Estelle Habouzit

He ´le `ne Richard

Herve ´sanchez

Nathalie Koulmann

Bernard Serrurier

Rachel Monnet

Rene ´e Ventura-Clapier

Xavier Bigard
email: xbigard@crssa.net

In the present study, we tested the hypothesis that chronic ANG I-converting enzyme (ACE) inhibition could improve the training-induced improvement in endurance exercise performance and that this could be related to enhanced skeletal muscle metabolic efficiency. Female Wistar rats were assigned to four groups comprising animals either maintained sedentary or endurance trained (Sed and Tr, respectively), and treated or not for 10 wk with an ACE inhibitor, perindopril (2 mg⅐kg Ϫ1 ⅐day Ϫ1 ) (Per and Ct, respectively) (n ϭ 8 each). Trained rats underwent an 8-wk treadmill training protocol that consisted of 2 h/day running at 30 m/min on a 8% decline. Before the start of and 1 wk before the end of experimental conditioning, the running time to exhaustion of rats was measured on a treadmill. The training program led to an increase in endurance time, higher in Tr-Per than in Tr-Ct group (125% in Tr-Ct vs. 183% in Tr-Per groups, P Ͻ 0.05). Oxidative capacity, measured in saponinpermeabilized fibers of slow soleus and fast plantaris muscles, increased with training, but less in Tr-Per than in Tr-Ct rats. The training-induced increase in citrate synthase activity also was less in soleus from Tr-Per than Tr-Ct rats. The training-induced increase in the percentage of the type IIa isoform of myosin heavy chain (MHC) (45%, P Ͻ 0.05) and type IIx MHC (25%, P Ͻ 0.05) associated with decreased type IIb MHC (34%, P Ͻ 0.05) was minimized by perindopril administration. These findings demonstrate that the enhancement in physical performance observed in perindopril-treated animals cannot be explained by changes in mitochondrial respiration and/or MHC distribution within muscles involved in running exercise. mitochondria; myosin heavy chain; angiotensin I-converting enzyme inhibitor; perindopril; running performance ENDURANCE TRAINING leads to molecular and cellular adaptations that mainly occur in the cardiovascular system and skeletal muscles and that improve performance during prolonged exercise. Skeletal muscle adaptation to endurance training includes quantitative and qualitative changes in mitochondria,

HIGHLIGHTED TOPIC Regulation of Protein Metabolism in Exercise and Recovery

Decreased muscle ACE activity enhances functional response to endurance training in rats, without change in muscle oxidative capacity or contractile phenotype marked development of muscle capillary network, and transition in isoforms of myosin heavy chain. Such training-induced changes contribute to improve aerobic capacity and reliance on oxidative metabolism to provide energy (for review, see Refs. 9 and 14). Individual responses to exercise training result from environmental factors (training program, altitude, nutrition) and are also obviously influenced by genetic factors [START_REF] Bouchard | Familial aggregation of V ˙O2 max response to exercise training: results from the HERITAGE Family Study[END_REF]. Recent advances in the sequencing of the human genome have strengthened efforts made to understand genetic differences that may explain individual responses to training, especially to endurance training [START_REF] Rankinen | The human gene map for performance and health-related fitness phenotypes: the 2001 update[END_REF][START_REF] Rankinen | The human gene map for performance and health-related fitness phenotypes: the 2003 update[END_REF]. Although it is unlikely that the genetic component of endurance performance phenotype might be explained by DNA sequence variations of only a few genes, some studies attempted to relate whether the polymorphism of candidate genes might be associated with some aspects of physical performance.

The most studied of the DNA sequence variations potentially associated with endurance performance and response to physical training is the I/D polymorphism of the ANG I-converting enzyme (ACE) gene (for review, see Ref. 11). The absence (deletion, D allele) of a 287-bp fragment in intron 16 of the ACE gene is associated with increased ACE activity, known to influence blood pressure through both the generation of the vasoconstrictor factor ANG II deriving from ANG I, and inactivation of the vasodilatator bradykinin. Conversely, the presence (insertion, I allele) of this fragment is associated with lower ACE activity in serum [START_REF] Rigat | An insertion/deletion polymorphism in the angiotensin I-converting enzyme gene accounting for half the variance of serum enzyme levels[END_REF]. Moreover, local reninangiotensin systems exist in many tissues including skeletal muscle, and these systems might affect muscle metabolic efficiency, likely through bradykinin availability [START_REF] Dragovic | Kininase II-type enzymes: their putative role in muscle energy metabolism[END_REF]. In human, the ACE genotype-dependent variation in ACE activity may thus influence endurance performance through a variety of mechanisms, including altered circulatory homeostasis [START_REF] Jones | Human performance: a role for the ACE genotype?[END_REF] and/or skeletal muscle metabolism [START_REF] Dragovic | Kininase II-type enzymes: their putative role in muscle energy metabolism[END_REF], two determinant factors of physical performance.

Although controversial, several human studies including elite groups of athletes suggested that the I allele and low circulating ACE levels were associated with endurance athletic performance (for review, see Ref. 11). Maximum exertional oxygen uptake (V ˙O2max ) is determined by changes in oxygen transport and consumption. Performance in endurance events is limited by V ˙O2max but also by the proportion of V ˙O2max that can be sustained and muscular efficiency, both factors determined by skeletal muscle characteristics (4). It has been shown that the putative association between the I allele, and thus low ACE activity in plasma and tissue, and endurance performance could be due rather to local muscle factors than central cardiovascular effects [START_REF] Roltsch | No association between ACE I/D polymorphism and cardiovascular hemodynamics during exercise in young women[END_REF][START_REF] Woods | Endurance enhancement related to the human angiotensin I-converting enzyme I-D polymorphism is not due to differences in the cardiorespiratory response to training[END_REF]. Moreover, the ACE I allele is mostly associated with an increased percentage of fibers with a low energy cost per unit of force, i.e., type I fibers in human skeletal muscle [START_REF] Zhang | The I allele of the angiotensin-converting enzyme gene is associated with an increased percentage of slow-twitch type I fibers in human skeletal muscle[END_REF]. Taken together, these findings suggest that low ACE activity could have positive effects on muscle efficiency during exercise. This hypothesis has been previously tested in sedentary rats and ACE inhibition failed to affect the endurance capacity of animals during running exercise and/or the maximal oxidative capacity of skeletal muscles [START_REF] Bahi | Does ACE inhibition enhance endurance performance and muscle energy metabolism in rats?[END_REF].

The putative advantage of the II genotype on endurance performance has never been reported in sedentary subjects. However, the hypothesis that the ACE gene I/D polymorphism could influence the adaptive responses to endurance training, especially muscular responses, has been previously tested. The mechanical efficiency of human muscle has been shown to increase in response to training, but only in subjects of the II genotype, those with low ACE activity [START_REF] Williams | The ACE gene and muscle performance[END_REF]. However, the cellular and molecular mechanisms responsible for the greater improvement in muscle efficiency reported in subjects with low ACE activity after endurance training remain to be elucidated. We thus studied the effects of ACE inhibition on training-related changes in the capacity of muscle mitochondria to oxidize energetic substrates via oxidative phosphorylation and on the shift toward type I fibers.

The purpose of this study was to examine whether or not muscular ACE activity influences the adaptive responses of skeletal muscle to endurance training. We hypothesized that chronic ACE inhibition combined with endurance training could improve endurance performance more than training alone and that this association could positively affect skeletal muscle metabolic efficiency. To address this issue, we measured the endurance performance of perindopril-treated rats, either maintained sedentary or submitted to 8-wk treadmill training. In addition to training-induced changes in running performance, we examined the associated alterations in muscle oxidative capacity and contractile phenotype. In the present study, oxidative capacity was assessed through respiratory parameters of the total mitochondrial population measured in situ in fresh saponin-skinned fibers as previously described [START_REF] Kuznetsov | Analysis of mitochondrial function in situ in permeabilized muscle fibers, tissues and cells[END_REF][START_REF] Veksler | Mitochondrial respiratory parameters in cardiac tissue: a novel method of assessment by using saponin-skinned fibers[END_REF].

MATERIALS AND METHODS

Animals. Female Wistar rats initially weighing 175-200 g were housed two per cage in a thermoneutral environment (22 Ϯ 2°C) on a 12:12-h light-dark period and were provided with food and water ad libitum throughout the study period. This study was performed in accordance with the Helsinki accords for humane treatment of laboratory animals and the European Convention for the Protection of Vertebrate Animals used for Experimental and other Scientific Purposes (Council of Europe no. 129, Strasbourg, 1985) and was approved by the Animal Ethics Committee of the Centre de Recherches du Service de Sante des Armees.

Experimental design. After 4 days of acclimatization to the animal facility, all animals were familiarized with rodent treadmill and running exercise. Rats were running onto a rodent treadmill with a stainless steel grid, 10 -25 m/min for 10 min/day, at 0 -8°grade. Mild electrical shock stimulation was applied, if necessary, only to maintain the running motivation.

To measure the maximal endurance performance all rats performed a bout of exhaustive running exercise. The animals were exercised to exhaustion at 25 m/min up a 8°gradient, 48 h after the last session of running acclimatization. Running time to exhaustion was measured when animals were unable to longer maintain the treadmill speed and lost the righting reflex.

After this first endurance test, animals were randomly assigned to one of four groups, designed as either sedentary control (Sed-Ct, n ϭ 8), sedentary perindopril-treated (Sed-Per, n ϭ 8), trained control (Tr-Ct, n ϭ 8), and trained perindopril-treated (Tr-Per, n ϭ 8). Perindopril was given orally (in drinking water, 2 mg ⅐ kg Ϫ1 ⅐ day Ϫ1 ) for 10 wk; all animals were weighted regularly to adjust perindopril dose. To begin the endurance training program with low ACE activity, running exercises started 2 wk after beginning the perindopril treatment and lasted 8 wk. Animals ran on a motorized treadmill 5 days/wk using an exercise program involving both progressive intensity and duration. Briefly, the rodents were initially running trained at 10 m/min for 15 min, and by the end of 2 wk, rats were running at 30 m/min on an 8% grade for 2 h/day. This intensity and duration were maintained for the remaining 6 wk of the training program. Electrical shocks were used sparingly to encourage the animals to run. This protocol was selected because it has been shown that an increase in mitochondrial oxidative activity.

One week before the end of the training program and/or perindopril administration, the running time to exhaustion of rats of all experimental groups was assessed. Trained animals stayed 48 h resting before exhaustive running exercise.

Surgical procedures. Forty-eight hours after the last endurance test, animals were anesthetized with pentobarbital sodium administered intraperitoneally (50 mg/kg body wt). Soleus (Sol) and plantaris (Pla) muscles of both hindlimbs were immediately excised. Right muscles were put in Krebs solution composed of 118 mM NaCl, 25 mM NaHCO 3, 4.7 mM KH2PO4, 1.2 mM MgSO4, and 4.7 mM KCl for mitochondrial respiration experiments. Left ones were quickly frozen in liquid nitrogen until biochemical assays were performed.

Determination of plasma and skeletal muscle ACE activity. The efficacy of perindopril administration was assessed on both plasma and muscle ACE activity. Perindopril-induced alterations in ACE activity were first tested through the accumulation of acetyl-Ser-Asp-Lys-Pro (AcSDKP), a tetrapeptide that is cleared from plasma by ACEmediated hydrolysis [START_REF] Azizi | High plasma level of N-acetyl-seryl-aspartyl-lysyl-proline. A new marker of chronic angiotensin-converting enzyme inhibition[END_REF]. Plasma AcSDKP levels were determined by a competitive enzyme immunoassay, as previously reported [START_REF] Azizi | High plasma level of N-acetyl-seryl-aspartyl-lysyl-proline. A new marker of chronic angiotensin-converting enzyme inhibition[END_REF].

Skeletal muscle ACE activity was determined in a crude muscle homogenate, using a modified sensitive fluorometric assay according to Morrel et al. [START_REF] Morrell | Right ventricular angiotensin converting enzyme activity and expression is increased during hypoxic pulmonary hypertension[END_REF]. Plantaris muscle samples were thawed and homogenized at 4°C in Tris (0.05 M, pH 7.4)-buffered saline (0.15 M NaCl), and homogenates were then centrifuged at 1,000 g for 15 min at 4°C. One-hundred microliters of 5 mM Phe-His-Leu substrate, 50 mM Tris, pH 7.5, and 150 mM NaCl were added to crude homogenates. The reaction was stopped after 60 min at 37°C by addition of 750 l 0.28 M NaOH. Fifty microliters of o-phthalaldehyde (1 mg in 100 l methanol) was added for 10 min, and the final reaction was then stopped by addition of 100 l 2 N HCl. The fluorescence of the samples was measured with a fluorescence spectrometer (Kontron Instrument Win25, Montigny le Bretonneux, France) at an emission wavelength of 424 nm and an excitation wavelength of 354 nm. Samples were run in duplicate, and blanks (all reagents except homogenates) and samples incubated in the presence of the ACE inhibitor captopril were included. The results were expressed as microunits (mU) per milligram of protein, where 1 mU represents the generation of 1 nmol His-Leu/min. The protein concentration was determined by using the Bradford method.

In situ study of mitochondrial respiration. Respiratory parameters of the total mitochondrial population were studied in situ in saponinskinned fibers, using the method described earlier [START_REF] Kuznetsov | Analysis of mitochondrial function in situ in permeabilized muscle fibers, tissues and cells[END_REF][START_REF] Veksler | Mitochondrial respiratory parameters in cardiac tissue: a novel method of assessment by using saponin-skinned fibers[END_REF]. Thin fibers bundles were excised from skeletal muscles and incubated with intense shaking for 30 min at 4°C in solution S (see below) containing saponin (2.5 mg/ml) and rinsed 10 min in solution S alone. Five-to 10-mg fiber bundles were then rinsed in solution R or solution R 0 (see below) to wash out adenine nucleotides and PCr. Respiration rates were determined using a Clark electrode in an oxygraphic cell (Hansatech Oxygraph Instrument, Norfolk, UK) containing 1.5 ml of solution R or R 0 at 22°C with continuous stirring. After measurements, fibers bundles were carefully removed, dried (10 min at 110°C), and weighed. Respiration rates and substrates consumption were expressed as micromoles O 2 per minute per milligram dry weight.

Solutions S, R, and R 0 were designed to mimic the intracellular milieu and were previously accurately described [START_REF] Veksler | Mitochondrial respiratory parameters in cardiac tissue: a novel method of assessment by using saponin-skinned fibers[END_REF]. The protocol was designed to determine the dependences of respiration on external [ADP] in the presence or in the absence of creatine (Cr) and with appropriate concentrations of glutamate and malate as substrates. Respiration rates were recorded in solution R. After addition of fibers, basal oxygen consumption (V ˙0) due to proton leak was measured. Then, respiration rate was accelerated by addition of successive [ADP] until it reached the maximal oxygen consumption (V ˙max). The ADP-stimulated respiration above V ˙0 was then measured under the presence of saturating amount of ADP as phosphate acceptor and glutamate-malate as mitochondrial substrates (V ˙GM) [START_REF] Veksler | Mitochondrial respiratory parameters in cardiac tissue: a novel method of assessment by using saponin-skinned fibers[END_REF]. Apparent K m for ADP was calculated using a nonlinear fitting of the Michaelis-Menten equation. The acceptor-to-control ratio (ACR) was calculated as V ˙GM/V ˙0 and represented the functional coupling between oxidation and phosphorylation.

Citrate synthase activity. Frozen tissue samples from soleus and plantaris muscles were weighed and homogenized into ice-cold buffer (50 mg/ml) containing 5 mM HEPES (pH ϭ 8.7), 1 mM EGTA, 0.1% Triton X-100, and 1 mM DTT. They were incubated for 60 min at 0°C to ensure complete enzyme extraction. Determination of citrate synthase (CS) activity was assayed at 30°C (pH ϭ 7.5), using coupled enzyme systems as previously described [START_REF] Srere | Citrate synthase[END_REF]. CS activity was given as micromoles per minute (i.e., international units, IU).

Analysis of myosin heavy chain proteins. Skeletal muscles were subjected to the analysis of myosin heavy chain (MHC) isoforms as described previously [START_REF] Talmadge | Electrophoretic separation of rat skeletal muscle myosin heavy-chain isoforms[END_REF]. A small section was taken from the midregion of the muscle. Tissue samples were minced with scissors, and myosin was extracted in seven volumes of an appropriate buffer. After 1 h of gentle shaking at 4°C, the mixture was centrifuged at 13,500 g for 15 min, and the supernatant containing myosin was diluted with one volume of glycerol. Extracts were stored at Ϫ20°C until the separation process. Electrophoresis was performed using a Mini Protean II system (Bio-Rad, Marne-la-Coquette, France). Separating gel solution contained 30% glycerol, 8% acrylamide-bis (50:1), 0.2 M Tris, 0.1 M glycine, and 0.4% sodium dodecyl sulfate (SDS). Stacking gel was composed of 30% glycerol, 4% acrylamide-bis (50:1), 70 mM Tris, 4 mM EDTA, and 0.4% SDS. Myofibril samples were denatured using a sample buffer containing 5% 2-␤-mercaptoethanol, 100 mM Tris-base, 5% glycerol, 4% SDS, and bromophenol blue. Gels were run at constant voltage (72 V) for 31 h and then silver stained [START_REF] Agbulut | Analysis of skeletal and cardiac muscle from desmin knockout and normal mice by high resolution separation of myosin heavy-chain isoforms[END_REF]. The MHC protein isoform bands were scanned and quantified using a densitometer system equipped with an integrator (GS-700, Bio-Rad).

Statistical analysis. Data were expressed as means Ϯ SE. A two-way ANOVA was used to determine the global effects of perindopril treatment and running training, and interactions between those factors. When appropriate, differences between groups were tested with a Newman-Keuls post hoc test. Within each experimental group, the comparison of K m values with and without creatine was performed using a Student's paired t-test. Statistical significance was set at P Ͻ 0.05.

RESULTS

Body weight and muscle mass. At the start of the study, animals in all groups had similar body weight values. Final body weight remained unaffected by the experimental conditioning (Table 1). Soleus muscle weight, expressed as either absolute values or relative to body weight, slightly increased after endurance training (global effect of training, P Ͻ 0.05), while plantaris as fast-twitch muscle remained unaffected. Perindopril treatment failed to affect muscle mass.

Plasma and muscle tissue ACE activity. No effect of endurance training was shown on plasma and muscle ACE activity. The effectiveness of perindopril administration was thus examined in plasma and muscle samples from both Sed and Tr animals. As expected, perindopril-treated rats showed plasma AcSDKP levels approximately four times higher than those observed in control nontreated animals (Fig. 1A).

Mean tissue ACE activity in plantaris muscle was 13 times lower in perindopril-treated rats than in nontreated animals (Fig. 1B). ACE activity in plantaris muscles from Per groups corresponded to a residual tissue activity level of 11% of that measured in muscles from control nontreated rats.

Exercise capacity. At the beginning of experimental conditioning, female rats ran ϳ180 min before becoming exhausted, without difference between groups (Table 2). Ten weeks after, the running time to exhaustion decreased significantly in sedentary rats (54 and 53% in Sed-Ct and Sed-Per groups, respectively, P Ͻ 0.01), without effect of perindopril administration. In contrast, the training program led to a marked increase in the duration of the exhaustive running exercise (119 and 154% in Tr-Ct and Tr-Per groups, respectively, P Ͻ 0.01). Interestingly, the running time to exhaustion was 14% higher in Tr-Per than in Tr-Ct group (P Ͻ 0.05).

CS activity. There was a global effect of endurance training that increased CS activity in both soleus and plantaris muscles (P Ͻ 0.001) (Table 3). The training-induced increase in CS activity occurred in both perindopril-treated animals (25% and 33% in soleus and plantaris muscles, respectively, P Ͻ 0.05) and control nontreated rats (39% and 46% in soleus and plantaris muscles, respectively, P Ͻ 0.05). However, CS ac- tivity in soleus muscle remained 19% lower in perindopriltreated than in control nontreated rats after endurance training (P Ͻ 0.05).

Mitochondrial function. In soleus muscle, basal respiration rate (V ˙0) increased by ϳ50% with endurance training, but only in the control nontreated group (P Ͻ 0.05) (Fig. 2). The training-induced increase in the basal respiration rate was not observed in perindopril-treated rats, so that V ˙0 values in Tr-Per rats were 26% lower than in Tr-Ct group (P Ͻ 0.05). The ADP-stimulated respiration of in situ mitochondria using glutamate-malate as substrates (i.e., V ˙GM ) was higher in soleus muscles from Tr-Ct than Sed-Ct animals (20%, P Ͻ 0.05). In contrast V ˙GM values were lower in trained perindopril-treated rats than in Tr-Ct animals (P Ͻ 0.05) (Fig. 2). The oxidation-phosphorylation coupling, expressed by ACR, slightly decreased with running training in both control and perindopril-treated rats (24% and 27%, respectively, P Ͻ 0.05) (Table 4). Moreover, repeated perindopril administration also decreased mean ACR values in both sedentary and trained animals (19% and 22%, P Ͻ 0.05). The mitochondrial affinity for ADP in the absence (K m ADP ϪCr ) or in the presence of creatine (K m ADP ϩCr ) remained unaffected either by endurance training or perindopril treatment.

In plantaris muscle, endurance training increased both basal and maximal ADP-stimulated respiration rates (V ˙0 and V ˙GM , respectively) in control nontreated animals (36% and 38%, respectively, compared with Sed-Ct rats P Ͻ 0.05) (Fig. 2). Surprisingly, perindopril administration alone significantly increased V ˙0 and V ˙GM in plantaris muscle taken from sedentary animals (P Ͻ 0.05). These alterations were not observed in trained animals. Like in soleus muscle, ACR values were slightly decreased by perindopril administration in both sedentary and physically trained rats (global effect P Ͻ 0.05) (Table 4). As expected in fast-twitch glycolytic muscles, values of the K m for ADP were low in the presence or absence of creatine. In the absence of creatine, K m values for ADP increased after 10 wk of perindopril administration in sedentary rats (P Ͻ 0.05), but values were still on the same order as those expected in fast-twitch glycolytic muscles. Endurance training failed to affect the mitochondrial affinity for ADP in either control or perindopril-treated rats.

MHC distribution. In Wistar rats, soleus muscle is known to comprise more than 95% type I MHC. The MHC distribution remained unaffected by either endurance training or repeated perindopril administration (data not shown).

In the fast-twitch plantaris muscles from sedentary rats, perindopril administration led to only a slight but significant decrease in the percentage of type IIb MHC (12%, P Ͻ 0.05) (Fig. 3). As expected, training alone in control nontreated rats induced an increase in the percentage of type IIa MHC (45%, P Ͻ 0.05) and type IIx MHC (25%, P Ͻ 0.05), at the expense of type IIb MHC (decrease by 34%, P Ͻ 0.05). Perindopril treatment associated with endurance training minimized the training-induced increase in type IIa MHC, and fully cancelled the expected increase in type IIx percentage and decrease in type IIb percentage, compared with Sed-Per animals. As a consequence, the relative contents of type IIa and type IIx MHC were lower, and the relative content of type IIb was higher in Tr-Per than in the Tr-Ct group (Fig. 3). No detectable change was shown in the percentage of type I MHC as a result of endurance training in either control or perindopril-treated rats.

DISCUSSION

Because the ACE genotype has never been associated with endurance performance in the untrained state, it has been suggested that the I allele and the decreased tissue ACE activity could be mainly related to greater enhancement in muscle metabolic efficiency in response to endurance training, rather than improving basal performances [START_REF] Jones | Human performance: a role for the ACE genotype?[END_REF][START_REF] Williams | The ACE gene and muscle performance[END_REF]. The present Table 2. Running time to exhaustion measured before and after the experimental protocol in sedentary control or perindopril-treated rats and trained control or perindopril-treated rats The effectiveness of perindopril administration on ACE activity was first examined. Because ACE is the main enzyme involved in AcSDKP metabolism, plasma AcSDKP was expected to increase after long-term perindopril administration [START_REF] Junot | Effect of angiotensin-converting enzyme inhibition on plasma, urine and tissue concentrations of hemoregulatory peptide acetyl-Ser-Asp-Lys-Pro in rats[END_REF]. It is now a well-known fact that AcSDKP accumulates in plasma during chronic ACE inhibitor administration (2). The perindopril-induced increase in plasma AcSDKP observed in the present study is consistent with this finding and similar to that previously reported after captopril administration [START_REF] Junot | Effect of angiotensin-converting enzyme inhibition on plasma, urine and tissue concentrations of hemoregulatory peptide acetyl-Ser-Asp-Lys-Pro in rats[END_REF]. Moreover, plantaris muscle ACE activity was markedly de-creased after chronic perindopril administration, to a residual level similar to that reported after administration of one of the most efficient ACE inhibitors on tissue ACE activity [START_REF] Fabre | Tissue inhibition of angiotensin-converting enzyme activity stimulates angiogenesis in vivo[END_REF]. There is thus evidence that chronic perindopril administration decreased tissue ACE activity to residual levels within skeletal muscle.

The present study revealed that decreased ACE activity after repeated administration of the ACE inhibitor perindopril in trained rats was associated with a higher increase in the running exercise capacity than in control nontreated trained animals. Female rats were used in the present study regarding their natural higher ability to run and lower body weight growth rate than male rats [START_REF] Titchenal | Exercise and food intake. What is the relationship?[END_REF]. Enhanced exercise tolerance in trained perindopril-treated rats could be related to direct effects of ACE inhibition on the acute responses to exercise and/or the extent of adaptive changes elicited by endurance training. However, the pharmacological inhibition of ACE activity failed to enhance physical performance in sedentary rats, a finding consistent with previous results (3). It is thus unlikely that the increased endurance capacity reported in the present study after endurance training in perindopril-treated rats could be related to more favorable acute responses to a bout of exercise. Rather than improving the acute responses to exercise, decreased ACE activity would enhance the adaptive changes in response to endurance training.

Many previous studies showed that ACE inhibitors such as perindopril have two primary effects, first a decrease in the conversion in ANG I to the vasoconstrictor ANG II, and second a decreased bradykinin degradation into inactive peptides, leading to increased bradykinin action [START_REF] Uehara | Effect on insulin sensitivity of angiotensin converting enzyme inhibitors with or without a sulphydryl group: bradykinin may improve insulin resistance in dogs and humans[END_REF]. It is thus likely that the cellular effects of ACE inhibition on skeletal muscle can involve both reduced ANG II action and increased bradykinin influence. A reduction in ANG II formation during exercise favors vasodilatation and then potential substrate delivery to active skeletal muscles [START_REF] Stroth | The renin-angiotensin system and its receptors[END_REF], while bradykinin likely has a synergistic role in mediating vasodilatation during muscular contraction [START_REF] Boushel | Combined inhibition of nitric oxide and prostaglandins reduces human skeletal muscle blood flow during exercise[END_REF].

Respiration rate in skinned fibers is a unique way to examine the muscle oxidative capacity by measuring oxygen consumption of the entire mitochondrial population within its cellular environment. Whether the pharmacological inhibition of ACE affects the mitochondrial function has been previously studied. Prolonged ACE inhibition completely prevented the decreased skeletal muscle oxidative capacity expected in patients with chronic heart failure [START_REF] Jondeau | Relation of functional improvement in congestive heart failure after quinapril therapy to peripheral limitation[END_REF] and in rats with heart failure secondary to myocardial infarction [START_REF] Yamaguchi | Improvement of exercise capacity of rats with chronic heart failure by long-term treatment with trandolapril[END_REF][START_REF] Zoll | ACE inhibition prevents myocardial infarction-induced skeletal muscle mitochondrial dysfunction[END_REF]. However, this protective effect of ACE inhibition remains controversial, and perindopril administration failed to restore muscle oxidative capacity and mitochondrial function in a rat model of heart failure induced by aortic stenosis [START_REF] Momken | Does angiotensin-converting enzyme inhibition improve the energetic status of cardiac and skeletal muscles in heart failure induced by aortic stenosis in rats?[END_REF] or in a rat model of type I diabetes [START_REF] Rouyer | Effect of angiotensin-converting enzyme inhibition on skeletal muscle oxidative function and exercise capacity in streptozotocin-induced diabetic rats[END_REF]. Taken together, these previous findings suggested that ACE inhibitors at best improve the mitochondrial dysfunction that characterizes skeletal muscle myopathy associated with chronic diseases. However, whether perindopril affects the muscle mitochondrial responses to endurance training in normal rats remained to be examined. The present study, for the first time, shows that the training-induced changes in CS activity and maximal mitochondrial respiration of saponinskinned fibers with glutamate-malate as substrates (V ˙GM ) were less in perindopril-treated rats than in control animals. These findings, together with the slight decrease in the functional coupling between oxidation and phosphorylation related to perindopril administration, disagree with the hypothesis that low ACE activity could improve the efficiency of mitochondrial respiration in skeletal muscle, and then endurance capacity (for review, see Ref. 12).

Prolonged administration of ACE inhibitors has been shown to alter the myosin content toward MHC isoforms associated with oxidative fatigue-resistant fibers [START_REF] Vescovo | Improved exercise tolerance after losartan and enalapril in heart failure. Correlation with changes in skeletal muscle myosin heavy chain composition[END_REF]. Moreover, population studies suggested that low ACE activity resulting from the presence of the I allele of the ACE gene was associated with an increased percentage of type I fibers in young untrained volunteers [START_REF] Zhang | The I allele of the angiotensin-converting enzyme gene is associated with an increased percentage of slow-twitch type I fibers in human skeletal muscle[END_REF]. However, the heterozygous disruption of the ace gene in mice (ACEϩ/Ϫ) did not affect the percent composition of soleus muscle fibers [START_REF] Zhang | Histochemical characteristics of soleus muscle in angiotensin-converting enzyme gene knockout mice[END_REF]. The direct effects of ACE activity on MHC expression remain thus controversial, while the results of the present study clearly suggest that the pharmacological inhibition of ACE activity failed to significantly affect the MHC composition of skeletal muscles from sedentary Moreover, results demonstrate that the shift toward MHC IIa expected with endurance training in fast-twitch muscle was less in perindopril-treated than in control animals. Under control conditions, the training-induced shift in MHC expression is consistent with low metabolic cost, high efficiency of the myosin molecule, and parallel increase in muscle oxidative capacity. In the present study, perindopril administration minimized both the training-induced shift toward more fatigueresistant MHC isoforms and the associated low response of mitochondrial function to endurance training. Although these negative effects of perindopril on the responses of contractile phenotype and muscle oxidative capacity to endurance training are finely coordinated, exact mechanisms remain to be elucidated. Taken together, all these findings strongly suggest that it is unlikely that changes in muscle metabolic efficiency and contractile characteristics contribute to explain the marked enhancement in physical performance observed in trained perindopril-treated animals.

In conclusion, the present study demonstrates for the first time that inhibition of skeletal muscle ACE activity using a nonsulfhydryl ACE inhibitor in rats improves the functional response to endurance training. The higher endurance capacity reported in trained perindopril-treated rats, compared with trained nontreated animals, was not related to detectable changes in muscle oxidative capacity or contractile phenotype. However, performance in endurance events is mainly limited by the ability of the cardiorespiratory system to transport oxygen to skeletal muscles. The degree of peripheral oxygenation through muscle capillary bed could also be involved to account for enhanced muscle efficiency. Whether the enhanced endurance capacity of perindopril-treated rats is related to improved angiogenic responses to endurance training needs to be addressed in future studies.

  are means Ϯ SE *Significantly different from values measured before the protocol, P Ͻ 0.05. †Significantly different from corresponding Sed group, P Ͻ 0.001. ‡Significantly different from Tr-Ct group, P Ͻ 0.05.

Fig. 1 .

 1 Fig. 1. Plasma acetyl-Ser-Asp-Lys-Pro (AcSDKP) levels (A) and plantaris muscle ANG I-converting enzyme (ACE) activity (B) measured in control (Ctr) and perindopril-treated animals (Per). Values are means Ϯ SE. *Significance vs. Ctr groups, P Ͻ 0.001.

Table 1 .

 1 Anatomic data of sedentary control or perindopril-treated rats, and trained control or perindopril-treated rats

		Sed-Ct (n ϭ 8)	Sed-Per (n ϭ 8)	Tr-Ct (n ϭ 8)	Tr-Per (n ϭ 8)	Global Effect of Tr	Global Effect of Per
	Final body wt, g	259Ϯ6	283Ϯ6	274Ϯ11	274Ϯ6	N S	N S
	Sol muscle weight, mg	112Ϯ7	128Ϯ4	134Ϯ9	138Ϯ10	P Ͻ 0.05	NS
	Pla muscle weight, mg	216Ϯ17	257Ϯ8	235Ϯ8	231Ϯ7	N S	N S
	Tibia length, cm	3.64Ϯ0.04	4.00Ϯ0.05	3.81Ϯ0.06	3.70Ϯ0.07	NS	NS
	Sol/body wt, mg/g	0.43Ϯ0.03	0.45Ϯ0.01	0.49Ϯ0.03	0.50Ϯ0.03	P Ͻ 0.05	NS
	Pla/body wt, mg/g	0.83Ϯ0.06	0.90Ϯ0.02	0.77Ϯ0.11	0.85Ϯ0.03	NS	NS

Values are means Ϯ SE. Sed-Ct, sedentary control rats; Sed-Per, sedentary perindopril-treated rats; Tr-Ct, trained control rats; Tr-Per, trained perindopriltreated rats; Sol, soleus muscle; Pla, plantaris muscle; NS, not significant.

Table 3 .

 3 CS activity for soleus and plantaris muscles of sedentary or trained rats, either control or perindopril treated SE. CS, citrate synthase; IU, international units. *Significantly different from values measured in corresponding Sed group, P Ͻ 0.05. †Significantly different from Tr-Ct group, P Ͻ 0.05. Values are means Ϯ SE. *Significance vs. corresponding Sed group, P Ͻ 0.05. $ Significance vs. corresponding Ct nontreated group, P Ͻ 0.05.

		Sed-Ct (n ϭ 8)	Sed-Per (n ϭ 8)	Tr-Ct (n ϭ 8)	Tr-Per (n ϭ 8)	Global Effect of Tr	Global Effect of Per
	CS activity, IU/g wet wt
	Sol	38.6Ϯ2.5	34.9Ϯ2.4	53.5Ϯ1.0*	43.5Ϯ2.1* †	P Ͻ 0.001	NS
	Pla	31.1Ϯ2.0	31.1Ϯ1.8	45.4Ϯ3.3*	41.4Ϯ2.6*	P Ͻ 0.001	NS
	Values are means Ϯ

Table 4 .

 4 Effects of endurance training and perindopril treatment on acceptor-to-control ratio (V ˙GM /V ˙0) and K m for ADP of in situ mitochondria from soleus and plantaris muscles SE. Km values were calculated by nonlinear fit of the Michaelis-Menten equation of respiration rates obtained for increasing ADP concentrations in absence (KmADPϪCr) or presence (KmADPϩCr) of creatine. ACR, acceptor-to-control ratio (V ˙GM/V ˙0, where V ˙0 is basal respiration rate and V ˙GM is maximal ADP-stimulated respiration rate with glutamate and malate as substrates). *Significantly different from values measured in corresponding Sed group, P Ͻ 0.05. †Significantly different from corresponding Ct group, P Ͻ 0.05. SE. *Significance vs. corresponding Sed group, P Ͻ 0.05. $ Significance vs. corresponding Ct nontreated group, P Ͻ 0.05.

		Sed-Ct(n ϭ 8)	Sed-Per(n ϭ 8)	Tr-Ct(n ϭ 8)	Tr-Per(n ϭ 8)	Global Effect of Tr	Global Effect of Per
	Sol muscle
	ACR	4.5Ϯ0.3	3.7Ϯ0.3 †	3.4Ϯ0.1*	2.7Ϯ0.2* †	P Ͻ 0.01	P Ͻ 0.05
	KmADPϪCr, M	338Ϯ64	291Ϯ46	303Ϯ37	231Ϯ35	NS	NS
	KmADPϩCr, M	76Ϯ9	6 4 Ϯ12	67Ϯ12	47Ϯ9	N S	N S
	Pla muscle
	ACR	4.5Ϯ0.3	2.5Ϯ0.2 †	4.3Ϯ0.3	3.0Ϯ0.3 †	NS	P Ͻ 0.05
	KmADPϪCr, M	15Ϯ3	4 6 Ϯ9 †	29Ϯ6	2 5 Ϯ9	N S	N S
	KmADPϩCr, M	16Ϯ4	2 6 Ϯ8	2 3 Ϯ4	4 2 Ϯ9	N S	N S
	Values are means Ϯ

on December 22, 2009 jap.physiology.org Downloaded from

J Appl Physiol• VOL 107 • JULY 2009 • www.jap.org on December 22, 2009 jap.physiology.org Downloaded from

ACKNOWLEDGMENTS

We thank Marielle Pasdeloup for expert technical assistance.

GRANTS

This study was supported by grants from Service de sante ´des arme ´es, Ministry of Defense, France.


Miranda Boutilier

Konstantin Brenner

Victorita Dolean

A Trefftz-like coarse space for the two-level Schwarz method on perforated domains

Introduction and Model Problem

Numerical modeling of overland flows plays an increasingly important role in predicting, anticipating and controlling floods, helping to size and position protective systems including dams, dikes or rainwater drainage networks. One of the challenges of the numerical modeling of urban floods is that the small structural features (buildings, walls, etc.) may significantly affect the flow. Luckily, modern terrain survey techniques including photogrammetry and Laser Imaging, Detection, and Ranging (LIDAR) allow to acquire high-resolution topographic data for urban areas as well as for natural (highly vegetated) media. For example, the data set used in this article has been provided by Métropole Nice Côte d'Azur (MNCA) and allows for the infra-metric description of the urban geometries [START_REF] Andres | L'apport de la donnée topographique pour la modélisation 3D fine et classifiée d'un territoire[END_REF].

From the hydraulic perspective, these structural features can be assumed to be essentially impervious, and therefore represented as perforations (holes) in the model domain. Our long term modelling strategy is based on the Diffusive Wave equation [START_REF] Alonso | On the diffusive wave approximation of the shallow water equations[END_REF]. However, understanding linear problems posed on perforated domains is a crucial preliminary step and the object of this contribution.

Let 𝐷 be an open simply connected polygonal domain in R 2 , we denote by Ω 𝑆,𝑘 𝑘 a finite family of perforations in 𝐷 such that each Ω 𝑆,𝑘 is an open connected polygonal subdomain of 𝐷. The perforations are mutually disjoint, that is Ω 𝑆,𝑘 ∩ Ω 𝑆,𝑙 = ∅ for any 𝑘 ≠ 𝑙. We denote Ω 𝑆 = 𝑘 Ω 𝑆,𝑘 and Ω = 𝐷 \ Ω 𝑆 , assuming that the family Miranda Boutilier Université Côte d'Azur, LJAD, e-mail: miranda.boutilier@univ-cotedazur.fr Konstantin Brenner Université Côte d'Azur, LJAD, CNRS, INRIA, e-mail: konstantin.brenner@univ-cotedazur.fr Victorita Dolean University of Strathclyde, Dept. of Maths and Stats and Université Côte d'Azur, LJAD, CNRS e-mail: work@victoritadolean.com Ω 𝑆,𝑘 𝑘 is such that Ω is connected. Note that the latter assumption implies that Ω 𝑆,𝑘 are simply connected.

Let 𝑓 ∈ 𝐿 2 (Ω), in this article we are interested in the boundary value problem

         -Δ𝑢 = 𝑓 in Ω, - 𝜕𝑢 𝜕n = 0 on 𝜕Ω ∩ 𝜕Ω 𝑆 , 𝑢 = 0 on 𝜕Ω \ 𝜕Ω 𝑆 . (1)
Depending on the geometrical complexity of the computational domain, the numerical resolution of (1) may become challenging. A typical data set that we are interested in, illustrated by Figure 3, may contain numerous perforations that are described on different scales. In this regard, our strategy relies on the use of a Krylov solver combined with domain decomposition (DD) methods. Generally, to achieve scalability with respect to the number of subdomains in overlapping Schwarz methods, coarse spaces/components are needed. Including a coarse space in a Schwarz preconditioner results in what is referred to as a two-level Schwarz preconditioner.

The model problem can be thought of as the extreme limit case of the elliptic model containing highly contrasting coefficients. Two-level domain decomposition methods have been extensively studied for such heterogeneous problems. There are many classical results for coarse spaces that are contructed so as to resolve the jumps of the coefficients; see [START_REF] Dryja | Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions[END_REF][START_REF] Dryja | Multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions[END_REF][START_REF] Mandel | Balancing domain decomposition for problems with large jumps in coefficients[END_REF] for further details. Approaches to obtain a robust coarse space without careful partitioning of the subdomains include spectral coarse spaces such as those given in [START_REF] Galvis | Domain decomposition preconditioners for multiscale flows in highcontrast media[END_REF][START_REF] Nataf | A coarse space construction based on local Dirichlet-to-Neumann maps[END_REF][START_REF] Spillane | Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps[END_REF]. Additionally, the family of GDSW (Generalized Dryja, Smith, Widlund) methods [START_REF] Dohrmann | Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions[END_REF] employ energy-minimizing coarse spaces and can be used to solve heterogeneous problems on less regular domains. These spaces are discrete in nature and involve both edge and nodal basis functions.

Alternatively, robust coarse spaces can be constructed using the ideas from multiscale finite elements methods (MsFEM) [START_REF] Aarnes | Multiscale domain decomposition methods for elliptic problems with high aspect ratios[END_REF][START_REF] Graham | Domain decomposition for multiscale PDEs[END_REF]. The combination of spectral and MsFEM methods can be found in [START_REF] Gander | Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods[END_REF]. Outside of the DD framework, specifically on domains with small and numerous perforations, the authors of [START_REF] Bris | An MsFEM type approach for perforated domains[END_REF][START_REF] Brown | A multiscale finite element method for Neumann problems in porous microstructures[END_REF] also introduced an enriched MsFEM-like method.

Here, we present an efficient and novel coarse space in the overlapping Schwarz framework inspired by the Boundary Element based Finite Element (BEM-FEM) method [START_REF] Weißer | BEM-based Finite Element Approaches on Polytopal Meshes[END_REF]. In contrast with the classical BEM-FEM approach, the local multiscale basis functions are computed numerically such as in MsFEM methods. This approach is motivated by our interest in nonlinear time dependent models for which the analytical expression of the fundamental solutions may not be easily available.

Discretization and Preliminary Notations

We introduce a coarse discretization of Ω which involves a family of polygonal cells Ω 𝑗 𝑗=1,..., 𝑁 , the so-called coarse skeleton Γ, and the set of coarse grid nodes that will be referred to by V.

The construction is as follows. Consider a finite nonoverlapping polygonal partitioning of 𝐷 denoted by 𝐷 𝑗 𝑗=1,..., 𝑁 and an induced nonoverlapping partitioning of Ω denoted by Ω 𝑗 𝑗=1,..., 𝑁 such that Ω 𝑗 = 𝐷 𝑗 ∩ Ω. We will refer to Ω 𝑗 𝑗=1,..., 𝑁 as the coarse mesh over Ω. Additionally, we denote by Γ the skeleton of the coarse mesh, that is Γ = 𝑗 ∈ {1,..., 𝑁 } 𝜕Ω 𝑗 \ 𝜕Ω 𝑆 .

Let vert(Ω 𝑗 ) denote the set of vertices of the polygonal domain Ω 𝑗 . The set of coarse grid nodes is given as V = 𝑗 ∈ {1,..., 𝑁 } vert(Ω 𝑗 ) ∩ Γ. The total number of coarse grid nodes is denoted by 𝑁 V .We refer to Figure 1 for the illustration of the coarse mesh entities. We discretize the model problem (1) with piecewise linear continuous finite elements on a triangular mesh of Ω. This mesh is conforming to the coarse polygonal Ω 𝑗 𝑗=1,..., 𝑁 ; an example of the triangulation for various numbers of coarse cells 𝑁 is given in Figure 2. The finite element discretization of ( 1 . With a coarse restriction matrix R 0 that will be specified later, the two-level discrete Additive Schwarz (ASM) preconditioner is given by

𝑀 -1 𝐴𝑆 𝑀 ,2 = R 𝑇 0 (R 0 AR 𝑇 0 ) -1 R 0 + 𝑁 ∑ 𝑗=1 R 𝑇 𝑗 ( ¯R 𝑗 AR 𝑇 𝑗 ) -1 R 𝑗 . (2)

Description of the Trefftz-like coarse space

Here we introduce the Trefftz-like coarse space spanned by the functions that are piecewise linear on the skeleton Γ and discrete harmonic inside the nonoverlapping subdomains Ω 𝑗 . For any node x 𝑠 ∈ V, we introduce the function 𝑔 𝑠 : Γ → R, which is continuous on Γ and linear on each edge of Γ. It is clear that 𝑔 𝑠 is fully defined by its values at the nodes x 𝑖 ∈ V, for which we set

𝑔 𝑠 (x 𝑖 ) = 1, 𝑠 = 𝑖, 0, 𝑠 ≠ 𝑖.
To illustrate the construction of the nodal basis of the coarse space, we consider the following set of boundary value problems. For all Ω 𝑗 and for all 𝑠 = 1, . . . , 𝑁 V , find

𝜙 𝑗 𝑠 ∈ 𝐻 1 (Ω 𝑗 ) such that 𝜙 𝑗 𝑠
is the weak solution to the following problem

           -Δ𝜙 𝑗 𝑠 = 0 in Ω 𝑗 , - 𝜕𝜙 𝑗 𝑠 𝜕n = 0 on 𝜕Ω 𝑗 ∩ 𝜕Ω 𝑆 , 𝜙 𝑗 𝑠 = 𝑔 𝑠 on 𝜕Ω 𝑗 \ 𝜕Ω 𝑆 . (3)
The finite element discretization of (3) results in the system of the form

A 𝑗 𝝓 𝑗 𝑠 = b 𝑗 𝑠 , where A
𝑗 is the local stiffness matrix and b 𝑗 𝑠 accounts for the Dirichlet boundary data in [START_REF] Andres | L'apport de la donnée topographique pour la modélisation 3D fine et classifiée d'un territoire[END_REF]. Let R 𝑗 denote the restriction matrices corresponding to Ω 𝑗 , and let 𝝓 𝑠 be a vector such that R 𝑗 𝝓 𝑠 = 𝝓 𝑗 𝑠 for all 𝑗 = 1, . . . , 𝑁. The coarse space is then defined as the span of the basis functions 𝜙 𝑠 , 𝑠 = 1, . . . , 𝑁 V , while the 𝑘th row of R 0 is given by 𝝓 𝑇 𝑘 for 𝑘 = 1, . . . , 𝑁 V .

Numerical Results

We present below the numerical experiments concerning the performance of the conjugate gradient (CG) method using two-level preconditioner (2) and the Trefftzlike coarse space introduced in Section 3. For the sake of comparison we also report the numerical results obtained using a more standard Nicolaides coarse space which is going to be detailed later.

The data sets used in this experiment have been kindly provided by Métropole Nice Côte d'Azur and reflect the structural topography of the city of Nice. Although this type of data is available for the whole city [START_REF] Andres | L'apport de la donnée topographique pour la modélisation 3D fine et classifiée d'un territoire[END_REF], we focus here on a relatively small special frame (see Figure 3). In this numerical experiment we consider two kinds of structural elements -buildings (and assimilated small elevated structures) and walls. We note that the perforations resulting from the data sets we use (especially the wall data) can span across multiple coarse cells, which is a challenging situation for traditional coarse spaces. In this numerical experiment we consider the problem (1) with the-right-hand side given by 𝑓 = 1. Figure 3 reports the finite element solution obtained for the data excluding and including walls. The figure also reflects the nonoverlapping partitioning into 𝑁 = 8 × 8 subdomains.

Figure 4 and Table 1 report the performance of the two-level preconditioner used in the PCG method, for varying number of subdomains 𝑁 and two relative overlap sizes. As the computational domain Ω remains fixed independently of 𝑁, the results of this experiment could be interpreted in terms of a strong scalability. However, we wish to stress that the fine-scale triangulation is obtained based on the nonoverlapping partitioning (Ω 𝑗 ) 𝑗=1,..., 𝑁 . Consquentially, the linear system Au = f changes from one coarse partitioning to another. Nevertheless we ensure that the dimension of the system is roughly constant throughout the experiment. Depending on the chosen 𝑁, the linear system involves about 60𝑘 (buildings alone) and 180𝑘 (buildings and walls) nodal unknowns.

For the sake of comparison, we also provide numerical results for the well-known Nicolaides coarse space [START_REF] Nicolaides | Deflation of conjugate gradients with applications to boundary value problems[END_REF], made of flat-top partition of unity functions associated with the overlapping partitioning. As the scalability provided by the Nicolaides space relies on the Poincaré inequality over the subdomains, we further partition (Ω 𝑗 ) 𝑗 into a family of connected regions for this space. In other words, let 𝑚 𝑗 denote the number of disconnected components for each overlapping subdomain Ω 𝑗 and let Ω 𝑗,𝑙 , 𝑙 = 1, . . . , 𝑚 𝑗 denote the corresponding disconnected component. Then our new overlapping partioning contains 𝑚 = 𝑁 𝑗=1 𝑚 𝑗 total subdomains and is given by

( Ω𝑘 ) 𝑘 ∈ {1,...,𝑚} = (Ω 𝑗,𝑙 ) 𝑙 ∈ {1,...,𝑚 𝑗 } 𝑗 ∈ {1,..., 𝑁 } .
Then, the Nicolaides coarse space is as follows. The 𝑘th row of R 0 and therefore the 𝑘th column of R 𝑇 0 is given by (R T 0 ) 𝑘 = R 𝑇 𝑘 D 𝑘 R 𝑘 1 for 𝑘 = 1, . . . , 𝑚, where R 𝑘 and D 𝑘 are the restriction and partition of unity matrices corresponding to Ω𝑘 and 1 is a vector full of ones. The partition of unity matrices are constructed such that 4 reports convergence histories of the preconditioned CG method using the Nicolaides and Trefftz-like coarse spaces for the data set including both walls and buildings. Table 1 summarizes the numerical performance for data sets including or excluding walls. In particular, for both preconditioners, it reports the dimensions of the coarse spaces, as well as the number of CG iterations required to achieve a relative 𝑙 2 error of 10 -8 .

I = 𝑚 𝑘=1 R 𝑇 𝑘 D 𝑘 R 𝑘 . Figure
The performance of the Trefftz-like coarse space appears to be very robust with respect to both 𝑁 and the complexity of the computational domain. The improvement with respect to the alternative Nicolaides approach is quite striking, especially in the case of the minimal geometric overlap. As expected, increased overlap in the first level of the Schwarz preconditioner provides additional acceleration in terms of iteration count. However, for the Trefftz-like space, the results with minimal geometric overlap appear to already be quite reasonable.

The dimensions and the relative dimensions of the two coarse spaces are reported in Table 1. Relative dimension refers to the would-be dimension of the coarse space in the case of a homogeneous domain with Ω 𝑆 = ∅, that is, the relative dimensions are computed as dim(𝑅 0 ) ( √ 𝑁 +1) 2 for the Trefftz-like space and as dim(𝑅 0 ) 𝑁 for the Nicolaides space. We observe that the Trefftz-like coarse space requires a much larger number of degrees of freedom, which naturally leads to a large coarse system to solve. We note that the contrast between the dimensions of two spaces reduces as 𝑁 grows. In general, the dimension of the Trefftz-like coarse space seems reasonable given the geometrical complexity of the computational domain.

Conclusions

In this work we presented a novel Trefftz-like coarse space for the two-level ASM preconditioner, specifically designed for problems resulting from elliptic PDEs in perforated domains. This coarse space is robust with respect to data complexity and number of subdomains on a fixed total domain size, and provides significant acceleration in terms of Krylov iteration counts when compared to a more standard Nicolaides coarse space. This improvement comes at the price of a somewhat larger coarse problem. Current work in progress involves coarse approximation error and stable decomposition estimates and is left to a future article by the same authors.

We are also planning to extend the presented two-level preconditioning strategy to nonlinear PDEs that model free-surface flows.

Fig. 1

 1 Fig. 1 Coarse grid cell Ω 𝑗 , nonoverlapping skeleton Γ (blue lines), and coarse grid nodes x 𝑠 ∈ V (red dots). Coarse grid nodes are located at Γ ∩ 𝜕Ω 𝑆 .

  ) results in the linear system Au = f. Let Ω 𝑗 𝑗=1,..., 𝑁 denote the set of overlapping subdomains of Ω. In practice, each Ω 𝑗 is constructed by propagating Ω 𝑗 by a few layers of triangles. Consider classical boolean restriction matrices R 𝑗 and corresponding extension matrices R 𝑇 𝑗 associated to the family of overlapping subdomains Ω 𝑗 𝑗=1,..., 𝑁

Fig. 2 :

 2 Fig. 2: Conforming triangulation for the same domain Ω with different numbers of coarse cells 𝑁. The coarse skeleton Γ is shown by the blue lines.

Fig. 3 :

 3 Fig. 3: Approximate solution over a computational domain divided in 𝑁 = 8 × 8 nonoverlapping subdomains.

Fig. 4 :

 4 Fig. 4: Convergence curves for the Trefftz-like (solid lines) and Nicolaides (dashed lines) coarse spaces for the data set involving both buildings and walls and two overlap sizes. Colors correspond to the number of subdomains as follows: 𝑁 = 16 (blue), 𝑁 = 64 (orange), 𝑁 = 256 (green), 𝑁 = 1024 (red).

Table 1 :

 1 PCG iterations, condition number, dimension, and relative dimension for the Trefftz-like and Nicolaides coarse spaces. Results are shown for minimal geometric overlap and 1 20 𝐻, where 𝐻 = max 𝑗 diam(Ω 𝑗 ). As the dimension of the Nicolaides space will change with respect to the overlap, its dimension is given as the average dimension over the two overlap values. ) 59 3060 13 1912 (6.6) 1024 no walls 120 83 341 149 1204 (1.2) 56 28 76 13 2500 (2.3) walls 362 174 3895 1310 2044 (2.0) 61 28 97 13 4253 (3.9)

			Nicolaides			Trefftz
		it.	cond.	dim. (rel)	it.	cond.	dim. (rel)
	N	min. 𝐻 20	min.	𝐻 20		min 𝐻 20	min. 𝐻 20
	16 no walls 149 51 581	82	21 (1.3)	52 28 59 11 170 (6.8)
		walls 348 70 6826 133	96 (6.0)	56 22 136 7 400 (16.0)
	64 no walls 164 78 567 119	85 (1.3)	50 28 50 12 433 (5.3)
		walls 359 132 5902 297 256 (4.0) 56 26 57	9 880 (10.9)
	256 no walls 136 81 273	89	312 (1.2) 56 27 54 10 1010 (3.5)
		walls 317 159 4575 12	719 (2.8

Acknowledgments

This work has been supported by ANR Project Top-up (ANR-20-CE46-0005). The high-resolution structural data has been provided by Métropole Nice Côte d'Azur. We warmly thank Florient Largeron, chief of MNCA's SIG 3D project, for his help in preparation of the data and for the multiple fruitful discussions.


Guillaume Métayer

Cécile Dagès

Jean-Stéphane Bailly

David Crevoisier

Marc Voltz

Assessment of the MHYDAS-Pesticide-1.0 model in simulating pesticide concentrations in surface waters at plot-scale continuously over decades

pesticide contamination of surface water 2

Assessment of the MHYDAS-

 (e.g. Mudgal et al., 2010, Baffaut et al., 2019)• Most studies < 1 year (e.g. Connolly et al., 2001, Malone et al. 2004) Poor knowledge about the simulation accuracy of pesticide concentrations in surface water

Materials & Methods

To assess performance in simulating intra-annual variability

Analysis of error structure

To assess global model performance

Statistical criteria NSE

Nash Sutcliffe efficiency

PBIAS (%)

Percent bias

Variables of interest

Satisfactory performance for C values and logC values

  Cécile Dagès, Jean -Stéphane Bailly, David Crevoisier, and Marc Voltz 1. Introduction Assessing the performance of a mechanistic model in reproducing pesticide concentrations measured in runoff at field-scale on a multi-year basis Objective of this study : Mechanistic model = cost-effective method compared to field studies Scarcity of long-term evaluations at field-scale, especially for Mediterranean context

  𝑠 𝑎 = 𝑲 𝑫 𝑐 𝒔 𝒂 : adsorbed-phase concentration (kg.m -3 ) 𝑲 𝑫 : adsorption partition coefficient C : solution-phase concentration (kg.m -3 ) 𝒌 : degradation rate coefficient (s -1 ) θ 𝒓𝒆𝒇 : reference water content (m 3 .m -3 ) 𝒕 𝟏/𝟐 : half-life (s) 𝒃 : Walker's function exponent Uniform mixing of runoff water with the first soil layers 𝑀 𝑟𝑢𝑛𝑜𝑓𝑓 + 𝑀 𝑠𝑜𝑖𝑙 = ℎ 𝑤𝑎𝑡𝑒𝑟 𝑐 + 𝒛 𝒎𝒊𝒙 𝑠 𝑎 𝑴 𝒓𝒖𝒏𝒐𝒇𝒇 : pesticide mass in runoff (kg.m -2 ) 𝑴 𝒔𝒐𝒊𝒍 : pesticide mass in soil (kg.m -2 ) 𝒉 𝒘𝒂𝒕𝒆𝒓 : water level (m) 𝒛 𝒎𝒊𝒙 : depth of the mixing layer (m) 𝑠 𝑎 = 𝑲 𝑫 𝑐 𝒔 𝒂 : adsorbed-phase concentration (kg.m -3 ) 𝑲 𝑫 : adsorption partition coefficient C : solution-phase concentration (kg.m -3 ) 𝒌 : degradation rate coefficient (s -1 ) θ 𝒓𝒆𝒇 : reference water content (m 3 .m -3 ) 𝒕 𝟏/𝟐 : half-life (s) 𝒃 : Walker's function exponent Uniform mixing of runoff water with the first soil layers 𝑀 𝑟𝑢𝑛𝑜𝑓𝑓 + 𝑀 𝑠𝑜𝑖𝑙 = ℎ 𝑤𝑎𝑡𝑒𝑟 𝑐 + 𝒛 𝒎𝒊𝒙 𝑠 𝑎 𝑴 𝒓𝒖𝒏𝒐𝒇𝒇 : pesticide mass in runoff (kg.m -2 ) 𝑴 𝒔𝒐𝒊𝒍 : pesticide mass in soil (kg.m -2 ) 𝒉 𝒘𝒂𝒕𝒆𝒓 : water level (m) 𝒛 𝒎𝒊𝒙 : depth of the mixing layer (m) 2. Materials & Methods Crevoisier et al., 2021 4 calibrated pesticide parameters : 𝒕 𝟏/𝟐 , 𝒃, 𝑲 𝑫 , 𝒛 𝒎𝒊𝒙 High inter and intra-

Results

Calibration period + validation period

Relative error isn't statistically time-dependent


est destinée au dépôt

What tools for agroecological diagnosis? Towards a methodological approach with Plantain (Musa paradisiaca).

Brunise

  Deloné-Louis Jeune, Gladys Loranger-Merciris, Harry Ozier-Lafontaine, Marc Dorel, Jean-Louis Diman, Patrick Lavelle To cite this version: Brunise Deloné-Louis Jeune, Gladys Loranger-Merciris, Harry Ozier-Lafontaine, Marc Dorel, Jean-Louis Diman, et al.. What tools for agroecological diagnosis? Towards a methodological approach with Plantain (Musa paradisiaca).. 47th Annual Scientific Meeting of the Caribbean Food Crops Society, Jul 2011, Lloyd Erskine Sandiford Centre, Bridgetown, Barbados. hal-04089209

Caribbean Agroecology Networking Symposium "Productive Agrosystems and Resources Conservation in an Island Environment: an Agroecological Challenge for Caribbean Sustainable Development"

47 th Annual Scientific Meeting of the Caribbean Food Crops Society, Lloyd Erskine Sandiford Centre, Bridgetown, Barbados, July 3-9, 2011

What tools for agroecological diagnosis ?

Towards a methodological approach with Plantain (Musa paradisiaca) B. DELONÉ-LOUIS JEUNE 1,2,3 , G. LORANGER-MERCIRIS 2,3 , H. OZIER-LAFONTAINE 3 , M. DOREL 4 , J.L. DIMAN 3 , P. LAVELLE 5 1 UEH-ENS ; 2 UAG ; 3 INRA-URASTRO ; 4 CIRAD-PERSYST ; 5 IRD-CIAT E-mail : dbrunise@yahoo.fr ; gladys.loranger-merciris@antilles.inra.fr ; harry.ozier-lafontaine@antilles.inra.fr ; marc.dorel@cirad.fr ; diman@antilles.inra.fr ; patrick.lavelle@ird.fr

Measuring different parameters to establish an agroecological diagnosis linking cropping practices and plantations environmental quality


Claire Forite

Alexandre Ogisma

Brunise Deloné-Louis-Jeune

Gladys Loranger-Merciris

Harry Ozier-Lafontaine

Jean-Louis Diman

An holistic

PLANTAIN CROPPING SYSTEM TYPOLOGY

Different levels of Cavendish export cropping management deviation depending on incomes and plant management

Caribbean Agroecology Networking Symposium "Productive Agrosystems and Resources Conservation in an Island

  Environment: an Agroecological Challenge for Caribbean Sustainable Development"47 th Annual Scientific Meeting of the Caribbean Food Crops Society, Lloyd Erskine Sandiford Centre, Bridgetown,Barbados, July 3-9, 2011 An holistic approach to cropping practices, no technical support for farmers about plantain production in Guadeloupe Plantain cropping systems are mainly improved in the Cavendish production area Cropping system duration is reduced to 1 or 2 years long because of telluric pathogens' pressure -Aim: Characterize plantain cropping system diversity in order to run an agroecological diagnosis Identify the factors influencing the cropping system sustainability -Hypothesis: Plantain cropping practices are influenced by Cavendish reference cropping system systems according to the production area (type 4: out of the main Cavendish production area) -Higher yields in more intensive systems REMAINING QUESTIONS: What are the investments and benefits for each cropping system? What is cropping systems sustainability?Necrotic area calculation -Type 1: higher yields than type 4 -Type 1: better homogeneity of bunches in "blanc" variety -Type 4:"Blanc" variety seems to be more productive -Higher nematodes' infestation in type 1 -Different nematodes species combination in type 1 & 4 -Less roots' necroses in type 1


Romain Delpoux
email: romain.delpoux@insa-lyon.fr

Thomas Huguet
email: thomas.huguet@toulouse-inp.fr

Federico Bribiesca
email: federico.bribiesca-argomedo@insa-lyon.fr

Loïc Queval
email: loic.queval@centralesupelec.fr

Jean-Yves Gauthier
email: jean-yves.gauthier@insa-lyon.fr

Zohra Kader
email: zohra.kader@laplace.univ-tlse.fr

Finite element dq-model for MTPA flux control of Synchronous Reluctance Motor (SynRM)

Keywords: MTPA, flux control, FEA, dynamic model

This article focuses on the modelling of SynRM including magnetic saturation and cross-saturation to obtain accurate dynamic simulations from models obtained by Finite Element Analysis (FEA). It is shown here that in the presence of nonlinear magnetic materials, the use of flux instead of current is of interest for both simulation and control design. From this observation, dynamic flux simulation model is obtained. The accuracy of the modelling is illustrated though the design of a Maximum Torque Per Ampere (MTPA) based flux control. Simulation results, close to the real SynRM obtained from FEA are conducted to validate the development and its practical usefulness.

I. INTRODUCTION

Permanent Magnet Synchronous Motors (PMSM), using rare-earth, are considered as the main candidate in most industrial applications due to their high performance. While these machines notably make it possible to offer vehicles that do not emit exhaust fumes and therefore reduce CO 2 emissions, environmental impacts of rare earth materials and supply risks, e.g. restricted access and price volatility, impacts their cost [START_REF]Critical materials strategy[END_REF], [START_REF] Rahman | History of interior permanent magnet motors[END_REF]. The development of applications using rare-earth-free motors is today a strategic priority. In this context, SynRM technology gains more and more interest since a few years and becomes a serious competitor [START_REF] Donaghy-Spargo | Synchronous reluctance motor technology : opportunities, challenges and future direction[END_REF], [START_REF] Lee | On-line mtpa control method for synchronous reluctance motor[END_REF]. Due to technical advances in computer science, electronics and power electronics, these machines seriously compete with asynchronous machines in terms of efficiency. New designs also continually improve them [START_REF] Bao | A novel concept of ribless synchronous reluctance motor for enhanced torque capability[END_REF], these machines are now tending to compete with the PMSM [START_REF] Boglietti | Induction and synchronous reluctance motors comparison[END_REF] in various industrial applications [START_REF] Ban | Synchronous reluctance machines: Theory, design and the potential use in traction applications[END_REF], [START_REF] Kärkkäinen | Technology comparison of induction motor and synchronous reluctance motor[END_REF], due to increased efficiency and reduced weight [START_REF] Rassolkin | Efficiency map comparison of induction and synchronous reluctance motors[END_REF]. In summary SynRM contributes towards more simple, resilient and robust electric motor to provide a cred-ible ecological alternative to permanent magnet synchronous motors.

To become even more competitive, control algorithm improvement for these machines is a challenge. Indeed, under rated condition magnetic flux saturation occurs heavily, causing large nonlinearities [START_REF] Lee | On-line mtpa control method for synchronous reluctance motor[END_REF]. Maximum Torque Per Ampere (MTPA) control is important to optimize energy efficiency of the motor. Recently, MTPA control for SynRM and interior permanent magnet motor drives has received a lot of attention as the recent literature review [START_REF] Tinazzi | Classification and review of mtpa algorithms for synchronous reluctance and interior permanent magnet motor drives[END_REF] can attest. In this article it is clearly stated that the presence of magnetic nonlinearities such as saturation and cross saturation makes the use of the notion of inductance complex, due to the difficulties about the precise knowledge of the differential inductances which intervene. The development of MTPA using apparent and differential inductances was developed in [START_REF] Bonifacio | On considering saturation and crosscoupling effects for copper loss minimization on highly anisotropic synchronous machines[END_REF]. These nonlinearities make Look Up Table (LUT) based solutions the most common in the scientific literature [START_REF] Cheng | Torque feedforward control technique for permanent-magnet synchronous motors[END_REF], [START_REF] Ge | Speed range extended maximum torque per ampere control for pm drives considering inverter and motor nonlinearities[END_REF], where LUT provides current controller references. Since inductances are nonlinear, it seems interesting to focus on the flux rather than on current and thus to get rid of the problems of differential inductances. While flux is difficult to measure, it is often obtained from LUTs as function of current. Flux controller for SynRM was notably proposed for SynRM in [START_REF] Hofmann | Stator-flux-oriented vector control of synchronous reluctance machines with maximized efficiency[END_REF], [START_REF] Vagati | High-performance control of synchronous reluctance motors[END_REF] using classical linear controller. Recent works deal with the use of flux vector control to achieve MTPA this without the need for tables [START_REF] Varatharajan | Direct flux vector control of synchronous motor drives: A small-signal model for optimal reference generation[END_REF], [START_REF] Varatharajan | Direct flux vector control of synchronous motor drives: Accurate decoupled control with online adaptive maximum torque per ampere and maximum torque per volts evaluation[END_REF].

The recent works discussed in the previous paragraph highlight the interest of making flux control laws for these motors, presenting strong nonlinearity. In this paper, the use of flux as state variable instead of current which allows to get rid of the notion of differential inductances. Moreover, it is shown that a feedback linearization strategy transforms the nonlinear model into a linear one which facilitates the control design. To evaluate the new control law, it is important to propose and carry out realistic dynamic simulations. Another contribution of the paper focusses on the realization of dynamic simulation of SynRM to perform flux control from FEA based motor design. While FEA tools provide nonlinear flux versus current characteristics, dynamic simulations of these machines requires the inverse. This paper presents a methodology to perform MTPA flux control on a SynRM from FEA nonlinear map obtained using the simulation software Finite Element Method Magnetics (FEMM). This work is a first step towards a joint optimization of the motor and its control.

The paper is divided into 3 sections. Section II presents the article motivations. Section III is dedicated to the modelling strategy from FEA, including MTPA characteristics. The model is used in Section IV to illustrates the advantage of using flux dynamics coupled with nonlinear control to ensure good performance at all operating points. II. MOTIVATIONS In this section, the dynamic model of the SynRM is presented and the practical problem that motivated our research is introduced. Figure 1 shows a schematic of the two-phase four poles SynRM used in this article. It is a 2 kW machine with maximum torque of 2 N.m. Two-phase machine (and not a commonly used three-phase) means that voltages are applied to the motor in the two-phase stationary reference frame called αβ. The rotating reference dq-frame is obtained thanks to the Park rotation matrix defined as:

(•) dq = P (θ) -1 (•) αβ (1)
with

P (θ) = cos(pθ) -sin(pθ) sin(pθ) cos(pθ) ,
where θ ∈ R is the rotor mechanical angle and p ∈ N + is the pole pairs number.

The dynamic of the windings in the dq reference frame is given by the state space system:

λdq (t) = v dq (t) -Ri dq (t) -pω(t)J λ dq (t) (2)
where λ dq (t) ∈ R 2 is the stator flux linkage, v dq (t) ∈ R 2 is the stator voltage, i dq (t) ∈ R 2 is the phase current and ω(t) ∈ R is the rotor angular velocity. The matrix R ∈ R + is the phase resistance diagonal matrix. The matrix J is a rotation matrix given by J

= [0 -1] [1 0] ⊺ .
The flux-current dependence is a static nonlinear map, where the nonlinearity is introduced by saturation and crosssaturation [START_REF] Preindl | Optimal state reference computation with constrained mtpa criterion for pm motor drives[END_REF], [START_REF] Woo | Flux saturation model including cross saturation for synchronous reluctance machines and its identification method at standstill[END_REF]. The apparent inductance is expressed as a nonlinear function ℓ dq (i dq ) : R 2 → R 2 yielding to the model:

λ dq = ℓ dq (i dq )i dq .
(

) 3
Electric machines produce electromagnetic torque τ em ∈ R when the current interact with the magnetic field, i.e. flux. The torque equation is:

τ em = p i ⊺ dq J λ dq (4)
Remark. Note that for three-phase motors the produced torque in the dq frame is multiplied by a factor 3/2 using Clarke transformation from three-phase abc to two-phase αβ coordinates. ⌟ Usually, since current is measured, it is interesting to look at the dynamic of the current rather than the flux, from (2) one has:

ℓ ′ dq (i dq (t))i dq (t) + ℓ dq (i dq (t))
di dq (t) dt = v dq (t) -Ri dq (t)pω(t)J ℓ dq (i dq (t))i dq (t).

(

) 5
where the term ℓ ′ dq (i dq ) is referred as differential inductances. This adds complexity, pushing us to use the flux as a state variable [START_REF] Ban | Synchronous reluctance machines: Theory, design and the potential use in traction applications[END_REF].

On the other hand, the use of model [START_REF] Ban | Synchronous reluctance machines: Theory, design and the potential use in traction applications[END_REF] for simulation requires the knowledge of the current from the flux, The inversion of the nonlinear function ℓ dq (i dq ) is required: The mathematical dynamic model of the motor being introduced in the previous section, this section is dedicated to obtaining the nonlinear flux-current map (3) of the SynRM including magnetic saturation and cross-saturation obtained thanks to finite element analysis (FEA). To perform dynamic simulation of the motor model, it is required to inverse the map given in equation ( 6), procedure detailed in this section. Finally MTPA trajectory is derived.

i dq = λ -1 dq (ℓ dq (i dq )i dq ). (6

A. FEA analysis

The FEA is conducted using FEMM software on a single pole of the machine using Anti-Periodic and Air Gap boundary conditions [START_REF]Rotor motion using an (anti)periodic air gap boundary condition[END_REF]. Fig. 2 shows the geometry and the mesh used. The different materials are air and copper (µ = µ 0 ) and M-19 Steel which is a nonlinear laminated silicon iron soft magnetic material. The simulation has been run for different couple (I s , γ) defined as follows:

I s = ||i dq || (7) γ = arg(i dq ) (8)
Fig. 2. Geometry and mesh of the FEA using FEMM software.

I s and γ take each 10 values in the interval [0 10] and [0 π/2] respectively, for a total of 100 combinations. For each combination, the simulation has been run for 10 mechanical positions of the rotor varying from 0 to 45 degree. This range of rotor positions is enough to capture a full period of the machine behavior. The FEA simulation then returns the average torque and average dq flux over the 10 mechanical positions for each (I s , γ) combination equivalent to different (i dq ) combinations.

The nonlinear characteristics that links current to flux modelled equation (3) obtained from FEA are plotted Fig. 3.

To highlight the impact of cross-saturation, i.e., the impact of current i k with k ∈ d, q on λ l for l ̸ = k, Fig. 4 represents the flux λ dq for minimal and maximal current.

Remark. Fig. 4 clearly shows saturation at low current on q axis due to flux barrier while saturation on the d axis appear for higher current due to larger quantity of iron in this axis. ⌟

B. Simulation model

From the previously obtained maps, simulation of the dynamic behavior of the motor flux modelled in (2) requires the inverse map given in equation ( 6 The inversion has been obtained numerically using the Matlab function fmincon which finds minimum of constrained nonlinear multivariable functions. For each pair λ dq , the objective was to find the pair i dq which minimize the objective function:

λ FEMM dq -ℓ dq (i dq )i dq ⊺ W λ FEMM dq -ℓ dq (i dq )i dq , (9)
with W = diag w 2 1 , w 2 2 and w 2 = 3w 1 in order to normalize the amplitudes factor of the different flux components λ dq as represented Fig. 4. Multiple initial conditions were given to the optimization algorithm through a multi-start procedure to avoid local minima in the function. The results for the nonlinear inversion is plotted on Fig. 6 For this simulation, the remaining parameters are p = 2, R = 0.52 Ω.

C. MTPA-LUT derivation

From the characteristics represented Fig. 3 and the electromagnetic torque given equation ( 4) one can obtain the torqueflux characteristic represented on Fig. 7. It illustrates the torque levels as a function of the flux which implies that a reference torque can be produced from an infinite combination of flux λ dq . The figure also highlights the necessity of producing larger flux on the d axis than on the q axis to produce torque.

The MTPA objective is to minimize the Joules losses, given by:

P Joules = Ri ⊺ dq i dq . ( 10
)
The criterion therefore consists in minimizing the norm of the current i dq i.e. ||i ⋆ dq || which are computed by minimizing the current magnitude. One poses:

i dq = I s cos(γ) sin(γ) (11)
where I s is the current magnitude and γ is the current angle [START_REF] Varatharajan | Direct flux vector control of synchronous motor drives: A small-signal model for optimal reference generation[END_REF]. The MTPA trajectory coincides with: which can be computed numerically from the LUT given Figure 7. The numerically computed MTPA is represented on the torque map Fig. 8. On the Fig. 9 representing the derivative of torque, MTPA corresponds, indeed to the bottom of the valley.

dτ em dγ = 0, (12)

IV. LUT BASED FLUX CONTROLLER FOR MTPA

The flux control objective is the regulation of the flux with reference flux provided by the MTPA trajectory represented in red on Fig. 7. Note that, in spite of Park transformation, the flux model given equation (2) remains non-linear. However, having access to the current measurement, speed measurement and flux reconstruction from LUT, it is possible to apply a feedback linearization. The proposed control algorithm is thus composed by two parts, a feedback linearization to compensate the nonlinearities and a feedback control to impose the closed loop dynamic. Similar approach can be found applied to the current controller of a permanent magnet motor in [START_REF] Bodson | High performance nonlinear feedback control of a permanent magnet stepper motor[END_REF]. Recent work proposes to do without LUTs [START_REF] Varatharajan | Direct flux vector control of synchronous motor drives: A small-signal model for optimal reference generation[END_REF], [START_REF] Varatharajan | Direct flux vector control of synchronous motor drives: Accurate decoupled control with online adaptive maximum torque per ampere and maximum torque per volts evaluation[END_REF]. The overall control scheme is represented Fig. 10. Note that the reference torque can be directly a torque reference or a reference computed from an outer speed control loop. This outer loop is out of the scope of this article. a) Feedback linearization: Choosing

v dq = u dq + Ri dq + pωJ λ dq , (13)
equation ( 2) with (13) becomes:

λdq = u dq (14)
It results in a time invariant linear system, in which the fluxes λ d and λ q are independent, leading to pure integrator. This is why this linearization is often called "decoupling".

Remark. The equivalent feedback linearization applied on the current dynamic (5) would require much more complicated calculation to become linear. ⌟ b) Dynamic control: One of the control objectives is to ensure a zero static error, that is why an integral action is added in the feedforward path between the error comparator and the plant. This control strategy generally called type 1 Servo system, since the plant has no integrator [14, p. 743]. The two flux equations being independent, for k ∈ {d, q}, one has:

ε k = (λ # k -λ k )dτ, (15)
the integrator output and then:

λk = u k , εk = λ # k -λ k , (16)
with the control:

u k = -k p λ k -k I ε k , (17)
with k p and k I the control gains to be tuned using Ackerman's formula to impose the dynamics described by a second order system with the classical characteristic polynomial equation

P (s) = s 2 + 2ζω n s + ω 2 n , (18)

V. CONCLUSION AND PERSPECTIVES

In this article was studied the modelling of SynRM including magnetic saturation and cross-saturation to obtain accurate dynamic simulations from models obtained by FEA. It was shown that in the presence of nonlinear magnetic materials, the use of flux instead of current is of interest for both simulation and control design. From this observation, dynamic flux model was proposed and an MTPA based flux control developed. Simulation results, close to the real SynRM obtain from FEA validate our development and its practical usefulness.

As future work, real-world experiments are on the way to evaluate the gap between simulation and experimentation and validate the methodology under real conditions, including the inverter nonlinearities such as dead-time effect and switching device voltage drop among others. Theoretical developments are also underway to propose a flux observer that would make it possible to do away with LUTs.

Fig. 1 .

 1 Fig. 1. Schematic of the SynRM.

  ) This article is dedicated to the flux dynamic modeling of a SynRM and the design of an MTPA based flux control based on data from FEA. The flux control allows the apply a simplified feedback linearization strategy and deal with linear flux equation only despite the nonlinearity (3). It highlights the usefulness of focusing on flux rather than current. The control objective is to produce the desired electromagnetic torque (4) from a reference torque τ # , with the aim of minimizing Joule losses at steady state. It is equivalent to compute flux references λ # dq from τ # . III. FLUX AND MTPA CHARACTERISTICS FROM FEA

Fig. 3 .

 3 Fig. 3. Nonlinear flux-current characteristic obtained from FEMM.

Fig. 4 .

 4 Fig. 4. Representation of the minimal and maximal flux λ dq with respect to current.

  ) has represented on the simulation schematic given Fig 5.

Fig. 5 .

 5 Fig. 5. Simulation model of the flux dynamics (2) .

Fig. 6 .

 6 Fig. 6. Nonlinear current-flux characteristic obtained from numerical inversion.

Fig. 7 .

 7 Fig. 7. Torque-flux characteristic and MTPA trajectory (red).

Fig. 8 .

 8 Fig. 8. Torque-flux characteristic and MTPA trajectory, obtained from FEMM.

Fig. 9 .Fig. 10 .

 910 Fig. 9. The derivative of torque with respect to the current angle and the MTPA trajectory obtained with FEMM.

  where, the parameters ω n and ζ are the desired closed loop pulsation and damping coefficient.The control objective is the obtaining of closed loop flux dynamic with 5% overshoot and time response equal to 3 ms leading to the characteristic polynomial given in[START_REF] Tinazzi | Classification and review of mtpa algorithms for synchronous reluctance and interior permanent magnet motor drives[END_REF] with w n = 100 rad/s and ζ = 0.7 and control gain k p = 140 and k i = -10000. The desired torque τ # is defined as a stair sequence for 0 to 1.8 Nm with a step of 0.2 Nm. Through the torque-flux characteristics, the flux reference is computed. The simulation are represented on Fig.11. The figure shows that indeed, for regular torque steps, the flux steps amplitudes varies, highlighting the nonlinear nature of the system. However, 5% time response for the different amplitudes remains equal, as represented on Fig.12were the step response for each step response errors are superposed, validating the linear behavior of the closed loop system. Finally, the resulting flux and torque are reported on the torque-flux characteristic plotted on a new figure, Fig.13, for clarity. The figure highlights again the performance of the proposed control strategy and shows that at steady state, the MTPA trajectory plotted in red is perfectly covered by the references and the controlled flux.

Fig. 11 .

 11 Fig. 11. Chronograph of the simulation results for the proposed approach.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Chronograph of the simulation errors for the proposed approach, each color representing a step of Fig. 11.