of a microstructurally short crack was not smooth but exhibited protruding and retreating parts (see arrows labeled 1, 2 and 3 on the figure). Although the crack propagation plane was on average perpendicular to the uniaxial fatigue stress, local deviations from this plane did occur. These appear as shadows on the image. The examination of several similar images revealed that these deviations were usually more pronounced in the interior of the specimen that at the surface where the crack remained, on average, more planar and perpendicular to the stress direction. Using the Gallium infiltration technique, it was possible to image the 3D shape of the grains surrounding the crack. The results of this analysis revealed that the irregularities observed on the crack front were always correlated with the presence of a grain boundary. Hence, in spite of the rather high stress level used for the tests, the local crystallography appeared to either promote or impede the crack propagation.

It has been suggested [START_REF] Cox | Monte Carlo Simulations of the Growth of Small Fatigue Cracks[END_REF] that this effect of local crystallography could in fact be balanced by a local modification of the stress intensity factor K along the crack front: a protruding part of the front leading to an increase in K and vice versa. In order to check this assumption, K calculations along the experimental crack fronts have been performed using the eXtended Finite Elements Method (XFEM). The authors are well aware that for such short fatigue cracks the use of the K parameter is probably not valid. However the calculation was performed to estimate, as a first approximation, the magnitude of variation of K along an experimental crack front compared with a crack of similar shape but showing a regular crack front. To perform the calculation the 3D crack shape was first obtained by thresholding the grey level tomographic images. The crack was then transformed into a 3D FE mesh.

The results of the calculation are shown in Fig. 2. They confirm that in the parts of the crack front which are propagating faster (eg for the part of the crack front comprised between the letters A and B on Fig. 2, a decrease in K is observed3 with respect to the crack with a regular shape. More precisely, the largest variations in K are induced by small values of the local radius of curvature (m 1 and m 2 on Fig. 2). The K values calculated at the surface of the sample (θ = 0 degrees and θ = 90 degrees) note were higher than those calculated for a crack with a regular shape indicating that sub-surface irregularities along the crack front (invisible from the surface) do affect the values of the stress intensity factor at the sample surface.

Al-Li alloy In the Al-Li material a crack initiated from the notch after 2000 cycles. Its propagation was monitored every 1000 cycles up to 24000 cycles. The observed crack shapes are shown in Fig. 3 for three different numbers of fatigue cycles. The crack fronts were very smooth and the plane of propagation was almost perfectly perpendicular to the cyclic stress direction. This is a direct consequence of the small grain size encountered in this material. The crack aspect ratio a/c (see definitions of a and c on Fig. 3 remained larger than 1 throughout the propagation, indicating a higher crack growth rate in the bulk.

Calculation of the K I values along the crack front was carried out using the same thresholding/meshing method described previously for the cast alloy. Fig. 4 shows examples of the calculated K I values. At the sample surface the K I values were obtained by extrapolating the values obtained immediately below the surface. As expected from the crack aspect ratio the K I values were found to be lower in the bulk of the sample. It is worth mentioning out that analytical values obtained with the classical Newman & Raju [9] formulas were in very good agreement with the FE values (see Fig. 4).

From the K=f(θ) curves, and the experimental values of the crack size at different number of cycles, da/dN=f(∆K) and dc/dN=f(∆K) curves could be obtained. They confirm that the crack growth rate is higher in the bulk of the material than at the surface, for a given ∆K value. The measured surface crack growth rate was very similar ( for ∆K larger than 3.5 MPa.m 1 2 ) to that measured on macroscopic CT samples.

The 3D propagation of the crack was simulated using the C and m parameters of the Paris law determined from the experimental da/dN=f(∆K) curve. This curve was chosen rather that the dc/dN curve because the calculation of K I values was assumed to be less problematic in the bulk (no extrapolation needed). An example of the simulation results is shown in Fig. 5. This figure shows that the experimental crack front cannot be simulated using a single propagation law; this was expected given the fact that, as mentioned before, different crack growth rates were observed in the bulk and at the surface.

The lower crack growth rate observed at the surface was assumed to be due to crack closure. In this material, Plasticity Induced Crack Closure (PICC) is considered to be the main mechanism responsible for closure. At the surface of the sample where plane stress prevails, a larger plastic zone is expected, leading to a closure level higher than in the bulk. This 3D aspect of closure has been reported various times in the literature (eg in reference [10]). A constant ratio of ∆K ef f /∆K of 0.4 was assumed at the sample surface. This value is typical of the closure mechanism in this material. The closure level was then assumed to vary from 0.4 at the surface to 0 in the bulk linearly. With this simple assumption, a very good correlation was found between the calculated experimental crack fronts all through the fatigue life of the sample. An example is shown in Fig. 5. More details on the simulation of crack propagation can be found in reference [START_REF] Ferrié | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method[END_REF] Conclusion The availability of powerful synchrotron X ray sources coupled with the development of new detectors has enabled the development of high resolution X ray microtomography setups which deliver images of internal features in optically opaque materials with a spatial resolution close to that of optical microscopy. In the work reported here, this technique has been applied to the field of fatigue. The 3D shape of fatigue cracks grown in situ can be determined, providing unique experimental data. For Al alloys, it has been shown that fatigue cracks with a size comparable to the grain size exhibit an irregular crack front which is caused by the presence of grains which impede or promote crack propagation. Those irregularities on the crack front induce variations in the stress intensity factor which tend to balance the effect of the microstructure. When the crack size is much larger than the grain size the crack shape becomes very planar and the crack front is very smooth. For the experimental conditions investigated the value of the crack growth rate was found higher in the bulk than at the surface. This anisotropy has been attributed to a 3D closure effect: the closure level being higher at the surface (plane stress) than in the bulk (plane strain). A good agreement between experimental and simulated crack fronts is found when the 3D correction of crack closure between bulk and surface is taken into account.

[ 9 ]Fig. 1 :Fig. 2 :Fig. 3 : 4 :

 91234 Fig. 1: 3D rendition of a small fatigue crack growing inside a cast Al alloy.

Fig. 5 :

 5 Fig. 5: Comparaison between experimental and simulated crack fronts with and without correction for 3D crack closure.

the crack being planar, only the K I (Mode I) factor is calculated

Acknowledgement The authors are grateful to Mr. Bultreys from FEI Europe for the FIB machining to Dr.Ian Sinclair (Southampton University) for fruitful discussions on the 3D nature of the closure phenomenon and to Prof. L. Edwards (Open University) for providing the Al-Li alloy. The ESRF and staff are acknowledged for their help and assistance during the tomographic imaging experiment.