Michel Latroche

Dr Annick Percheron-Guégan

Prof Volodymyr Yartys
email: volodymyr.yartys@ife.no

Dr. Fermin Cuevas
email: fermin.cuevas@cnrs.fr

Hydrogen-based energy: Status and prospects. Special Issue dedicated to the memory of Dr

Michel Latroche, internationally recognized as an outstanding solid-state chemist and a pillar of the hydride community, left us suddenly. This special issue of the Journal of Alloys and Compounds entitled "Hydrogen-based energy: Status and prospects" is dedicated to his memory. It aims at reviewing the role of hydrides as key materials for energy storage and unveiling recent advances and novel perspectives in this long-standing and ever-challenging research field. The 13 selected papers included in this issue present original results and give comprehensive reviews, and were prepared by Michel Latroche's close collaborators. Many of them, as shown in the group photo below, attended the specially organized International workshop on Hydrides and Energy Storage that took place at the East Paris Institute of Chemistry and Materials Science (ICMPE), Thiais, France, on June 13, 2022. This workshop, organized by the group Interaction of Hydrogen and Matter at East Paris Institute of Chemistry and Materials Science (ICMPE), comprised high-level scientific presentations in the fields of solid-state chemistry, hydrogen storage and electrochemical energy storage. The international research community, ICMPE colleagues, CNRS and UPEC officers enjoyed active participation of the speakers. Michel's dearest family and relatives attended the workshop and were pleased to observe a great appreciation of his work by the French and international scientists, his colleagues and friends.

His main scientific interests were related to metallic hydrides. His research works dealt with structural, thermodynamic and electrochemical properties of rare earths and transition metal-based intermetallic compounds and their hydrides, as well as hybrid intermetallic alloyporous materials. His research activities were devoted to the fundamental aspects of metallic hydrides as related to their applications, such as negative electrodes of the Nickel-Metal Hydride (Ni-MH) and Li-ion batteries or storage of hydrogen gas for the use in fuel cells. He published over 270 articles in peer-reviewed journals and filled 11 patents.

Major scientific achievements of Michel Latroche were gained during the investigations of intermetallic compounds as negative electrodes of the Ni-MH batteries [1,2], comprising of the studies of AB 5 [3,[START_REF] Latroche | Structural studies of LaNi 4 CoD 6[END_REF], AB 2 [START_REF] Joubert | Improvement of the electrochemical activity of Zr-Ni-Cr Laves phase hydride electrodes by secondary phase precipitation[END_REF], AB [START_REF] Cuevas | Influence of polymorphism on the electrochemical properties of[END_REF] and AB 5 -AB 2 superlattice alloys [7][START_REF] Monnier | Identification of a new pseudo-binary hydroxide during calendar corrosion of (La, Mg) 2 Ni 7 -type hydrogen storage alloys for Nickel-Metal Hydride batteries[END_REF][9]. He succeeded in determining their crystallographic properties and reaction mechanisms by using manyfold diffraction techniques [10][11][START_REF] Latroche | Mechanistic and Kinetic Study of the Electrochemical Charge and Discharge of La 2 MgNi 9 by in Situ Powder Neutron Diffraction[END_REF][START_REF] Latroche | Structural properties of two deuterides LaY 2 Ni 9 D 12.8 and CeY 2 Ni 9 D 7.7 determined by neutron powder diffraction and X-ray absorption spectroscopy[END_REF][START_REF] Latroche | Influence of stoichiometry and composition on the structural and electrochemical properties of AB 5+y -based alloys used as negative electrode materials in Ni-MH batteries[END_REF][START_REF] Latroche | Intrinsic behaviour analysis of substituted LaNi 5 -type electrodes by means of in-situ neutron diffraction[END_REF], often in-situ or operando, and with particular emphasis on the contribution of the rare earths chemistry to the observed behaviors [START_REF] Iosub | Optimisation of MmNi 5-x Sn x (Mm=La, Ce, Nd and Pr; x=0.27[END_REF][START_REF] Latroche | Crystallographic and hydriding properties of the system La 1-x Ce x Y 2 Ni 9 (x Ce =0, 0.5 and 1)[END_REF][START_REF] Latroche | Structure determination of Y0.95Ni2 by X-ray powder diffraction[END_REF][START_REF] Paul-Boncour | In situ XAS study of the hydrogenation of AB 5 compounds[END_REF]. Many of these intermetallic compounds and related derivatives were also investigated as reversible solid-state hydrogen storage materials when extending the studies to TiFe [START_REF] Dematteis | Fundamental hydrogen storage properties of TiFe-alloy with partial substitution of Fe by Ti and Mn[END_REF][START_REF] Guéguen | Influence of the addition of vanadium on the hydrogenation properties of the compounds TiFe 0.9 V x and TiFe 0.8 Mn 0.1 V x (x=0, 0.05 and 0.1[END_REF], V-type alloys [START_REF] Challet | Hydrogenation properties and crystal structure of the single BCC (Ti 0.355 V 0.645 ) 100-x M x alloys with M=Mn[END_REF][START_REF] Massicot | Hydrogenation properties of Fe-Ti-V bcc alloys[END_REF], Mg-based compounds [START_REF] Janot | Influence of crystallinity on the structural and hydrogenation properties of Mg 2 X phases (X=Ni, Si, Ge, Sn)[END_REF][START_REF] Yartys | Hydrogenassisted phase transition in a trihydride MgNi 2 H 3 synthesized at high H 2 pressures: Thermodynamics, crystallographic and electronic structures[END_REF] and tritium storage [START_REF] Evin | Investigation by STEM-EELS of helium density in nanobubbles formed in aged palladium tritides[END_REF]. Hydrogen storage using getter alloys was also considered [START_REF] Janot | Development of a hydrogen absorbing layer in the outer shell of high pressure hydrogen tanks[END_REF][START_REF] Prigent | Hydrogen trapping properties of Zr-based intermetallic compounds in the presence of CO contaminant gas[END_REF].

More recently the research activities of Michel Latroche extended beyond intermetallic compounds and their hydrides. He led pioneering works on hydrogen storage in metal-organic frameworks [START_REF] Latroche | Hydrogen Storage in the Giant-Pore Metal-Organic Frameworks MIL-100 and MIL-101[END_REF] and hybrid materials formed by metal nanoparticles embedded in porous scaffolds [START_REF] Cachet-Vivier | Hydrogen sorption properties of Pd nanoparticles dispersed on graphitic carbon studied with a cavity microelectrode[END_REF][START_REF] Campesi | Hydrogen storage properties of Pd nanoparticle/carbon template composites[END_REF][START_REF] Latroche | Structural Instability in R 1-x Ni 2 Compounds and Their Hydrides (R = Y, Rare Earth)[END_REF][START_REF] Zlotea | Pd Nanoparticles Embedded into a Metal-Organic Framework: Synthesis, Structural Characteristics, and Hydrogen Sorption Properties[END_REF][START_REF] Zlotea | Role of nanoconfinement on hydrogen sorption properties of metal nanoparticles hybrids[END_REF]. His interests were also focused on the complex hydrides and Mg-based nanocomposites first synthesized via mechanochemical synthesis routes [START_REF] Cuevas | Synthesis, structural and hydrogenation properties of Mg-rich MgH 2 -TiH 2 nanocomposites prepared by reactive ball milling under hydrogen gas[END_REF][START_REF] Huot | Mechanochemical synthesis of hydrogen storage materials[END_REF][START_REF] Rizo-Acosta | Hydrides of early transition metals as catalysts and grain growth inhibitors for enhanced reversible hydrogen storage in nanostructured magnesium[END_REF][START_REF] Zhang | Highlighting of a Single Reaction Path during Reactive Ball Milling of Mg and TM by Quantitative H 2 Gas Sorption Analysis To Form Ternary Complex Hydrides (TM = Fe, Co, Ni)[END_REF] and then through their applications in Li-ion batteries as anode materials [START_REF] Monnier | Hydrides compounds for electrochemical applications[END_REF][START_REF] Sartori | Metal hydrides used as negative electrode materials for Li-ion batteries[END_REF][START_REF] Teprovich | Li-Driven Electrochemical Conversion Reaction of AlH 3 , LiAlH 4 , and NaAlH 4[END_REF][START_REF] Zhang | XAS investigations on nanocrystalline Mg 2 FeH 6 used as a negative electrode of Liion batteries[END_REF] and as electrolytes [START_REF] Ha Dao | Improvement of the ionic conductivity on new substituted borohydride argyrodites[END_REF]. These studies resulted in the elaboration of the first concept of solid-state Li-ion battery using hydride compounds both as anode and electrolyte [START_REF] Latroche | Full-cell hydride-based solid-state Li batteries for energy storage[END_REF][START_REF] López-Aranguren | An all-solid-state metal hydride -Sulfur lithium-ion battery[END_REF]. Last but not least, he also made important contributions to the development of Si-containing intermetallic composites as high-capacity anodes of Li-ion batteries [START_REF] Azib | Ni-Sn intermetallics as an efficient buffering matrix of Si anodes in Li-ion batteries[END_REF][START_REF] Edfouf | Nanostructured Si/Sn-Ni/C composite as negative electrode for Li-ion batteries[END_REF].

Michel Latroche widely disseminated his research outcome at French and international forums and networks through high-level seminars and conferences as well as in the university lectures and in the media communication. He regularly participated in the national (Matériaux, Colloque Energie, JDN, GFECI) and international (E-MRS, ICNS/ECNS, ICFE) conferences. Remarkably, he attended all International Symposia (from 1990) and Gordon Research Conferences (from 1993) on Metal-Hydrogen Systems.

Michel Latroche was a creative, sincere, tireless and passionate scientist, and was gifted with extraordinary competence and capacity for creative work. He was an insightful researcher who was always ready to share his knowledge with colleagues and students. His main research collaborators at the Laboratoire de Chimie Métallurgique des Terres Rares in France (J.-C. Crivello, F. Cuevas, J.-M. Joubert, E. Leroy, J. Monnier, V. Paul-Boncour, A. Percheron-Guégan, J.-X. Zhang, C. Zlotea) and hydride researchers abroad (M. Baricco, R.-C. Bowman, R. Černý, M. Dornheim, B.C. Hauback, M. Hirscher, J. Huot, T.R. Jensen, L.J. Jiang, P. de Jongh, P.H.L. Notten, V.A. Yartys, K. Yvon) strongly enjoyed his enthusiasm and research skills. He created and preserved a large network of collaborators extending all over the world (Australia, Canada, China, many European countries, India, Japan, Marocco, Tunisia, USA). He deserved a recognition as a stalwart of the metal-hydride community, and more generally of materials research for sustainable energy applications. Michel's legacy will remain in our memories to guide a progress for a brighter and better energy future.

Critical reading by Jean-Marc Joubert and Valérie Paul-Boncour of Michel Latroche's biography is acknowledged.

List of contributions to the to the Special Issue of Journal of Alloys and Compounds. "Hydrogen-based energy: Status and prospects". Editors Prof. V.A. Yartys and Dr. F. Cuevas.

After a post-doctoral position at ) when working in the group of Prof. J. Ibers, in 1989 he joined the CNRS in Meudon (FranceExecutive Officer at the CNRS French Research Network on Hydrogen Energy, Expert member at the International Energy Agency IEA-HIA Task 40, Vice-Chair of the Gordon Research Conference Hydrogen-Metal Systems and Secretary General of the International Steering Committee of the International Symposia on Metal-Hydrogen Systems, the world largest conference on hydrogen interaction with materials.


domain: chim spi

text:
Cyriaque Amerein

Utsab Banerjee

Zhenfeng Pang

Wenqing Lu

Vanessa Pimenta

Kong Ooi Tan

Kong Ooi Tan
email: kong-ooi.tan@ens.psl.eu

In-House Fabrication of 1.3 to 7 mm MAS Drive Caps using Desktop 3D Printers

Keywords: Magic-Angle Spinning, Solid-State NMR, 3D Printing, Additive Manufacturing 2

Introduction

Solid-state NMR is a powerful characterization technique that has been widely applied to study solid-state samples ranging from small molecules, biological macromolecules, to inorganic materials. 1 Although solid-state samples usually have broad peaks due to anisotropic interactions such as dipolar couplings and chemical-shift anisotropy, they can be averaged out by performing magic-angle spinning (MAS) experiments, 2,3 where the sample is spun at an angle of 54.74 with respect to the static magnetic field. It is essential that the MAS frequency (r) should be sufficiently larger than the size of the anisotropic interactions to achieve an efficient line-narrowing effect, i.e., the rotors can spin up to r=160 kHz depending on their size (Table 1). 4,5 To achieve such fast MAS frequencies without significantly heating the NMR sample, the rotors are pneumatically spun with gas to reduce frictional heating. Besides achieving faster MAS frequencies, maintaining a stable spinning frequency (usually within a deviation of few Hz) is important over the course of experiments, especially for 2D or 3D NMR experiments that could last several days or week(s).

Hence, the choices of material for both the rotor and MAS drive cap are important as they must be mechanically rigid (high tensile or flexural strength), and easily fabricated-especially the drive cap with an intricate spiral feature. Some of the materials that fulfill these criteria are ZrO2, Macor, boron nitride, Kel-F, Vespel, and Torlon. Among these choices, ZrO2 and Vespel are popular materials for the drive cap because ZrO2 has superior mechanical strength and a low thermal expansion coefficient, which is preferred for low-temperature dynamic nuclear polarization (DNP) experiments. [6][7][8] However, machining ZrO2 is challenging and, hence, expensive, i.e. a 3.2mm ZrO2 cap costs ~1k USD (CortecNet). Although the Vespel drive cap is more affordable (~130 USD for 3.2 mm), it is still a non-negligible expense because the delicate fins on the drive cap wear out easily if they are mishandled or after repeatedly fitted/removed from the rotor. These effects are noticeably noted for smaller rotors, i.e., the 0.7 mm drive cap is usually for one-time use only, and it has to be replaced with a new piece when packing new samples into the same rotor. To minimize the expenditure on these consumables (MAS drive caps), it has been recently demonstrated that the MAS drive caps can be fabricated using relatively low-cost 3D printing technology. [START_REF] Xu | Schmedt auf der Günne[END_REF]10 There are two main categories of 3D printing technology: fused deposition modelling (FDM) and stereolithography (SLA). Although the FDM method has, in general, a poorer resolution, it is more affordable and easier to use relative to the resin-based SLA method. Both 3D printing technologies have been exploited in various NMR applications, ranging from fabricating MAS sample eject systems, 11 RF coils, [12][13][14][15] NMR sample tubes, 16 parahydrogen generator, 17 NMR sample degassing apparatus, 18,19 , NMR probeheads, 20 etc. Besides these examples, Xu et al. have recently demonstrated that it is possible to fabricate MAS stator and drive caps using FDM, but only for > 3.5 mm system. [START_REF] Xu | Schmedt auf der Günne[END_REF] High-resolution SLA technology has also been exploited for constructing MAS spinning system. Notable examples include the fabrication of > 4 mm spherical-rotor stators using a benchtop 3D printer by the Barnes' group, 21,22 and a 3.2 mm cylindrical-rotor stator and drive caps produced by dedicated manufacturing companies reported by Banks et al. 10 Although outsourcing the manufacturing process can save labour resources, we note that the quality of the parts varies across different batches. This could be because the printing and/or post-processing steps performed by the company are not necessarily the same for each batch. Consequently, the lack of consistent results could lead to a more timeand cost-consuming process. Moreover, the same publication commented that the benchtop 3D printer (Form3 with 25 µm XY resolution) was tested to be incapable of producing the 3.2 mm drive caps. 10 In this publication, we re-examine this statement by showing that not only the Form3+ 3D printer can fabricate robust 3.2 mm drive caps, but also smaller 2.5-and 1.3-mm drive caps (with slightly compromised performances). In particular, the 3.2 mm 3D-printed drive caps (Fig. 1) can spin up to 𝜈 r = 22 kHz  1.3 Hz (Table 1), which has similar performances as the commercial versions, but only costs < 1 US cents and 5 minute/piece to be 3D-printed (assuming a simultaneous printing of 100 pieces in a single print job). Additionally, the in-house fabricated 3D-printed drive caps exhibit excellent spinning stability that is sufficient for practical multidimensional MAS NMR experiments.

Experimental Methods

3D Printing and Post-Processing Methods

The drive caps were designed in Autodesk Inventor (Autodesk Inc, San Rafael) following a similar design described in Banks et al. 10 The drive caps were printed using standard Clear v4 resin in a Form 3+ (Formlabs) printer with a 25 µm XY resolution and layer thickness. The 3D printer costs ~4 k USD, and the resin costs ~140 USD/litre. The printing supports were placed on the bottom of the stem with a contact point size of 0.3 mm. It takes ~4 hours to simultaneously 3D-print 100 pieces of drive caps in a single print job. Note that the build platform of the 3D printer can accommodate more than 100 pieces in a single print (if needed) without significantly increasing the printing hours. After the printing job finished, the parts were first rinsed in an isopropanol-water mixture (9:1 by volume ratio) to remove the uncured resin and irregularities on the surface. Then, the supports were removed and rinsed again before being cured under UV light (Elegoo Mercury) for 1 h (2x30 min, with another rinsing step between the two curing sessions) to enhance the mechanical properties. Note that the caps should be completely dry after every rinsing step. It is known that the SLA-printed parts gradually shrink in size over time, and the long curing time (1 hr) is a crucial step to accelerate the shrinking process and, hence, more stable in the long term. To account for the change in dimension after the post-processing steps, we have iteratively compensated for the changes by redesigning and reprinting the parts. In addition to the Form 3+ 3D printer, we have also tested another benchtop 3D printer (Elegoo Saturn), which has a lower XY resolution (50 µm). We have used the Elegoo 3D printer to print 4 mm caps (with Elegoo standard grey resin), and they show similar performances as the commercial (Bruker) version.

Artefact Compensation by Iterative Design Modification

Note that the dimensions of the 3D-printed pieces could be slightly different from the computeraided design (CAD) due to the properties of the resins, printing environment (temperature), postprocessing steps, printing orientations, etc. Nevertheless, some of these undesired changes could be compensated by minor modifications (often iterative) in the CAD. An example of this is the 'overhang' feature that occurs when the resin tends to bulge downward due to gravity (Fig. 2a).

The artefact has resulted in a small gap between the drive cap and the rotor and, hence, compromised the spinning performances, i.e., it does not spin up beyond 22 kHz. Although this feature can be partially compensated by reversing the printing orientation, it, unfortunately, affects the form of the fins (Fig. 2b). Thus, we introduced a concave curvature to the horizontal slab (Fig. 2c). Consequently, the printed part (Fig. 2d) did result in a better fit and a higher spinning frequency to ~26 kHz. Besides that, we have also made the fins on the 1.3 mm drive cap sharper (Fig. 3d ande) to compensate for the broader-than-expected fins (Fig. 3a andb) due to the finite size of the laser beam used in the Form 3+ 3D printer (85 µm). The corrected drive caps (Fig. 3e) were able to achieve the same spinning frequency with less drive gas (vide infra, Fig. 6a), and the maximum r was improved from 37 to 45 kHz. Additionally, we would like to highlight that the diameters of the stem (the component that fits into the rotor) and the slab (Fig. 2c) of the drive caps require very tight tolerances, i.e., they must be fabricated within a ~ 10 µm accuracy for optimal performances. At last, we emphasize that such iterative compensation is possible only because the 3D printing protocol yields very consistent and reproducible results. We have fabricated many batches of the drive caps over several months, and they yield very similar results.

MAS Experiments using 3D-Printed Caps

The 3D-printed caps were fitted into standard ZrO2 rotors for MAS experiments in Bruker MAS probes or a benchtop spinning station at ambient temperature. A Bruker MAS II unit was used to regulate the drive and bearing gas pressure to achieve a stable spinning frequency. We have tested at least 10 pieces of each design for at least 15 minutes to evaluate the MAS performance. mode were logged using TopSpin (Bruker). We have tested the 3.2 mm drive caps on both ZrO2 and sapphire rotors. All zirconia rotors fitted with 3D-printed caps spun > 20 kHz without major issues. However, the sapphire rotor crashed when spun above 18 kHz (rated for 15 kHz at RT).

For the 1.3 mm drive caps printed with an industrial 3D printer (Boston Micro Fabrication (BMF)),

we have tested only 2 out of the 3 ordered pieces. The 13 C-15 N L-Isoleucine sample was purchased from CortecNet and used without further purification. For the Metal-Organic Framework (MOF-801) sample, 23 the MOF was synthesized through an established procedure and washed with D2O using ultrasounds. 23 The D2O washing step was performed to promote an exchange of the hydroxyl proton (Zr-OH) with deuterium, which helps unambiguously assign the Zr-O 1 H NMR peak observed in an H2Owashed MOF sample. The D2O-washed MOF sample was then activated in a vacuum drying oven for 12 hours at 100 °C to remove excess solvent. To minimize the exposure of the dried MOF sample to the moist in the atmosphere, the sample was packed into a 3.2 mm rotor closed with a 3D-printed drive cap sealed with silicone-filled (Bluestar CAF 1) groove inside an argon-filled glove box.

NMR Samples and Experiments

The NMR spectra were acquired using an 800 MHz spectrometer equipped with an Avance Neo console (Bruker). We have used either a 1.3 mm, 2.5 mm, or 3.2 mm (E-free) triplechannel MAS probe to perform MAS NMR experiments. The 1 H and 13 C chemical shifts were referenced to the adamantane signal at 38.5 ppm on the tetramethylsilane (TMS) scale. The RFDR experiment (Fig. 4a) was performed using the 1.3 mm probe at 45 kHz MAS frequency with a ~ 2 ms RFDR mixing time. The DARR experiment (Fig. 4b) was performed using the 2.5 mm probe at r = 20 kHz with a 100 ms DARR mixing time. The HETCOR spectra (Fig. 4c & d) were acquired using a 3.2 mm probe at r = 10 kHz. The experimental time for each 2D

experiment is ~2 hr. The magic angle and field B0 homogeneity were optimized with KBr and adamantane, respectively. The low-temperature MAS experiment (Fig. 7) was performed using a 3.2 mm HXY MAS DNP probe (Bruker). The sapphire rotor was packed with MOF-801, the surface of which was wetted with solutions of DNP juice (d8-glycerol: D2O: H2O in 6:3:1 by volume) doped with 5 mM TinyPol. We filled the groove on the 3D-printed drive cap with red silicone sealant so that the cap does not become loose as it shrinks at low temperatures. 11 Prior to the MAS experiment, the rotor was immersed in liquid nitrogen at ~ 77 K to ensure that the 3D-printed cap remained intact and well-fitted to the rotor. We initially planned to perform DNP experiments with the 3.2 mm system, but the gyrotron was not operational during the experiment. Nevertheless, we performed conventional NMR experiments at low temperatures without microwaves, which allows the sample to reach a colder temperature (~105 K) due to the absence of microwave heating. The stator temperature was maintained at ~ 109 K (measured at stator after the mixing of the variable temperature (VT), bearing, and drive gas) by regulating the temperature of the three spinning gases at the probe base to be ~95 K. Hence, we estimated the actual sample temperature to be ~100 K.

Results and Discussions

A statistical summary of the MAS tests performed on the 1.3 to 7 mm drive caps is tabulated in Table . 1. The max r is defined as the maximum spinning frequency we have tested, and it is intentionally chosen to be slightly below the specification of the commercial (Bruker) caps to avoid rotor-crash incidents. We found that the in-house 3D-printed drive caps for sizes  3.2 mm have almost equivalent MAS performances as the commercial caps, while the cost of 3D printing is almost negligible. Nevertheless, the smaller 3D-printed drive caps (1.3 mm and 2.5 mm) yield less satisfactory spinning performances than the machined caps. Hence, we improvised the original CAD to compensate for the printing inaccuracies (vide supra), which moderately improved the maximum r to > 70 % of those specified for the commercial caps, i.e., the 2.5-and 1.3-mm 3Dprinted caps can spin up to 26 and 46 kHz, respectively. Despite the lower max r, the 3D-printed caps still achieve excellent spinning stability ( 2 Hz), which is practical for MAS NMR experiments.

To further assess the transfer efficiency of the momentum from the gas to the printed drive, we have recorded the spinning profiles (Fig. 6) of the smaller caps as a function of the drive gas pressure. We noted that the 3D-printed 1.3 mm caps are less efficient and require more drive gas to reach the same spinning frequency as the commercial caps. 24,25 We hyphothesized that the 1.3 mm caps could spin faster if they could be more accurately made. To test our hyphothesis, we have 3D-printed the 1.3 mm caps using a state-of-the-art industrial 3D printer (BMF) with HTL resin, which has a ~12x higher XY resolution (2 µm) than our desktop Form 3+ printer. Not only the BMF cap exhibits sharper fins (circled in Fig. 3f) than those made by Form 3+ (Fig. 3e), but also those machined Vespel caps (Fig. 3c). Indeed, those industrial-printed caps were able to achieve r = 60 kHz  3 Hz (Fig. 5b). Note that we have only tested two out of three available pieces. Both pieces achieve similar spinning frequencies with the same bearing and drive gas, but the second piece broke when spun above 50 kHz. This issue could be mitigated if harder materials (alumina ceramics or HT 200 from BMF) are used for printing the caps, but they are not in-house fabricated and are beyond the scope of this work. Although these industrial 3D-printed caps cost > 20 USD / piece (for a large-quantity order), they are still an order of magnitude cheaper than the machined Vespel caps.

On the printing consistency, which we defined as the ratio of the number of pieces that spin to the total number of printed parts, we have achieved very consistent results, i.e., all printed drive caps can be reliably reproduced with > 90 % consistency. Note that it is necessary to spin-test the printed drive caps  2.5 mm, as the defective drive caps cannot be easily identified with visible features. For  3.2 mm drive caps, they are remarkably near 100 %. For the most used 3.2 mm system, we have repeatedly fabricated more than 100 pieces over few months using the same optimized protocol, and they have all worked. We have also tested both freshly printed and several-month-old pieces, and they yield equal performances. It is noteworthy to be aware that we did not reuse any old caps because the cost is almost negligible.

To assess if the 3D-printed drive caps are practical for long MAS NMR experiments, we have performed various 2D NMR correlation experiments on the 1.3 mm, 2.5 mm, and 3.2 mm drive caps on standard compounds. Among these experiments, the RFDR dipolar recoupling 1.

experiment performed on the 1.3 mm rotor requires the rf pulses to be rotor-synchronized to yield an efficient correlation between the 13 C peaks. 26 The fact that the 2D spectra show excellent linewidths and all the expected cross peaks, we conclude that the homebuilt 3D-printed drive caps are practical for actual MAS NMR experiments.

In addition to the standard drive cap designs, we also prototyped new models that could be useful for special NMR applications. For instance, we have added a groove to the stem (Fig. 2c) so that the rotor could be sealed using epoxy or silicone sealant, which is useful for studying airor moisture-sensitive NMR samples. As proof of principle, we have applied such a protocol on a MOF-801 sample packed in a 3.2 mm rotor (see Section 2.4). MOF-801 is built-up from Zr6O4(OH)4 oxo-clusters and fumaric acid molecules. The labile proton in the -OH groups could, in principle, exchange with deuterium. To test the hypothesis, we recorded some 2D 1 H- 13 C HETCOR spectra (Fig. 4c &d) of MOF-801 prepared under various conditions. The 1 H peak intensity at ~2.4 ppm is significantly lower than in the D2O-washed sample relative to the H2Owashed counterpart. This implies that labile 1 H in Zr-OH could be exchanged with D2O to form Zr-OD, which attenuates the 1 H peak intensity. The experimental results allow us to unambiguously assign the observed 1 H peak to Zr-OH, which was previously stated in literature with only DFT calculations. 27 To examine if the sealed 3D-printed cap is sufficiently robust for studying moisturesensitive samples, we put the rotors packed with D2O-washed samples inside Eppendorf vials filled with a few drops of H2O for ~1 week. For the rotor fitted with a silicone-sealed 3D-printed cap, the HETCOR spectrum (Fig. 4d) is nearly identical to the spectrum acquired before exposure to water vapor. In contrast, the rotor without a cap showed significant growth in the Zr-O 1 H peak. Hence, the results confirm that the 3D-printed cap with silicone seals provides a robust airtight condition suitable for studying air-or moisture-sensitive NMR samples. Moreover, emptying the rotors fitted with silicone-or epoxy-sealed drive caps-which could become unusable if destructively removed-is more cost-efficient if the caps were 3D-printed in-house.

Another main concern about the 3D-printed drive caps is their feasibility for low-temperature applications. It was reported by Banks et al. that the 3D-printed caps fabricated by an industrial 3D printer (Protolabs) crack at ~ 100 K. 10 To examine our printed drive caps printed (Formlabs), we fitted a 3D-printed cap into a sapphire rotor packed with MOF-801 wetted with DNP juice (see As the exact chemical composition of the commercial silicone sealant is not known, we tentatively assign the anomalous peak to originate from the silicone sealant. In the 2D spectrum, we have observed peaks of both silicone and ligands of MOF 801 (fumarate) but not a correlation between the two sites.

Experimental Section) and sealed with silicone sealant (Fig. 7c), before spinning it at ~100 K using a 3.2 mm 800 MHz MAS DNP probe. Figure 7a shows the histogram of regulated MAS frequency of r = 8 kHz  2 Hz at a sample temperature of ~ 100 K over ~ 17 hours. The recorded spinning frequency at low temperatures is slightly less stable than the room-temperature operation (Fig. 5d), possibly due to a larger standard deviation in stator temperature (see Fig. 7b and Experimental Section). Following that, we recorded a 2D 1 H-1 H double-quantum filtered (DQF) spectrum (Fig. 7d) using a symmetry-based C72 1 sequence, which is a rotor-synchronized dipolar recoupling sequence that requires a stable spinning frequency for efficient performance. 28 Hence, the 2D experiment acquired at ~ 100 K demonstrates that the 3D-printed drive cap is robust for low-temperature (including MAS DNP) applications.

Besides that, we have also modelled 4 mm drive caps with a center hole (Fig. 8), which could be useful for in-situ monitoring of chemical reactions during MAS experiments, or those NMR experiments that require in-situ illumination of light or lasers. [29][30][31][32] For the first case, similar instrumentation and experiments were demonstrated in the literature, [33][34][35][36] where a reactant gas was directed into a 7 mm rotor via an injection tube through the drive cap with an axial hole. The

Conclusion

We have demonstrated that it is possible to fabricate 1.3-7 mm drive caps using an affordable desktop 3D printer in-house. The moderate-accuracy but high-precision 3D printing method allows us to fabricate highly consistent (> 90 %) MAS drive caps with similar results as the commercial caps for the  3.2 mm system. The smaller ( 2.5 mm) caps can spin up to ~ 70 % as fast as the commercial caps, and yet with a high spinning stability (< 2 Hz). The high spinning stability of the drive caps allows practical multidimensional NMR experiments to be performed. Additionally, we have exploited the high-resolution (2 µm) industrial 3D printer (BMF) to fabricate the 1.3 mm caps, which yield near equivalent MAS performances as the commercial caps. We foresee that it could be possible to 3D print smaller ( 0.7 mm) 4,5,37,38 caps with similar or harder materials offered by the same company. In contrast to traditional machining, which becomes more challenging and expensive to fabricate smaller drive caps, 3D printing becomes cheaper as the size of the drive cap decreases-because more pieces could be simultaneously printed in a single print job on the platform. The relatively low cost and short fabrication time have allowed us to prototype new designs for new or custom MAS NMR applications. For instance, it costs less than 1 USD and 6 hours to fabricate 100 pieces of the MAS drive caps in-house. Not only does the silicone-filled groove on the 3D-printed cap provides an airtight environment suitable for studying air-or moisture-sensitive materials, but also allows low-temperature MAS experiment (including DNP) at ~ 100 K. As the 3D printing technology continues to flourish, we expect higher-resolution 3D printers will become more easily accessible by laboratories, and new tools could be fabricated inhouse for novel NMR experiments.

Figure 1 :

 1 Figure 1: Image of (a) a commercial (Bruker) Vespel and (b) a 3D-printed 3.2 mm MAS drive cap.

Figure 2 :

 2 Figure 2: Images of 2.5 mm MAS drive caps. Note the undesirable overhang effect present in the (a) upright and (b) reversed orientation. The artefacts were compensated by designing (c) a concaved (curved inward) slab, which yields the (d) corrected cap that improves the maximum spinning frequency from 22 to 26 kHz.

Figure 3 :

 3 Figure 3: 1.3 mm MAS drive caps. Images of the (a and d) CAD and (b and e) 3D-printed pieces (a and b) before and (d and e) after the design compensation. Images of the (c) machined and (f) industrial 3Dprinted (BMF) drive caps.

Figure 4 :

 4 Figure 4: 2D NMR spectra acquired with (a) 1.3 mm, (b) 2.5 mm, or (c) 3.2 mm 3D-printed caps. (a) 13 C-13 C RFDR spectrum of L-Isoleucine in a 1.3 mm rotor at r = 45 kHz. (b) 13 C-13 C DARR spectrum of L-Isoleucine in a 2.5 mm rotor with r = 20 kHz. (c) 1 H-13 C HETCOR spectrum of MOF-801 in a 3.2 mm rotor with r = 10 kHz. * denotes the spinning sidebands. Note that there are two isoleucine isomers that give rise to two C2 peaks, where the second peak is labelled as C2'. 39 (d) 1D projection showing the 1 H dimension (integrate along the grey area in (c)) for MOF-801 washed in H2O (blue), D2O (red), D2O-washed MOF exposed to saturated water vapour for one week with (yellow) or without (green) the 3D-printed cap sealed with silicone.

Figure 5 :

 5 Figure 5: Histograms of MAS frequencies for (a and b) 1.3 mm, (c) 2.5 mm, (d) 3.2 mm, (e)4 mm, and (f) 7 mm drive caps that are either 3D-printed (blue) or machined (red, Bruker). All 3D-printed caps were made in-house except (b), which was outsourced to BMF. All caps were tested in a regulated mode using MAS II for at least 15 minutes. Only the standard deviation in MAS frequency for the 3D-printed caps is shown. The results are summarized in Table1.

Figure 6 :

 6 Figure 6: MAS frequencies of the (a) 1.3 mm and (b) 2.5 mm drive caps as a function of the drive gas pressure with a constant bearing gas pressure of 3 bar. The dashed lines (--) show fitted data with function 𝜈 r = 𝐴𝑃 + 𝐵√𝑃 + 𝐶, where A, B, and C are numerical constants.

Figure 7 :

 7 Figure 7: (a) Histogram of MAS frequency of a 3.2 mm 3D-printed drive cap at ~100 K. (b)The stator and exhaust temperature were measured to be ~109.4 K and ~166.5 K, respectively. The exhaust temperature was measured at the probe base. (c) Photo of a 3D-printed drive cap with a silicone-filled groove to prevent the cap from becoming loose at low temperatures. (d) A 2D 1 H-1 H DQ-SQ (single quantum) spectrum was acquired with a C7 sequence at ~ 100 K and 18.8 T. As the exact chemical composition of the commercial silicone sealant is not known, we tentatively assign the anomalous peak to originate from the silicone sealant. In the 2D spectrum, we have observed peaks of both silicone and ligands of MOF 801 (fumarate) but not a correlation between the two sites.

4

  mm drive cap (without an injection tube) could spin till 𝜈 r = 7 kHz  0.3 Hz, which is slightly slower than the normal cap (~ 10 kHz).

Figure 8 :

 8 Figure 8: Image of the 4 mm drive cap (a) without or (b) with a center hole. The drive caps were printed using a desktop 3D printer (Elegoo Saturn). Their MAS performances are specified in the image.

Table 1 :

 1 The pressure of each gas inlet and spinning frequency as a function of time in the automated Summary of rotor sizes, spinning frequencies, and other specifications of both commercial (Vespel) and 3D-printed MAS drive caps. We have tested at least 10 pieces for each rotor size, except for the industrial-printed (BMF) 1.3 mm drive cap, where only two out of three available pieces were tested. We have only examined the 3.2 mm cap at ~ 100 K only once for ~ 17 hours. RT refers to room temperature.

	Diameter	Volume	Max r of machined	Max r of 3D-printed cap	Printing
	(mm)	(l)	Vespel cap (kHz)	(kHz)	consistency
	7	240	7	6 kHz  0.5 Hz	<100 %
	4	70	15	10 kHz  1.6 Hz	<100 %
				(Elegoo, no hole)
				7 kHz  0.3 Hz
				(Elegoo, with hole)
	3.2	30	24 (RT)	22 kHz  1.3 Hz. (RT)	<100 %
			12 (~100 K)	8 kHz  2 Hz (~100 K)
	2.5	12	35	26 kHz  2.1 Hz	95 %
	1.3	2.5	67	46 kHz  1.1 Hz (Form3+)	92 %
				60 kHz  3.4 Hz (BMF)

Acknowledgements

We acknowledge Dr. Nicolas Birlirakis, Dr. Dan Banks, and Natalie Golota for helpful discussions. This work was supported by fundings obtained from the French National Research Agency: PulsedDNP (ANR-20-ERC9-0008) and HFPulsedDNP (ANR-21-CE29-0019), as well as RESPORE (n° 339299). Wenqing Lu acknowledges the China Scholarship Council for providing the funding (n° 201906880002).

Supporting Information Available:

The designs of the drive caps (1.3, 2.5, 3.2, 4, and 7 mm) are available in the '.stl' and '.ipt' (Autodesk Inventor) format. In addition to 3D printing, we have machined a simple cap-removal tool to remove the drive caps from the rotors. A movie that shows the spinning of 3D-printed cap is available.

Supporting Information

Design and Fabrication of Cap Removal Tool via CNC Machining

We have designed and fabricated a cap-removal tool (Fig. S1) for a 7 mm rotor system. The tool was designed on Autodesk Inventor (see supplementary data for the model), and it can be easily adapted for any rotor sizes. The tool consists of only two pieces, which were held together with a screw and a nut. We used Fusion360 as the Computer-Aided Manufacturing (CAM) software for simulating the toolpath. The g-code is later loaded into the Grbl software, which guides the CNC machine (Genmitsu PROVerXL 4030). We machined the cap-removal tool on a 5 mm aluminum sheet using a 3 mm titanium nitride (TiN) coated end mill loaded to an ER 11 collet on a 300 W spindle capable of spinning up to 10,000 rpm. The manufacturing time was ~ 3 hr per piece, and it has successfully removed several drive caps from the 7 mm rotor.

Design of MAS Drive Caps

We have included the '*.ipt' (Autodesk Inventor) '*.stl' files for the 1.3, 2.5, 3.2, 4, and 7-mm drive caps, as well as the cap-removal tool for the 7 mm system (adaptable to other dimensions). Note that the dimensions of all drive caps were reiteratively optimized to be printed using Clear V4 resins and Form 3+ (Formlabs), except the 4 mm, which was printed using Elegoo Saturn. Moreover, note that the shape of the slab in the 2.5 mm cap and the fins of the 1.3 mm were also optimized for better MAS performances. Note that the shape of the printed parts depends on the ambient environment surrounding the 3D printer, i.e., temperature, ambient light condition, etc.

Hence, some minor alterations to the design might be necessary to yield satisfactory results.


Laureline Barbie
email: barbie@lma.cnrs-mrs.fr

Isabelle Ramiere

Jerome Sercombe
email: jerome.sercombe@cea.fr

AMR METHODS IN SOLIDS MECHANICS FOR PELLET-CLADDING INTERACTION MODELLING

Keywords: Adaptive Mesh Refinement (AMR) techniques, Local Defect Correction (LDC) method, Solids mechanics, Elastic behaviour, Pellet-Cladding Interaction (PCI)

In this study, an Adaptive Mesh Refinement (AMR) method is applied in order to conduct reliable solids mechanics studies with acceptable computing times when local phenomena are under consideration. The Local Defect Correction (LDC) method is performed, which enables us to validate the AMR philosophy for homogeneous linear elastic behaviour. The test case under study outcomes from an industrial situation: the pellet-cladding interaction in pressurized water reactors. A simplified model focused on the cladding response is used.

INTRODUCTION

For now, AMR methods are widely used in fluid mechanics but not yet very applied in solids mechanics. However, when localized phenomena are under consideration, these techniques are very efficient . For that reason, the goal of our work is to extend AMR techniques to tridimensional non-linear solids mechanics (non-linear constitutive equations, contact, etc.). This work is applied to a specific industrial test case: the pellet-cladding interaction (PCI) in pressurized water reactors. PCI is characterized by concentrated stress on the cladding, and the problem is fully tridimensional. As computing complete 3D simulations is limited by current computer capacities, AMR techniques are full of promise. As a first step, we only consider the 2D response of the cladding, supposed to have a linear elastic behaviour. The effect of the pellet is then represented thanks to the boundary conditions. 1 Laureline Barbie, Isabelle Ramiere, lt<' rederic Lebon and Jerome Sercombe

ADAPTIVE MESH REFINEMENT APPROACH

General principle

Adaptive Mesh Refinement (AMR) methods are devoted to take into account localized phenomena and significant variations of scale. To guarantee the stability and the uniformity of the mesh, our approach is based on hierarchical multigrid architecture [1]. The regular structured AMR meshes are then generally not adapted to the local phenomena of the physical problem but located around these phenomena.

We consider a set of nested domains D 1 , 0 :s; l :s; l*, containing the local phenomena of interest [START_REF] Kuss | Error estimation and mesh adaptation for Signorini-Coulomb problems using E-FEM, Computers and Structures[END_REF], such that D 0 = D (original physical domain) and D 1 C D 1 _ 1 . Nested structured regular sub-grids G 1 are defined on each D 1 , such that h 1 < h 1 _ 1 . Then G 0 corresponds to the initial coarse grid defined on the original physical domain. The core of a two-grid AMR process is composed by two steps:

1. A prolongation step from the coarse grid G 1 _ 1 to the fine grid G 1 , which consists in defining the boundary conditions on the fine grid. For l -/=-0 the boundary conditions on 8D 1 are defined by:

-on the part of the boundary common with 8D, a restriction of the boundary conditions defined on 8D.

-on the other part of the boundary, a projection of the former coarser solution obtained on G1-1- 2. A restriction step from the fine grid G 1 to the coarse grid G 1 _ 1 , which consists in correcting the solution on the coarse grid.

These two steps are repeated until the convergence of the coarse solution.

A local multigrid process is the generalization of the two-grid method. The two-grid algorithm is then repeated recursively on more sub-grids, as we can see figure 1 where S is the solver (which may be different on each grid), P the prolongation step and R the restriction step.

Local Defect Correction (LDC) method

This method, introduced by Hackbusch [START_REF] Hackbusch | Local Defect Correction Method and Domain Decomposition Techniques[END_REF], consists in using a local defect correction in the restriction step of the AMR algorithm. We defin e the sets A 1 (nodes of the grid G 1 strictly included in the fin e grid G 1 + 1 ) and .A. 1 (nodes of G 1 such that the discretization scheme on these nodes only implies nodes of A 1 ).

If uf is the displacement vector at level l and. iteration k, Lz is the stiffness matrix at level l and H is the loadings vector at level l and iteration k, the restriction step writes:

-Compute the restricted solution uf (x) = (Rf+ 1 uf+ 1 )(x) Vx E A 1 , where Rf+i is the restriction operator ( or fin e-to-coarse operator) ,

-Compute the local defect rf (u)( x) = (fz° -L 1 uf)(x) Vx E .A. 1 , -Solve the corrected problem on G 1 with f~1 = Jg 1 - x;i.
1 rf(u), where xis the char- acteristic function.

In our case, we use an interpolation op erator as restriction operator.

Due to the definition of A 1 and .A. 1 , we have to use rather wide grids so that the areas where the restriction is applied are not top small. In our case, supposing that the boundary conditions are reliable, we preset Az = A 1 •

INDUSTRIAL TEST CASE

Introduction to Pellet-Cladding Interaction (PCI)

The t est case under study outcomes from an industrial situation, the pellet-cladding interaction in pressurized water reactors [START_REF] Michel | 3D fuel cracking modelling in pellet cladding mechanical interaction[END_REF]. During the irradiation, the fu el pellet swells and the cladding cree ps and shrinks inducing contact. Furthermore, fu el pellet is submitted to a non-isothermal loading during irradiation, which leads to thermal gradient in the pellet and, consequently, to high stresses which usually exc ee d the t ensile strength of the material. As the pellet is fragmented (figure 2(a)) , the contact betwee n fu el pellet and cladding is discontinuous. This phenomenon is very localized ( about micromet ers for a structure about 10 millimet ers of diamet er). Moreover, as the axial dimension of the fu el pellet is finit e, the thermal gradient associated to fu el fragmentation leads to close the pellet-cladding gap first at the inter-p ellet plane (hourglass shape of the pellet , figur e 2(b)). This phenomenon is also very localized (about 0,1 millimet ers for a structure of 14 millimet ers of heigth).

Then to model PCI in PWRs, it is necessary to account for various non-linear physical phenomena which take place in the fu el pellet and the cladding, furthermore on a wide range of scales [START_REF] Bailly | Le combustible nucleaire des reacteurs a eau sous pression et des reacteurs a neutrons rapides[END_REF]. Modelling precisely this phenomenon is important, as it concerns the integrity of the cladding which is the first confinement barrier for the irradiated fu el. That's why research and development on this item are still undertaken worldwide in order Complete 3D simulations are currently limited because of the required unstructured and irregular mesh, inducing an ill-conditionned system with an important number of degrees of freedom. AMR approach seems then well suited for this application.

In this study, a simplified model focused on the cladding response is used. Its behaviour is supposed elastic and the contact with the pellet is represented by a discontinuous pressure on the internal radius.

2D modelling

As first modelling, we consider separately the two phenomena occuring in the PCI and described before: the Hourglass phenomenon and the pellet fragmentation. Each of these phenomena can be completely described by 2D models as a first approximation. The meshes used to obtain reference solutions are adapted to the problem geometry and the pressure discontinuity, whereas those used in the AMR approach are regular structured meshes, not adapted to the singularity but localised around it. For the two types of approaches, a Ql FEM solver is used. In this article, 'classical FEM' denotes the FEM applied on meshes adapted to the physics of the problem while 'AMR approach' denotes the FEM applied within an AMR process on nested regular meshes. In the AMR approach, between two consecutive nested grids, we choose to divide the mesh step by two in each direction.

Hourglass shape phenomenon: 2D(r,z) modelling

Modelling hypothesis

Our model is axisymmetric and the hourglass shape of the fuel pellet is represented by a pressure peak on the internal radius of the cladding. Thanks to symmetrical considerations, we only consider a half-pellet height of the cladding ( figure 3 (a)).

------mid-pe ll et plane pellet pressure ----+ <------+ claddi ng s--const ant ----+ <-ext ernal - ---+ ----+ ----+ pe llet pressure -------, peak due to ====: To validate our approach, we first compute a reference solution adapted to the discontinuity, obtained with a fine uniform mesh (with a step of 5 micrometers while the discontinuity is 160 micrometers from the inter-pellet plane).

hourglass shape __ _ <--pressure -----------inter-pell et plane

Results of the AMR approach

For the AMR approach, we use nested meshes whose an exemple is given on figure 3(b). The interest of this approach is that these meshes are not adapted to the geometry of the discontinuity. We compare the results obtained with the reference mesh and with the AMR algorithm applied on nested meshes. For the AMR approach, the pressure gap is imposed from the closest upper node of the physical discontinuity.

We compare the Von Mises stress on the cladding, as presented on the figure 4. The Von Mises stress fi elds are very similar, whiclh confirms our approach accuracy. To further appreciate the performance of our LDC method, we study several error norms (L 2 norm, L 00 norm or max norm) calculated from the reference solution. For each simulation, it is the final coarse solution (obtained at ,convergence) that is considered in the error calculation. The AMR errors will be represented according to the initial distance between the singularity and its approximation ( represented by d on the figure 3 (b)). We observe that without refin ement (0 sub-grid) , the error dec reases linearly with the singularity approximation: a first-order accuracy is then obtained. The loss of an order of convergence (the classical Ql FEM is second-order acc urat e for the L 2 -norm) is due to the use of meshes non-adapted to the discontinuity. Moreover, what ever the number of sub-grids used, for the same fin er grid size we obtain the same error after convergence : then the method is of first-order with respec t to the fin est grid [START_REF] Ramiere | A fictitious domain approach with spread interface for elliptic problems with general boundary conditions[END_REF]. Lastly, we can observe a plat eau for some sub-grids. This plat eau is due to the fact that in this case, betwee n two sub-grids, even if the mesh size is divided by two , the singularity approximation is not improved. As we observe t he same error for a global fin e grid t han for nested local sub-grids with t he same fin est mesh size, we can conclude t hat our AMR approach is validated for t he 2D linear elastic behaviour.

Laureline Barbie, Isabelle Ramiere, lt<'rederic Lebon and Jerome Sercombe

Contact with a fractured pellet: 2D(r,0) modelling

Modelling hypothesis

We focus here on the mid-pellet plane (in order to use generalized plane strains hypothesis). It does not allow to account hourglass shape but it enables us to describe fuel crack opening.

Here, we assume that the fuel pellet is initially fragmented in eight symmetrical parts, as often observed after base irradiation [START_REF] Non | Differential PCI behaviour of PWR fuel rods under transient conditions[END_REF]. Thanks to symmetrical considerations, we only represented a sixteenth of the cladding (figure 6). As the fractures are open at the external radius of the pellet, there is no contact between the pellet and the cladding in the fracture plan. Then, in our symmetrical representation, under the fuel pellet fracture, no pressure is applied. For the AMR approach, the pressure gap is imposed from the closest upper node of the physical discontinuity. We study the influence of the singularity position. We study a pressure gap of 8 and 10 micrometers, which is very localized as the cladding has an internal diameter of 8 millimeters and a thickness of 0,6 millimeters.

5 Figure 1 :

 51 Figure 1: Local multigrid iteration represented by V-cycles

  Laureline Barbie, Isabelle Ramiere, lt<' rederic Lebon and Jerome Sercombe (a) Pellet Fragmentation (b) Hourglass shape

Figure 2 :

 2 Figure 2: Pellet behaviour during irradiation

  (a) Representation of the 2D(r,z) model

Figure 3 :

 3 Figure 3: Hourglass shape phenomenon

  Laureline Barbie, Isabelle Ramiere, lt<' rederic Lebon and Jerome Sercombe

Figure 4 :Figure 5

 45 Figure 4: Von Mises stresses for the 2D(r,z) model, interpolated on the coarser grid

Fig ure 5 :

 5 Fig ure 5 : Res ults for the 2D(r ,z) model

Figure 6 :

 6 Figure 6: Contact with fractured pellet

Singularity thickness of 8 µm

Due to the circular geometry, our meshes are approximations of the global domain studied. The reference solution is adapted to the discontinuity, which implies a really non-uniform mesh (cf. example figure 6(a) whose mesh step varies from 5µm to 50µm). The AMR nested meshes are not stricly uniform but are as regular as possible ( cf. figure 6(b)). Figure 7 validates the AMR approach as we obtain almost the same result on a global fine grid than on local nested grids with the same finest discretization step.

Singularity thickness of 10 µm

We study the case where the singularity approximation is fastly improved between two mesh sizes (see figure 8 between mesh (40x80) and mesh (80x160)).

When the error decreases abruptly, the correction on local grid could be not sufficient. Thus, when this phenomenon comes on the second or third sub-grid, we see on the figure 8 that the correction is not as effective as if it happens on the first sub-grid. This weakness could be explained by the fewer zones where the correction is done. However, the AMR algorithm still brings partial correction.

CPU time and memory space required for each approach

We also compare CPU time required by the simulation with regard to the precision obtained (see figure 9), the results without refinement being FEM computations on a non-adapted mesh (without any sub-grid).

The more the precision expected is constraining, the more the AMR approach is advantageous. In the same way, the more the precision expected is constraining, the more the use of several sub-grids on the AMR approach is interesting. AMR techniques are also interesting for saving memory space, as on each sub-grid we only use a few number of nodes distributed equitably.

CONCLUSION AND PERSPECTIVES

To conclude, this work was a first step to validate AMR techniques for solids mechanics problems. For the 2D(r,z) model, the geometry of the problem is very simple and can be represented perfectly by rectangular meshes. In this case, we obtained strictly the same result for global fine grid or AMR approach with a finest grid with the same mesh size. For the 2D(r,0) model, the geometry of the problem is not so simple but AMR approach is still efficient . Using AMR techniques allows us to save CPU time and memory space during computation, which could allows to compute problems whose computation is impossible using classical FEM approachs, such as 3D PCI modelling. Meshes are not adapted to the singularity but we saw here that we obtain satisfactory results (validated in 2D). However, there are some limitations, especially when the approximation varies abruptly between two consecutive nested meshes for example. Now, the validation of this technique has to be extended to solids mechanics specificities, such as material non-linearity, contact ...