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DOE OSTI2021

Anisotropy evolution of elastomeric foams during uniaxial compression measured via in-situ X-ray computed tomography

We have characterized the three-dimensional evolution of microstructural anisotropy of a family of elastomeric foams during uniaxial compression via in-situ X-ray computed tomography. Flexible polyurethane foam specimens with densities of 136, 160 and 240 kg/m 3 were compressed in uniaxial stress tests both parallel and perpendicular to the foam rise direction, to engineering strains exceeding 70%. The uncompressed microstructures show slightly elongated ellipsoidal pores, with elongation aligned parallel to the foam rise direction. The evolution of this microstructural anisotropy during deformation is quantified based on the autocorrelation of the image intensity, and verified via the mean intercept length as well as the shape of individual pores. Trends are consistent across all three methods. In the rise direction, the material remains transversely anisotropic throughout compression. Anisotropy initially decreases with compression, reaches a minimum, then increases up to large strains, followed by a small decrease in anisotropy at the largest strains as pores collapse. Compression perpendicular to the foam rise direction induces secondary anisotropy with respect to the compression axis, in addition to primary anisotropy associated with the foam rise direction. In contrast to compression in the rise direction, primary anisotropy initially increases with compression, and shows a slight decrease at large strains. These surprising non-monotonic trends and qualitative differences in rise and transverse loading are explained based on the compression of initially ellipsoidal pores. Microstructural anisotropy trends reflect macroscopic stress-strain and lateral strain response. These findings provide novel quantitative connections between three-dimensional microstructure and anisotropy in moderate density polymer foams up to large deformation, with important implications for understanding complex three-dimensional states of deformation.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Correlations Between Fission Fragment and Neutron Anisotropies in Neutron-Induced Fission

Several sources of angular anisotropy for fission fragments and prompt neutrons have been studied in neutron-induced fission reactions. These include kinematic recoils of the target from the incident neutron beam and the fragments from the emission of the prompt neutrons, preferential directions of the emission of the fission fragments with respect to the beam axis due to the population of particular transition states at the fission barrier, and forward-peaked angular distributions of pre-equilibrium neutrons which are emitted before the formation of a compound nucleus. In addition, there are several potential sources of angular anisotropies that are more difficult to disentangle: the angular distributions of prompt neutrons from fully accelerated fragments or from scission neutrons, and the emission of neutrons from fission fragments that are not fully accelerated. Here, we study the effects of the first group of anisotropy sources, particularly exploring the correlations between the fission-fragment anisotropy and the resulting neutron anisotropy. While kinematic effects were already accounted for in our Hauser-Feshbach Monte Carlo code cgmf, anisotropic angular distributions for the fission fragments and pre-equilibrium neutrons resulting from neutron-induced fission on 233,234,235,238 U, 239,241 Pu, and 237 Np have been introduced for the first time. The effects of these sources of anisotropy are examined over a range of incident neutron energies, from thermal to 20 MeV, and compared with experimental data from the Chi-Nu liquid scintillator array. The anisotropy of the fission fragments is reflected in the anisotropy of the prompt neutrons, especially as the outgoing energy of the prompt neutrons increases, allowing for an extraction of the fission-fragment anisotropy to be made from a measurement of the neutrons.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Evolution of anisotropy with saturation and its implications for the elastoplastic responses of clay rocks

Many clay rocks have distinct bedding planes. Experimental studies have shown that their mechanical properties evolve with the degree of saturation (DOS), often with higher stiffness and strength after drying. For transversely isotropic rocks, the effects of saturation can differ between the bed-normal (BN) and bed-parallel (BP) directions, which gives rise to saturation-dependent stiffness and strength anisotropy. Accurate prediction of the mechanical behavior of clay rocks under partially saturated conditions requires numerical models that can capture the evolving elastic and plastic anisotropy with DOS. In this study, we present an anisotropy framework for coupled solid deformation-fluid flow in unsaturated elastoplastic media. Here we incorporate saturation-dependent strength anisotropy into an anisotropic modified Cam-Clay (MCC) model and consider the evolving anisotropy in both the elastic and plastic responses. The model was calibrated using experimental data from triaxial tests to demonstrate its capability in capturing strength anisotropy at various levels of saturation. Through numerical simulations, we demonstrate the role of evolving stiffness and strength anisotropy in the mechanical behavior of clay rocks. Plane strain simulations of triaxial compression tests were also conducted to demonstrate the impacts of material anisotropy and DOS on the mechanical and fluid flow responses.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Illuminating Ligand Field Contributions to Molecular Qubit Spin Relaxation via T 1 Anisotropy

Electron spin relaxation in paramagnetic transition metal complexes constitutes a key limitation on the growth of molecular quantum information science. However, there exist very few experimental observables for probing spin relaxation mechanisms, leading to a proliferation of inconsistent theoretical models. Here we demonstrate that spin relaxation anisotropy in pulsed electron paramagnetic resonance is a powerful spectroscopic probe for molecular spin dynamics across a library of highly coherent Cu(II) and V(IV) complexes. Here, neither the static spin Hamiltonian anisotropy nor contemporary computational models of spin relaxation are able to account for the experimental T 1 anisotropy. Through analysis of the spin-orbit coupled wavefunctions, we derive an analytical theory for the T 1 anisotropy that accurately reproduces the average experimental anisotropy of 2.5. Furthermore, compound-by-compound deviations from the average anisotropy provide a promising approach for observing specific ligand field and vibronic excited state coupling effects on spin relaxation. Finally, we present a simple density functional theory workflow for computationally predicting T 1 anisotropy. Analysis of spin relaxation anisotropy leads to deeper fundamental understanding of spin-phonon coupling and relaxation mechanisms, promising to complement temperature-dependent relaxation rates as a key metric for understanding molecular spin qubits.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Superstructure magnetic anisotropy in Fe 3 O 4 nanoparticle chains

Magnetic anisotropy is essential for many applications of ferromagnetic/ferrimagnetic materials, including permanent magnets and magnetic recording media. Attempts have been made recently to build up 3-D nanoparticle and quantum dot assemblies, however, it is not understood yet if a nanoparticle assembly can possess high magnetic anisotropy with low anisotropic materials. In this article, we report our discovery of high magnetic anisotropy resulted from Fe 3 O 4 nanoparticle chains. We started with closely-packed nanoparticle assemblies of spherical Fe 3 O 4 nanoparticles that exhibit low magnetocrystalline anisotropy and shape anisotropy, and corresponding negligible coercivity. When the nanoparticle assemblies are compressed under pressure, they form bundles or arrays that consist of Fe 3 O 4 chains with a length scale of several hundred nanometers. Magnetic measurements show that these Fe 3 O 4 chain arrays possess a high uniaxial magnetic anisotropy (K eff ~ 2.9×10 5 J/m 3 ) and significant magnetic coercivity. Our simulations reveal that interparticle magnetic dipolar interactions contribute to this type of superstructure magnetic anisotropy. This study demonstrates the feasibility and approaches to create “patterned” high magnetic anisotropy in nanoparticle superstructures/assemblies.

Mohapatra, Jeotikanta (ORCID:0000000151968626)↗

DOE OSTI2021

The Water-Fe-Pressure dependent single-crystal elastic properties of wadsleyite: Implications for the seismic anisotropy in the upper Mantle Transition Zone

Recent seismic studies suggest an anisotropic Mantle Transition Zone (MTZ) in areas adjacent to subducted slabs. Wadsleyite is the main anisotropy contributor in the upper MTZ, therefore the interpretation of these seismic observations requires the knowledge of single-crystal elastic moduli (C ij s) and the deformation-induced lattice preferred orientation (LPO) of wadsleyite. Wadsleyite can host up to 3 wt% water in its crystal structure as point defects in the form of hydroxyl groups, however, the combined effect of water content, Fe content, and pressure on the C ij s of wadsleyite remains unclear. In this study, we measured the high-pressure single-crystal elasticity of a synthetic hydrous Fe-bearing wadsleyite (0.14(5) wt% water, Fe#=9.4, Fe 3+ /ΣFe=0.3) up to 18.2 (2) GPa. In combination with previous experimental data, we separated the effects of pressure, water, and Fe contents on the Cijs and intrinsic elastic anisotropy of wadsleyite. Overall, our results suggest that the intrinsic elastic anisotropy of wadsleyite decreases with pressure, water, and Fe contents. At 15 GPa, increasing the water content by 0.1 wt% or Fe# by 1 decreases the V P and V S anisotropy of wadsleyite by ~1.1-1.3%, and ~0.8-1.3% in average, respectively. Combined with the LPO determined in previous deformation experiments, we modeled the seismic anisotropy in the upper MTZ generated by a sub-vertical mantle flow near cold subducted slabs and a sub-horizontal mantle flow in the ambient mantle. In both scenarios, the LPO of wadsleyite leads to V SV (vertically polarized shear wave velocity) > V SH (horizontally polarized shear wave velocity). Our results suggest that wadsleyite may account for a weak anisotropic MTZ (<1%) on the global scale. Considering the fact that water decreases the elastic anisotropy but promotes LPO of wadsleyite, seismic anisotropy may not be a good water sensor.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Optical anisotropy of CsPbBr3 perovskite nanoplatelets

Abstract The two-dimensional CsPbBr 3 nanoplatelets have a quantum well electronic structure with a band gap tunable with sample thicknesses in discreet steps based upon the number of monolayers. The polarized optical properties of CsPbBr 3 nanoplatelets are studied using fluorescence anisotropy and polarized transient absorption spectroscopies. Polarized spectroscopy shows that they have absorption and emission transitions which are strongly plane-polarized. In particular, photoluminescence excitation and transient absorption measurements reveal a band-edge polarization approaching 0.1, the limit of isotropic two-dimensional ensembles. The degree of anisotropy is found to depend on the thickness of the nanoplatelets: multiple measurements show a progressive decrease in optical anisotropy from 2 to 5 monolayer thick nanoplatelets. In turn, larger cuboidal CsPbBr 3 nanocrystals, are found to have consistently positive anisotropy which may be attributed to symmetry breaking from ideal perovskite cubes. Optical measurements of anisotropy are described with respect to the theoretical framework developed to describe exciton fine structure in these materials. The observed planar absorption and emission are close to predicted values at thinner nanoplatelet sizes and follow the predicted trend in anisotropy with thickness, but with larger anisotropy than theoretical predictions. Dominant planar emission, albeit confined to the thinnest nanoplatelets, is a valuable attribute for enhanced efficiency of light-emitting devices.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2021

On the interplay between magnetic field and anisotropy in holographic QCD

We investigate the combined effects of anisotropy and a magnetic field in strongly interacting gauge theories by the gauge/gravity correspondence. Our main motivation is the quark-gluon plasma produced in off-central heavy-ion collisions which exhibits large anisotropy in pressure gradients as well as large external magnetic fields. We explore two different configurations, with the anisotropy either parallel or perpendicular to the magnetic field, focusing on the competition and interplay between the two. A detailed study of the RG flow in the ground state reveals a rich structure where depending on which of the two, anisotropy or magnetic field, is stronger, intermediate geometries with approximate AdS 4 × $\mathbb{R}$ and AdS 3 × $\mathbb{R}$ 2 factors arise. This competition is also manifest in the phase structure at finite temperature, specifically in the dependence of the chiral transition temperature on anisotropy and magnetic field, from which we infer the presence of inverse magnetic and anisotropic catalyses of the chiral condensate. Finally, we consider other salient observables in the theory, including the quark-antiquark potential, shear viscosity, entanglement entropy and the butterfly velocity. We demonstrate that they serve as good probes of the theory, in particular, distinguishing between the effects of the magnetic field and anisotropy in the ground and plasma states. We also find that the butterfly velocity, which codifies how fast information propagates in the plasma, exhibits a rich structure as a function of temperature, anisotropy and magnetic field, exceeding the conformal value in certain regimes.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Strong Magnetocrystalline Anisotropy Arising from Metal–Ligand Covalency in a Metal–Organic Candidate for 2D Magnetic Order

Layered metal–organic frameworks are promising candidates for new two-dimensional magnets, as their synthetic programmability of these materials can provide a route to diverse structural and electronic properties. However, such framework materials typically lack the heavy elements that engender magnetocrystalline anisotropy in the monolayer ferromagnets reported to date. Alternative sources of magnetic anisotropy are therefore needed in these materials. Here, we report the synthesis of single crystals of the framework material (NMe 4 ) 2 [Fe 2 L 3 ] (H 2 L = 3,6-dichloro-2,5-dihydroxybenzoquinone) and evaluate the angular dependence of its magnetic properties. Oriented-crystal magnetization measurements reveal strong uniaxial anisotropy, where the easy axis is aligned with the crystallographic c axis. While the spin carriers of this structure are isotropic $S = ^5/_2$ Fe III metal centers and $S = ^1/_2$ organic linkers, the anisotropy energy of the framework material is comparable to that of reported 2D ferromagnets. Density functional theory calculations indicate that the observed magnetocrystalline anisotropy arises from ligand-to-metal charge transfer that enhances the magnetic anisotropy of the otherwise isotropic Fe centers, suggesting that metal–ligand covalency can be utilized as a general additive for the development of 2D magnets. These results in this study show the possibility for (NMe 4 ) 2 [Fe 2 L 3 ] to retain magnetic order down to the 2D monolayer limit. In addition, the combination of large magnetic anisotropy and semiconducting character in (NMe 4 ) 2 [Fe 2 L 3 ] highlights its potential as a new 2D magnetic semiconductor.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Intrinsic Elastic Anisotropy of Westerly Granite Observed by Ultrasound Measurements, Microstructural Investigations, and Neutron Diffraction

Westerly granite (WG) has been generally accepted as an isotropic homogeneous rock. Here, we return to WG and observe significant elastic anisotropy using multidirectional ultrasonic sounding on spherical samples at pressures up to 400 MPa. Thermal treatment of WG leads to formation of microcracks that reduce elastic wave velocities and increase its elastic anisotropy. The 3D distribution of P-wave velocities at low pressure is close to orthorhombic symmetry. Application of hydrostatic pressure closes most of thermally induced microcracks and decreases elastic anisotropy of WG, but at high pressure the anisotropy is practically reversed compared to low pressure: maximum P-wave velocity direction at low pressures is near minimum velocity direction at high pressure and vice versa. To understand this effect, microstructures of the rock were investigated by optical and scanning electron microscopy. Preferred orientations of four major rock-forming minerals—quartz, orthoclase, plagioclase, and biotite—were measured by time-of-flight neutron diffraction, which confirms significant crystal alignment. All these data were used to numerically model anisotropic elastic properties of WG. It is shown that WG possesses weak intrinsic elastic anisotropy related mainly to the preferred orientation of feldspars formed during igneous crystallization. Observed microcracks are mostly related to the cleavage planes of feldspars and biotite, and thus also demonstrate preferred orientation. Higher preheating temperatures produce larger quantity of longer microcracks. A numerical model shows that these microcracks act against the weak intrinsic elastic anisotropy of WG, and define the elastic anisotropy at low pressures.

58 GEOSCIENCES↗

DOE OSTI2022

In-situ tunable giant electrical anisotropy in a grating gated AlGaN/GaN two-dimensional electron gas.

Materials with in-plane electrical anisotropy have great potential for designing artificial synaptic devices. However, natural materials with strong intrinsic in-plane electrical anisotropy are rare. We introduce a simple strategy to produce extremely large electrical anisotropy via grating gating of a semiconductor two-dimensional electron gas (2DEG) of AlGaN/GaN. We show that periodically modulated electric potential in the 2DEG induces in-plane electrical anisotropy, which is significantly enhanced in a magnetic field, leading to an ultra large electrical anisotropy. This is induced by a giant positive magnetoresistance and a giant negative magnetoresistance under two orthogonally oriented in-plane current flows, respectively. This giant electrical anisotropy is in situ tunable by tailoring both the grating gate voltage and the magnetic field. Our semiconductor device with controllable giant electrical anisotropy will stimulate new device applications, such as multi-terminal memtransistors and bionic synapses.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Solar Wind Turbulence from 1 to 45 au. III. Anisotropy of Magnetic Fluctuations in the Inertial Range Using Voyager and ACE Observations

We examine both Voyager and Advanced Composition Explorer magnetic field measurements at frequencies that characterize the inertial range and evaluate the anisotropy of the fluctuations as they relate to both the compressive component and underlying wavevector anisotropy of the turbulence. The magnetic fluctuation anisotropy as it relates to the compressive component is directly dependent upon both the plasma beta of the thermal proton component and the ratio of magnetic fluctuation magnitude to the strength of the mean magnetic field. This has been seen before at 1 au. The magnetic fluctuation anisotropy in the plane perpendicular to the mean magnetic field, which is a measure of the anisotropy of the underlying wavevector distribution, should depend on the angle between the mean magnetic field and the radial direction and should be confined to values between one and the index of the power spectrum, which is typically 5/3. Our results show that the average of this anisotropy exceeds the value of the spectral index and is out of bounds with the theory. Although the results are suggestive of past analyses, we find that spherical expansion of the turbulence may offer at least a partial explanation of the apparent amplification of this measured anisotropy.

heliosphere↗

DOE OSTI2026

Characterizing the Velocity Anisotropy of the Milky Way’s Stellar Halo

Abstract Modeling the Milky Way stellar halo requires well-determined density and velocity anisotropy profiles. However, it has been challenging to gather a large sample of stars with six-dimensional data that extend beyond 40 kpc to map the outer halo. Our work investigates the velocity anisotropy in the Milky Way stellar halo with more than 10,000 blue horizontal-branch stars, combining Gaia astrometric data and spectroscopic data from SEGUE, DESI, and LAMOST. This large sample allows us to obtain a detailed profile of up to ∼50 kpc. Radial velocities are predominant in the inner halo (<30 kpc), and the anisotropy presents a smooth decrease before rapidly dropping to negative values, becoming dominated by tangential dispersion velocities. Removing the main known accreted structures of the Milky Way makes the anisotropy profile radially dominated at all radii. Our profile clearly shows an increase in the anisotropy from the center of the Galaxy, in accordance to the simulations. We also investigate the correlation of anisotropy with metallicity and with color. The lack of correlation between metallicity and anisotropy in our clean sample reinforces that this relation is driven by merger events. The initial exploration with color indicates a relation between kinematics and age, showing that older stars are dynamically colder and present less radial orbits than younger stars in the inner halo.

Barbosa, F. O. [Universidade de São Paulo] (ORCID:↗

DOE OSTI2023

Large ordered moment with strong easy-plane anisotropy and vortex-domain pattern in the kagome ferromagnet Fe 3 Sn

We report the magnetic anisotropy of kagome bilayer ferromagnet Fe 3 Sn probed by the bulk magnetometry and magnetic force microscopy (MFM) on high-quality single crystals. The dependence of magnetization on the orientation of the external magnetic field reveals strong easyplane magnetocrystalline anisotropy and anisotropy of the saturation magnetization. The leading magnetocrystalline anisotropy constant shows a monotonous increase from $\mathrm{K_1$≈ –1.0 \times 10^6 J/m^3}$ at 300 K to $\mathrm{–1.3 \times 10^6 J/m^3}$ at 2 K. Our ab initio electronic structure calculations yield the value of total magnetic moment of 7.1$μ_Β$/$f.u.$ and a magnetocrystalline anisotropy energy density of –0.57 $\mathrm{meV/}$$f.u.$ $\mathrm{(–1.62 \times 10^6J/m^3)}$ both being in reasonable agreement with the experimental values. The MFM imaging reveals micrometer-scale magnetic vortices with weakly pinned cores that vanish at the saturation field of ~3T applied perpendicular to the kagome plane. The observed vortex-domain structure is well reproduced by the micromagnetic simulations, using the experimentally determined value of the anisotropy and exchange stiffness.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Exploring magnetic anisotropy and robustness of the J eff = 1/2 state under substantial orthorhombic distortion in Sr 2 IrO 4 thin films

Here, we present a comprehensive study revealing the intricate interplay of the magnetic anisotropy and orthorhombic distortion in thin films of Sr 2 IrO 4 through a Ca 3 Ru 2 O 7 substrate. By inducing a pronounced orthorhombic distortion along the direction of oxygen octahedral edges, we effectively modulated the uniaxial magnetic anisotropy in the system. Remarkably divergent responses along the easy and hard magnetic axes were unveiled through x-ray magnetic circular dichroism (XMCD) measurements under magnetic fields. Specifically, the spin flop transition observed when the magnetic field aligns with the hard axis allows us to estimate the magnetic anisotropy energy, which is around 14.2 µeV, close to that estimated from the single magnon peak measured via Raman spectroscopy. The observed anisotropy energy remains notably lower than the linear estimates derived from the strain-anisotropy energy relationship outlined in H.-H. Kim et al., Nat. Commun. 13, 6674 (2022). This underscores the enduring preservation of the isotropic character of the J eff = 1/2 states. This is also supported by the negligible XMCD intensity ratio at the L 2 edge compared to that of the L 3 edge. Furthermore, the branching ratio determined from x-ray absorption spectroscopy shows that the expectation value of the spin-orbit coupling is similar to that of bulk Sr 2 IrO 4 single crystals. Our findings indicate that even under a substantial anisotropic biaxial distortion, Sr 2 IrO 4 remains remarkably proximate to the J eff = 1/2 state. This study not only provides valuable information in understanding the interplay between magnetic anisotropy and strain but also the robustness of the J eff = 1/2 state under octahedral distortion within materials exhibiting emergent quantum phenomena.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Giant magnetic anisotropy of Pb atoms in 3 d -based magnets

Electronic structure analysis is performed to study the properties of several Pb-containing 3⁢ d intermetallics. Our study reveals that binary metastable C⁢o 3 ⁢Pb and F⁡e 3 ⁢Pb intermetallic compounds exhibit very attractive intrinsic magnetic properties. We primarily focus on the magnetic anisotropic properties arising from the high spin-orbit coupling of the Pb atom. Decomposing the total anisotropy into intra- and interatomic contributions reveals a significant deviation from single-ion anisotropy model with strong symmetric anisotropic pair interactions present. Furthermore, we consider magnetic properties of ternary Pb-based 3⁢⁢ d intermetallics, which recently have been reported as stable or metastable. Giant magnetic anisotropy is found on Pb atoms in these systems. The origin of such strong anisotropy in L⁢a 18⁢ C⁢o 28 ⁢P⁢b 3 appears from two sources: spin-orbit and interelectronic Breit couplings. Additionally, the significance of Breit interaction for magnetic anisotropy in bulk systems is reported. It is expected that Breit coupling-induced anisotropy is dominating in magnetic Pb-based magnets with lower dimensionality including thin films.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2020

Extreme Elasticity Anisotropy in Molecular Glasses

Abstract Glasses are generally assumed to be isotropic and there are no literature reports of elastic anisotropy for molecular glasses. However, as glasses formed by physical vapor deposition can be structurally anisotropic, it is of interest to investigate the elastic anisotropy in these materials. Micro‐Brillouin light spectroscopy is used in several experimental geometries to determine the elastic stiffness tensors of three glass films of itraconazole vapor‐deposited at substrate temperatures ( T sub ) of 330, 315, and 290 K, respectively. Significant elastic anisotropy is observed and, in these glasses, the elastic anisotropy shows a strong correlation with the molecular orientation. The out‐of‐plane and in‐plane Young's moduli of the high T sub (330 K) sample, which features a predominantly vertical molecular orientation, exhibit a high anisotropy ratio of 2.2. The observed elastic anisotropy is much larger than those previously observed in liquid crystals and even many crystalline solids.

Cang, Yu↗

DOE OSTI2026

Overlapping Schwarz Methods Are Not Anisotropy‐Robust Multigrid Smoothers

We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. Here, a natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter 𝒪⁡(𝜖 −1/2 ) for anisotropy ratio 𝜖 ∈(0,1] .

97 MATHEMATICS AND COMPUTING↗