Johann Lavaud
email: lavaud@biologie.ens.fr

Bernard Rousseau

Anne-Lise Etienne

GENERAL FEATURES OF PHOTOPROTECTION BY ENERGY DISSIPATION IN PLANKTONIC DIATOMS (BACILLARIOPHYCEAE) 1

Keywords: diadinoxanthin, diatoxanthin, non-photochemical fluorescence quenching, photoinhibition, photoprotection, planktonic diatoms, xanthophyll cycle Chl a, chlorophyll a, DD, diadinoxanthin, DT, diatoxanthin, LHC, lightharvesting complex, NPQ, non-photochemical fluorescence quenching, PS II, Photosystem II

Planktonic diatoms (Bacillariophyceae) have to cope with large fluctuations of light intensity and periodic exposure to high light. After a shift to high light, photoprotective dissipation of excess energy characterized by the non-photochemical quenching of fluorescence (NPQ) and the concomitant de-epoxidation of diadinoxanthin to diatoxanthin (DT) were measured in four different planktonic marine diatoms (Bacillariophyceae): Skeletonema costatum (Greville) Cleve, Cylindrotheca fusiformis Reimann et Lewin, Thalassiosira weissflogii (Grunow) Fryxell et Hasle and Ditylum brightwellii (West) Grunow in comparison to the model organism Phaeodactylum tricornutum Böhlin. Upon a sudden increase

of light intensity, de-epoxidation was rapid and de novo synthesis of DT also occurred. In all species, NPQ was linearly related to the amount of DT formed during high light. In this report, we focused on the role of DT in the dissipation of energy which takes place in the LHC. In S. costatum for the same amount of DT, less NPQ was formed than in P. tricornutum and as a consequence the photoprotection of PS II was less efficient. The general features of photoprotection by harmless dissipation of excess energy in planktonic diatoms described here partly explain why diatoms are well adapted to light intensity fluctuations.

INTRODUCTION

Diatoms are ubiquitous primary producers accounting for approximately 40% of the primary carbon production in the oceans (Tréguer et al. 1995[START_REF] Field | Primary production of the biosphere: integrating terrestrial and oceanic components[END_REF]. Planktonic diatoms (Bacillariophyceae) are the dominant group of phytoplankton. They have to cope with large fluctuations in light intensity and periodic exposures to high light [START_REF] Harris | Phytoplankton Ecology: Structure, Function, and Fluctuation[END_REF][START_REF] Fogg | The phytoplanktonic ways of life[END_REF][START_REF] Long | Photoinhibition of photosynthesis in nature[END_REF]). To survive a sudden increase in light intensity, photoprotective mechanisms are essential [START_REF] Niyogi | Arabidopsis mutants define a central role for the xanthophyll cycle in the regulation of photosynthetic energy conversion[END_REF][START_REF] Niyogi | Safety valves for photosynthesis[END_REF]. The non-radiative dissipation of excess energy is an important short-term process for the photoprotection of photosystem (PS) II against light-induced damage [START_REF] Müller | Non-photochemical quenching. A response to excess light energy[END_REF]. Photoprotective dissipation is attributed to rapid modifications within the light-harvesting complex (LHC) of PS II leading to a nonphotochemical chlorophyll (Chl) a fluorescence quenching (NPQ). NPQ is induced by the formation of a proton gradient across the thylakoid membrane (pH) and is associated with the operation of a xanthophyll cycle, which converts epoxidized to de-epoxidized forms of xanthophylls. It has been largely documented in higher plants and green migroalgae (for reviews see [START_REF] Pfündel | Regulation and possible function of the violaxanthin cycle[END_REF][START_REF] Horton | Regulation of light harvesting in green plants[END_REF][START_REF] Eskling | The xanthophyll cycle, its regulation and components[END_REF][START_REF] Gilmore | Mechanistics aspects of xanthophyll cycle-dependent photoprotection in higher plant chloroplasts and leaves[END_REF][START_REF] Bassi | LHC proteins and the regulation of photosynthetic light harvesting function by xanthophylls[END_REF][START_REF] Müller | Non-photochemical quenching. A response to excess light energy[END_REF]) but much less so in diatoms.

The photosynthetic apparatus of diatoms differs in many respects from that of green algae and higher plants. Thylakoid membranes are loosely appressed and organized in extended layers of three without grana stacking; PS I and PS II are randomly distributed [START_REF] Pyszniak | Immunocytochemical localisation of photosystem I and the fucoxanthin-chlorophyll a/c light-harvesting complex in the diatom Phaeodactylum tricornutum[END_REF]. Diatoms have a specific set of pigments with Chl c, fucoxanthin and diadinoxanthin (DD) (which can be de-epoxidized to diatoxanthin, DT), in addition to the Chl a and -carotene found in all photosynthetic organisms [START_REF] Brown | Photosynthetic pigment organization in diatoms (Bacillariophyceae)[END_REF]. The xanthophyll concentration relative to Chl can be two to four times more than in higher plant LHC (Wilhelm 1990[START_REF] Passaquet | Lightharvesting complexes of brown algae -Biochemical characterization and immunological relationships[END_REF]. These pigments are bound to the LHC subunits that are composed of several highly homologous proteins encoded by a multigene family (fucoxanthin chlorophyll proteins, FCP, [START_REF] Bhaya | Characterization of gene clusters encoding the fucoxanthin chlorophyll proteins of the diatom Phaeodactylum tricornutum[END_REF]. The exact organization of LHC subunits in diatoms, and especially the location of DD and its de-epoxidation within these complexes, is currently being studied [START_REF] Lavaud | Enrichment of the light-harvesting complex in diadinoxanthin and implications for the nonphotochemical fluorescence quenching in diatoms[END_REF]. No obvious orthologs of some of the important components of LHC of higher plant PS II (e. g. the minor antennae CP 26, CP 29 and CP 22 (PsbS protein)) have been found in Phaeodactylum tricornutum Böhlin or within the genome sequence of Thalassiosira pseudonana Hustedt (Hasle and Heimdal) (Müller et al. 2001, C. Bowler, personal com.).

Studies of the photoprotective dissipation of excess energy in diatoms have mostly been performed using P. tricornutum as a model organism [START_REF] Ting | Photochemical and non-photochemical fluorescence quenching processes in the diatom Pheaodactylum tricornutum[END_REF][START_REF] Arsalane | Influence of the pool size of the xanthophyll cycle on the effects of light stress in a diatom: Competition between photoprotection and photoinhibition[END_REF], Olaizola et al. 1994[START_REF] Casper-Lindley | Fluorescence quenching in four unicellular algae with different light-harvesting and xanthophyll-cycle pigments[END_REF][START_REF] Goss | Stimulation of the diadinoxanthin cycle by UV-B radiation in the diatom Phaeodactylum tricornutum[END_REF][START_REF] Mewes | Supplementary ultraviolet-B radiation induces a rapid reversal of the diadinoxanthin cycle in the strong light-exposed diatom Phaeodactylum tricornutum[END_REF]. This choice results in part from the fact that this species is easy to grow and maintain in artificial conditions in dense cultures. In P. tricornutum, we have previously shown that NPQ indeed has a photoprotective effect for the PS II reaction centers under an excess of light (Lavaud et al. 2002a). NPQ depends on the size of the DD pool and can reach much larger values than in higher plants (Lavaud et al. 2002a, Li et al. 2002c). NPQ and DT were linearly related and if DT is not present, NPQ cannot be formed (Lavaud et al. 2002a, Lavaud et al. 2002b). Nevertheless, the ecological relevance of P. tricornutum is weak compared to other diatom species. In order to define some general features of photoprotection in diatoms, we compared the effects of high light illumination on four species of planktonic marine diatoms: Cylindrotheca fusiformis Reimann et Lewin, Ditylum brightwellii (West) Grunow and two ecologically relevant species Skeletonema costatum (Greville) Cleve, Thalassiosira weissflogii (Grunow) Fryxell et Hasle. The photoprotective dissipation of excess energy (kinetics of DD de-epoxidation and of NPQ formation) was characterized in each species as well as in P. tricornutum as a control. For the species with the smallest and the largest NPQ (S. costatum and P. tricornutum, respectively), the kinetics of PS II photoinhibition were compared. [START_REF] Guillard | Studies of marin planktonic diatoms. 1. C. nana (Hustedt) and D. confervacea (Cleve)[END_REF]. Non-axenic cultures of 300 mL were incubated at 18C in airlifts continuously flushed with sterile air. They were illuminated at a light intensity of 40 µmol photons.m -2 .s -1 with white fluorescent tubes (Claude, Blanc Industrie, France) with a 16 h light/8 h dark photoperiod. Cells were harvested during the exponential phase of growth, centrifuged at 3000 g for 10 min and resuspended in their culture medium to a final concentration of 10 µg Chl a.mL -1 . The concentrated suspension was then continuously stirred at 18C under low light before use.

MATERIALS AND METHODS

Cultures

Pigment content.

Pigment analyses were performed by HPLC as previously described [START_REF] Arsalane | Influence of the pool size of the xanthophyll cycle on the effects of light stress in a diatom: Competition between photoprotection and photoinhibition[END_REF][START_REF] Lavaud | Enrichment of the light-harvesting complex in diadinoxanthin and implications for the nonphotochemical fluorescence quenching in diatoms[END_REF]. Cells collected from the Clark electrode adapted to the PAMfluorometer (see below) were deposited on a filter and frozen in liquid nitrogen. Pigments were extracted with a methanol:acetone (70:30, v/v) solution. Published extinction coefficients for Chl [START_REF] Berkaloff | Subunit organization of PSI particles from brown algae and diatoms: polypeptide and pigments analysis[END_REF]) and for diadinoxanthin (DD) and diatoxanthin (DT) [START_REF] Johansen | Carotenoids of the Dinophyceae[END_REF] were used. Cell counts were performed with a Thoma hematocymeter, using the public domain NIH Image program (US National Institute of Health).

Chl fluorescence yield and non-photochemical fluorescence quenching (NPQ).

Chl fluorescence yield was monitored with a modified PAM-101 fluorometer (Walz, Effeltrich, Germany) as described previously [START_REF] Ritz | pH-dependent fluorescence quenching and its photoprotective role in the unicellular red alga Rhodella violacea[END_REF]. A Clark electrode was adapted to the PAM light guides in order to record simultaneously oxygen and fluorescence. Fluorescence was excited by a very weak (non-actinic) modulated 650 nm light. After 20 min dark-adaptation, continuous actinic light of adjustable intensity was applied. 600 ms pulses of white light (4000 µmol photons.m -2 .s -1 ) were admitted by an electronic shutter (Uniblitz, Vincent, USA, opening time 2 ms) placed in front of a KL-1500 quartz iodine lamp continuously switched on in order to monitor NPQ evolution. The average fluorescence (acquisition time 33 µs) measured during the last 400 ms of the pulse was taken as F m or F m '. Data were recorded with a computer through a 12 bit analogue-digital interface and the system was driven by home-made software [START_REF] Arsalane | A new fluorometric device to measure the in vivo chlorophyll a fluorescence yield in microalgae and its use as a herbicide monitor[END_REF]. For each experiment, 2 mL of cell suspension were used. Sodium bicarbonate was added at a concentration of 4 mM from a freshly prepared 0.2 M stock solution in distilled water to prevent any limitation of the photosynthetic rate by carbon supply. [START_REF] Krause | Chlorophyll fluorescence and photosynthesis: the basics[END_REF]. NPQ quantified by the 'Stern-Volmer' expression is proportional to the concentration of the deepoxidized xanthophyll zeaxanthin or DT [START_REF] Gilmore | Mechanistics aspects of xanthophyll cycle-dependent photoprotection in higher plant chloroplasts and leaves[END_REF]Yamamoto 1991, Lavaud et al. 2002a). Non-photochemical quenching also occurs on the minimal fluorescence level F o and is also proportional to the concentration of DT (Lavaud et al. 2002a). It is expressed as SV o = F o /F o '-1 where F o ' is the PS II fluorescence yield of light-adapted cells. It is always smaller than NPQ (or SV m ) (Lavaud et al. 2002a). While NPQ is frequently used because it is the most easy to determine, F o ' is also very useful to determine the quantum yield of photochemistry in the presence and absence of DT with the following relationship: (Lavaud et al. 2002a).

 Q [DT=0] /  Q [DT] = F o / F o '

O 2 yield per flash

The relative O 2 yield produced per flash during a sequence of single-turnover saturating flashes at a frequency of 2 Hz was measured at 18C with a rate electrode as described by Lemasson and Etienne 1975. Short (5 µs) saturating flashes were produced by a Strobotac (General Radio Co., Concord, MA). For the control sequences, cells were first darkadapted for 20 min and then deposited on the electrode. Both control and illuminated samples taken from the PAM-fluorimeter were allowed to settle on the electrode for 7 min in darkness before measurement. The steady-state O 2 yield per flash (Y SS ) was attained for the last flashes of a sequence of twenty flashes when the classical four-step oscillations due to the S-states cycle [START_REF] Kok | Cooperation of charges in photosynthetic O2 evolution-I. A linear four step mechanism[END_REF] were fully damped. Y SS was used to evaluate the relative number of oxygen producing PS II reaction centers (see Lavaud et al. 2002a).

RESULTS

Characteristics of the five species of diatoms.

Under the same growth conditions, growth rate, cell size and Chl a content per cell of the five species were different but, as can be seen in Table 1, the relative pigment concentrations were rather similar, with a large amount of fucoxanthin, and 5 to 10 times less DD and Chl c. The amount of DD ranged between 7 and 9 mol/100 mol Chl a and at the light intensities used for culturing (40 µmol photons.m -2 .s -1 ) no DT was present. -carotene, known to be bound to the core complexes of both photosystems, was also present in similar quantities. The quantum yield of PS II estimated by the ratio F v /F m was similar for all species.

Characteristics of the xanthophyll cycle.

After the light shift, de-epoxidation of DD to DT occurred rapidly and after a few minutes (at most), DD reached a minimal value (Fig. 1). The rate of de-epoxidation can be estimated by fitting the decrease of DD with an exponential ([DD t = (DD initial -DD minimal ) x e - kt ]). The values found for the rate constant are shown in Table 2. They vary between 2.2 and 1.2 min -1 , the fastest de-epoxidation being in P. tricornutum. The maximum de-epoxidation observed was around 50 %. It was smaller for T. weissflogii and D. brightwellii (Table 2). If the duration of illumination exceeded 15 min at the highest intensity used (2000 µmol photons.m -2 .s -1 ), which is equivalent to full sunlight [START_REF] Long | Photoinhibition of photosynthesis in nature[END_REF], then a slow synthesis of DT independent of DD de-epoxidation (de novo DT synthesis, Olaizola et al. 1994) could be observed. The rate of de novo DT synthesis (estimated from the slope of the slow DT increase independent of DD de-epoxidation) varied by a factor of two between all species (Table 2). The three parameters shown in Table 2 were all light dependent, as illustrated for P. tricornutum using a 1 h illumination duration and two different light intensities of 450 and 2000 µmol photons.m -2 .s -1 . The light dependency observed for all species. The de-epoxidation rate, degree of de-epoxidation and rate of de novo DT synthesis were larger at higher light intensities (Table 3).

Manuscript 03-026 9 Correlation between DT accumulation and NPQ.

We have previously shown that in P. tricornutum grown under low light intensities, NPQ and DT were linearly related (Lavaud et al. 2002a, Lavaud et al. 2002b). In the five strains, the values of DT and NPQ were concomitantly determined in two different ways: i) by varying the illumination time from 0.5 to 60 min at 2000 µmol photons.m -2 .s -1 (Fig. 1 and 2) and ii) by varying the light intensity from 50 to 2000 µmol photons.m -2 .s -1 for a fixed illumination time of 5 min (not shown). These data were used to define the correlation between NPQ and DT (Fig. 3A andB). The regression coefficient of the linear relationship was close to one for all species except for S. costatum, in which it was lower (0.5). In S. costatum, for the same amount of DT, two times less NPQ was formed. In T. weissflogii, for values above 3 mol DT/100 mol Chl a, the regression coefficient between NPQ and DT was smaller (Fig. 3B).

Comparison of PS II photoinhibition and recovery kinetics for P. tricornutum and S. costatum.

During an illumination at 2000 µmol photons.m -2 .s -1 , the variation in the relative concentration of active PSII reaction centers was determined, as a function of the duration of illumination up to 45 min, by measuring the steady-state oxygen yield per flash for the strains with the largest (P. tricornutum) and the lowest (S. costatum) NPQ. Under excess light illumination photoinactivation of PS II reaction centers was faster in S. costatum than in P. tricornutum, as shown in Figure 4. In vivo, photoinhibition occurs in at least two steps: 1) inactivation of PS II reaction centers activity which can be reversed in the dark, 2) inhibition irreversible in the dark [START_REF] Kirilovsky | Protection from photoinhibition by low temperature in Synechocystis 6714 and in Chlamydomonas reinhardtii: detection of an intermediary state[END_REF]). To estimate the irreversible part of dark photoinhibition, the number of active PS II reaction centers was measured after 30 min of Manuscript 03-026 10 dark relaxation for the different illumination times at 2000 µmol photons.m -2 .s -1 (Fig. 4). In both species, a partial relaxation was observed but the difference between P. tricornutum and S. costatum remained: photoinhibition was more pronounced in S. costatum than in P. tricornutum.

DISCUSSION

DD de-epoxidation characteristics and two different pathways for DT synthesis.

The rapid kinetics of de-epoxidation of DD into DT which takes place in all the strains may be due to the fact that a weak proton gradient is sufficient to activate the DD deepoxidase [START_REF] Jakob | Unusual pH-dependence of diadinoxanthin deepoxidase activation causes chlororespiratory induced accumulation of diatoxanthin in the diatom Phaeodactylum tricornutum[END_REF]. DD de-epoxidation thus begins rapidly after the onset of high light and the rate is light intensity-dependent. Part of DD molecules may be constitutive within the LHC [START_REF] Lohr | Xanthophyll synthesis in diatoms: quantification of putative intermediates and comparison of pigment conversion kinetics with rate constants derived from a model[END_REF] inaccessible to the de-epoxidase, and thus nonconvertible, as observed for violaxanthin in higher plants [START_REF] Ruban | Determination of the stoichiometry and strength of binding of xanthophylls to the photosystem II light harvesting complexes[END_REF]). This might be the main reason for the incomplete de-epoxidation of DD (50-55 % in vivo). The pool of the convertible population of DD can vary in size and is most likely weakly bound at the periphery of the pigment-protein complexes [START_REF] Lavaud | Enrichment of the light-harvesting complex in diadinoxanthin and implications for the nonphotochemical fluorescence quenching in diatoms[END_REF] or even in the lipid membrane of the thylakoid, as suggested for violaxanthin in higher plants [START_REF] Eskling | The xanthophyll cycle, its regulation and components[END_REF] and citations within).

For all species examined, exposure to excess light beyond the first 15-30 min leads to further DT accumulation not resulting from DD de-epoxidation. Such DT synthesis has already been reported and denoted de novo DT synthesis (Olaizola et al. 1994, Olaizola and[START_REF] Olaizola | Short-term response of the diadinoxanthin cycle and fluorescence yield to high irradiance in Chaetoceros muelleri (Bacillariophyceae)[END_REF]. It corresponds to a direct conversion of violaxanthin into DT without a DD intermediate [START_REF] Lohr | Xanthophyll synthesis in diatoms: quantification of putative intermediates and comparison of pigment conversion kinetics with rate constants derived from a model[END_REF] because it is insensitive to dithiothreitol, an inhibitor of DD de-epoxidation (Olaizola et al. 1994). The rate of de novo DT synthesis varies between species and with light intensity and duration. With the accumulation of DT, a concomitant formation of NPQ is observed. The direct linear correlation between NPQ and DT concentration, previously found in P. tricornutum (Lavaud et al. 2002a, Lavaud et al. 2002b), exists also in the other species studied. All regressions show a slope of 1, except S. costatum for which it is 0.5. For T. weissflogii, the slope changes for DT values above 3 mol/100 mol Chl a. According to Figure 1, it could mostly correspond to de novo synthesized DT molecules but this may not be true at other light intensities. We tentatively propose that the smaller slope observed with T. weissflogii for high values of DT is the result of a combination between a slope 1 and a horizontal line, indicating that a fraction of the DT molecules does not bind to the NPQ locus in the LHC because it becomes saturated. In S. costatum, the slope is smaller from the start, possibly indicating that the amount of DT involved in NPQ is a smaller fraction of the total pool. It has been shown in higher plants that an increase in the pool of violaxanthin did not influence the amplitude of NPQ [START_REF] Davidson | Overexpression of -carotene hydroxylase enhances stress tolerance in Arabidopsis[END_REF]. Also in S. costatum the amount of the specific component(s) of LHC involved in NPQ could be lower than in other species. Li et al. 2002c recently demonstrated that, other parameters being unchanged, especially the degree of deepoxidation, NPQ was doubled by increasing the expression of the PsbS protein. The molecules of DT which are not directly involved in NPQ could nevertheless play a different photoprotective role by preventing thylakoid membrane lipids from photo-oxidation during prolonged exposure to high light as shown for its analogue zeaxanthin in higher plants [START_REF] Havaux | The violaxanthin cycle protects plants from photooxidative damage by more than one mechanism[END_REF]. In this case, de-epoxidized xanthophylls are located in the lipid matrix of thylakoid membrane (Eskling et al. 1997 and citations within).

Manuscript 03-026 12 Protection against PS II photoinhibition by NPQ.

NPQ has been shown to account for the efficiency of photoprotection of PS II against photoinactivation in P. tricornutum cultures with different NPQ capabilities [START_REF] Arsalane | Influence of the pool size of the xanthophyll cycle on the effects of light stress in a diatom: Competition between photoprotection and photoinhibition[END_REF], Lavaud et al. 2002a). Such a correlation had not been tested for different diatoms with different NPQ characteristics. We have shown previously in P. tricornutum that energy dissipation involving DT equally decreases the quantum yield of oxygen evolution and the quantum yield of fluorescence emission. We have now shown in this report that for the same light treatment and for the same DD/DT pool size, there is two times less NPQ (or SV m ) formed in S. costatum than in P. tricornutum. As excepted, SV o is also smaller in S. costatum than in P. tricornutum for the same DT concentration (data not shown) and therefore the decrease of the quantum yield of photochemistry is smaller for S. costatum than for P. tricornutum. As a consequence, kinetics of PS II photoinhibition are faster in S. costatum than in P. tricornutum and the irreversible part of photoinhibition is larger in S. costatum than in P.

tricornutum.

In summary, in the present report, we have shown that what has been found in the model organism P. tricornutum is also valid for other marine planktonic diatoms. They react rapidly to a sudden increase in light intensity protecting their PS II reaction centers (the main target of photoinhibition) by dissipation of excess energy in the antenna complex of PS II. DT plays a crucial role in this mechanism. A second role for DT which could be prevention of lipid peroxidation needs to be verified. Even though the amplitude and regulation of energy dissipation are species dependent, the following general features can be drawn: (1) a fast DD de-epoxidation and concomitant formation of NPQ (within seconds); (2) a direct linear relationship between DT accumulation and NPQ development; (3) a de novo synthesis of DT that accounts for supplementary photoprotection under prolonged illumination; and (4) all parameters (especially those concerning the xanthophyll cycle) are regulated by the light regime. Because recent studies in higher plants show clearly that PsbS plays a major role for energy dissipation and photoprotection (Li et al. 2002a, Li et al. 2002b[START_REF] Aspinall-O'dea | In vitro reconstitution of the activated zeaxanthin state associated with energy dissipation in plants[END_REF], Li et al. 2002c), it is intriguing that an ortholog has not been found in diatoms. The possibility that another type of protein plays a similar role has not yet been explored.

  Phaeodactylum tricornutum Böhlin (Laboratoire Arago algal collection, Banyuls-sur-Mer, France), Skeletonema costatum UTEX-LB2308 (Greville) Cleve (University of Texas Collection (UTEX), Austin), Cylindrotheca fusiformis CCMP343 Reimann et Lewin (Provasoli-Guillard National Center for Culture of marine Phytoplankton (CCMP), West Boothbay Harbor, Maine), Thalassiosira weissflogii CCMP1051 (Grunow) Fryxell et Hasle, and Ditylum brightwellii CCMP359 (West) Grunow cells were grown photoautotrophically in sterile natural seawater F/2 medium

  Standard fluorescence nomenclature was used (van Kooten and Snel 1990). F o and F m are defined as the minimum PS II fluorescence yield of dark-adapted cells and the maximum PS II fluorescence yield reached in such cells during a saturating pulse of white light, respectively. The quantum yield of excitation trapping by PS II is the ratio F v /F m where F v is the variable part of the fluorescence emission and is equal to F m -F o . Non-photochemical fluorescence quenching (NPQ or SV m ) is quantified by the 'Stern-Volmer' expression F m /F m '-1, where F m ' is the maximum PS II fluorescence yield of light-adapted cells

  light inducedDT and NPQ.

Figure 1 .

 1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Non-photochemical quenching development during 60 min illumination at 2000 µmol photons.m -2 .s -1 ; (insert) enlargement of the first 5 min. P. tricornutum (circles),

Figure 3 .

 3 Figure 3. Correlation of NPQ and DT concentration. (A) P. tricornutum (circles),

Figure 4 .

 4 Figure 4. Loss of active PS II reaction centers during illumination at 2000 µmol photons.m - 2 .s -1 (solid symblos) in P. tricornutum (circles) and S. costatum (triangles), for different

Figure

  Figure 1-Lavaud, Rousseau and Etienne

Table 1 .

 1 Quantum yield of PS II (F v /F m ) and pigment content of the five planktonic marine

	diatoms examined.
	Species	F v /F m	a	Fucoxanthin	Chl c	DD	DT	-carotene
	( 0.03)		(mol.100 mol Chl a -1 )
	P. tricornutum	0.77		67.1  1.3	13.1  0.6	9.2  1.7	0	7.4  0.3
	S. costatum	0.77		73.3  1.8	15.3  1.2	8.3  2.1	0	6.4  0.5
	C. fusiformis	0.79		71.4  1.9	17.5  0.9	6.6  0.6	0	5.7  0.4
	T. weissflogii	0.79		53.6  1.6	11.1  0.5	9.4  0.9	0	8.3  1.5
	D. brightwellii	0.82		81.3  1.8	22.7  0.4	7.5  0.4	0	6.9  0.5
	(

a ) F v /F m (F v =F m -F o )

is the quantum yield of PS II. Data ( SD) are the average of two to four measurements.

Table 2 :

 2 Characteristics of the DD de-epoxidation and DT de novo synthesis measured under 2000 µmol photons.m -2 .s -1

	Species	k a	Degree of de-epoxidation b	Rate of de novo DT synthesis
		(min -1 )	(%)	(mmol.mol Chl a -1 .min -1 )
	P. tricornutum	2.2  0.6	50  1	0.54  0.05
	S. costatum	1.7  0.3	50  4	0.59  0.09
	C. fusiformis	1.2  0.1	47  5	0.26  0.08
	T. weissflogii	1.4  0.1	33  6	0.36  0.04
	D. brightwellii	1.2  0.1	40  1	0.24  0.05
	(

a ) k is the rate constant of DD de-epoxidation (see equation in the Results section). ( b ) The degree of DD de-epoxidation is [DT/(DD+DT)x100] measured before DT de novo synthesis starts (30 min for P. tricornutum, S. costatum and C. fusiformis, 15 min for T. weissflogii and D. brightwelii). Data ( SD) are the average of two to three measurements.

Table 3 .

 3 Effect of light intensity on the DD de-epoxidation and DT de novo synthesis in P. tricornutum cells.

	Light intensity	k a	Degree of de-epoxidation b	Rate of de novo DT synthesis
	(µmol photons.m -2 .s -1 )	(min -1 )	(%)	(mmol.mol Chl a -1 .min -1 )
	450 c	1.4	32	0.27
	2000	2.8	51	0.50

corresponds to full sunlight (well above saturation).Lavaud, Rousseau and EtienneManuscript 03-026

ACKNOWLEDGMENTS

We thank G. Vétion from the Laboratoire Arago, Banyuls-sur-Mer, France for kindly providing us the Skeletonema costatum strain. We thank Dr C. Bowler and Dr. J. Houmard for critical reading and improvement of the manuscript. J. L. thanks the French Ministry of National Education, Research and Technology for financial support.


Laurent Pierre
email: laurent.pierre@bordeaux.ensam.fr

Denis Teissandier
email: denis.teissandier@u-bordeaux1.fr

Jean Pierre Nadeau
email: jean-pierre.nadeau@bordeaux.ensam.fr

Copyright of IDMME -Virtual Concept Qualification of turbine architectures in a multiphysical approach: application to a turboshaft engine

Keywords: geometric variability, tolerancing analysis, thermomechanics, performance criteria

The performance of a turboshaft engine derives essentially from the performance of the turbine, which in turn is closely correlated with rotor/stator clearance at the blade tips. In this article we propose to define criteria for qualifying turbine architectures based on a geometric model which integrates variability due to the processes of obtaining parts, the assembly processes and the thermomechanical behaviour of the turbine. The geometric model proposed here integrates thermomechanical strains in 3D dimension-chains formalised by operations on polytopes (Minkowski sum and intersection).

1-Introduction

Controlling the behaviour and the energy yield of turboshaft engines for each of the different operating regimes is essential to ensure that the desired power is achieved. One way to improve the performance of these turboshaft -engines is to control the geometric variability of the turbine, and more particularly the clearance between the blade tips and the stator.

In the preliminary design phase, several alternative turbine architectures are envisaged. These alternatives are often based on different component shapes and dimensions, with several technical solutions being proposed for joints between components and different materials. In this article we propose a model which will define clearance between blade tips and stator for different turbine architectures, taking the following variabilities into account:

-processes for obtaining parts, -processes for assembling parts, -thermomechanical behaviour of the turbine.

Two performance criteria are formulated with which to qualify the proposed solutions: risk of touching and leakage section between rotor and stator. This work is part of a general series of studies into decision support systems to assist the designer in choosing a qualified turbine architecture that performs better than any other.

In the first part, the procedure for modelling different geometric variabilities is described. In the second part the qualification criteria for turbine architectural solutions are presented. In the third part, we describe an application of this work to a sub-unit of a turbine of turboshaft engine.

Finally, after drawing the principal conclusions, future developments for this work are presented.

2-Modelling geometric variability

The geometric variability in processes for obtaining parts and in assembly processes are taken into account by 3D dimension chain simulation tools. However, most of these tools model the different parts as infinitely rigid solids, so to make up for this, the thermomechanical strains on the parts must be integrated into 3D dimension chain simulations. This is essential in order to control clearance at the tip of the turbine blade in different operating phases in a turboshaft engine [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF].

-Variability due to manufacturing and assembly processes

The geometric models used in 3D dimension chains are generally based on the following hypotheses: no defect in the shape of the real surfaces, no local strain on surfaces in contact and no flexible parts. The limits for geometric defects in a part (defined by specification), and acceptable limits for relative displacement between two surfaces in contact (defined by clearance) are formalised mathematically by a small displacement torsor [BB1] and [START_REF] Clément | A Study of optimalcriteria identification based on the small-displacement screw model[END_REF], a matrix [T1], deviation hulls [START_REF] Giordano | Clearance Space and Deviation Space, Application to three-dimensional chains of dimensions and positions[END_REF] and clearance [START_REF] Giordano | Tolerance analysis and synthesis by means of deviation domains, axisymetric cases[END_REF] or T-Maps [START_REF] Mujezinovi | A new mathematical model for geometric tolerances as applied to round faces[END_REF], [START_REF] Jian | Tolerance Analysis and Allocation using Tolerance-Maps for a Power Saw Assembly[END_REF]. In the following part of the article, hulls will

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be modelled using polytopes [START_REF] Teissandier | Operations on polytopes: application to tolerance analysis[END_REF].

-Variability due to thermomechanical behaviour

Several studies have been carried out to manage compliant structure: [START_REF] Hu | Modeling and Control of Compliant Assembly Systems[END_REF], [START_REF] Stewart | Variation simulation of fixtured assembly for compliant structures using piecewise-linear analysis[END_REF], [START_REF] Söderberg | Computer-aided robustness analysis for compliant assemblies[END_REF] and [START_REF] Xie | Variation propagation analysis on compliant assemblies considering contact interaction[END_REF]. These works take into account geometric variations induced by the assembly process and manufacturing dispersions. [START_REF] Mandil | Computational methodology for the prediction of functional requirement variations across the product life-cycle[END_REF] and [START_REF] Mandil | Parametric Approach for Geometrical Requirement Calculation Along the Product Life-Cycle[END_REF] show the interest of considering functional requirement variations along the various phases of the product life cycle. Simulation of the thermomechanical behaviour of the system studied is based on finite element digital simulations.

Let us consider the 3D dimension chain system which characterises a functional requirement, where the parts are all considered as infinitely rigid (reference behaviour). This system includes intersections and Minkowski sums for polytopes which define acceptable limits of deviation in the parts and acceptable limits of displacement between two parts potentially in contact [START_REF] Pierre | Integration of multiple physical behaviours into a geometric tolerancing approach[END_REF]. This system is referred to as the reference system in the reference behaviour [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF].

Our hypothesis is that the topological structure of the contacts graph in the reference behaviour and all thermomechanical behaviours remain the same. This means that there is no extra contact and no suppression of contact between the two behaviours. On the other hand, the different parameters that determine contact (minimal clearance, maximal clearance and nature of contact) may change.

Each thermomechanical behaviour is characterised by a specific 3D dimension chain system. A 3D dimension chain system for a given behaviour is deduced from the reference system:

-by updating the clearance hulls, taking into account changes in the different contact parameters from rigid behaviour (see Figure 1) to thermomechanical behaviour (Figure 2).

-by updating deviation hulls to incorporate displacements caused by thermomechanical strains and the acceptable limits for geometrical defects by superimposition [START_REF] Pierre | Integration of multiple physical behaviours into a geometric tolerancing approach[END_REF] (see Figure 3).

Features :

-Name : 1,2/2,2 -Type : Cylindrical Pair (CP) -Situation element : line (P, x)

-Nature : floating contact -Clearance : J max >J min ≥ 0

x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 P J L 1, 2 max 2,2 J L + max 2,2 J L - max 2 J + max 2 J - P y ε z ρ
Clearance space : Features :

-Name : 1,2/2,2 -Type : Cylindrical Pair (CP) -Situation element : line (P, x)

-Nature : floating contact -Clearance : J max >J min ≥ 0 x y z x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 P J L 1, 2 max 2,2 J L + max 2,2 J L - max 2 J + max 2 J - P y ε z ρ P y ε z ρ Clearance space :

Deviation induced by thermomechanical strains

A y

ε z ρ 1,1 ,1,1/1,2th 2 A y t ε -+ 1,1 ,1,1/1,2th 2 A y t ε + + ,1,1/1,2th A y ε 1,1/1,2thz ρ 1,1 1,1/1,2th 1,1 z t L ρ - + 1,1 1,1/1,2th 1,1 z t L ρ - +

Deviation induced by thermomechanical strains

A y

ε z ρ 1,1 ,1,1/1,2th 2 A y t ε -+ 1,1 ,1,1/1,2th 2 A y t ε + + ,1,1/1,2th A y ε 1,1/1,2thz ρ 1,1 1,1/1,2th 1,1 z t L ρ - + 1,1 1,1/1,2th 1,1 z t L ρ - +

Figure 3: Deviation hull incorporating thermomechanical strains

Thermomechanical stresses on the parts lead to strains in the free surfaces, i.e. the surfaces not in contact with other surfaces. Deviations of thermomechanical origin in the shape of the free surfaces are taken into account in the 3D dimension chain modelling. A deviation in shape will be characterised by the local distance between a point on the real surface and the associated substitute surface.

3-Turbine performance criteria

-Correlation of performance criteria with geometrical criteria

The correct functioning of a turbine relies on two performance criteria: energy yield and the risk of a blade touching the stator. These two performance criteria depend on the clearance between the turbine blades and the stator. Turbine energy yield is correlated with the flow of gas between the blade tips and the stator. This flow rate depends on the leakage section, the difference between the rotor and the stator sections; in order to guarantee optimal yield, it must not exceed a maximal value: max S .

To maximise the turbine's energy yield, rotor-stator clearance should be as small as possible while ensuring that in all of the turbine's operational phases the blades are not too close to the stator (which could damage the turbine and risk damaging the engine in operation). The risk of touching is correlated with a minimal clearance value that must be respected: min C .

-Geometrical criteria

Local clearance between the blades and the stator ( )

C θ should be sufficient to avoid touching any part of the periphery of the turbine (see Figure 4).

Paper Number: IDMME_P65

-3- is the minimum clearance that guarantees that the turbine functions correctly.

Copyright IDMME -Virtual Concept
( ) [ [ 0; 2 min C C θ θ π > ∈ (1)
The leaking of gas between the stator and the rotor is determined by the leakage section S (see Figure 4).

The criterion for power in a turbine is correlated with the leakage section. To maximise turbine power the leakage section must be minimised, as defined in the following equation:

( ) ( ) ( ) D , , . 2 2 stator d M M dev M θ θ θ   = + + +     δ e n (3)
The preceding equation is defined as follows:

-( ) By integrating equation (3) into equation (1), the touch criterion can be defined by the following equation:

( ) ( ) ( ) ( ) [ [ D , . 2 2 , avec : 0;2 rotor d M dev M M θ θ θ θ θ π    + + +    =     -  ∈ e n C δ (4)
By combining equation (3) with equation (2), which describes the stator, the energy yield criterion is then defined as:

( )

2 2 0 1 D . , . 2 2 2 rotor d S dev M d S π θ θ   = + + -     ∫ (5)
The deviations in dimension and shape are very small compared with the nominal diameter of the stator section. The integral which formalises the area of the stator section in equation ( 5) can then be linearised:

( )

2 2 0 D D , . 2 2 2 stator d S dev M d π π θ θ     = × + × +         ∫ (6)
4-Application to a simplified turbine model

-Description of the simplified turbine

The simplified model of the turbine (Figure 6) consists of two sub-units: a stator (parts labelled 1 and 2) and a rotor (part labelled 3), with a turning pair contact around x: see Figure 6. Part 3 forms a revolution, where the largest diameter corresponds to the diameter of the blade tips. When the engine is in operation, a flow of hot gases (1000°C approx.) is created by the combustion chamber (not shown in Figure 6) and as a result parts 1, 2 and 3 undergo strain. In this example, only the thermomechanical strains on part 1 are

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taken into account when the engine is in operation.

In the following part of the article, parts 2 and 3 are by assumption considered as infinitely rigid and geometrically perfect. Moreover, the turning pair contact between part 2 and part 3 is considered to have no defects and no clearance.

The criteria defined in §2 will be studied for two different turbine architectures. Two variants of a technical solution will be considered, the aim being to produce a clamping contact between part 1 and part 2 (see Figure 7):

-Architecture 1: planar pair contact with five short centring plugs located on a diameter D and distributed equiangularly. -Architecture 2: planar pair contact with one short centring plug.

-Defining parameters and formulating performance criteria

According to the hypotheses defined in the previous paragraph, the geometric variabilities of the two technical solutions considered depend only on the processes of obtaining part 1, and assembling parts 1 and 2 together, and the thermomechanical behaviour of part 1. Equation (5) defines the touch criterion; this then becomes:

( ) ( ) ( ) ( ) ( ) ( ) 0 0 avec : . , 2 D , . 2 stator C C C d C M dev M C M θ θ θ θ θ θ θ = ∆ + ∆ = + + = - e n δ n (7)
Equation ( 6) defining the energy yield criterion then becomes:

( )

0 2 0 2 0 D avec : , . 2 2 D et : 2 rotor S S S d S dev M d S S π θ θ π = ∆ +   ∆ = × +       = × -     ∫ (8) ( ) C θ ∆
and S ∆ are dependent on the architecture variants. 0 C and 0 S are variables that are independent of the architectures.

-Geometric characterisation

In the first phase, the turbine is modelled in a reference behaviour where all the solids are infinitely rigid: only manufacturing defects and clearance in the different contacts is taken into account. In phase two, a thermomechanical behaviour is considered, where there are strains due to the flow of hot gases from the combustion chamber at a point when the turbine is operating in a steady regime. Thermomechanical strains are then integrated into the reference model as described in §2.2.

Characterisation of the variability of production and assembly processes

Figure 8 describes the two architectures considered. For architecture 1, the clamping contact consists of a planar pair contact and five ball and cylinder pair contacts, with clearance, by means of five centring slugs. For architecture 2, the clamping contact consists of a planar pair contact and one ball and cylinder pair contact, with a clamping screw.

Architecture 1 2 3 1 TP 2,2 3,2 PP A C BCP1-i B D ( ) C θ 3,1 1,1 Architecture 2 2 3 1 TP 2,2 3,2 BCP2 B D PP A C ( ) C θ 3,1 1,1 Features of BCP1-i (1 ≤ ≤ ≤ ≤ i ≤ ≤ ≤ ≤ 5
C θ 3,1 1,1 Architecture 1 2 3 1 TP 2,2 3,2 PP A C BCP1-i B D ( ) C θ 3,1 1,1 2 3 1 2 2 3 3 1 1 TP 2,2 3,2 PP A C BCP1-i B D ( ) C θ 3,1 1,1 TP 2,2 3,2 TP 2,2 3,2 2,2 2,2 3,2 3,2 PP A C PP A C A A C C BCP1-i B D BCP1-i B D B B D D ( ) C θ 3,1 1,1 ( ) C θ ( ) C θ 3,1 3,1 1,1 1,1 Architecture 2 2 3 1 TP 2,2 3,2 BCP2 B D PP A C ( ) C θ 3,1 1,1 Architecture 2 2 3 1 TP 2,2 3,2 BCP2 B D PP A C ( ) C θ 3,1 1,1 2 3 1 2 2 3 3 1 1 TP 2,2 3,2 BCP2 B D PP A C ( ) C θ 3,1 1,1 TP 2,2 3,2 TP 2,2 3,2 2,2 2,2 3,2 3,2 BCP2 B D BCP2 B D B B D D PP A C PP A C A A C C ( ) C θ 3,1 1,1 ( ) C θ ( ) C θ 3,1 3,1 1,1 1,1 Features of BCP1-i (1 ≤ ≤ ≤ ≤ i ≤ ≤ ≤ ≤ 5) -Name : 1,i/2,i -
In both architectures, bolts are used to keep parts 1 and 2 in position. They are not considered in the modelling process as they do not affect the relative positioning of parts 1 and 2.

Determination of: ( ) ( ).

e M M = e n θ is based on a 3D dimension chain formalised by operations on the hulls (intersections and Minkowski sums).

The hull that defines the displacement limits of surface 1,1 in relation to surface 3,1 is expressed by the following Minkowski sum:

1,1/ 3,1 1,1/ / / 2,2 2,2 / 3,2 3,2 / 3,1 AB AB CD CD = + + + + D D D D D D (9)
Paper Number: IDMME_P65

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Copyright IDMME -Virtual Concept Architecture 1 Architecture 2 y z x B ∅ D y z x C 1,2 1,1 1,3 A L 1,1 L 1,7 A φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t y z x N 1 1,6 1,5 N 5 N 4 1,4 N 3 1,3 N 2 1,2 ∅t 1 B 5×∅D 1 ±d1 ∅ D y z x 1,7 1,1 C A A L 1,1 L 1,7 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 Architecture 1 Architecture 2 y z x B ∅ D y z x C 1,2 1,1 1,3 A L 1,1 L 1,7 A φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t y z x B ∅ D y z x y z x B ∅ D B B ∅ D ∅ D y z x C 1,2 1,1 1,3 A L 1,1 L 1,7 A φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t y z x C 1,2 1,1 1,3 A y z x y z x C C 1,2 1,2 1,1 1,1 1,3 1,3 A A L 1,1 L 1,7 A φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t L 1,1 L 1,7 L 1,1 L 1,7 A A φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t B A φ 1,1 t y z x N 1 1,6 1,5 N 5 N 4 1,4 N 3 1,3 N 2 1,2 ∅t 1 B 5×∅D 1 ±d1 ∅ D y z x 1,7 1,1 C A A L 1,1 L 1,7 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 y z x N 1 1,6 1,5 N 5 N 4 1,4 N 3 1,3 N 2 1,2 ∅t 1 B 5×∅D 1 ±d1 ∅ D y z x y z x N 1 1,6 1,5 N 5 N 4 1,4 N 3 1,3 N 2 1,2 N 1 1,6 N 1 1,6 1,5 N 5 1,5 N 5 N 4 1,4 N 4 1,4 N 3 1,3 N 3 1,3 N 2 1,2 N 2 1,2 ∅t 1 B 5×∅D 1 ±d1 ∅ D ∅t 1 B 5×∅D 1 ±d1 ∅t 1 B 5×∅D 1 ±d1 ∅t 1 ∅t 1 B B 5×∅D 1 ±d1 ∅ D ∅ D y z x 1,7 1,1 C A A L 1,1 L 1,7 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 y z x 1,7 1,1 C A y z x y z x 1,7 1,7 1,1 1,1 C C A A A L 1,1 L 1,7 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 A A L 1,1 L 1,7 L 1,1 L 1,7 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1 B A φ 1,1 t B A φ 1,1 t φ +d1,1_ma_max +d1,1_ma_min D1,1

D

is the clearance hull defined by the planar pair contact and the five ball and cylinder pair contacts for architecture 1 and the one ball and cylinder pair contact for architecture 2. 1,1/ AB D is the deviation hull characterised by the location of surface 1,1 in relation to reference system AB according to ISO standards [I3], [I1] and [I2]. for the two architectures (see Figure 9). Hulls

D

characterises the clearance hull for the turning pair contact between parts 2 and 3.

According to the hypotheses advanced in §4.1, equation (9) then becomes:

1,1/ 3,1 1,1/ / AB AB CD = + D D D (10)
Deviation hulls 1,1/ 3,1 D , resultants from the Minkowski sums defined by equation (10), are shown in Figure 10, where max J represents clearance in the five ball and cylinder pair contacts and 1,1 t the dimension of the tolerance zone for the location specification of surface 1,1.

Architecture 1 Architecture 2

ε ,1,1/ 3,1 A z ρ 1,1/ 3,1y + + 1,1 max 2 2 t J - -1,1 max 2 2 t J -1,1 1,1 t L -1,1 1,1 t L ρ 1,1/3,1y + 1,1 1,1 t L -1,1 1,1 t L + 1,1 2 t -1,1 2 t ε ,1,1/3,1 A z Architecture 1 Architecture 2 ε ,1,1/ 3,1 A z ρ 1,1/ 3,1y + + 1,1 max 2 2 t J - -1,1 max 2 2 t J -1,1 1,1 t L -1,1 1,1 t L ε ,1,1/ 3,1 A z ρ 1,1/ 3,1y + + 1,1 max 2 2 t J - -1,1 max 2 2 t J -1,1 1,1 t L -1,1 1,1 t L ρ 1,1/3,1y + 1,1 1,1 t L -1,1 1,1 t L + 1,1 2 t -1,1 2 t ε ,1,1/3,1 A z ρ 1,1/3,1y + 1,1 1,1 t L -1,1 1,1 t L + 1,1 2 t -1,1 2 t ε ,1,1/3,1 A z ρ 1,1/3,1y + 1,1 1,1 t L -1,1 1,1 t L + 1,1 2 t -1,1 2 t ε ,1,1/3,1 A z
Figure 10: Deviation hull of location of surface 1,1 at point A.

From Figure 10 we are able to determine the values of the location deviation between surfaces 1,1 and 2,1. Equations ( 11) and ( 12) define the minimal values of this deviation for architecture 1 and architecture 2 respectively:

( )

1,1 1,1 max max stator _ 1 2 2 2 2 A t t J J e M - - ≤ ≤ + + (11)
( )

1,1 1,1 stator _ 2 2 2 A t t e M -≤ ≤ + (12)
As seen in the definition drawing for part 1, the dimension deviations for surface 1,1 in both architectures are given in equations ( 13) and ( 14).

1,1_ma_min stator_A1 1,1_ma_max d d d ≤ ≤ (13) 1,1_ma_min stator_A2 1,1_ma_max d d d ≤ ≤ (14)
By definition, deviations in shape due to manufacture are not taken into account, which gives:

( ) ( ) stator_A1 stator_A2 , = , =0 dev M dev M θ θ (15)

Characterisation of thermomechanical variability

Thermomechanical strains caused by a flow of gases from the combustion chamber and illustrated in Figure 11 have to be integrated into the geometric variability of the stator.

A preliminary thermal study can determine any change in the joint between part 1 and part 2. As described by [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF] and [START_REF] Pierre | Integration of multiple physical behaviours into a geometric tolerancing approach[END_REF], the nature of the contacts and the clearances may change from one behaviour to another. In the case of architecture 1, the radial displacement of the slugs in their housing uses up all the clearance (see Figure 12). A thermomechanical analysis of part 1 was carried out with the thermal limit conditions defined in Figure 11 and the mechanical limit conditions defined in Figure 8.

y z 1,5 x N 1 N 2 N 3 N 4 N 5 2,3 1,4 1,3 1,2 V3 V4 V5 V1 V2 1,1 2,2 2,1 2,4 2,5 2,7 1,7 y z 1,5 x N 1 N 2 N 3 N 4 N 5 2,3 1,4 1,3 1,2 V3 V4 V5 V1 V2 1,1 2,2 2,1 2,4
After a thermomechanical calculation of finite elements using the Samcef software, the geometry of the deformation of surface 1,8 was studied in section for each of the architectures (see Figure 13). From Figure 13 we can determine location deviation, dimension deviation and shape deviation, due to thermomechanical strain for both architectures.

When changes in contacts, changes in geometry, and the different deviations defined in equations ( 11) to (15) are incorporated, this gives:

( )

Application to two alternative architectures

All the parameters defining the variability of clearance at the blade tips as a function of geometric and architectural variability have been defined. These different parameters are then exploited to compare the two architectures one with another in terms of thermomechanical situation.

From equations ( 22) and ( 23), for thermomechanical behaviour, we can compare variations in minimal clearance for the two architectures.

Figure 1 :-P-PFigure 2 :

 12 Figure 1: Cylindrical pair contact in rigid behaviour

Figure 4 :

 4 Figure 4 : Sections of the rotor blades and the stator. The touch criterion can be expressed by the following equation, where min Cis the minimum clearance that guarantees that the turbine functions correctly.

Figure 5 :

 5 Figure 5: Parameters of a nominal circular surface The position ( ) , M θ δ is written:

  M e : location deviation of the centre of the substitute circle in relation to the reference point -D : nominal diameter d : dimension deviation between the nominal diameter and the substitute circle diameter -( ) , dev M θ : shape deviation of the real profile in relation to the substitute circle.

Figure 6 :

 6 Figure 6: Simplified geometric model of a turbine.

Figure 7 :

 7 Figure 7: Turbine architectures.

  ) -Name : 1,i/2,i -Type : Ball and Cylinder Pair (BCP) -Situation element : point N i , line (N i , x)

-Figure 8 :

 8 Figure 8: Graphs of contacts.

Figure 9 :

 9 Figure 9: Definitions of parts 1.

Figure 11 :

 11 Figure 11: Thermal marginal conditions

Figure 12 :

 12 Figure 12 : Radial displacements of the slugs The planar pair contact / short centring, used in the technical solution for architecture 2, remains without clearance.

Figure 13 :

 13 Figure 13: Thermomechanical strains of surface 1,1 in plane ( ) , A x

Where the risk of touching predominates over energy yield, architecture 2 is the most efficient.

Where energy yield predominates over the risk of touching, then architecture 1 performs best.

Thus the preponderance of one or other of the two performance criteria determines which is the more efficient solution.

5-Conclusions-Perspectives

In this article we have defined the geometrical variability of a turbine using a multiphysical approach, and including the variability of processes associated with obtaining parts, assembly processes and the thermomechanical behaviour of the turbine. Variants of technical solutions for joints between the ring and the turbine housing have been formalised. Two performance criteria were used, risk of touching the stator and energy yield, to qualify the solutions envisaged.

One of the future perspectives for this work will be to combine variants of technical solutions for joints with geometrical variants (shape and nominal dimensions of parts). The multiphysical geometrical analysis used in this article can be applied to all types of mechanical systems.

6-References

[BB1] Bourdet P. and Ballot E. Geometrical behavior laws for computer aided tolerancing. In 4th CIRP Seminar on Computer Aided Tolerancing (ed. F. Kimura): [143][144][145][146][147][148][149][150][151][152][153]1995


L Pierre

D Teissandier

J P Nadeau

Integration of multiple physical behaviours into a geometric tolerancing approach

Keywords: Tolerancing analysis, flexible parts, thermomechanic, behaviour, lifecycle, helicopter engine

The performance of a helicopter engine depends essentially on the performance of the high-pressure turbine, closely correlated with the rotor/stator clearance at the tip of the blades. The engine's thermodynamic cycle causes elastic or plastic strains on some of the high-pressure turbine components, causing considerable geometric variations, some of which are irreversible.

In this article we propose to integrate the notion of behaviour into the evaluation of clearance in a highpressure turbine by defining the physical phenomena that are influential during certain operational phases and more generally over the lifetime of the turbine. Modelling the variations in the geometric deviations of the parts and the contacts is based on deviation hulls and clearance hulls.

INTRODUCTION

Controlling the behaviour and the energy yield of helicopter engines for each of their different operational regimes is clearly essential in order to guarantee that the desired power is achieved. One way to improve the performance of these turboshaft engines is to optimise the energy yield of the different constituent components, and in particular to control clearance between the tips of the high-pressure turbine blades and the stator. In this article we propose a method that will take into account different behaviours in order to characterise the physical phenomena that are influential during certain phases in the operational cycle. In the first part, three behaviours are introduced for the system studied: behaviour I, where the system is new and the parts are at 20°C; behaviour II where some components in the system are subject to thermal flux, creating thermomechanical strains; behaviour III at 20°C , subsequent to behaviour II, where certain parts retain the memory of earlier plastic strains. In part two we show how changes in the contacts between the different behaviours and strains in the parts themselves are taken into account when modelling the dimension chains by operations on the hulls (Minkowski sum and intersection). In part three, we present an application of this work to a sub-unit of a high-pressure helicopter engine turbine. Lastly, after describing the main conclusions, we present future developments for this work.

TAKING INTO ACCOUNT SEVERAL BEHAVIOURS

IN 3D DIMENSION-CHAINS Three different behaviours are considered when modelling the operating cycle of a system (see figure 1): -behaviour I: the system has just been assembled and has not yet been operational. All the parts are at 20°C and are considered to be infinitely rigid.

-behaviour II: some parts of the system that have been subjected to thermal flows undergo thermomechanical strains. These strains are then taken into account in the 3D dimension-chains.

-behaviour III: the system is at 20°C. Some parts that have been subjected to thermal flows for a specified duration t during behaviour II have undergone thermomechanical strains, which can create plastic strains. These plastic strains, if any exist, are then taken into account in the 3D dimension-chains. In this article, deviation hulls [START_REF] Giordano | Clearance Space and Deviation Space, Application to three-dimensional chains of dimensions and positions[END_REF] are used to define the tolerated limits of the geometric defects of a part (defined by a specification), and clearance hulls to define the accepted limits of relative displacement between two surfaces in contact (defined by a clearance level), [START_REF] Giordano | Tolerance analysis and synthesis by means of deviation domains, axi-symetric cases[END_REF].

The concept of hull introduced by [START_REF] Fleming | Geometric relationships between toleranced features[END_REF] is also developed in [START_REF] Mujezinovi | A new mathematical model for geometric tolerances as applied to round faces[END_REF], [START_REF] Jian | Tolerance Analysis and Allocation using Tolerance-Maps for a Power Saw Assembly[END_REF] and [START_REF] Teissandier | Operations on polytopes: application to tolerance analysis[END_REF]. Studies which integrate strains on the parts caused by the assembly process are currently underway: [START_REF] Stewart | Variation simulation of fixtured assembly for compliant structures using piecewise-linear analysis[END_REF], [START_REF] Söderberg | Computer-aided robustness analysis for compliant assemblies[END_REF] and [START_REF] Cid | Geometrical study of assembly behaviour, taking into accounts rigid components' deviations, actual geometric variations and deformations[END_REF].

Behaviour I: new system, temperature: 20°C

Let us consider the system shown in figure 2, consisting of two parts 1 and 2 in contact via a cylindrical pair joint along x and a ball and plane pair joint along x. The surface j of the part i is indicated by i,j. One functional condition FC limiting translation d along y of the cylindrical surface 1,1 in relation to the axis of surface 2,1 at point A is to be fulfilled. This FC is defined in the following equation: min

max d d d ≤ ≤ (1)
Thus

1,1/ 2,1

D

is the deviation hull characterising the relative displacement of the axis of the cylindrical surface 1,1, in relation to the axis of surface 2,1 at point A.

We transferred the FC to parts 1 and 2 in order to determine the ISO geometrical specifications [START_REF]Toleranced characteristics and symbols --Examples of indication and interpretation[END_REF], [START_REF]Technical drawings --Geometrical tolerancing --Datums and datum-systems for geometrical tolerances[END_REF] ,

[12] that guaranteed that the functional condition FC was fulfilled [START_REF] Ballu | Choice of functional specifications using graphs within the frame work of education[END_REF], [START_REF] Dantan | Tolerance synthesis: quantifier notion and virtual boundary[END_REF]: see figure 2.

transfer transfer

x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 A x y z x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 A 1 1,2 1,3 1,1 2 2,2 2,3 2,1
∅D 1,2 FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 CP BPP ∅D 2,2 ∅D 1,2 FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 CP BPP FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 BPP CP FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 BPP
CP A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A L 1,1 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2 A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A L 1,1 A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A ∅t 1, 1 A L 1,1 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B ∅t 2, 2 B ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2 transfer transfer transfer transfer x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 A x y z x y z 1 1,2 1,3 1,1 2 2,2 2,3 2,1 A 1 1,2 1,3 1,1 2 2,2 2,3 2,1
∅D 1,2 FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 CP BPP ∅D 2,2 ∅D 1,2 FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 CP BPP FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 BPP CP FC 1,2 1,1 1 2,1 2,2 2 1,3 2,3 BPP
CP A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A L 1,1 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2 A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A L 1,1 A ∅D 1, 2 ±d 1,2 ∅t 1, 1 A ∅t 1, 1 A L 1,1 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2 ∅t 2, 2 B ∅D 2,2 ±d 2,2 B ∅t 2, 2 B ∅t 2, 2 B ∅D 2,2 ±d 2,2 B L 2,2

D

of the coaxialities of parts 1 and 2 respectively between surfaces 1,1 and 1,2 and surfaces 2,1 and 2,2 (see figure 3).

L t L L   + + ×     2,2 2,2 1,1 2,2 2 L t L L   - + ×     ,2,1/2,2 A y ε 2,1/2,2z ρ 1,1 1,1 t L + 1,1 1,1 t L - 1,1 2 t + 1,1 2 t - ,1,1/1,2 ε A y 1,1/1,2z ρ 2,2 2,2 t L + 2,2 2,2 t L + 2,2 2,2 1,1 2,2 2 L t L L   + + ×     2,2 2,2 1,1 2,2 2 L t L L   - + ×     ,2,1/2,2 A y ε 2,1/2,2z ρ 2,2 2,2 t L + 2,2 2,2 t L + 2,2 2,2 1,1 2,2 2 L t L L   + + ×     2,2 2,2 1,1 2,2 2 L t L L   - + ×     ,2,1/2,2 A y ε 2,1/2,2z ρ 2,2 2,2 t L + 2,2 2,2 t L + 2,2 2,2 1,1 2,2 2 L t L L   + + ×     2,2 2,2 1,1 2,2 2 L t L L   - + ×     ,2,1/2,2 A y ε 2,1/2,2z ρ 1,1 1,1 t L + 1,1 1,1 t L - 1,1 2 t + 1,1 2 t - ,1,1/1,2 ε A y 1,1/1,2z ρ 1,1 1,1 t L + 1,1 1,1 t L - 1,1 2 t + 1,1 2 t - ,1,1/1,2 ε A y 1,1/1,2z ρ 1,1 1,1 t L + 1,1 1,1 t L - 1,1 2 t + 1,1 2 t - ,1,1/1,2 ε A y 1,1/1, 2
J L + max 2,2 J L - ( ) max 2,2 2,2 1,2 1,2 D +d D -d J = - 2,2 max 1,1 2,2 2
L J L L   - + ×     ,1,2/2,2 A y ε 1,2/2,2z ρ 2,2 max 1,1 2,2 2
L J L L   + + ×     max 2,2 J L + max 2,2 J L - ( ) max 2,2 2,2 1,2 1,2 D +d D -d J = - 2,2 max 1,1 2,2 2
L J L L   - + ×     ,1,2/2,2 A y ε 1,2/2,2z ρ max 2,2 J L + max 2,2 J L - ( ) max 2,2 2,2 1,2 1,2 D +d D -d J = - 2,2 max 1,1 2,2 2
L J L L   - + ×     ,1,2/2,2 A y ε 1,2/2,2z ρ max 2,2 J L + max 2,2 J L - ( ) max 2,2 2,2 1,2 1,2 D +d D -d J = - 2,2 max 1,1 2,2 2
L J L L   - + ×     ,1,2/2,2 A y ε 1,2/2,2z ρ 2,2 max 1,1 2,2 2
L J L L   + + ×    

D

(see figure 5).

1,1/ 2,1 1,1/ 1,2 1,2 / 2,2 2,2 / 2,1 = + + D D D D (2) 1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   + + + ×           1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   -- + ×           1,1 2,2 max 1,1 2,2 2,2 t t J L L L + + + 1,1 2,2 max 1,1 2,2 2,2 t t J L L L - - - 1,1/2,1z ρ ,1,1/2,1 A y ε 1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   + + + ×           1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   -- + ×           1,1 2,2 max 1,1 2,2 2,2 t t J L L L + + + 1,1 2,2 max 1,1 2,2 2,2 t t J L L L - - - 1,1/2,1z ρ ,1,1/2,1 A y ε 1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   + + + ×           1,1 2,2 2,2 max 1,1 2,2 2 2 t L t J L L   +   -- + ×           1,1 2,2 max 1,1 2,2 2,2 t t J L L L + + + 1,1 2,2 max 1,1 2,2 2,2 t t J L L L - - - 1,1/2,1z ρ ,1,1/2,1 A y ε

Behaviour II: system subjected to thermal flux

Consider figure 6 which describes the system studied in behaviour II. By hypothesis, part 2 remains inflexible: only part 1 undergoes thermomechanical strain. These thermomechanical strains are the result of a steep temperature gradient along direction y in part 1. The result is a camber in part 1: see figure 6. D is reduced to a point at the origin of the hull area: see figure 7. .

P A 1 1,2 1,3 1,1 2
2,2 2,3 2,1 x y z P A 1 1,2 1,3 1,1 2 2,2 2,3 2,1 P A 1 1,2 1,3 1,1 2 2,2 2,3
,1,2/2,2 A y ε 1,2/2,2z ρ ,1,2/2,2 A y ε 1,2/2,2z ρ ,1,2/2,2 A y ε 1,2/2,
II 1,1 1,1/ 1,2 1,1 z t L ρ + II 1,1 1,1/ 1,2 1,1 z t L ρ - II 1,1 ,1,1/ 1,2 2 A t ε + y II 1,1 ,1,1/ 1,2 2 A t ε - y ,1,1/ 1,2 A y ε 1,1/ 1,2z ρ II ,1,1/ 1,2 A ε y II 1,1/ 1,2 z ρ II 1,1 1,1/ 1,2 1,1 z t L ρ + II 1,1 1,1/ 1,2 1,1 z t L ρ - II 1,1 ,1,1/ 1,2 2 A t ε + y II 1,1 ,1,1/ 1,2 2 A t ε - y ,1,1/ 1,2 A y ε 1,1/ 1,2z ρ II ,1,1/ 1,2 A ε y II 1,1/ 1,2 z ρ Figure 8: Deviation hull 1,1/1,2 D at point A in behaviour II.
The Minkowski sum (2) describes hull

1,1/ 2,1 D which is
shown graphically for behaviour II in figure 9.

IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε 1,1 2,2 1,1 2,2 x t t L L + + ρ 1,1 2,2 1,1/ 1,2II 1,1 2,2 z t t L L - - ρ 1 1 2 1 , / , ρ z ,1,1/ 2,1 A y ε IIy ,1,1/ 1,2 A ε 1,1/ 1,2IIz ρ IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε 1,1 2,2 1,1 2,2 x t t L L + + ρ 1,1 2,2 1,1/ 1,2II 1,1 2,2 z t t L L - - ρ 1 1 2 1 , / , ρ z ,1,1/ 2,1 A y ε IIy ,1,1/ 1,2 A ε 1,1/ 1,2IIz ρ IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε 1,1 2,2 1,1 2,2 x t t L L + + ρ 1,1 2,2 1,1/ 1,2II 1,1 2,2 z t t L L - - ρ 1 1 2 1 , / , ρ z ,1,1/ 2,1 A y ε IIy ,1,1/ 1,2 A ε 1,1/ 1,2IIz ρ

Behaviour III: system at 20°C previously

subjected to thermal flux for X hours Now we consider figure 10. Part 1 has undergone thermomechanical strain in the context of behaviour II for a specified time of X hours, and has not returned to its initial state. It undergoes a state of plastic strain called a decambering phenomenon, which is then taken into account in the 3D dimension-chains. By hypothesis, the plastic strain in part 1 maintains null clearance in the joint 1,2/2,2. Moreover, the decambering of part 1 modifies the deviation hull of this part at point A defined in §2.1 via translation: see figure 11. The Minkowski sum for the deviation hulls of parts 1 and 2 and for the clearance hull between parts 1 and 2 in behaviour III is illustrated in figure 12.

III 1,1 1,1/ 1,2 1,1 z t L ρ + 1,1 1,1/ 1,2 2 A y t ε + 1,1/ 1,2 A y ε 1,1/ 1,2z ρ 1,1 1,1/ 1,2 2 A y t ε - III 1,1 1,1/ 1,2 1,1 z t L ρ - III ,1,1/ 1,2 A ε y III 1,1/ 1,2 z ρ III 1,1 1,1/ 1,2 1,1 z t L ρ + 1,1 1,1/ 1,2 2 A y t ε + 1,1/ 1,2 A y ε 1,1/ 1,2z ρ 1,1 1,1/ 1,2 2 A y t ε - III 1,1 1,1/ 1,2 1,1 z t L ρ - III ,1,1/ 1,2 A ε y III 1,1/ 1,2 z ρ III 1,1 1,1/ 1,2 1,1 z t L ρ + 1,1 1,1/ 1,2 2 A y t ε + 1,1/ 1,2 A y ε 1,1/ 1,2z ρ 1,1 1,1/ 1,2 2 A y t ε - III 1,1 1,1/ 1,2 1,1 z t L ρ - III ,1,1/ 1,2 A ε y III 1,1/ 1,2 z ρ
IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L + + ρ z III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L - - ρ z 1,1/ 2,1z ρ 1,1/ 2,1 A y ε III 1,1/ 1,2 ρ z IIIy ,1,1/ 1,2 A ε IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L + + ρ z III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L - - ρ z 1,1/ 2,1z ρ 1,1/ 2,1 A y ε III 1,1/ 1,2 ρ z IIIy ,1,1/ 1,2 A ε IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   + + + ×     ε IIIy 1,1 2,2 2,2 ,1,1/ 1,2 1,1 2,2 2 2 A t L t L L   -- + ×     ε III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L + + ρ z III 1,1 2,2 1,1/ 1,2 1,1 2,2 t t L L - - ρ z 1,1/ 2,1z ρ 1,1/ 2,1 A y ε III 1,1/ 1,2 ρ z IIIy ,1,1/ 1,2 A ε
t L t J L L t L t J L L   +   + + + × ≤             +   -- + × ≥           Behaviour II IIy IIy 1,1 2,2 2,2 ,1,1/1,2 1,1 max 2,2 1,1 2,2 2,2 ,1,1/1,2 1,1 min 2,2 d 2 2 d 2 2 A A t L t L L t L t L L   + + + × ≤       -- + × ≥     ε ε Behaviour III IIIy IIIy 1,1 2,2 2,2 ,1,1/1,2 1,1 max 2,2 1,1 2,2 2,2 ,1,1/1,2 1,1 min 2,2 d 2 2 d 2 2 A A t L t L L t L t L L   + + + × ≤       -- + × ≥     ε ε

APPLICATION TO A HIGH-PRESSURE TURBINE

Description of the turbine

The turbine is made up of two sub-units: a rotor (consisting of the part numbered 5) and a stator (consisting of the parts numbered 1, 2, 3 and 4) in a turning pair joint along x by the intermediary of a ball bearing numbered 6 and a cylindrical roller bearing numbered 7: see figure 13. Part 5 has a revolution shape where the largest diameter corresponds to the diameter of the heads of the blades. Clearance at the top of the blade between the vanes of part 5 and part 1 of the stator is defined as the difference in diameter between parts 1 and 5: see figure 13.

In order to maximise turbine power, this clearance should be minimised while still ensuring that in all turbine operational phases part 5 does not come into contact with part 1. 3

Thermal Flux Thermal Flux

Figure 13: Simplified geometric model of a high-pressure turbine. When the turbine is operational, a flux of hot gases (about 1000°C) is created in the combustion chamber (not shown in figure 13) which results in strain on parts 1, 3 and 5: see figure 13. In this application, we restrict our study to a sub-unit of the turbine made up of parts 1 and 2: see figure 14. The functional condition ensuring control of clearance C at the top of the blades between the rotor and the stator (see figure 13) corresponds to an alignment FC of the axes of surfaces 1,8 and 2,8: see figure 14. FC is defined in equation ( 3) where e represents the distance between the axes of surfaces 1,8 and 2,8 in a direction orthogonal to x at point A:

min max e e ≤ ≤ e (3)
In this example, the thermomechanical strains in part 1 are taken into account solely when operational. Part 2 is considered to be infinitely rigid and geometrically perfect: see figure 14.

The specifications required to satisfy the FC in the three different behaviours defined in §2 are studied in turn. First, the turbine is modelled in behaviour I where only manufacturing defects and clearance in the different contacts are taken into account. Next, behaviour II is considered, where strains due to the flux of hot gases from the combustion chamber are also integrated, at a point where the turbine is functioning in a stationary regime. Finally, in a third phase, we study behaviour III after the turbine has been functioning for X hours, when only plastic strains caused by the thermomechanical strains from behaviour II are taken into account.

y z=v 1 v 5 v 4 v 3 v 2 1,2 ⁄ 2,2 2,7 N 1 N 5 N 3 N 4 N 2 A 1,1⁄ 2,1 1,7 1,8 1,3 ⁄ 2,3 1,4 ⁄ 2,4 1,5 ⁄ 2,5 y z x A 2,8 C ∅ D x y z=v 1 v 5 v 4 v 3 v 2 1,2 ⁄ 2,2 2,7 N 1 N 5 N 3 N 4 N 2 A 1,1⁄ 2,1 1,7 1,8 1,3 ⁄ 2,3 1,4 ⁄ 2,4 1

Behaviour I: new turbine, temperature: 20°C

Part 1 is positioned via a support plane contact with part 2, by the intermediary of surfaces 1,7 and 2,7. Contacts between surfaces 1,1/2,1; 1,2/2,2; 1,3/2,3; 1,4/2,4; 1,5/2,5 modelled by ball and cylinder pair joints, complete the positioning between these two parts. Five clamping screws ensure that parts 1 and 2 are held in place. The graph of the joints corresponding to the system described in figure 14 is shown in figure 15 with the parameters of the different joints. -Nature : floatting contact

-Clearance : 0 ≤ ≤ ≤ ≤ J min ≤ ≤ ≤ ≤ J max FC PP BCP5 BCP4 BCP3 BCP2 BCP1 2

D

is formalised by the intersection of the clearance hulls of the joint between surfaces 1,7/2,7 and the five joints 1,i/2,i:

( ) AB / CD 1,7 / 2,7 1,1/ 2,1 1,2 / 2,2 1,3 / 2,3 1,4 / 2,4 1,5 / 2,5 = ∩ ∩ ∩ ∩ ∩ D D D D D D D (5)
For each slug maximal clearance max J which is the difference between the maximum diameter of the hole and the minimum diameter of the shaft, corresponds to the most unfavourable configuration for the FC. The clearance hull AB / CD D , resultant of the intersection of the clearance hulls for the five ball and cylinder pairs and the support plane joint, is shown in figure 17, in 2D modelling in plane ( ) D is defined at point A. This clearance characterising the location specification (see figure 16) is shown in figure 18. D , / , at point A in behaviour I.

A,x . ,AB/CD A z ε AB/CDy ρ max 2 J + max 2 J - ,AB/CD A z ε AB/CDy ρ max 2 J + max 2 J -
From this we can deduce in particular expression [START_REF] Teissandier | Operations on polytopes: application to tolerance analysis[END_REF], formalising respect for the worst case FC in behaviour I.

 ≥ + +     ≤ - -   (6)
3.3 Behaviour II: turbine subjected to thermal flux (flux of hot gases at 1000 °C) The thermomechanical strains induced by a flux of gases from the combustion chamber illustrated in figure 13 should be integrated into the dimension chains verifying that the FC are respected. As described in §2.2, the nature of the contacts and the clearances may evolve from one behaviour to another [START_REF] Pierre | Integration of thermomechanical strains into tolerancing analysis[END_REF]. A preliminary thermomechanical study was carried out to determine changes in the contacts. The thermal marginal conditions were provided by an expert engineer working in this field. They consisted of two fluxes: a convection flux due to the flux of hot gas from the combustion chamber, situated on the internal face of part 1; a cooling convective flux on the external face of part 1: see figure 20. Figure 20: Thermal marginal conditions. The only mechanical marginal conditions considered were the contact conditions of the support plane joint 1,7/2,7. This enabled us to carry out an initial analysis on the behaviour of surfaces 1,1 to 1,5. This initial thermomechanical study shows that the radial displacement of the slugs is much greater than the maximal clearances present in each BCP type joint. Hence the nature of the contact between the five ball and cylinder pair joints changes from floating to fixed. Moreover, the strain on part 1 adds a forced contact to each BCP type joint along direction i v : see figure 21. Thermomechanical analysis of part 1 was carried out with the thermal marginal conditions defined in figure 20 and with the mechanical marginal conditions that conformed to the parameters of the different joints in behaviour II.

y z 1,5 x N 1 N 2 N 3 N 4 N 5 2,3 1,4 1,3 1,2 V 3 V 4 V 5 V1 V2 1,1 2,2
From the strain on surface 1,8 from thermomechanical stresses we were able to determine the results obtained for figure 11 Three results in particular were characteristic of the strain in plane ( )

A,x see figure 23: -the translation deviation at point A in the centre of the least square circle in relation to the centre of the nominal circle: A , 1 , 8 / 2 , 8 t h ε = 0

-the diameter of the least square circle of the deformed section:

+ th th D D d 1,8 1,8 1,8 =
-the deviation in the form of the deformed section:

1,8 ( )

th dev θ
The thermomechanical distortion of surface 1,8 is not axisymmetrical.

Using a similar process to that used in §2.2, figure 24 shows a representation of hull

+ ± = + + th A z th d d b dev θ ε θ (8)
3.4 Behaviour III: turbine at 20°C after functioning for X hours The gas fluxes from the combustion chamber during behaviour II (see §2.3) lead to plastic strains when the turbine is at rest and when the temperature of all the parts is 20°C. The geometric deviations caused by these plastic strains are quantified by taking tridimensional measurements from part 1 after the turbine has been dismantled. The thermomechanical deformation of surface 1,8 is not axisymmetrical.

y z 1,5 x N 1 N 2 N 3 N 4 N 5 2,3 1,4 1,3 1,2 V 3 V 4 V 5 V 1 V 2 1,1 2,2 2,1 2,4 2,5 2,7 1,7 y z 1,5 x N 1 N 2 N 3 N 4 N 5 2,3 1,4 1,3 1,2 V 3 V 4 V 5 V 1 V 2 1,1 2,2 2,1 2,4
We can deduce the Minkowski sum for hull For behaviour situation I, see equation [START_REF]Toleranced characteristics and symbols --Examples of indication and interpretation[END_REF]. For behaviour situation II, see equation [START_REF]Technical drawings --Geometrical tolerancing --Datums and datum-systems for geometrical tolerances[END_REF]. For behaviour situation III, see equation (12).

CONCLUSIONS AND FUTURE PROSPECTS

We have shown how deviation hulls and clearance hulls can take into account thermomechanical and plastic strains in 3D dimension-chains. This was put into application on a high-pressure turbine from a helicopter engine. For a specific behaviour, the intervention of thermomechanical and plastic strains is in relation to a reference definition of the variability of geometric defects which corresponds to the behaviour of the system when new, integrating the effects of permanent strain which occur during the previous behaviour. One of the main objectives of this study was to put forward qualification criteria for technical solutions envisaged in the design cycle of a high-pressure turbine using geometric specifications.

Another area for future study is to show the usefulness of considering different physical phenomena such as wear in certain joints and damage to some components, thus extending the definition of clearance to the tip of the blade and to the entire lifetime of a turbine.

Figure 1 :

 1 Figure 1: Operational cycle.In this article, deviation hulls[START_REF] Giordano | Clearance Space and Deviation Space, Application to three-dimensional chains of dimensions and positions[END_REF] are used to define the tolerated limits of the geometric defects of a part (defined by a specification), and clearance hulls to define the accepted limits of relative displacement between two surfaces in contact (defined by a clearance level),[START_REF] Giordano | Tolerance analysis and synthesis by means of deviation domains, axi-symetric cases[END_REF].

  Ball and Plane Pair CP : Cylindrical Pair FC : Functional Condition : influent cycle BPP : Ball and Plane Pair CP : Cylindrical Pair FC : Functional Condition ∅D 2,2

  Ball and Plane Pair CP : Cylindrical Pair FC : Functional Condition : influent cycle BPP : Ball and Plane Pair CP : Cylindrical Pair FC : Functional Condition ∅D 2,2

Figure 2 :.

 2 Figure 2: Transfer of FC

Figure 4

 4 Figure 4: Clearance hull 1,2 / 2,2 D of joint 1,2/2,2 at point A. Lastly, using the Minkowski sum (2) we are able to deduce hull 1,1/ 2,1

Figure 5 : 1 D

 51 Figure 5: Minkowski sum illustrating the deviation hull 1,1/ 2,1 D at point A in behaviour I.

Figure 6 :

 6 Figure 6: Behaviour II. The cambering of part 1 within part 2 eliminates all degrees of freedom in the joint between surfaces 1,2 and 2,2. Clearance between parts 1 and 2 becomes null. Clearance hull 1,2 / 2,2 D is reduced to a point at the origin

2z ρFigure 7 2 D

 2z72 Figure 7: Clearance hull 1,2 / 2,2 D at point A in behaviour II. The thermomechanical strain in part 1 results in the translation of the deviation hull 1,1/1,2 D defined in §2.1: see figure 8.

Figure 9 :

 9 Figure 9: Minkowski sum illustrating deviation hull 1,1/ 2,1 D

Figure 10 :

 10 Figure 10: Behaviour III. By hypothesis, the plastic strain in part 1 maintains null clearance in the joint 1,2/2,2. Moreover, the decambering of part 1 modifies the deviation hull of this part at point A defined in §2.1 via translation: see figure 11.

Figure 11 :

 11 Figure 11: Deviation hull 1,1/1,2 D at point A in behaviour III.

Figure 12 : 1 D

 121 Figure 12: Minkowski sum illustrating deviation hull 1,1/ 2,1 D at point A in behaviour III.

, 5 ⁄ 2 Figure 14 :

 5214 Figure 14: Sub-unit of high-pressure turbine.

7 -Features 7 -Features

 77 Type : Planar Pair (PP) -Situation element : line (C, x) -Nature : sliding contact -Clearance : J max =J min = 0 Ball and Cylinder Pair (BCP) -Situation element : point N i , line (N i , x) Type : Planar Pair (PP) -Situation element : line (C, x) -Nature : sliding contact -Clearance : J max =J min = 0 Ball and Cylinder Pair (BCP) -Situation element : point N i , line (N i , x)

Figure 15 :Figure 16 :D

 1516 Figure 15: Graph of joints.The CD datum reference frame on part 2 is defined in the same way as for part 1. The primary surface designated C corresponds to surface 2,7. The secondary surface designated D corresponds to the group of 5 surfaces 2,1 to 2,5.

Figure 17 :

 17 Figure 17: Clearance hull AB / CD D at point A. The deviation hull 1,8 / AB

  W.m -2 .K -1 T 2 = 1000 °C H = 2000 W.m -2 .K -1

Figure 21 :AB 2 , 8 DFigure 22 :

 212822 Figure 21: Radial displacement of the slugs. For this reason clearance hull

  Figure 23: Thermomechanical strain of surface 1,8 in plane ( ) A,x .

Figure 25 :AB 2 , 8 DFigure 26 :Figure 27 :-

 25282627 Figure 25: Radial displacement of the slugs. Changes in the clearances in the five ball and cylinder pair joints can be checked by a simple visual inspection of the parts. The plastic strain of part 1 adds a forced contact to each BCP type joint along direction i v : see figure 25. As a result, clearance hull

  the translation deviation 1,8 ( ) b θ from a point on the cylindrical surface 1,8 in relation to its nominal position takes plastic strains into account.

Table 1

 1 summarises the different results obtained for behaviours I, II and III in figures 5, 9 and 12 respectively in order to fulfil the FC.

	Behaviour I	1,1 2	1,1	2,2 2	2,2	2,2	max	max d
		1,1 2	1,1	2,2 2	2,2	2,2	max	min d

Table 1 :

 1 Synthesis of results of FC transfer.

  Dcharacterising the displacement limits for surface 1,8 in relation to surface 2,8 for behaviour III: see figure28. Equation (9), formalising fulfilment of the worst case FC condition in behaviour III, is the same as in behaviour II.

					1,8 / 2,8
					ε ε ε	A A A	,1,8/2,8 z ,1,8/2,8 z ,1,8/2,8 z
				+ + +	1,8 2 t 1,8 2 t 1,8 2 t
	---	1,8 1,8 L t 1,8 1,8 t L 1,8 1,8 t L			+ + +	1,8 1,8 L t 1,8 1,8 t L 1,8 1,8 t L
					ρ ρ	1,8/2,8y 1,8/2,8y
		---	1,8 2 t 1,8 2 t 1,8 2 t
	Figure 28: Hull 1,8 / 2,8 D	at point A in behaviour III.
	1,8
	max
	1,8
	min

  In behaviour I, II and III, the limits of ,1,8 / 2,8

	1,8 b θ ε ( ) =	,1,8 / 2,8 A	z	+	1,8 D	2 ±	1,8 d										(10)
	1,8 b	( ) θ ε =	,1,8 / 2,8 A	z	+	1,8 D	+	1,8 2 d	th	±	1,8 d	+	dev	1,8	th	( ) θ	(11)
	1,8 b	( ) θ ε =	,1,8 / 2,8 A	z	+	1,8 D	+	1,8 2 plast d	±	1,8 d	+	dev	1,8	plast	( ) θ	(12)
																ε	A	z	are given
	by the hulls of figures 19, 24 and 28 respectively.

Acknowledgements: This work was carried out within the framework of ERT 1070, joint research team of TREFLE-Turbomeca. Thanks to J.L. Breining, J. G. Senger and R. Thomas, of Turbomeca for their advice and technical assistance in carrying out this work.