Romain Gautier
email: romain.gautier@cemes.fr

Armin Rajabzadeh
email: armin.rajabzadeh@pwc.ca

Melvyn Larranaga
email: melvyn.larranaga@cemes.fr

Nicolas Combe
email: nicolas.combe@cemes.fr

Frédéric Mompiou
email: frederic.mompiou@cemes.fr

Marc Legros
email: marc.legros@cemes.fr

Samuel Forest

Shear-coupled migration of

Keywords: . Grain boundaries, Disconnections, plasticity, mechanical behavior

come

grains sont des obstacles connus pour les dislocations, mais lorsqu'ils atteignent des dimensions nanométriques, les cristallites deviennent généralement vides de dislocations, ce qui impose une contrainte supplémentaire pour développer la plasticité. Comprendre les mécanismes de déformation basés sur les joints de grains est devenu un enjeu majeur de la métallurgie physique. Ces mécanismes peuvent être multiples, impliquant des processus conservatifs et diffusifs qui sont mal compris. Une première approche qui consiste à transposer aux petites dimensions des mécanismes documentés à grande échelle comme le fluage de Coble, s'est avérée très limitée. Au contraire, la croissance des grains assistée par la contrainte ou la migration des joints de grains couplée au cisaillement, récemment observées dans les matériaux à petits grains à température ambiante, peuvent fournir une clé pour comprendre pleinement la "plasticité sans dislocation" dans les nanocristaux. Comme il s'agit d'un domaine relativement nouveau avec beaucoup plus de degrés de liberté, un effort de recherche continu doit être mené pour relier les propriétés mécaniques des nanocristaux à ces processus de plasticité basés sur les joints de grains.

Introduction

Origin of hardening in metals

As the main vector of plastic deformation in crystals, the mobility of dislocations directly influences the mechanical properties. This is why hardening a metal mainly consists in blocking or hindering dislocation movements. Among the many ways to do so, the oldest one lies in multiplying obstacles, often embodied by extrinsic atoms present in metallic alloys [START_REF] Cahn | Physical Metallurgy[END_REF][START_REF] Laughlin | Physical Metallurgy[END_REF]. Strain (or Taylor-) hardening is caused by the development of a dense dislocation microstructure where dislocations themselves become their own obstacles. The most prominent example of such hardening mechanism is fatigue where dislocations self-arrange in so-called "walls" that grow impermeable to further plasticity, delimiting "channels". As the channels become narrower, mobile dislocation are forced to curve tighter [START_REF] Legros | Fatigue of single crystalline silicon: Mechanical behaviour and TEM observations[END_REF], requiring additional stress to maintain plastic deformation. Similar confinement-driven hardening is observed in Severe Plastic Deformation processes, that lead to the formation of dislocation cells, low-then high-angle grain boundaries (HAGB) upon increasing strain [START_REF] Isik | Grain Refinement Mechanism and Evolution of Dislocation Structure of Co-Cr-Mo Alloy Subjected to High-Pressure Torsion[END_REF][START_REF] Liu | Microstructure evolution and dislocation configurations in nanostructured Al-Mg alloys processed by high pressure torsion[END_REF][START_REF] Scheriau | Influence of grain size on orientation changes during plastic deformation[END_REF]. Such top-down methods usually manage to reduce the grain size to a few hundreds of nm, but going further down requires bottom-up methods such as physical or chemical deposition, crystallite condensation and agglomeration [START_REF] Legros | Microsample tensile testing of nanocrystalline metals[END_REF][START_REF] Meyers | Mechanical properties of nanocrystalline materials[END_REF]. In fact, the microstructure of pure metals becomes thermodynamically unstable when they contain too many grain boundaries, and grain growth or recrystallization is observed at room temperature [START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF][START_REF] Ames | Unraveling the nature of room temperature grain growth in nanocrystalline materials[END_REF][START_REF] Haber | Kinetic Instability of Nanocrystalline Aluminum Prepared by Chemical Synthesis; Facile Room-Temperature Grain Growth[END_REF].

Mechanical behavior of nc-metals

Reducing the grain size of a polycrystalline metal or alloy to increase their yield stress has been rationalized through the Hall-Petch relation that states that the yield stress of steels goes up as the inverse of the square root of the average grain size [START_REF] Hall | The Deformation and Ageing of Mild Steel: II Characteristics of the Lüders Deformation[END_REF][START_REF] Armstrong | The influence of polycrystal grain size on several mechanical properties of materials[END_REF][START_REF] Petch | The cleavage strength of polycrystals[END_REF]. This empirical law has then been verified for a number of metals and alloys, mainly with fcc structures [START_REF] Dunstan | The scaling exponent in the size effect of small scale plastic deformation[END_REF][START_REF] Dunstan | Grain size dependence of the strength of metals: The Hall-Petch effect does not scale as the inverse square root of grain size[END_REF]. But as with other systems where dislocations are confined in a given volume (thin films on substrates, pillars and nanowires [START_REF] Kraft | Plasticity in Confined Dimensions[END_REF]), the yield stress increase due to the path limitation of mobile dislocations ultimately saturates when entering the sub-micron or nanocrystalline regime [START_REF] El-Awady | Unravelling the physics of size-dependent dislocation-mediated plasticity[END_REF]. In this so-called "Hall-Petch breakdown regime", plateaus or decrease of the yield stress have been reported [START_REF] Schuh | Hall-Petch breakdown manifested in abrasive wear resistance of nanocrystalline nickel[END_REF]. In such nanocrystalline (nc-) materials, the proportion of atoms involved in grain boundaries (GB) can reach several percents, and the dislocation mean free path can be reduced to the grain size [START_REF] Van Swygenhoven | Deformation in nanocrystalline metals[END_REF]. The plastic deformation of such nanocrystals would require the cooperative motion of several dislocations in most grains [START_REF] Saada | Deformation mechanisms of nanograined metallic polycrystals[END_REF], which is practically impossible, especially when experimental observations return an intragranular dislocation content nearing zero [START_REF] Legros | Microsample tensile testing of nanocrystalline metals[END_REF]. As a result, nc-metals generally show very low to almost no ductility [START_REF] Legros | Microsample tensile testing of nanocrystalline metals[END_REF]. However, in some occurrences, nanocrystalline metals can exhibit significant ductility [START_REF] Lu | High-tensile ductility in nanocrystalline copper[END_REF][START_REF] Gianola | Stress-assisted discontinuous grain growth and its effect on the deformation behavior of nanocrystalline aluminum thin films[END_REF], and many experiments or simulations [START_REF] Farkas | Grain boundary migration during room temperature deformation of nanocrystalline Ni[END_REF][START_REF] Yamakov | Deformation-mechanism map for nanocrystalline metals by molecular-dynamics simulation[END_REF] have pointed to "GB-based plasticity" to explain it. This generic appellation encompasses many possible mechanisms, going from GB sliding to grain rotation [START_REF] Yu | In situ TEM nanomechanics[END_REF]. These mechanisms would also be responsible for the Hall-Petch breakdown regime, and could be based on diffusion [START_REF] Balluffi | Mechanism for diffusion induced grain boundary migration[END_REF] or conservative processes [START_REF] Guillopé | A model for stress-induced migration of tilt grain boundaries in crystals of NaCl structure[END_REF]. Direct observations of these GB-based mechanisms are very difficult to obtain for reasons that will be developed later, but early reports of their occurrences in the mid 2000's eventually made headlines [START_REF] Shan | Grain Boundary-Mediated Plasticity in Nanocrystalline Nickel[END_REF] or sparked controversy [START_REF] Chen | Comment on "Grain Boundary-Mediated Plasticity in Nanocrystalline Nickel[END_REF].

Shear-coupled grain boundary migration

Among all the GB-based plasticity processes, the shear-coupled GB migration (SCGBM) particularly captured attention because it may produce significant shear, it is a conservative mechanism, and also because it probably relates to another extensively observed phenomenon in ncmetals : shear-assisted grain growth [START_REF] Rupert | Experimental Observations of Stress-Driven Grain Boundary Migration[END_REF][START_REF] Legros | In situ TEM observations of fast grain-boundary motion in stressed nanocrystalline aluminum films[END_REF][START_REF] Zhang | Stress-driven grain growth in ultrafine grained Mg thin film[END_REF][START_REF] Zhang | Rapid stress-driven grain coarsening in nanocrystalline Cu at ambient and cryogenic temperatures[END_REF]. This later mechanism is clearly different from traditional recrystallization or grain growth as both these microstructural changes occur without shape change. The mechanism of shear-coupled grain boundary migration is not as intuitively understandable as for dislocation shear. Figure 1 illustrates how it occurs phenomenologically, and how it is measured through the quantity β, that is called the coupling factor: under a shear stress, the grain boundary (GB) separating grain 1 and grain 2 moves along its normal (either upwards or downwards). This migration over a distance h is accompanied by a shear displacement d. β, the coupling factor, is the ratio of d over h.

Shear-coupled grain boundary migration was first theorized by Read and Shockley who considered low-angle grain boundaries as arrays of perfect lattice dislocations. These dislocations could simultaneously move by application of an external shear stress, resulting in the migration C. R. Physique -Online first, 21st June 2021 of the low-angle grain boundary [START_REF] Read | Dislocation models of crystal grain boundaries[END_REF]. Experimental observations came shortly after in zinc bicrystals [START_REF] Bainbridge | Recent observations on the motion of small angle dislocation boundaries[END_REF][START_REF] Li | Stress-induced movement of crystal boundaries[END_REF]. The topological dislocation content of any grain boundary can be obtained from the Frank-Bilby equation [START_REF] Yang | Use of the Frank-Bilby equation for calculating misfit dislocation arrays in interfaces[END_REF] and this served as the basis of the Cahn, Mishin, Suzuki, and Taylor (CMST) model which extended the low-angle grain boundary model from Read and Shockley to any [001] tilt boundary in fcc structures [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. It should be noted that a similar model for NaCl-type cubic structures was published by Guillope and Poirier in 1980 [START_REF] Guillopé | A model for stress-induced migration of tilt grain boundaries in crystals of NaCl structure[END_REF]. Logically, this conservative model predicts that the larger the misorientation, the larger the shear carried by a given GB (modulo the 90°symmetry of this specific tilt axis). Experiments carried out at medium-toelevated temperatures (300-400°C) on Al bi-crystals tended to support this model [START_REF] Gorkaya | Stress-driven migration of symmetrical 〈100〉 tilt grain boundaries in Al bicrystals[END_REF] (Figure 2), although some scatter in the results existed and some of them were discarded when only segments of the GB coupled for example [START_REF] Molodov | Migration of the Σ7 tilt grain boundary in Al under an applied external stress[END_REF].

This model, appealing by its simplicity, suffers however, from several drawbacks:

• The dislocation content of high-angle grain boundaries is only virtual as their core overlap once a 10-15°misorientation is surpassed so that their existence as single defects when the boundary misorientation is high becomes impossible. • The model only applies to a specific type of boundaries, and its extension to general boundaries such as the ones present in real crystals may be facing limitations. The first one being that twist boundaries shouldn't be able to shear-migration couple. • The main limitation is that only perfect lattice dislocations are considered as intrinsic GB defects while it has been shown that dislocations with smaller Burgers vectors (DSC-Displacement Symmetry Conserving -see discussion section) exist and might be activated at lower energy cost (owing to their shorter Burgers vector). • Finally, if one only considers GB misorientation, the coupling factor predicted by the CMST model is often very high and incompatible with experimental observations in polycrystals.

In the Cahn, Mishin, Suzuki, and Taylor (CMST) model [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF], as the GB misorientation is a direct function of its virtual dislocation content, β goes up with the GB misorientation, and coupling factors as high as 50 to 100% are theoretically possible for symmetric tilt boundaries in the range of 30°to 40°misorientations. Such amounts, seldom seen in specific bicrystal experiments [START_REF] Gorkaya | Stress-driven migration of symmetrical 〈100〉 tilt grain boundaries in Al bicrystals[END_REF] and simulations [START_REF] Ivanov | Dynamics of grain boundary motion coupled to shear deformation: An analytical model and its verification by molecular dynamics[END_REF] have never been reported in polycrystals. In early experiments on ultra fine Al grains [START_REF] Mompiou | Grain boundary shear-migration coupling-I. In situ TEM straining experiments in Al polycrystals[END_REF], it was found that the shear produced by mobile grain boundaries seems poorly connected to their misorientation. Molecular dynamics-based simulations [START_REF] Homer | Phenomenology of shear-coupled grain boundary motion in symmetric tilt and general grain boundaries[END_REF] and paper cutsbased topological calculations [START_REF] Caillard | Grain-boundary shear-migration coupling. II. Geometrical model for general boundaries[END_REF] revealed that the coupling factor associated with a given grain boundary could be manyfold, and in particular that much lower coupling factors could be found for symmetric <001> GBs. In the following, we will show how complementary approaches (experimental and computational) are used to quantify shear-coupled grain boundary migration in small-grain polycrystals and to unveil the atomic processes at play when the mechanism is active.

Methods and results

In-situ TEM

Ultra Fine Grain (UFG) pure and alloyed (1-3wt% Mg) aluminum was processed by high pressure torsion in ESI, Leoben, resulting in ø25 mm, 4mm thick discs with an average grain size of 160 nm (Al1%Mg) and 110 nm (Al3%Mg). Alloying Al with Mg allows to stabilize a small grain size as pure UFG Al experiences grain growth even at room temperature. 1 x 3 x.5 mm rectangles are then cut with an electro-discharge machine and mechanically polished down to 20-50 µm thickness using SiC grinding papers of diminishing grids. A circular electron-transparent zone is then created in the middle of these rectangles using a 33% nitric acid methanol solution in a Tenupol electro-polishing unit at T=-10°C. The in-situ TEM straining experiments start by gluing the samples on copper straining grids using epoxy glue or zirconia-based cement. A first step consists in initiating a micro-crack at room temperature. This is followed by a mild annealing at 220°C for 5 minutes inside the TEM on a home-made heating-straining holder to remove most of intra-granular dislocations. ASTAR-ACOM (Automatic Crystallographic Orientation Mapping) is then performed to assess the new average grain size (270 nm for Al1%Mg and 145 nm for Al3%Mg) and in the specific zone ahead of the crack tip on a CM20 FEG TEM. The reason UFG Al is chosen over nc-Al is that e -transparent regions are typically below 400 nm thickness in these TEM foils. Having grain sizes much lower than that (typically in the 10-50 nm range) would cause multiple grains to diffract simultaneously, hampering their indexation. Grain-boundary migrations are monitored during more extensive in-situ straining experiments carried out on the CEMES-custom-made high temperature straining holder fitting a JEOL 2010 TEM. Once several SCGBM events are recorded, the experiment is stopped and the sample is returned to the CM20 FEG TEM to perform a post-straining ASTAR-ACOM crystal orientation map. The shear induced by grain boundary migration is assessed through digital image correlation analysis (open source Open CV software [START_REF] Bradski | The OpenCV library[END_REF]) from images captured from videos recorded through a MegaView III CCD camera (SIS-Olympus) side-mounted on the JEOL 2010 TEM.

C. R. Physique -Online first, 21st June 2021

In-situ TEM results

Figure 3 is an example of the results obtained using the procedure that has been elaborated at CEMES to quantify the coupling factor on known grain boundaries in UFG metals. Figures 3 (a (tensile stress, vertical on the pictures). The stress state is not known here, as a slowly opening crack tip is located a few microns away (upper right corner) from the mobile GB. An evaluation of the stress was made from a single spiral source operating near a crack-tip (zone similar to those where SCGBM occurs). The line tension approximation τ= µb/2R, where R is the radius of curvature of the dislocation and µ the shear modulus of Al at 400 K (22 GPa) returned a resolved shear stress τ = 19 ± 3 MPa.

As in the example above, most of the GB movements are usually smooth and proceed by alternating periods of activity and usually longer periods where they are immobile. Collapsing of smaller grains usually accelerates until their complete annihilation, which suggests that surface tension effects take over when the GB size goes below 50-100 nm. In examples such as in Figure 3, flat segments of GB are moving, excluding capillarity effects and attesting of the shear-coupling mechanism.

Similar GB movements have been observed regularly in our in-situ TEM experiments even if they are complicated to fully capture: GB migration under stress occurs among groups of grains and may cease after a few seconds to resume in another location. These movements are then limited in time and space and, to date, impossible to predict. The migration of GBs is almost never jerky, but may proceed by successive rapid movements of segments while others stay put. Speeds on the order of 1-2 nm/s are regularly observed. To analyze the shear produced by the migration, simple digital image correlation (DIC) using a custom script is performed on two sets of about 30 square zones (white squares in Figure 3 (c) located ahead and aside from the moving GB. Each square zone contains several markers (surface contamination) and the produced shear is the average value returned by the square box displacement in the direction parallel to the GB plane divided by the migration distance. In this case, this yields 3% in average (±1%). This average is obtained after measuring shear migration couples at 29, 57 and 87 nm migrations. At each of these steps, the shear was 0.8, 0.8 and 3 nm, respectively. Despite the DIC returning a sub-pixel resolution, we cannot claim a resolution better than 1 nm, which may explain why the first two measurements, being in the error bar of the technique, give the same value. Moreover, the marker displacements are not everywhere parallel to the GB plane, which indicates that the overall strain field resulting from deformation in neighboring grains is more complicated than a simple shear. Overall, these small values are however typical of our experiments where coupling factors are low in most cases: a significant migration is therefore needed to obtain measurable coupling factors. Referring to the misorientation map of Figure 4 (c), the minimal misorientation between grains 1 and 2 is 26.9°, around a [450 -240 162] (close to [START_REF] Legros | Fatigue of single crystalline silicon: Mechanical behaviour and TEM observations[END_REF][START_REF] Isik | Grain Refinement Mechanism and Evolution of Dislocation Structure of Co-Cr-Mo Alloy Subjected to High-Pressure Torsion[END_REF][START_REF] Liu | Microstructure evolution and dislocation configurations in nanostructured Al-Mg alloys processed by high pressure torsion[END_REF][START_REF] Scheriau | Influence of grain size on orientation changes during plastic deformation[END_REF][START_REF] Legros | Microsample tensile testing of nanocrystalline metals[END_REF][START_REF] Meyers | Mechanical properties of nanocrystalline materials[END_REF][START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF][START_REF] Ames | Unraveling the nature of room temperature grain growth in nanocrystalline materials[END_REF][START_REF] Haber | Kinetic Instability of Nanocrystalline Aluminum Prepared by Chemical Synthesis; Facile Room-Temperature Grain Growth[END_REF][START_REF] Hall | The Deformation and Ageing of Mild Steel: II Characteristics of the Lüders Deformation[END_REF][START_REF] Armstrong | The influence of polycrystal grain size on several mechanical properties of materials[END_REF][START_REF] Petch | The cleavage strength of polycrystals[END_REF][START_REF] Dunstan | The scaling exponent in the size effect of small scale plastic deformation[END_REF][START_REF] Dunstan | Grain size dependence of the strength of metals: The Hall-Petch effect does not scale as the inverse square root of grain size[END_REF][START_REF] Kraft | Plasticity in Confined Dimensions[END_REF][START_REF] El-Awady | Unravelling the physics of size-dependent dislocation-mediated plasticity[END_REF][START_REF] Schuh | Hall-Petch breakdown manifested in abrasive wear resistance of nanocrystalline nickel[END_REF][START_REF] Van Swygenhoven | Deformation in nanocrystalline metals[END_REF][START_REF] Saada | Deformation mechanisms of nanograined metallic polycrystals[END_REF]) axis. According to the CMST model, β should be equal to 47% if the (100) mode is activated or 122% for the (110) mode and considering a pure tilt GB. The orientation maps seen in Figures 4 (a)-(b) attest that a massive grain growth is taking place under stress. This is visible in Figure 4 (c), where only zones that are not affected by GB migrations are in black. In this figure, 7.85 µm 2 out of the 17.44 µm 2 (45%) of the map are affected by grain migration. This surface change corresponds to a grain size increase of 48%, significantly larger than the one only obtained by annealing (110 to 145 nm), corresponding to a 32% increase for the same temperature. To account for the rigid-body rotation due to the crack propagation (not seen on the pictures) and the subsequent tilt affecting the TEM sample, this average rotation is subtracted to the orientations in the map of Figure 4 (c). This map represents the minimum misorientation angle between the two maps of Figures 4 (a),(c), corrected from rigid-body rotation due to nearby crack propagation and obtained using a custom script [START_REF] Mompiou | Quantitative grain growth and rotation probed by in-situ TEM straining and orientation mapping in small grained Al thin films[END_REF]. One can then see that some grains are growing at the expense of smaller grains. The sequence seen in Figures 4 (a)-(b) occurs between grains Gr1 and Gr2 labeled in Figure 3 (b) (after the growth of grain 1 towards grain 2 across the yellow grain in between -Figure 3 (a)).

Although seldom evidenced, moving lines bound to migrating or sliding GBs can be seen (Figure 5) when imaging conditions are favorable. GBs have to be sufficiently inclined from their edge-on position for instance. In this sequence, these lines were observed to steadily move from a surface defect pointed by the red arrow in Figure 5 (a)(towards the upper left triple junction TJ, a second TJ is also visible on the lower right side of the figure). Since these linear defects are strictly attached to the GB and because they show a dislocation-like behavior, we think they are disconnections (see Figure 6, Figure 8), although we cannot attest their step character in this case (wrong orientation of the foil and insufficient resolution). No stress concentration can clearly be seen at triple junctions, but many pieces of evidence show that this specific location of polycrystals serve both as source and sink for disconnections. The difficulty to obtain strong contrasts of disconnections is probably due to their position between two diffracting/nondiffracting grains, and also to the fact that many of them possess a shorter Burgers vector (DSC dislocations) compared to lattice dislocations. As a result the distortion of the crystal around these defects is limited and their diffraction contrast faint. In most of our observations, trains of disconnections move along segments of grain boundaries and seem to have similar Burgers vector as their contrast, even light, is similar (Figure 5).

Simulation: Nudge Elastic Band (NEB) method

In the Molecular Dynamics simulations carried out by Mishin and co workers, the shear coupled migration is observed to occur by successive and sudden jumps of the GB [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. We have investigated one of the systems they studied: a copper bicrystal containing a planar symmetric coincident Σ13 (320) tilt grain boundary, with a misorientation of 67.38°around a [001] rotation axis (Figure 6).

We use an EAM Cu potential [START_REF] Mishin | Structural stability and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations[END_REF], and the LAMMPS simulation package [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. Performing a shear loading until GB migration and then an unloading, we determined two stable states of the bicrystal corresponding to configurations before and after the first migration of the GB (blue points c i and c f respectively, in Figure 6 (c)), for a given position of the slabs (or a given strain). In order to determine the Minimum Energy Path (MEP) between the c i and c f configurations i.e. the lowest energy curve that connects them in the energy landscape, we performed NEB calculations: for a N atoms system, an extended phase space (dimension 3Nm) is defined using intermediate configurations of the system (or replicas) as R i atomic arrangements (3N coordinates), i = (0, . . . , m -1), R 0 being the initial state before the GB jump (configuration ci) and R m-1 the final one after the jump(configuration c f ). Harmonic potentials couple replica in order to form an elastic band and atoms in a replica interact through the EAM potential. The NEB method applies a judicious dynamics to this system in order to drive it to the MEP (more details can be found in [START_REF] Rajabzadeh | Experimental and theoretical study of the shear-coupled grain boundary migration mechanism[END_REF][START_REF] Sheppard | Optimization methods for finding minimum energy paths[END_REF]). Figure 7 shows such a path as a function of a reaction coordinate (RC). The RC is here defined as the cumulative distance (normalized by the total cumulative distance) between adjacent replicas in the configuration space of dimension 3N. In Figure 7, the cell size is 10.3 x 6.5 x 1.4 nm (X x Y x Z), and the cell contains 12 structural units such as the ones in Figure 6.

Figure 7 shows multiple local minima which atomic configurations are displayed in the figure. The SCGBM mechanism occurs through the nucleation (from states (a) to (b) in Figure 7), motion (from states b to j in Figure 7) and recombination (from states j to k in Figure 7) of GB steps. We have identified these GB steps as disconnections by performing a Burgers circuit analysis around them. The two GB steps have opposite Burgers vectors and step height: respectively b=0.1 nm and h=0.25 nm (see also Figure 8 for disconnection description). At the end of the reaction (RC= 1), the grain boundary moved down by the step height h and the bicrystal is sheared by an amount of b= 0.1 nm. The coupling factor is directly derived from this process as β = b/h = 0.4. The nucleation and propagation of a disconnection along a GB constitute therefore the basic mechanism of the shear-coupled GB migration. This migration can then be seen as a continuous process, at variance of the stick-slip motion previously considered. Finally, the process also gives physical sense to the coupling factor, that is simply the Burgers vector divided by the step height of the moving disconnections.

Discussion

To fully assess for the deformation mechanisms at play in nanocrystalline metals and alloys, GBbased plasticity has to be better understood. Consistently with previous observations, we have shown that stress-assisted grain growth occurs through the migration of the grain boundaries, and that this migration can occur without involving lattice dislocations [START_REF] Khater | The disconnection mechanism of coupled migration and shear at grain boundaries[END_REF]. In the example reported here (Figure 3), it should be noted, that at variance from recrystallization processes, no intragranular dislocations are involved, and that the moving GB is almost flat, which eliminates the elastic energy gradient and curvature-driven migration that are often invoked for such C. R. Physique -Online first, 21st June 2021 GB rearrangements. As in previous studies [START_REF] Rajabzadeh | The role of disconnections in deformation-coupled grain boundary migration[END_REF][START_REF] Mompiou | Stress assisted grain growth in ultrafine and nanocrystalline aluminum revealed by in-situ TEM[END_REF][START_REF] Mompiou | Grain Boundary Based Plasticity: In-Situ TEM Experiments and Modelling[END_REF], we report small shear migration coupling factors, on the order of a few percents, a quantity that is in line with the applied strain. Also consistently with similar studies in nanocrystalline Cu, all the types of GB migrate, irrelevant to the misorientation they carry [START_REF] Rottmann | Experimental quantification of mechanically induced boundary migration in nanocrystalline copper films[END_REF][START_REF] Kobler | Combination of in situ straining and ACOM TEM: A novel method for analysis of plastic deformation of nanocrystalline metals[END_REF][START_REF] Izadi | Grain rotations in ultrafine-grained aluminum films studied using in situ TEM straining with automated crystal orientation mapping[END_REF]. In all these studies, no specific stress concentration that could result from incompatible GB processes are reported. This supports the possibility of GBs with various misorientations to adopt different coupling factors so that grain growth happening through SCGBM does not create low shear/high shear zones. One common belief or approach in GB-based plasticity is that moving GBs would act similarly to dislocations, but at a larger scale and dimension. The corollary of this approach is to consider the defining parameters of the GB, and especially the misorientation as an intrinsic value from which the coupling factor will depend [START_REF] Gottstein | Grain boundary migration: misorientation dependence[END_REF]. Low coupling factors would be the result of a judicious combination of two coupling modes [START_REF] Molodov | Grain boundary mediated plasticity: On the evaluation of grain boundary migration -shear coupling[END_REF].

A more recent approach, that we would like to promote here, considers each grain boundary as a network with its own translations and symmetries that differ from the crystal lattices of its two adjacent grains. In particular, this "coincident site lattice" (CSL) possesses translational invariants (and thus possible GB Burgers vectors) that have a much smaller amplitude than those of the adjacent grains. These possible shear vectors are called DSC, for "displacement shift complete" or "displacement symmetry conserving" [START_REF] Pond | The symmetry and interfacial structure of bicrystals[END_REF]. When a perfect dislocation reacts with a GB, a dissociation into these smaller Burgers vectors is energetically favorable [START_REF] Pond | On the absorption of dislocations by grain boundaries[END_REF]. Similarly, if these shear vectors are involved in the SCGBM process instead of perfect grain lattice Burgers vectors, they will lead to smaller shears and thus smaller coupling factors (for a given step height). The main contribution of these DSC vectors is to provide many more combinations with interplanar distances (possible steps) and therefore a much larger number of coupling factors for a given GB, compared to models where deformation carriers are limited to intrinsic dislocations. It explains why this diversity of coupling factor has been found in purely topological models [START_REF] Caillard | Grain-boundary shear-migration coupling. II. Geometrical model for general boundaries[END_REF] and extensive MD simulations [START_REF] Homer | Phenomenology of shear-coupled grain boundary motion in symmetric tilt and general grain boundaries[END_REF]. In fact, the CMST model may provide an upper bound for the various coupling factors found in these other models, although the SMIG model [START_REF] Caillard | Grain-boundary shear-migration coupling. II. Geometrical model for general boundaries[END_REF] has no energetic considerations and is therefore not imposing a maximum β. The defect combining one dislocation (perfect or DSC) and a step is called a disconnection (Figure 8) [START_REF] Hirth | Steps, dislocations and disconnections as interface defects relating to structure and phase transformations[END_REF]. A disconnection is specific to a GB and cannot escape from it unless it reacts to give rise to a regular lattice dislocation.. Inversely, lattice dislocations may enter GBs and decompose in several disconnections as their Burgers vector can be significantly smaller. Aside from diffusive process [START_REF] Swiatnicki | Investigation of grain boundary diffusion in polycrystals by means of extrinsic grain boundary dislocations spreading rate[END_REF], lattice dislocations decomposing into disconnections may accelerate their absorption by GBs [START_REF] Mompiou | In-situ TEM observations of reverse dislocation motion upon unloading in tensile-deformed UFG aluminium[END_REF]. This explains in part why most real grain boundaries in materials contain disconnections. More generally, real GBs are rarely symmetric nor flat, nor do they have a pure tilt or pure twist character, and even without preliminary deformation, they contain disconnections of various characters [START_REF] Rajabzadeh | The role of disconnections in deformation-coupled grain boundary migration[END_REF]. This C. R. Physique -Online first, 21st June 2021 defect has been known for decades in phase transformation and twinning [START_REF] Pond | Special interfaces: military transformations[END_REF][START_REF] Howe | The role of disconnections in phase transformations[END_REF][START_REF] Hirth | Steps, dislocations and disconnections as interface defects relating to structure and phase transformations[END_REF][START_REF] Serra | Computer simulation of the structure and mobility of twinning disclocations in H.C.P. Metals[END_REF], but has only recently been considered as the basic brick of shear-coupled GB migration [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF][START_REF] Serra | A model for simulating the motion of line defects in twin boundaries in HCP metals[END_REF], in metals with bcc and hcp structures too [START_REF] Molodov | On the twinning shear of {101 2 } twins in magnesium -Experimental determination and formal description[END_REF][START_REF] Serra | On the common topological conditions for shear-coupled twin boundary migration in bcc and hcp metals[END_REF]. One reason for this late acceptance is that the observation of moving disconnections is complicated experimentally, as in-situ TEM is almost the only way to do it [START_REF] Zhu | In situ atomistic observation of disconnection-mediated grain boundary migration[END_REF] and that many atomistic simulations considered perfect GBs where the disconnection nucleation step overshadows the other mechanisms such as their propagation. Disconnections moving concomitantly to GBs in polycrystals have been seldom observed [START_REF] Rajabzadeh | Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration[END_REF], as the obtention of a good contrast to image disconnections moving at an interface between two grains requires specific orientation conditions [START_REF] Marukawa | A new method of Burgers vector identification for grain boundary dislocations from electron microscopic images[END_REF]. The coupling factor of a given GB would therefore not be the result of its macroscopic features (misorientation, habit plane), but its atom-scaled content, the disconnections. As many different types of disconnections can populate a given GB, the resulting β will be a result of the migration of these defects. Forecasting the behavior of a given GB would then require to know its disconnection content, which is probably extremely complicated to assess because of the number of possible combinations of diverse defects that would concur to the many degrees of freedom of a GB segment.

Such gap between disconnection-containing GBs in real materials and those used in atomistic simulations that are often perfect and separating only two grains (bicrystals) may explain why the observed coupling factors (low) differ from the simulated ones (high). In fact, atomistic simulations of real GBs are only beginning, addressing the possible influence of defects present prior to deformation such as vacancies [START_REF] Larranaga | Role of sessile disconnection dipoles in shear-coupled grain boundary migration[END_REF] or immobile disconnections [START_REF] Combe | Heterogeneous disconnection nucleation mechanisms during grain boundary migration[END_REF]. A second reason for the current mismatch between the observed and simulated coupling factors may be the complex stress tensor acting locally on real GBs in small-grained metals whereas atomistic simulations involve perfectly defined and often homogenous stress states [START_REF] Combe | Shear-coupled grain-boundary migration dependence on normal strain/stress[END_REF]. These two factors may explain why many more disconnections could be activated in real metals at lower stresses and coupling factors while the nucleation step and strictly imposed strain directions favor a given disconnection that may have a large coupling factor in simulations.

Stress discrepancies are frequently observed between experimental and simulated plasticity studies [START_REF] Shin | Possible origin of the discrepancy in Peierls stresses of fcc metals: First-principles simulations of dislocation mobility in aluminum[END_REF], and the same goes for the SCGBM: the stress required to activate this mechanism remains poorly known. Atomistic simulations suggest levels of stress to move a perfect GB in bicrystals over 1 GPa in Al at 0 K [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF] or in Cu at 400 K [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. When defects are artificially introduced into these virtual GBs, the stress threshold to nucleate the first moving disconnection may be significantly lowered (about a factor 2 in [START_REF] Combe | Heterogeneous disconnection nucleation mechanisms during grain boundary migration[END_REF]), but remains very high, especially when compared to experimental values. For instance, the Aachen team, specialized in Al bicrystals, was able to trigger SCGBM (generally done around 500-600 K) typically below 1 MPa [START_REF] Gorkaya | Experimental method for true in situ measurements of shearcoupled grain boundary migration[END_REF]. This is more than 3 orders of magnitude lower, and it is arguable that the temperature and strain rate differences between virtual and actual experiments may not explain the whole story. In the polycrystal experiments reported here, there is no instrumented holder at play, so local stress levels can only be approximated using curved dislocations that are rarely observed as we take care of eliminating most of them prior to test the UFG samples. From a single spiral source operating near an opening crack-tip (zone similar to those where SCGBM occurs), we have determined that the plastic deformation occurred at a resolved shear stress of about 19 MPa, corresponding to an applied tensile stress of nearly 250 MPa if we consider the low (0.07) apparent Schmid factor, calculated in our case from the macroscopic tensile axis. This probably overestimates the applied stress as the stress tensor near a crack tip and close to grain boundaries may be strongly affected. Nonetheless, this stress value is in the order of what is found for UFG commercially pure Al at room temperature [START_REF] Tsuji | Strength and ductility of ultrafine grained aluminum and iron produced by ARB and annealing[END_REF]. In other UFG / nanocrystalline metals studies, threshold values of 250-300 MPa have been reported at room temperature [START_REF] Izadi | Grain rotations in ultrafine-grained aluminum films studied using in situ TEM straining with automated crystal orientation mapping[END_REF] (although it is not clear if a real plastic regime has been reached). More divergent values (between 100 and 400 MPa) have also been published for similar films, also at RT, with a strong effect on the grain size [START_REF] Gianola | Stress-assisted discontinuous grain growth and its effect on the deformation behavior of nanocrystalline aluminum thin films[END_REF]. In this C. R. Physique -Online first, 21st June 2021 later work, a clear GB-based plastic deformation was however observed slightly below 200 MPa. This very large range of stresses needed to trigger SCGBM, and the strong effect of temperature and grain size suggest that the mechanism is thermally activated. The temperature increase may be compensated by smaller grain size, which also suggest that diffusion is important, although we haven't considered it here, as disconnection mechanisms (glide) are conservative. The importance of diffusion was appreciable in the fact that triple junctions were observed to be both sources and sinks for disconnections [START_REF] Mompiou | Quantitative grain growth and rotation probed by in-situ TEM straining and orientation mapping in small grained Al thin films[END_REF][START_REF] Rajabzadeh | Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration[END_REF]. As diffusion and temperature directly influence the steps fluctuation in GBs [START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF], they must strongly influence the mobility of the GBs, but probably not their coupling with shear.

Conclusion

In absence of dislocations, a mechanism called shear-coupled grain boundary migration (SCGBM) seems the most capable to alleviate for plasticity in nanocrystalline and small-grained metals. Current models and many simulations suggest that the coupling factor (shear over migration distance) depends directly on the misorientation carried by a given GB, and as such reaches values on the order of tens of percents. In-situ TEM experiments where migrating GBs under stress have been monitored and quantified suggest that the coupling factor β is generally close to a few percents, typically the amount of total deformation reached during the tests. No stress concentrations resulting from GBs migrating with very different β were observed. Convergent experiments (real and computed) point to disconnections as the basic brick of GB migration, which have very fundamental consequences: as a defect that possesses a step and dislocation character, each disconnection produces its own coupling factor. The coupling factor of a GB is the net sum of the shears and step heights carried by its operating disconnections, which, in turn implies that a given GB may have different coupling factors. In short, the coupling factor of a GB is not a unique function of its misorientation [START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF]. More generally, it may be foreseen that the GB properties such as its mobility or coupling factor are not intrinsic, but depend on the properties of its defects, namely the disconnections. The density of these defects, provided they are not all geometrically necessary along a finite length of GB, may also affect its energy. A key step in the coming years to understand the plastic behavior of small-grained metals is therefore to completely incorporate the disconnections into the problem. This defect is also probably a part of the solution to predict the mechanical behavior of this class of material as disconnections and dislocations share the same formalism. GB migration, but also GB transmission could be seen as a disconnection/dislocation problem.

Figure 1 .

 1 Figure 1. Principle of shear coupling grain boundary migration. Under shear (bold blue arrows), a GB migrates by a distance h from the position on the left figure to its position on the right figure. A shear (d) occurs along the GB plane. The coupling factor defining the amplitude of the shear associated with this process is β = d/h.

Figure 2 .

 2 Figure 2. Coupling factor as a function of misorientation for [100] coherent tilt boundaries in fcc metal. The two red curves β = 2 tan(θ/2) and β = 2 tan(π/4 -θ/2) are derived from the CMST model [39]. The dots are experimental points from bicrystals experiments by Gorkaya et al. [40].

Figure 3 .

 3 Figure 3 is an example of the results obtained using the procedure that has been elaborated at CEMES to quantify the coupling factor on known grain boundaries in UFG metals. Figures3 (a), (b) are video stills extracted from a video sequence where a grain boundary migrates under stress

Figure 4 .

 4 Figure 4. Stress-assisted grain growth in Al3%Mg at 220°C. (a), (b) ASTAR-ACOM crystallography mapping of a zone ahead of a propagating crack tip before (a) and after (b) stressinduced GB migration. In (c), map of the minimum misorientation angle between the two maps (a)-(b) corrected from rigid-body rotation due to crack propagation. The black pixels represent grain interiors that haven't changed during grain growth (55% of the total surfacesee text for details).

Figure 5 .

 5 Figure 5. Video frame captured during an in-situ TEM deformation experiment of an UFG pure Al at RT. (a) Disconnections with very faint contrast move from a surface defect (below the small red arrow in a) to a triple junction (TJ, upper left). (b) Two images separated by 7.5 s are subtracted to enhance the contrast of the moving disconnections (_|-). Red arrows point to some of them.

Figure 6 .

 6 Figure 6. (a) Configuration of the simulation box containing a planar symmetric coincident Σ13 (320) tilt grain boundary, with a misorientation of 67.38°around a [001] rotation axis. The shear is applied horizontally on atoms contained in both slabs (1.5 nm thick) on top and bottom of the bicrystal. (b) Equilibrium atomic configuration of the GB, showing 2 structural units ("kite" shape). Black (red) and grey (pink) atoms are on different z coordinates. (c) Shear and elastic energy as a function of the slab displacement at 0 K. The black curve represents the elastic loading, followed by a GB jump. The red curve corresponds to the atomic configuration of the simulation box after the migration of the GB (including its unloading).

Figure 7 .

 7 Figure 7. Energy landscape during the migration of a simulated perfect coincident Σ13 (320) tilt GB at zero K under stress in an artificial Cu bicrystal containing 12 structural units (after [50]). (See text for details).

Figure 8 .

 8 Figure 8. (a) Topological construction of a disconnection joining two incompatible stepped grains G1 and G2. b G1 (resp. b G2 ) and h G1 (resp. h G2 ) are Burgers vector and step belonging to grain 1 (resp. grain 2). The resulting disconnection has a Burgers vector b = b G1 -b G2 and a step height h = (1/2)(h G1 + h G2 ). (b) Example of a disconnection (located on the green atomic row) in a Σ3[110](111) grain boundary in a fcc crystal. The disconnection is identified using a Burgers circuit between grain 1 and grain 2 and reported in the dichromatic pattern in (c). Here, the disconnection is one interplane distance high with b = 1/6[112] (in G1). In (b), the atoms in each grain are colored according to their z coordinates.

Funding. This work has been financially supported by the Agence Nationale de la Recherche through the "Rodin" project, ANR-17-CE08-0007.


H Barucq

R Djellouli
email: rabia.djellouli@csun.edu

A Saint-Guirons
email: anne-gaelle.saint-guirons@univ-pau.fr

Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries

Keywords: Absorbing boundary conditions, prolate spheroidal coordinates, Dirichlet-to-Neumann operator, scattering problems

We propose a new class of approximate local DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior ellipsoidal-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroid coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values.

Introduction

Exterior Helmholtz problems are classical mathematical models for studying scattering problems arising in many applications such as sonar, radar, geophysical exploration, nondestructive testing, etc... Despite their simplicity, this class of problems is not completely solved particularly from a numerical point of view. For example, the computation of the solutions of these problems requires first to limit it to a finite domain. This is often achieved by surrounding the given scatterer(s) (or radiator) by an artificial boundary that is located at some distance (measured in multiples of wavelength of interest) from its surface. A so-called "nonreflecting" boundary condition is then prescribed on the artificial boundary to represent the "far-field" behavior of the scattered field. The challenge here is the development of a simple but reliable as well as cost-effective computational procedure for representing the far-field behavior of the scattered. The quest for such conditions is ongoing (see, e.g., the recent review by Turkel in the book [START_REF] Turkel | Iterative methods for the exterior Helmholtz equation including absorbing boundary, conditions[END_REF]).

We propose in this work new three-dimensional approximate local DtN boundary conditions to be employed on prolate spheroid boundaries that are primary candidates for surrounding elongated scatterers. The idea for constructing such conditions is driven by several considerations chief among them the following two reasons. First, the widely-used second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel for sphericalshaped boundaries [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF] performs poorly when it is expressed in elliptical coordinates and applied to prolate spheroid boundaries in the low frequency regime [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]. The accuracy deteriorates significantly for large eccentricity values of the boundaries as observed in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]. The damping effect introduced to this condition [START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF] improves the performance for small eccentricity values. However, the modified BGT2 still performs poorly for eccentricity values larger than 0.6 in the (relatively) low frequency regime (see Figure 15 in [START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF]). Hence, there is a need for constructing local absorbing boundary conditions (ABC) that extend the range of satisfactory performance. Second, the three-dimensional approximate local DtN conditions designed for spherical-shaped boundaries [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] perform very well for low wavenumber values as reported in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]. We recall that, in R 3 , the second-order DtN boundary condition and the BGT2 condition are identical [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]. Nevertheless, using these conditions on sphericalshaped exterior boundaries when solving scattering problems by elongated scatterer often leads to larger than needed computational domains, which hampers computational efficiency. This suggests that approximate local DtN boundary conditions designed for prolate spheroidal-shaped boundaries is an attractive alternative for improving the computational performance.

Given that, this work is devoted to the construction of theses conditions and to the assessment of their performance when employed on prolate spheroidal-shaped boundaries for solving three-dimensional radiating and scattering problems. The idea of constructing three-dimensional approximate local DtN boundary conditions is not new. Indeed, as stated earlier, such conditions have been already derived for spherical-shaped boundaries [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] (see also reference [START_REF] Turkel | Iterative methods for the exterior Helmholtz equation including absorbing boundary, conditions[END_REF] and references therein). The construction procedure adopted in [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] is based on the localization of the truncated global DtN boundary condition [START_REF] Keller | Exact nonreflecting boundary conditions[END_REF]. The key ingredient of this procedure is the trigonometric identities that express high order derivatives of sine and cosine functions. However, this property is not satisfied by the angular spheroidal wave functions [START_REF] Flammer | Spheroidal Functions[END_REF]. Consequently, the procedure used in [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] is no longer applicable to the truncated global DtN boundary operator when expressed in elliptical coordinates [START_REF] Grote | On nonreflecting boundary conditions[END_REF][START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. Hence, the construction methodology we propose for deriving the class of approximate local DtN boundary conditions in prolate spheroidal coordinates can be viewed as an inverse-type approach. More specifically, we start from a Robin-type boundary condition with unknown coefficients. Unlike the case of spherical coordinates, these coefficients depend on the angles (ϕ, θ) of the prolate spheroidal coordinates. Such dependence is necessary to preserve the symmetry and local nature of the resulting boundary conditions. Then, we require that the considered condition to be an exact representation of the first modes. Consequently, the coefficients are the unique solution of a linear algebraic system.

We assess mathematically and numerically the performance of the constructed approximate local DtN boundary conditions. More specifically, we analyze the effect of low wavenumber and the eccentricity on the performance of these conditions in the case of three-dimensional radiating and scattering problems. We adopt the on-surface radiation condition formulation (OSRC) [START_REF] Kriegsmann | A new formulation of electromagnetic wave scattering using an on-surface radiation boundarycondition approach[END_REF] in order to perform analytically this investigation. We note that such formulation is not appropriate for high frequency regime as observed previously in [START_REF] Antoine | Fast approximate computation of a time-harmonic scattered field using the on-surface radiation condition method[END_REF]. The main interest in the following analyses is to evaluate the performance of the proposed approximate local DtN conditions at low wavenumber to see if relatively small computational domains can be employed in order to avoid excessive computational cost. The OSRC formulation must be viewed as an extreme case while an exterior ellipsoidal-shaped boundary surrounding an elongated scatterer would be less "demanding" on the boundary condition. The analysis herein shows that the constructed second-order local DtN condition retains a good level of accuracy in the low frequency regime for all eccentricity values of the elliptical-shaped boundaries. We must point out that we have also performed a similar investigation when these conditions are employed for solving exterior three-dimensional radiating problems. However, because of space limitation, we do not report in this paper on the results obtained in this case. These results can be found however in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF].

Preliminaries

Throughout this paper, we use the prolate spheroidal coordinates (ξ, ϕ, θ) which are related to the Cartesian coordinates (x, y, z) by the following transformation:

x = b sin ϕ cos θ, y = b sin ϕ sin θ, z = a cos ϕ

where ϕ ∈ [0, π), θ ∈ [0, 2π). The parameters a and b respectively represent the major and the minor axes and are given by:

a = f cosh ξ, b = f sinh ξ (2)
where ξ is a strictly positive real number and f is the interfocal distance and is defined by:

f = √ a 2 -b 2 (3)
We also define the eccentricity e on a prolate spheroid at ξ = ξ 0 , by:

e = 1 cosh ξ 0 = 1 - b 2 a 2 (4)
The eccentricity e characterizes the slenderness of the surface. It satisfies 0 < e < 1. Note that when e → 0, the prolate spheroid degenerates into a sphere and when e → 1, the spheroid degenerates into a line with length 2f on the z-axis.

Furthermore, any incident plane wave u inc can be expressed in this coordinate system as follows:

u inc = e ikf cosh ξ(cos ϕ cos ϕ 0 +tanh ξ sin ϕ sin ϕ 0 cos θ) (

where the wavenumber k is a positive number and ϕ 0 is the incident angle.

We consider in this work three-dimensional acoustic scattering problems by ellipsoidal-shaped scatterers. Such a restriction allows us however to adopt the OSRC formulation and set the ellipsoidal-shaped artificial boundary on the boundary of the obstacle. In addition, we assume, for simplicity, that the scatterers are sound-soft. Consequently, the acoustic scattered field u scat solution of a sound-soft elliptical-shaped scatterer can then be expressed in terms of spheroidal wave functions as follows [START_REF] Senior | Scalar diffraction by a prolate spheroid at low frequencies[END_REF]:

u scat = -2 ∞ m=0 ∞ n=m (2 -δ 0m )A mn i n R (3) mn (kf, cosh ξ)S mn (kf, cos ϕ) cos mθ (6)
where

A mn = R (1)
mn (eka, e -1 )S mn (eka, cos ϕ 0 )

N mn R (3) mn(eka, e -1 ) (7)
and N mn represents the normalization factor of the angular spheroidal wave functions [START_REF] Flammer | Spheroidal Functions[END_REF] and δ 0m is the Kronecker delta symbol. We recall that R (j) mn represents the spheroidal wave functions of the j th type (see p. in Reference [START_REF] Flammer | Spheroidal Functions[END_REF]), S mn denotes the angular wave functions.

We recall that the mnth prolate spheroidal mode is given by [START_REF] Senior | Scalar diffraction by a prolate spheroid at low frequencies[END_REF]:

u mn = R (3)
mn (kf, cosh ξ)S mn (kf, cos ϕ) cos mθ (8)

3 Three-dimensional approximate l ocal DtN boundary conditions in prolate spheroid coordinates

The three-dimensional first-order (DtN1) and second-order (DtN2) local Dirichletto-Neumann boundary conditions, defined on the ellipsoidal-shaped boundary ξ = ξ 0 , are given by:

DtN1 : ∂ u ∂ξ = √ 1 -e 2 e R 00 u (9)
DtN2 :

∂ u ∂ξ = √ 1 -e 2 (λ 01 -λ 00 ) e λ 01 R 01 -λ 00 R 00 -(R 00 -R 01 ) (eka) 2 cos 2 ϕ u + (R 00 -R 01 ) ∆ Γ u ] (10
) where λ mn represents the spheroidal eigenvalues [START_REF] Flammer | Spheroidal Functions[END_REF] and the coefficients R mn depend on the radial spheroidal wave functions of the third kind R (3) mn , the wavenumber ka, and the eccentricity e as follows:

R mn = R ′(3) mn (eka, e -1 ) R (3) mn(eka, e -1 ) (11)
and ∆ Γ denotes the Laplace Beltrami operator, which reads in prolate spheroidal coordinates (ξ, ϕ, θ) as:

∆ Γ = 1 sin ϕ ∂ ∂ϕ sin ϕ ∂ ∂ϕ + 1 sin 2 ϕ ∂ 2 ∂θ 2 (12)
We are going to shortly describe the procedure used for constructing these two local approximate DtN boundary conditions. More details can be found in [START_REF] Barucq | Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems[END_REF] and [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF].

As stated earlier in the introduction, the construction methodology we propose for deriving the class of approximate local DtN boundary conditions in prolate spheroidal coordinates can be viewed as an inverse-type approach. More specifically, we start from a Robin-type boundary condition with unknown coefficients. Hence, in the case of DtN2 condition, we set:

∂u ∂ξ = A u + B (∆ Γ -(eka) 2 cos 2 ϕ)u (13)
where A and B are constant (independent of ϕ) to be determined. Note that, unlike DtN2 boundary condition for the spherical-shaped boundaries, the coefficients of this condition depend on the angular variable ϕ. Such dependence is necessary for constructing a symmetric boundary condition since the angular spheroidal wave functions satisfied differential equation [START_REF] Flammer | Spheroidal Functions[END_REF]. Then, we observe that all radiating modes u mn given by Eq. ( 8) satisfy

∆ Γ u mn = -λ mn + (eka) 2 cos 2 ϕ u mn (14)
Hence, in order to determine the constants A and B, we assume that, at ξ = ξ 0 , we have:

∂u mn ∂ξ = A u mn +B ∆ Γ -(eka) 2 cos 2 ϕ u mn ; m = 0 and n = 0, 1 (15)
Then, using Eq. ( 14), it follows that (A, B) is the unique solution of the folllowing 2 × 2 linear system:

               A -B λ 00 = √ 1 -e 2 e R 00 A -B λ 01 = √ 1 -e 2 e R 01 (16)
where the coefficients R mn are given by Eq. ( 11). The DtN2 boundary condition given by Eq. ( 10) is a direct consequence of solving the system ( 16) and substituting the expressions of (A, B) into Eq. ( 13).

In the simpler case of DtN1 condition, we follow the same procedure which leads to figure out the coefficient C such that:

∂u 00 ∂ξ = C u 00 (17)
where u 00 has been defined at Eq. ( 8). Then we get C = √ 1e 2 e R 00 and we obtain condition [START_REF] Grote | On nonreflecting boundary conditions[END_REF].

The following two remarks are noteworthy:

• First, the boundary conditions ( 9), [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] are called local DtN conditions because they result from a localization process of the truncated global DtN boundary operator defined in [START_REF] Grote | On nonreflecting boundary conditions[END_REF][START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. The local feature of these conditions is of a great interest from a numerical view point. Indeed, the incorporation of these conditions in any finite element code introduces only mass-and stiffness-type matrices defined on the exterior boundary. The coefficients λ mn and R mn can be computed once for all at the preprocessing level. • Second, it must be pointed out that when e = 0 (the prolate spheroid becomes a sphere), conditions ( 9) and ( 10) are identical to the three-dimensional local DtN conditions designed for spherical shaped boundaries (see Eqs. ( 12)-( 13) in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]). This property can be easily established using the asymptotic behavior of the radial spheroidal wave functions of the third kind R (3) mn and the prolate spheroidal eigenvalues λ mn .

Moreover, the boundary conditions ( 9) and ( 10) satisfy, by construction, the following property that one can easily verify: Lemma 1 The first-order DtN boundary condition (DtN1) given by ( 9) is an exact representation of the first mode u 00 given by Eq.( 8). The second-order DtN boundary condition (DtN2) given by ( 10) is an exact representation of the first mode u 00 and the second mode u 01 given by Eq.( 8).

Performance analysis

In the following, we assess analytically and numerically the performance of the approximate local DtN1 and DtN2 boundary conditions given by ( 9) and [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF]. More specifically, we analyze the effect of low wavenumber ka and the eccentricity e on the performance of DtN1 and DtN2 in the case of sound-soft scattering problems. We adopt the on-surface radiation condition formulation (OSRC) [START_REF] Kriegsmann | A new formulation of electromagnetic wave scattering using an on-surface radiation boundarycondition approach[END_REF] in order to perform this investigation analytically. As in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF][START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF][START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF], we assess the performance of the ABCs DtN1 and DtN2 using the specific impedance introduced in [START_REF] Geers | Doubly asymptotic approximations for transient motions of submerged structures[END_REF][START_REF] Geers | Third-order doubly asymptotic approximations for computational acoustics[END_REF] as a practical tool for measuring the efficiency of ABCs in the context of the OSRC formulation. This non-dimensional quantity measures the effect of the truncated medium in physical terms. It provides a convenient indicator of the performance of a given approximate representation. In the elliptical coordinate system, the specific impedance can be expressed as follows:

Z = i √ 1 -e 2 ka u scat ∂ ∂ξ (u scat )| ξ=ξ 0 (18)
Therefore, the specific impedance Z exact corresponding to the exact solution for three-dimensional sound-soft acoustic scattering problems can be computed analytically using u scat = -u inc (see Eq. ( 5)), and the Fourier series given by Eq. ( 6) to evaluate ∂ ∂ξ (u scat ) | ξ=ξ 0 (see, for more details, [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF][START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF][START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]).

Mathematical results

The following lemma states the expression of the approximate specific impedances on the boundary of a sound-soft ellipsoidal-shaped scatterer at ξ = ξ 0 . These expressions can be easily derived from Eq. ( 18) by using u scat = -u inc (see Eq. ( 5)), along with substituting u = -u inc into the DtN conditions ( 9) and ( 10).

Lemma 2

The approximate specific impedance (Z DtN1 ) corresponding to the first order DtN boundary condition (DtN1) is given by:

Z DtN1 = ieka R 00 (19)
The approximate specific impedance (Z DtN2 ) corresponding to the second order DtN boundary condition (DtN2) is given by:

Z DtN2 = λ 01 -λ 00 (λ 01 R 00 -λ 00 R 01 ) -2iαka -(ka) 2 τ -(eka) 2 cos ϕ (20)
where

α = cos ϕ cos ϕ 0 + √ 1 -e 2 sin ϕ sin ϕ 0 cos θ, τ = ∂α ∂ϕ 2 + 1 sin 2 ϕ ∂α ∂θ 2 (21)
and R 00 and R 01 have been defined in Eq. [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF].

Remark 3 Note that when e = 0, that is the prolate spheroid degenerates to sphere, the approximate DtN specific impedances given by Eqs. ( 19) and (20) are identical to the ones obtained in the case of spherical-shaped scatterers for an incident angle ϕ 0 = 0 (see Eq. (136) of Lemma 5.2.1 in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]). Now, we recall (see Eq. (91), p. 3655 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]), that the asymptotic behavior of the exact specific impedance Z ex3 of the scattered field on the surface of a prolate spheroid as ka → 0 is given by:

Z ex3 ∼ (ka) 2 -ika (22)
Proposition 4.1 The asymptotic behavior of the approximate specific impedance Z DtN1 as ka → 0 is given by:

Z DtN1 ∼ (ka) 2 -ika (23)
The asymptotic behavior of the approximate specific impedance Z DtN2 as ka → 0 is given by:

Z DtN2 ∼ (1 -α) (ka) 2 -ika (24)
where α = cos ϕ cos ϕ 0 + √ 1e 2 sin ϕ sin ϕ 0 cos θ.

Proof of Proposition 4.1. This proof is detailed in [START_REF] Barucq | Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems[END_REF] and [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF].

Remark 4 First, we note that the asymptotic behavior of Z DtN1 is identical to the behavior of the exact specific impedance Z ex3 (see Eq. ( 22) and Eq. ( 23)). Second, Eq. ( 24) indicates that the asymptotic behavior of Z DtN2 depends on the eccentricity as well as on the observation angle ϕ. This dependence is comparable to the asymptotic behavior of the BGT2 approximate specific impedance (see Eq. ( 93), p. 3657 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]). Last, when e = 0, the asymptotic of both Z DtN1 and Z DtN2 are identical to the case of spheres (see Eq. ( 140) p. 47 in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]).

Illustrative numerical results

We have performed several experiments to investigate numerically the effect of the wavenumber and the slenderness of the boundary on the performance of the second-order DtN boundary condition DtN2 given by Eq. ( 10) when solving sound-soft scattering problems in the OSRC context. We have evaluated the relative error

||Z ex3 -Z DtN2 || 2 ||Z ex3 || 2 .
We have compared the results to the ones obtained with BGT2 condition (see Eq. (60) p. 3645 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]) when applied on prolate spheroid boundaries (see Figs. ( 22) to (33) in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]). We report in here a sample of the results obtained for illustration. More numerical results can be found in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. Moreover, since the impedances in this case depend on ϕ ∈ [0, π) and θ ∈ [0, 2π), we have reported 3D plots of obtained results. These results have been obtained for three eccentricity values e = 0.1 (corresponding to a prolate spheroid boundary very close to a sphere), e = 0.6 (corresponding to a "regular" prolate spheroid boundary), and e = 0.9 (corresponding to a very elongated prolate spheroid boundary), for two low wavenumber values ka = 0.1 (on the left side), ka = 1 (on the right side) and for an incident angle )). One can observe that the DtN2 boundary condition delivers an excellent level of accuracy in the low frequency regime for all eccentricity values. Indeed, the results depicted in Fig. [START_REF] Abramovitz | Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables[END_REF] indicate that the relative error on the approximate DtN2 specific impedance is always below 2%, while the BGT2 accuracy deteriorates significantly for e ≥ 0.6 (the relative error is larger than 40%).

ϕ 0 = π 4 (see Fig. ( 1

Conclusion

We have designed a new class of approximate local ABCs to be applied on ellipsoidal-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are exact for the first modes, they are easy to implement and to parallelize, and they preserve the local structure of the computational finite element scheme. The analysis reveals that in the case of the radiator, DtN2 boundary conditions, by construction, outperforms the BGT2 conditions while performing similarly for small values of e and the DtN2 boundary condition outperforms BGT2 for lower single modes. However, for higher modes both conditions performs poorly. In the case of the scattering problem, the situation is different. Indeed, the BGT2 boundary conditions performs poorly for eccentricity values e ≥ 0.6 (the relative error ≥ 40%), while the DtN2 boundary condition delivers an excellent level of accuracy (the relative error ≤ 2%) in the low frequency re gime for all eccentricity values. We are currently investigating analytically and numerically the effect of large values of the wavenumber on the accuracy of the DtN2 condition. We plan to report the results of this ongoing analysis in a separate note.

Fig. 1 . 4 , and for ka = 0. 1 (

 141 Fig. 1. Relative error of the absolute value of the specific impedance for the DtN2 (black) and the BGT2 (dark grey) for the incident angle ϕ 0 = π 4 , and for ka = 0.1 (left) and ka = 1 (right).

Acknowledgments

The authors acknowledge the support by INRIA/CSUN Associate Team Program. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of INRIA or CSUN.