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1
Diner, Çağrı. "The Structure of Moment Tensors in Transversely Isotropic Focal Regions." Bulletin of the Seismological Society of America 109, no. 6 (September 24, 2019): 2415–26. http://dx.doi.org/10.1785/0120180316.
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Abstract Full moment tensor inversion has become a standard method for understanding the mechanisms of earthquakes as the resolution of the inversion process increases. Thus, it is important to know the possible forms of non–double‐couple (non‐DC) moment tensors, which can be obtained because of either the different source mechanisms or the anisotropy of the focal regions. In this study, the form of the moment tensors of seismic sources occurring in transversely isotropic (TI) focal regions is obtained using the eigendecomposition of the elasticity tensor. More precisely, a moment tensor is obtained as a linear combination of the eigenspaces of TI elasticity tensor in which the coefficients of the terms are the corresponding eigenvalues multiplied with the projection of the potency tensor onto the corresponding eigenspaces. Moreover, the eigendecomposition method is also applied to obtain the three different forms of moment tensors in isotropic focal regions, in particular, for the shear source, tensile source, and for any type of potency tensor whose rank is three. This linear algebra point of view makes the structure of the moment tensors more apparent; for example, a shear source tensor is an eigenvector of isotropic elasticity tensor, and hence the resulting moment tensor is proportional to its shear source tensor. Moreover, a geometric interpretation for the scalar seismic moment, which is the norm of the moment tensor, for anisotropic focal regions is achieved through the eigendecomposition method. This method also gives a simple way to quantify the percentage of the isotropic component of the moment tensor of shear sources in TI focal regions. Hence, the complexities in the moment tensor introduced by the anisotropy of the focal region and by the source mechanism can be differentiated.
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Boitz, Nepomuk, Anton Reshetnikov, and Serge A. Shapiro. "Visualizing effects of anisotropy on seismic moments and their potency-tensor isotropic equivalent." GEOPHYSICS 83, no. 3 (May 1, 2018): C85—C97. http://dx.doi.org/10.1190/geo2017-0442.1.
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Radiation patterns of earthquakes contain important information on tectonic strain responsible for seismic events. However, elastic anisotropy may significantly impact these patterns. We systematically investigate and visualize the effect of anisotropy on the radiation patterns of microseismic events. For visualization, we use a vertical-transverse-isotropic (VTI) medium. We distinguish between two different effects: the anisotropy in the source and the anisotropy on the propagation path. Source anisotropy mathematically comes from the matrix multiplication of the anisotropic stiffness tensor with the source strain expressed by the potency tensor. We analyze this effect using the corresponding radiation pattern and the moment tensor decomposition. Propagation anisotropy mathematically comes from the deviation between the polarization and the propagation direction of a quasi P-wave in an anisotropic medium. We investigate both effects separately by either assuming the source to be anisotropic and the propagation to be isotropic or vice versa. We find that both effects have a significant impact on the radiation pattern of a pure-slip source. Finally, we develop an alternative visualization of source mechanisms by plotting beach balls proportional to their potency tensors. For this, we multiply the potency tensor with an isotropic elasticity tensor having the equivalent shear modulus [Formula: see text] and [Formula: see text]. In this way, we visualize the tectonic deformation in the source, independently of the rock anisotropy.
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Yao, Yi, and Yibo Wang. "Seismic radiation analyses in anisotropic media based on general dislocation source model." Journal of Geophysics and Engineering 18, no. 2 (April 2021): 231–40. http://dx.doi.org/10.1093/jge/gxab006.
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Abstract Anisotropy affects the focal mechanism and makes it complicated. A shear motion generates a pure double-couple (DC) source in isotropic media. While in anisotropic media, it will produce non-DC components, which contain isotropic (ISO) and compensated linear vector dipole (CLVD) components. Besides, coupled with the diversity of fault motion, the source may become extremely complicated. In this paper, the seismic moment tensor is obtained based on the dislocation model, and then a variety of analyses are performed with the moment tensor, including moment tensor decomposition, radiation pattern, radiated energy ratio and seismic propagation characteristics. Since the anisotropy of the medium also influences seismic wave propagation, a hypothesis is made that the source region is minimal and anisotropic, but the propagation path is isotropic. The research gives some interesting conclusions. It is found that the anisotropy mainly affects the focal mechanism under low slope angle while high slope angle has little effect on the polarity. In terms of the moment tensor decomposition, if only one of ISO or CLVD exists, it can be asserted that the source region is anisotropic because ISO components are accompanied by CLVD components in isotropy media. As for the DC component, the results indicate it is one of the most important factors for determining the ratio of radiant energy. This paper presents some valuable findings of the focal mechanism of the general dislocation source under anisotropy, which helps to recognise the source characteristics of the earthquake and build solid foundations for the subsequent inversion of the focal mechanism.
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Menke, William, and Joshua B. Russell. "Non-Double-Couple Components of the Moment Tensor in a Transversely Isotropic Medium." Bulletin of the Seismological Society of America 110, no. 3 (May 5, 2020): 1125–33. http://dx.doi.org/10.1785/0120190319.
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ABSTRACT The non-double-couple (non-DC) components of the moment tensor provide insight into the earthquake processes and anisotropy of the near-source region. We investigate the behavior of the isotropic (ISO) and compensated linear vector dipole (CLVD) components of the moment tensor for shear faulting in a transversely ISO medium with an arbitrarily oriented symmetry axis. Analytic formulas for ISO and CLVD depend on the orientation of the fault relative to the anisotropy symmetry axis as well as three anisotropic parameters, which describe deviations of the medium from isotropy. Numerical experiments are presented for the preliminary reference Earth model. Both ISO and CLVD components are zero when the axis of symmetry is within the fault plane or the auxiliary plane. For any orientation in which the ISO component is zero, the CLVD component is also zero, but the opposite is not always true (e.g., for strong anisotropy). The relative signs of the non-DC components of neighboring earthquakes may help distinguish source processes from source-region anisotropy. We prove that an inversion of ISO and CLVD components of a set of earthquakes with different focal mechanisms can uniquely determine the orientation and strength of anisotropy. This study highlights the importance of the ISO component for constraining deep slab anisotropy and demonstrates that it cannot be neglected.
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Menke, William. "Analytic Solution to the Moment Tensor—Anisotropy Inverse Problem." Pure and Applied Geophysics 177, no. 7 (March 23, 2020): 3119–33. http://dx.doi.org/10.1007/s00024-020-02468-2.
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Yahia, Eman, William Schupbach, and Kannan N. Premnath. "Three-Dimensional Central Moment Lattice Boltzmann Method on a Cuboid Lattice for Anisotropic and Inhomogeneous Flows." Fluids 6, no. 9 (September 10, 2021): 326. http://dx.doi.org/10.3390/fluids6090326.
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Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.
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Вилков, Е. А., Г. М. Михайлов, С. А. Никитов, А. Р. Сафин, М. В. Логунов, V. N. Korenivskii, С. Г. Чигарев, and Л. А. Фомин. "Динамика пространственно неоднородной спиновой поляризации неравновесных электронов проводимости в магнитных переходах." Физика твердого тела 61, no. 6 (2019): 1021. http://dx.doi.org/10.21883/ftt.2019.06.47674.279.
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AbstractThe equation of the dynamics of the magnetic moment motion that has been averaged over the ensemble of nonequilibrium spin-injected electrons in a ferromagnetic transition in the presence of exchange interaction, interaction with an external electromagnetic field, and a thermostat has been obtained taking into account the spatial inhomogeneity of the charge carrier distribution. It has been shown that taking into account the spatial inhomogeneity of the charge carrier distribution in the equation of the dynamics of the magnetic moment motion allows for describing the various processes of electron transfer in the magnetic transition. The probability of quantum transitions of electrons with opposite spin directions, which determine spin relaxation in interaction with a thermostat, has been estimated. It has been shown that the anisotropy of the radiation medium is determined not only by the anisotropy of the sd -exchange tensor, but also by the additional anisotropy that is caused by the electron impulse derivatives of this tensor. The considered phenomena have a great potential for the detection of new effects and development of new devices based on them, including compact tunable radiation sources in the terahertz frequency range, which is obviously difficult to generate.
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Saha, Saikat, and Meheboob Alam. "Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method." Journal of Fluid Mechanics 757 (September 19, 2014): 251–96. http://dx.doi.org/10.1017/jfm.2014.489.
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AbstractThe non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $, pressure $p$, first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $, (ii) restitution coefficient $e$, (iii) dimensionless shear rate $R$, (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle $\phi $ between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier–Stokes order, recovering the known exact transport coefficients from the present analysis in the limit $\eta ,\phi \to 0$, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of $R$, $\eta $ and $\phi $ with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ($e \ll 1$). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$, $\mu $ and ${\mathcal{N}}_1$ over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ($e\sim 0.9$). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.
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Bondarev, I. V., and S. A. Kuten. "Positronium Quadrupole Interactions in Crystals." Zeitschrift für Naturforschung A 49, no. 1-2 (February 1, 1994): 439–44. http://dx.doi.org/10.1515/zna-1994-1-266.
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Abstract It is shown that, due to the hyperfine interaction between the electron and the positron, positronium atoms in a crystal can acquire an effective quadrupole moment and a tensor polarizability. In such a case the effective quadrupolar interaction with intracrystalline fields leads to a quadrupolar splitting of the triplet level of positronium and also to an anisotropy of its magnetic quenching in the presence of an external magnetic field. The possibilities of observating this anisotropy experimentally are discussed.
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Brot, Claude. "Liquides de toupies symétriques: leur tenseur de polarisabilité optique intrinsèque via deux expériences sous champ statique: CH3CN." Canadian Journal of Chemistry 68, no. 9 (September 1, 1990): 1490–93. http://dx.doi.org/10.1139/v90-228.
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NMR line splitting under a strong static electric field (EF-NMR) allows one to measure the average alignment (second spherical harmonic) of the molecule studied under the influence of the field. This average alignment (per unit squared field) can be formulated theoretically by an expression involving the permanent dipole moment, the anisotropy of the static polarizability, etc. and possibly orientational intercorrelations within the liquid. On the other hand the Kerr constant per molecule can be represented by the same expression, multiplified by a factor that is independent of any static electrical property of the molecule (for a gas this factor would be simply proportional to the anisotropy of the optical polarizability). Consequently the ratio of the above experimental results contains information bearing only on the optical properties. The formalism of the "tensor of polarizability increment" developed by the author for liquids made up of ellipsoidal molecules is employed. It is shown that by combining the above ratio with the constitutive equation for the refractivity of the liquid, the two distinct principal elements of the tensor of polarizability increment of the symmetric top molecules constituting the liquid are immediately obtained. Using then the molecular volume, a van der Waals estimate for the elongation of the molecule, and the index of refraction of the liquid, one can calculate the elements of the optical polarizability tensor invacuo. The method is illustrated by neat acetonitrile for which the necessary experimental data exist in the literature. One finds, in units 1040 C2 m2 J−1 5.06 for the mean polarizability and 2.52 for its anisotropy. A published experimental study using depolarized light scattering in the gas phase indicates that the corresponding figures are 4.96 and 2.49 respectively at λ = 633 nm. Finally and accessorily, it is suggested that dipolar intercorrelations are weak in acetonitrile. Indeed, using an analogous formalism for the static permittivity of the liquid, a value for the permanent moment of the molecule is deduced. This value is within 1% of the experimental gas value. Keywords: polarizability tensors, liquids, Kerr constant, NMR.
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Böttcher, O., and D. H. Sutter. "The 14N Nuclear Quadrupole Coupling Tensors, the Tensors of the Molecular Magnetic Susceptibility and the Molecular Electric Quadrupole Moment in Pyrazole: A High Resolution Rotational Zeeman Effect Study." Zeitschrift für Naturforschung A 45, no. 11-12 (December 1, 1990): 1248–58. http://dx.doi.org/10.1515/zna-1990-11-1205.
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AbstractHigh resolution zero field and Zeeman rotational spectra of 1 D-pyrazole have been studied by microwave Fourier-transform spectroscopy. The zero field hfs patterns allowed to improve the quadrupole coupling constants for both 14N nuclei. From the high field Zeeman multiplets the diagonal elements of the g-tensor were obtained as gaa= -0.1178(2),.gbb=-0.0762(2) and gcc = 0.0608 (2). The two independent components of the molecular magnetic susceptibility anisotropy in units of 10-6 erg G-2 mole- 1 are 2 ξaa - ξbb - ξcc= 52.69(32) and 2 ξbb - ξcc - ξaa = 39.32(29) were, a, b, c denote the molecular principal inertia axes. From these values the components of the molecular electric quadrupole moment tensor in units of 10-26 esu cm2 follow as Qaa = 5.84(22), Qbb= -0.58 (21) and Qcc= -5.27(38). Comparison with corresponding values for the undeuterated species leads to the complete tensors including their orientation with respect to the nuclear frame.
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Krause, H., and D. H. Sutter. "The Molecular 0-Tensor, the Magnetic Susceptibility Anisotropy, and the Molecular Electric Quadrupole Moment Tensor of Monofluoroacetonitrile, a Rotational Zeeman Effect Study." Zeitschrift für Naturforschung A 46, no. 12 (December 1, 1991): 1049–54. http://dx.doi.org/10.1515/zna-1991-1209.
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Abstract The rotational Zeeman effect of H2F12C12C14N was observed at fields up to 20 kG (2 Tesla). The observed vibronic ground state expectation values for the molecular ^-values, the magnetic susceptibility anisotropies and the molecular electric quadrupole moments, all referred to the molecular principal inertia axes system, are: gaa= -0.03572 (11), gbb= -0.03438 (7), gcc= -0.03988 (6). 2ξaa- ξbb - ξcc = - 14.58 (10) • 10-6 erg/(G2 mole), 2ξaa- ξbb - ξcc= 1.60 (11) • 10-6 erg/(G2 mole), Qaa= -9.13 (6) DÅ, Qbb = 4M (7) DÅ, and Qcc = 4.96 (9) DÅ, respectively. The latter are in close agreement with the results of a restricted Hartree Fock self consistent field calculation with a basis of TZVP quality, which was carried out at the partial restructure determined earlier. Therefore the RHF/TZVP-value for the second electronic moment perpendicular to the heavy atom plane, <0|Σ c2c|0)RHF, was used as additional input to predict the molecular bulk susceptibility and the individual components of the magnetic susceptibility tensor
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Yang, S. L., B. D. Peschke, and K. Hanjalic. "Second-Moment Closure Model for IC Engine Flow Simulation Using Kiva Code1." Journal of Engineering for Gas Turbines and Power 122, no. 2 (August 31, 1999): 355–63. http://dx.doi.org/10.1115/1.483213.
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The flow and turbulence in an IC engine cylinder were studied using the SSG variant of the Reynolds stress turbulence closure model. In-cylinder turbulence is characterized by strong turbulence anisotropy and flow rotation, which aid in air-fuel mixing. It is argued that solving the differential transport equations for each turbulent stress tensor component, as implied by second-moment closures, can better reproduce stress anisotropy and effects of rotation, than with eddy-viscosity models. Therefore, a Reynolds stress model that can meet the demands of in-cylinder flows was incorporated into an engine flow solver. The solver and SSG turbulence model were first successfully tested with two different validation cases. Finally, simulations were applied to IC-engine like geometries. The results showed that the Reynolds stress model predicted additional flow structures and yielded less diffusive profiles than those predicted by an eddy-viscosity model. [S0742-4795(00)00101-0]
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Meyer, V., and D. H. Sutter. "A Broadband Microwave Fourier Transform Spectrometer, Especially Designed for Stark Effect Investigations of Almost Nonpolar Molecules; the Electric Dipole Moment and the Anisotropy in the Static Electric Polarizability Tensor of 1,1-Dideuteroallene, D2C = C = CH2." Zeitschrift für Naturforschung A 48, no. 5-6 (June 1, 1993): 725–32. http://dx.doi.org/10.1515/zna-1993-5-626.
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Abstract A bridge type microwave Fourier transform spectrometer, equipped with flat oversized Stark cells and an operational range from 8 GHz to 40 GHz is described. As application we report the experimental determination of the vibronic ground state dipole moment and of the anisotropy in the static polarizability of 1,1-dideuteroallene. Our experimental values are: μa = 0.0053(2) D for the dipole moment, and (α∥ - α⊥) = 4.26(6) • 10-24 cm3 , for the polarizability anisotropy.
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Cho, Junghun, Dong Zhou, Youngwook Kee, Pascal Spincemaille, and Yi Wang. "Quantitative Susceptibility Mapping of Magnetic Quadrupole Moments." Concepts in Magnetic Resonance Part A 2019 (August 6, 2019): 1–14. http://dx.doi.org/10.1155/2019/7174937.
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We modeled the magnetic field up to the quadrupole term to investigate not only the average susceptibility (dipole), but also the susceptibility distribution (quadrupole) contribution. Expanding the magnetic field up to the 2nd order provides the quadrupole (0th: monopole, 1st: dipole). Numerical simulations were performed to investigate the quadrupole contribution with subvoxel nonuniformity. Conventional dipole and our dipole + quadrupole models were compared in the simulation, the phantom and human brain. Furthermore, the quadrupole field was compared with the anisotropic susceptibility field in the dipole tensor model. In a nonuniformity case, numerical simulations showed a nonnegligible quadrupole field contribution. Our study showed a difference between the two methods in the susceptibility map at the edges; both the phantom and human studies showed sharper structural edges with the dipole + quadrupole model. Quadrupole moments showed contrast mainly at the structural boundaries. The quadrupole moment field contribution was smaller but nonnegligible compared to the anisotropic susceptibility contribution. Nonuniform and uniform source distributions can be separately considered by quadrupole expansion, which were mixed together in the dipole model. In the presence of nonuniformity, the susceptibility maps may be different between the two models. For a comprehensive field model, the quadrupole might need to be considered along with susceptibility anisotropy and microstructure effects.
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Meyer, V., D. H. Sutter, and B. Vogelsanger. "Molecular g-tensor, molecular magnetic susceptibility anisotropy, and molecular electric quadrupole moment tensor of allene from a rotational Zeeman effect study of 1,1-dideuteroallene." Journal of Molecular Spectroscopy 148, no. 2 (August 1991): 436–46. http://dx.doi.org/10.1016/0022-2852(91)90399-u.
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Caimmi, R. "R3 fluids." Serbian Astronomical Journal, no. 174 (2007): 13–33. http://dx.doi.org/10.2298/saj0774013c.
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The current paper is aimed at getting more insight on three main points concerning large-scale astrophysical systems, namely: (i) formulation of tensor virial equations from the standpoint of analytical mechanics; (ii) investigation on the role of systematic and random motions with respect to virial equilibrium configurations; (iii) determination of extent to which systematic and random motions are equivalent in flattening or elongating the shape of a mass distribution. The tensor virial equations are formulated regardless of the nature of the system and its constituents, by generalizing and extending a procedure used for the scalar virial equations in presence of discrete subunits (Landau and Lifchitz 1966). In particular, the self potential-energy tensor is shown to be symmetric with respect to the exchange of the indices, (Epot)pq = (Epot)qp. Then the results are extended to continuous mass distributions. The role of systematic and random motions in collisionless, ideal, self-gravitating fluids is analysed in detail including radial and tangential velocity dispersion on the equatorial plane, and the related mean angular velocity, ?, is conceived as a figure rotation. R3 fluids are defined as ideal, self-gravitating fluids in virial equilibrium, with systematic rotation around a principal axis of inertia, taken to be x3. The related virial equations are written in terms of the moment of inertia tensor, Ipq, the self potential-energy tensor, (Epot)pq, and the generalized anisotropy tensor, ?pq (Caimmi and Marmo 2005, Caimmi 2006a). Additional effort is devoted to the investigation of the properties of axisymmetric and triaxial configurations. A unified theory of systematic and random motions is developed for R3 fluids, taking into consideration imaginary rotation (Caimmi 1996b, 2006a), and a number of theorems previously stated for homeoidally striated Jacobi ellipsoids (Caimmi 2006a) are extended to the more general case of R3 fluids. The effect of random motion excess is shown to be equivalent to an additional real or imaginary rotation, respectively, inducing flattening (along the equatorial plane) or elongation (along the rotation axis). Then it is realized that a R3 fluid always admits an adjoint configuration with isotropic random velocity distribution. In addition, further constraints are established on the amount of random velocity anisotropy along the principal axes, for triaxial configurations. A necessary condition is formulated for the occurrence of bifurcation points from axisymmetric to triaxial configurations in virial equilibrium, which is independent of the anisotropy parameters. A particularization of general relations is made to the special case of homeoidally striated Jacobi ellipsoid, and some previously known results (Caimmi 2006a) are reproduced. .
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MEYER, V., D. H. SUTTER, and B. VOGELSANGER. "ChemInform Abstract: Molecular g-Tensor, Molecular Magnetic Susceptibility Anisotropy, and Molecular Electric Quadrupole Moment Tensor of Allene from a Rotational Zeeman Effect Study of 1,1-Dideuteroallene." ChemInform 22, no. 42 (August 22, 2010): no. http://dx.doi.org/10.1002/chin.199142046.
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Dovhaliuk, Volodymyr, Y. Chоvniuk, M. Shyshyna, and A. Moskvitina. "Functional analysis of thermal conductivity and viscosity of quasi-solid capillary-porous bodies under varying air environment conditions during museum storage." Ventilation, Illumination and Heat Gas Supply 34 (September 24, 2020): 7–15. http://dx.doi.org/10.32347/2409-2606.2020.34.7-15.
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Fundamental analysis of the thermal conductivity and viscosity of quasi-solid capillary-porous bodies (CPBs), which are museum exhibits’ materials, is presented. The air environment parameters change leads to a temperature gradient in the CPBs. Non-uniform heating of the solid medium, in particular, quasi-solid CPB, is not accompanied by convection, and heat transfer is carried out only due to the mechanism of thermal conductivity. In order to create a mathematical model of this process in CPB, a system of partial differential equations in time and space coordinates is obtained. The resulting system adequately describes the thermal conductivity process in quasi-solid CPBs. The anisotropy of CPB’s thermal parameters, especially, its coefficients of thermal expansion and thermal conductivity, is also taken into account. Theoretically, the deformation process during motion in quasi-solid CPB is taken as reversible. In real conditions, the process is thermodynamically reversible only when it occurs at an infinitesimal speed. Then at each point in time, the CPB is able to establish a thermodynamic equilibrium state. Real motion occurs at a finite velocity, the CPB is not in an equilibrium state at any given moment, so there are endogenous processes that try to get it into a balanced condition. The occurrence of these processes causes the irreversibility of motion, which acts, in particular, through the dissipation of mechanical energy, which eventually turns into heat. The energy dissipation is caused by irreversible processes of thermal conductivity and processes of internal friction or viscosity. The dissipative function for isotropic and anisotropic cases was determined in order to analyze the viscosity of quasi-solid CPBs. The viscosity in the equations of motion can be considered by replacing the stress tensor with a tensor, which additionally takes into account the "dissipative" stress tensor.
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Carciumaru, Dana, Roberto Ortega, Jorge Castillo Castellanos, and Eduardo Huesca-Pérez. "Crustal Characteristics in the Subduction Zone of Mexico: Implication of the Tectonostratigraphic Terranes on Slab Tearing." Seismological Research Letters 91, no. 3 (February 26, 2020): 1781–93. http://dx.doi.org/10.1785/0220190117.
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Abstract During the past years, significant work has been done for studying the crustal anisotropy and state of stress of the Mexican subduction zone. At the same time, there is new evidence of the geometry of the subducted slab proposing subduction tearing. Here, we present a study of the Earth crust using three different methods: azimuthal anisotropy based on ambient noise, shear-wave splitting of tectonic tremors, and moment tensor inversions of the earthquakes of 7 September 2017 Mw 8.2 Tehuantepec, Mexico. This earthquake initiated a seismic sequence that triggered shallow seismicity and aftershocks. The shallow earthquakes fall into a region where there were few published focal mechanism higher than Mw 4.5. Two slab tearings: in the Michoacán–Guerrero border and in central Oaxaca, best represent the slab geometry of the Mexican subduction zone. At the Michoacán–Guerrero, the subducted slab is subhorizontal, whereas in central Oaxaca the plate is characterized by northeast vergence. We interpret that the mantle’s flow in this part of the subducted slab produces multiple alignments in the crust and differentiates the tectonostratigraphic terranes of the southern region of Mexico.
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Yeung, P. K., James G. Brasseur, and Qunzhen Wang. "Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views." Journal of Fluid Mechanics 283 (January 25, 1995): 43–95. http://dx.doi.org/10.1017/s0022112095002230.
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As discussed in a recent paper by Brasseur & Wei (1994), scale interactions in fully developed turbulence are of two basic types in the Fourier-spectral view. The cascade of energy from large to small scales is embedded within ‘local-to-non-local’ triadic interactions separated in scale by a decade or less. ‘Distant’ triadic interactions between widely disparate scales transfer negligible energy between the largest and smallest scales, but directly modify the structure of the smallest scales in relationship to the structure of the energy-dominated large scales. Whereas cascading interactions tend to isotropize the small scales as energy moves through spectral shells from low to high wavenumbers, distant interactions redistribute energy within spectral shells in a manner that leads to anisotropic redistributions of small-scale energy and phase in response to anisotropic structure in the large scales. To study the role of long-range interactions in small-scale dynamics, Yeung & Brasseur (1991) carried out a numerical experiment in which the marginally distant triads were purposely stimulated through a coherent narrow-band anisotropic forcing at the large scales readily interpretable in both the Fourier- and physical-space views. It was found that, after one eddy turnover time, the smallest scales rapidly became anisotropic as a direct consequence of the marginally distant triadic group in a manner consistent with the distant triadic equations. Because these asymptotic equations apply in the infinite Reynolds number limit, Yeung & Brasseur argued that the observed long-range effects should be applicable also at high Reynolds numbers.We continue the analysis of forced simulations in this study, focusing (i) on the detailed three-dimensional restructuring of the small scales as predicted by the asymptotic triadic equations, and (ii) on the relationship between Fourier- and physical-space evolution during forcing. We show that the three-dimensional restructuring of small-scale energy and vorticity in Fourier space from large-scale forcing is predicted in some detail by the distant triadic equations. We find that during forcing the distant interactions alter small-scale structure in two ways: energy is redistributed anisotropically within high-wavenumber spectral shells, and phase correlations are established at the small scales by the distant interactions. In the numerical experiments, the long-range interactions create two pairs of localized volumes of concentrated energy in three-dimensional Fourier space at high wavenumbers in which the Fourier modes are phase coupled. Each pair of locally phase-correlated volumes of Fourier modes separately corresponds to aligned vortex tubes in physical space in two orthogonal directions. We show that the dynamics of distant interactions in creating small-scale anisotropy may be described in physical space by differential advection and distortion of small-scale vorticity by the coherent large-scale energy-containing eddies, producing anisotropic alignment of small-scale vortex tubes.Scaling arguments indicate a disparity in timescale between distant triadic interactions and energy-cascading local-to-non-local interactions which increases with scale separation. Consequently, the small scales respond to forcing initially through the distant interactions. However, as energy cascades from the large-scale to the small-scale Fourier modes, the stimulated distant interactions become embedded within a sea of local-to-non-local energy cascading interactions which reduce (but do not eliminate) small-scale anisotropy at later times. We find that whereas the small-scale structure is still anisotropic at these later times, the second-order velocity moment tensor is insensitive to this anisotropy. Third-order moments, on the other hand, do detect the anisotropy. We conclude that whereas a single statistical measure of anisotropy can be used to indicate the presence of anisotropy, a null result in that measure does not necessarily imply that the signal is isotropic. The results indicate that non-equilibrium non-stationary turbulence is particularly sensitive to long-range interactions and deviations from local isotropy.
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Dumon, B., J. Berrue, A. Chave, and A. Barreau. "Diffusion dépolarisée Rayleigh par des molécules anisotropes: calcul analytique des contributions propres, collisionnelles et croisées pour divers potentiels." Canadian Journal of Physics 67, no. 5 (May 1, 1989): 525–31. http://dx.doi.org/10.1139/p89-096.
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In a preceding paper, a new analytical calculation of the integrated Rayleigh light scattered intensities, taking into account the permanent anisotropy in the collisionnal polarizability tensor, has been developped. The contributions of orientational pair correlation collision induced light scattering and the cross term have been evaluated in the first order dipole induced dipole approximation, at low density in the center–center polarizability scheme. In this article, from these analytical expansions, we calculate contribution values in N2, CO2, Cl2, and O2 to the scattered intensities. The intermolecular potential is represented by a two (or three)-site Lennard–Jones model without (or with) a quadrupolar moment. We show that these new analytical calculations lead to important corrections of up to 30% for the cross term in O2, and that the potential choice plays an essential role. These results are compared with experimental data and to other studies.
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Grechka, Vladimir. "Moment tensors of double-couple microseismic sources in anisotropic formations." GEOPHYSICS 85, no. 1 (November 18, 2019): KS1—KS11. http://dx.doi.org/10.1190/geo2019-0471.1.
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Moment tensors of kinematically double-couple microseismic events triggered in anisotropic formations are known to exhibit non-double-couple focal mechanisms. The weak anisotropy approximation of these mechanisms reveals the combinations of anisotropy coefficients of vertically transversely isotropic and orthorhombic focal regions responsible for the deviations of moment tensors from double couples. Numerical examples for models of typical unconventional shales indicate the non-double-couple components of moment tensors to be sufficiently large to cause misinterpretation of the nature of ruptures associated with hydraulic fracturing.
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Wong, Henry, Didier Subrin, and Daniel Dias. "Convergenceconfinement analysis of a bolt-supported tunnel using the homogenization method." Canadian Geotechnical Journal 43, no. 5 (May 1, 2006): 462–83. http://dx.doi.org/10.1139/t06-016.
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The behaviour of tunnels reinforced with radially disposed fully grouted bolts is investigated in this paper. Perfect bonding and ideal diffusion of bolt tension are assumed, so that the bolt tension can be assimilated to an equivalent uniaxial stress tensor. An analytical model of the convergenceconfinement type is proposed that accounts for the delayed action of bolts due to ground decompression prior to bolt installation. This factor leads to nonsimultaneous yielding, and more generally, a different stress history for each constituent, requiring special treatments in the incremental elastoplasticity calculations. Nonetheless, the resulting model remains sufficiently simple, and an analytical solution is still accessible. Charts are provided to allow for parametric studies and quick preliminary designs. Comparisons with 3D numerical calculations show that the model gives precise results if the correct convergence at the moment of bolt installation is used as an "external" input parameter, validating the homogenization approach. An approximate methodology based on previous works is proposed to determine this parameter to render the proposed model "self-sufficient." Its predictions are again compared to 3D numerical computations, and the results are found to be sufficiently accurate for practical applications.Key words: reinforcement, anisotropy, analytical, lining, yield, elastoplasticity.
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Caimmi, R. "R fluids." Serbian Astronomical Journal, no. 176 (2008): 23–35. http://dx.doi.org/10.2298/saj0876023c.
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A theory of collisionless fluids is developed in a unified picture, where nonrotating (?f1 = ?f2 = ?f3 = 0) figures with some given random velocity component distributions, and rotating (?f1 = ?f2 = ?f3 ) figures with a different random velocity component distributions, make adjoint configurations to the same system. R fluids are defined as ideal, self-gravitating fluids satisfying the virial theorem assumptions, in presence of systematic rotation around each of the principal axes of inertia. To this aim, mean and rms angular velocities and mean and rms tangential velocity components are expressed, by weighting on the moment of inertia and the mass, respectively. The figure rotation is defined as the mean angular velocity, weighted on the moment of inertia, with respect to a selected axis. The generalized tensor virial equations (Caimmi and Marmo 2005) are formulated for R fluids and further attention is devoted to axisymmetric configurations where, for selected coordinate axes, a variation in figure rotation has to be counterbalanced by a variation in anisotropy excess and vice versa. A microscopical analysis of systematic and random motions is performed under a few general hypotheses, by reversing the sign of tangential or axial velocity components of an assigned fraction of particles, leaving the distribution function and other parameters unchanged (Meza 2002). The application of the reversion process to tangential velocity components is found to imply the conversion of random motion rotation kinetic energy into systematic motion rotation kinetic energy. The application of the reversion process to axial velocity components is found to imply the conversion of random motion translation kinetic energy into systematic motion translation kinetic energy, and the loss related to a change of reference frame is expressed in terms of systematic motion (imaginary) rotation kinetic energy. A number of special situations are investigated in greater detail. It is found that an R fluid always admits an adjoint configuration where figure rotation occurs around only one principal axis of inertia (R3 fluid), which implies that all the results related to R3 fluids (Caimmi 2007) may be ex- tended to R fluids. Finally, a procedure is sketched for deriving the spin parameter distribution (including imaginary rotation) from a sample of observed or simulated large-scale collisionless fluids i.e. galaxies and galaxy clusters.
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Гололобов, Петр, Peter Gololobov, Прокопий Кривошапкин, Prokopy Krivoshapkin, Гермоген Крымский, Germogen Krymsky, Владислав Григорьев, Vladislav Grigoryev, Сардаана Герасимова, and Sardaana Gerasimova. "Distribution of tensor anisotropy of cosmic rays near the neutral current sheet." Solar-Terrestrial Physics 3, no. 2 (August 9, 2017): 16–19. http://dx.doi.org/10.12737/stp-3220173.
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We analyze time profiles of isotropic intensity, components of vector and tensor anisotropies of cos-mic rays (CR) when Earth crosses the neutral sheet of the interplanetary magnetic field (IMF) in solar activity cycles 23–24. The moments of the crossings are de-termined from Wilcox Observatory synoptic charts and IMF data. Periods of Forbush decreases and ground level enhancements are excluded from the analysis. The events are analyzed for the epochs of positive and negative signs of the Sun’s general magnetic field. During each epoch, the crossings from the positive sector to the negative one and vice versa are separated. In total, 213 crossing events have been selected. The first two spherical harmonics of the angular CR-distribution are obtained using the global survey method. In each case, the average number of stations is equal to 32. The analysis shows that the temporal change of the isotropic component is caused by a magnetic mirror. For the first time, the zonal harmonics are reliably distinguished, and the existence of the antisymmetric diurnal CR-variation in a low energy range, which is oriented along IMF, is recognized. We compare our results with those obtained earlier.
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Melrose, D. B. "Scintillations in a magnetized plasma. Part 1. The mutual coherence function." Journal of Plasma Physics 50, no. 2 (October 1993): 267–81. http://dx.doi.org/10.1017/s0022377800027069.
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A standard theory of strong scintillations in an isotropic medium is extended to the case of polarized radiation propagating through a weakly anisotropic, randomly inhomogeneous, magnetized plasma. A hierarchy of moments of the radiationfield is defined, with an nth-order moment being an nth-rank polarization tensor, and the tensor equation for the evolution of an arbitrary moment is derived. Emphasis isplaced on the mutual coherence function (a second moment), which is rewritten in terms of the Stokes parameters. The evolution of the polarized radiation through the randomly inhomogeneous, weakly anisotropic medium is described in terms ofa matrix equation for the Stokes vector. It is shown that the theory implies that a depolarization occurs as a result ofsuch propagation. This corresponds to a decrease in the degree of linear polarization for propagation through a weakly anisotropic plasma. The rate of depolarization is estimated, and an interpretation is suggested. The polarization dependence of the angular size of the apparent image is determined. Two counter-intuitive results are found: that the image canhave a circularly polarized component even for an unpolarized source, and that the angular size of the linearly polarized source can decrease. These are interpreted in terms of random variations in the ray path with opposite signs for the two natural modes, resulting in a separation of the centroids of the images in the two circular polarizations.
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Kushch, Volodymyr I., Igor Sevostianov, and Albert Giraud. "Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2207 (November 2017): 20170472. http://dx.doi.org/10.1098/rspa.2017.0472.
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An accurate semi-analytical solution of the conductivity problem for a composite with anisotropic matrix and arbitrarily oriented anisotropic ellipsoidal inhomogeneities has been obtained. The developed approach combines the superposition principle with the multipole expansion of perturbation fields of inhomogeneities in terms of ellipsoidal harmonics and reduces the boundary value problem to an infinite system of linear algebraic equations for the induced multipole moments of inhomogeneities. A complete full-field solution is obtained for the multi-particle models comprising inhomogeneities of diverse shape, size, orientation and properties which enables an adequate account for the microstructure parameters. The solution is valid for the general-type anisotropy of constituents and arbitrary orientation of the orthotropy axes. The effective conductivity tensor of the particulate composite with anisotropic constituents is evaluated in the framework of the generalized Maxwell homogenization scheme. Application of the developed method to composites with imperfect ellipsoidal interfaces is straightforward. Their incorporation yields probably the most general model of a composite that may be considered in the framework of analytical approach.
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Sayad, D., C. Zebiri, S. Daoudi, and F. Benabdelaziz. "Analysis of the Effect of a Gyrotropic Anisotropy on the Phase Constant and Characteristic Impedance of a Shielded Microstrip Line." Advanced Electromagnetics 8, no. 5 (December 17, 2019): 15–22. http://dx.doi.org/10.7716/aem.v8i5.946.
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In this work, we present an analytical modeling of a highly complex medium-based shielded microstrip line. The study aims at a numerical evaluation of the characteristic impedance and the dispersion characteristics of the dominant hybrid mode in the microstrip line printed on an anisotropic medium. The newly considered complex anisotropy has a full 3×3 tensor form of permittivity and permeability. The study is based on the derivation of the Green's functions of the general complex-medium-based structure in the Fourier domain. The spectral Method of Moments (MoM) and the Galerkin's procedure are combined to solve the resulting homogeneous system of equations. The effect of the gyrotropic anisotropy on the phase constant and the characteristic impedance is particularly investigated. Original and interesting numerical results are obtained and discussed. Our results are found to be in good agreement with available isotropic case data reported in literature.
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Arntsen, Børge, and José M. Carcione. "A new insight into the reciprocity principle." GEOPHYSICS 65, no. 5 (September 2000): 1604–12. http://dx.doi.org/10.1190/1.1444848.
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Reciprocity is usually applied to wavefields associated with concentrated point forces and point receivers, but, reciprocity has a much wider application potential. In many cases, however, it is not used at its full potential because (1) a variety of source and receiver types are not considered or (2) its implementation is not well understood. We obtain reciprocity relations for inhomogeneous, anisotropic, viscoelastic solids and for distributed sources and receivers, and test these relations with a full‐wave numerical modeling algorithm. The theory and the numerical experiments show that, in addition to the usual relations involving directional forces, (1) the diagonal components of the strain tensor are reciprocal for dipole sources (single couple without moment), (2) the off‐diagonal components of the stress tensor are reciprocal for double couples with moments, (3) the dilatation due to a directional force is reciprocal to the particle velocity due to a dilatational source, and (4) some combinations of the off‐diagonal strains are reciprocal for single couples with moments.
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Minakov, Alexander, and Viktoriya Yarushina. "Elastoplastic source model for microseismicity and acoustic emission." Geophysical Journal International 227, no. 1 (May 31, 2021): 33–53. http://dx.doi.org/10.1093/gji/ggab207.
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SUMMARY The microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can further clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: (i) it accounts for incipient faulting as a microseismicity source mechanism, (ii) it does not require a pre-defined fracture geometry, (iii) it accounts for both shear and volumetric source mechanisms, (iv) it is valid for general heterogeneous and anisotropic rocks and (v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2-D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments. We consider four special cases corresponding to ‘dry’ elastically homogeneous and elastically heterogeneous isotropic rocks, ‘dry’ transversely isotropic rocks and ‘wet’ isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters.
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Dalaq, Ahmed S., and Shivakumar I. Ranganathan. "Invariants of mesoscale thermal conductivity and resistivity tensors in random checkerboards." Engineering Computations 32, no. 6 (August 3, 2015): 1601–18. http://dx.doi.org/10.1108/ec-08-2014-0162.
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Purpose – The purpose of this paper is to study the statistics of thermal conductivity and resistivity tensors in two-phase random checkerboard microstructures at finite mesoscales. Design/methodology/approach – Microstructures at finite scales are generated by randomly sampling an infinite checkerboard at 50 percent nominal fraction. Boundary conditions that stem from the Hill-Mandel homogenization condition are then applied as thermal loadings on these microstructures. Findings – It is observed that the thermal response of the sampled microstructures is in general anisotropic at finite mesoscales. Based on 1,728 boundary value problems, the statistics of the tensor invariants (trace and determinant) are obtained as a function of material contrast, mesoscale and applied boundary conditions. The histograms as well as the moments (mean, variance, skewness and kurtosis) of the invariants are computed and discussed. A simple analytical form for the variance of the trace of mesoscale conductivity tensor is proposed as a function of individual phase conductivities and the mesoscale. Originality/value – A rigorous methodology to determine the evolution of the invariants of thermal conductivity (and resistivity) tensors across a variety of length scales (microscale to macroscale) is presented. The objective is to enable setting up of constitutive equations applicable to heat conduction that are valid across all length scales.
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CAI, Xiao-Gang, Chen YAO, and Xiao-Fei CHEN. "Seismic Moment Tensors in Anisotropic ATI Media: Shear Faulting." Chinese Journal of Geophysics 54, no. 4 (July 2011): 448–59. http://dx.doi.org/10.1002/cjg2.1628.
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Chapman, C. H., and W. S. Leaney. "A new moment-tensor decomposition for seismic events in anisotropic media." Geophysical Journal International 188, no. 1 (November 28, 2011): 343–70. http://dx.doi.org/10.1111/j.1365-246x.2011.05265.x.
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Shi, Peidong, Doug Angus, Andy Nowacki, Sanyi Yuan, and Yanyan Wang. "Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation." Surveys in Geophysics 39, no. 4 (March 19, 2018): 567–611. http://dx.doi.org/10.1007/s10712-018-9466-2.
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Ridier, Karl, Béatrice Gillon, Arsen Gukasov, Gregory Chaboussant, Ana Borta, Olga Iasco, Dominique Luneau, Hiroshi Sakiyama, Masahiro Mikuriya, and Makoto Handa. "Polarized Neutron Diffraction study of the molecular magnetic anisotropy." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C278. http://dx.doi.org/10.1107/s2053273314097216.
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The magnetic anisotropy is a prerequisite for a metal complex to behave as a single-molecule magnet (SMM). Unfortunately, today we do not fully understand the relationships between the local structural parameters and the magnetic anisotropy that results at the molecular level. This is an issue that has become recursive in this area. Out of the synthesis work which is still important, but generates a multiplication of SMMs with frustrating properties, there are various studies to understand these relationships among which most are theoretical studies. In this context, we believe that polarized neutron diffraction (PND) can provide an experimental and complementary point of view to these theoretical studies. PND is indeed well known to allow an accurate determination of the spin density in magnetic compounds and in the field of molecular magnetism it has provided unique information on the pathways and the nature of intra- or intermolecular magnetic coupling [1]. In the case of highly anisotropic paramagnetic materials, where local magnetic moments cannot be aligned by an external magnetic field, that is more tricky, but a method based on local magnetic susceptibility tensor, has been recently developed that allows now analysing the data in this case and obtaining the magnetization distribution [2]. This approach was first used for inorganic compounds. Our idea has been to use this approach to go beyond the reconstruction of spin density to study the magnetic anisotropy in molecular systems. In this paper, we present the results of such an approach applied for the first time to metal complexes that are simple mono and dinuclear cobalt(II) complexes.
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Calzetta, Esteban, and Alejandra Kandus. "A hydrodynamic approach to the study of anisotropic instabilities in dissipative relativistic plasmas." International Journal of Modern Physics A 31, no. 35 (December 18, 2016): 1650194. http://dx.doi.org/10.1142/s0217751x16501943.
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We develop a purely hydrodynamic formalism to describe collisional, anisotropic instabilities in a relativistic plasma, that are usually described with kinetic theory tools. Our main motivation is the fact that coarse-grained models of high particle number systems give more clear and comprehensive physical descriptions of those systems than purely kinetic approaches, and can be more easily tested experimentally as well as numerically. Also they make it easier to follow perturbations from linear to nonlinear regimes. In particular, we aim at developing a theory that describes both a background nonequilibrium fluid configurations and its perturbations, to be able to account for the backreaction of the latter on the former. Our system of equations includes the usual conservation laws for the energy–momentum tensor and for the electric current, and the equations for two new tensors that encode the information about dissipation. To make contact with kinetic theory, we write the different tensors as the moments of a nonequilibrium one-particle distribution function (1pdf) which, for illustrative purposes, we take in the form of a Grad-like ansatz. Although this choice limits the applicability of the formalism to states not far from equilibrium, it retains the main features of the underlying kinetic theory. We assume the validity of the Vlasov–Boltzmann equation, with a collision integral given by the Anderson–Witting prescription, which is more suitable for highly relativistic systems than Marle’s (or Bhatnagar, Gross and Krook) form, and derive the conservation laws by taking its corresponding moments. We apply our developments to study the emergence of instabilities in an anisotropic, but axially symmetric background. For small departures of isotropy we find the dispersion relation for normal modes, which admit unstable solutions for a wide range of values of the parameter space.
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Jarillo Michel, Oscar, and Ilya Tsvankin. "Waveform inversion for microseismic velocity analysis and event location in VTI media." GEOPHYSICS 82, no. 4 (July 1, 2017): WA95—WA103. http://dx.doi.org/10.1190/geo2016-0651.1.
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Waveform inversion (WI), which has been extensively used in reflection seismology, could provide improved velocity models and event locations for microseismic surveys. Here, we develop an elastic WI algorithm for anisotropic media designed to estimate the 2D velocity field along with the source parameters (location, origin time, and moment tensor) from microseismic data. The gradient of the objective function is obtained with the adjoint-state method, which requires just two modeling simulations at each iteration. In the current implementation the source coordinates and velocity parameters are estimated sequentially at each stage of the inversion to minimize trade-offs and improve the convergence. Synthetic examples illustrate the accuracy of the inversion for layered VTI (transversely isotropic with a vertical symmetry axis) media, as well as the sensitivity of the velocity-analysis results to noise, the length of the receiver array, errors in the initial model, and variability in the moment tensor of the recorded events.
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Chan, T. K., Tom Theuns, Richard Bower, and Carlos Frenk. "Smoothed particle radiation hydrodynamics: two-moment method with local Eddington tensor closure." Monthly Notices of the Royal Astronomical Society 505, no. 4 (June 12, 2021): 5784–814. http://dx.doi.org/10.1093/mnras/stab1686.
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ABSTRACT We present a new smoothed particle hydrodynamics-radiative transfer method (sph-m1rt) that is coupled dynamically with sph. We implement it in the (task-based parallel) swift galaxy simulation code but it can be straightforwardly implemented in other sph codes. Our moment-based method simultaneously solves the radiation energy and flux equations in sph, making it adaptive in space and time. We modify the m1 closure relation to stabilize radiation fronts in the optically thin limit. We also introduce anisotropic artificial viscosity and high-order artificial diffusion schemes, which allow the code to handle radiation transport accurately in both the optically thin and optically thick regimes. Non-equilibrium thermochemistry is solved using a semi-implicit sub-cycling technique. The computational cost of our method is independent of the number of sources and can be lowered further by using the reduced speed-of-light approximation. We demonstrate the robustness of our method by applying it to a set of standard tests from the cosmological radiative transfer comparison project of Iliev et al. The sph-m1rt scheme is well-suited for modelling situations in which numerous sources emit ionizing radiation, such as cosmological simulations of galaxy formation or simulations of the interstellar medium.
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Shin, Jong-Keun, Jeong-Sik Seo, and Young-Don Choi. "Elliptic-blending second-moment turbulence closure using an algebraic anisotropic dissipation rate tensor model." Fluid Dynamics Research 41, no. 3 (January 23, 2009): 035501. http://dx.doi.org/10.1088/0169-5983/41/3/035501.
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Pavlova, A. "Modelling of the wave fields by the modification of the matrix method in anisotropic media." Solid Earth Discussions 6, no. 1 (February 3, 2014): 467–85. http://dx.doi.org/10.5194/sed-6-467-2014.
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Abstract. The modification of the matrix method of construction of wave field on the free surface of an anisotropic medium is presented. The earthquake source represented by a randomly oriented force or a seismic moment tensor is placed on an arbitrary boundary of a layered anisotropic medium. The theory of the matrix propagator in a homogeneous anisotropic medium by introducing a "wave propagator" is presented. It is shown that, for an anisotropic layered medium, the matrix propagator can be represented by a "wave propagator" in each layer. The matrix propagator P (z, z0 = 0) acts on the free surface of the layered medium and generates stress-displacement vector at depth z. The displacement field on the free surface of an anisotropic medium is obtained from the received system of equations considering the radiation condition and that the free surface is stressless. The approbation of the modification of the matrix method for isotropic and anisotropic media with TI symmetry is done. A comparative analysis of our results with the synthetic seismic records obtained by other methods and published in foreign papers is executed.
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LEE, SEUNG-HYUN, and HYUNG JIN SUNG. "Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall." Journal of Fluid Mechanics 584 (July 25, 2007): 125–46. http://dx.doi.org/10.1017/s0022112007006465.
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The effects of surface roughness on a spatially developing turbulent boundary layer (TBL) are investigated by performing direct numerical simulations of TBLs over rough and smooth walls. The Reynolds number based on the momentum thickness was varied in the range Reθ = 300 ∼ 1400. The roughness elements were periodically arranged two-dimensional spanwise rods, and the roughness height was k = 1.5θin, where θin is the momentum thickness at the inlet, which corresponds to k/δ = 0.045 ∼ 0.125, δ being the boundary layer thickness. To avoid generating a rough-wall inflow, which is prohibitively difficult, a step change from smooth to rough was placed 80θin downstream from the inlet. The spatially developing characteristics of the rough-wall TBL were examined. Along the streamwise direction, the friction velocity approached a constant value, and self-preserving forms of the turbulent Reynolds stress tensors were obtained. Introduction of the roughness elements affected the turbulent stress not only in the roughness sublayer but also in the outer layer. Despite the roughness-induced increase of the turbulent Reynolds stress tensors in the outer layer, the roughness had only a relatively small effect on the anisotropic Reynolds stress tensor in the outer layer. Inspection of the triple products of the velocity fluctuations revealed that introducing the roughness elements onto the smooth wall had a marked effect on vertical turbulent transport across the whole TBL. By contrast, good surface similarity in the outer layer was obtained for the third-order moments of the velocity fluctuations.
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Tsai, Y. L., and T. Chen. "Transformation Media in Acoustics with Constant Bulk Modulus or Constant Density Tensor." Journal of Mechanics 32, no. 3 (August 19, 2015): 313–23. http://dx.doi.org/10.1017/jmech.2015.65.
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AbstractThis work is to present a formulation of cloaking or concentrating device in acoustics in which the transformed material could be either having uniform bulk modulus or having homogeneous density tensor. The transformed material parameters, depending on the mapping of physical and virtual coordinates, are often position-varying and anisotropic. This often adds substantial complexity in practical implementation. Here we present a theoretical algorithm that allows us to design a transformation field that could have either uniform bulk modulus or constant density tensor. For cloaking devices with constant bulk modulus, analytical and numerical results are presented for circular as well as for non-circular cloaking devices. Specifically, elliptical and twin-cloak devices are exemplified. To achieve the effect of constant density tensor, we consider only circular geometry. Devices with cloaking or concentrating effects can be exactly formulated. We note, however, that it seems unlikely at this moment to have a transformation device that has constant bulk modulus and constant density tensor at the same time. Nevertheless, we remark the present results are of still sufficient merit in that the uniform material parameters, in either set of material parameters, indeed greatly simplify the practice in real implementations.
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Rendell, J. C. T., D. S. Zimmerman, A. J. van der Est, and E. E. Burnell. "Orientational order of 1,3-dichloro-2-ethenylbenzene in liquid-crystal solvents." Canadian Journal of Chemistry 75, no. 8 (August 1, 1997): 1156–61. http://dx.doi.org/10.1139/v97-138.
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The order parameter matrices of a molecule that has no symmetry, 1,3-dichloro-2-ethenylbenzene, dissolved in two different nematic liquid-crystal solvents are analyzed in terms of various models for the intermolecular mean-field potential. In a mixture of liquid crystals for which the interaction between the molecular quadrupole moment tensor and the average electric field gradient of the nematic solvent has been minimized, the orientational order is best described by models for the short-range anisotropic potential. The most successful potentials are written in terms of anisotropic interactions between the solute surface and the liquid-crystal mean field. This represents a strong test of such models because they simultaneously fit the five independent orientational parameters obtained from the same solute, thus removing the problems associated with comparing results among different solutes in either the same or a different sample tube. Keywords: liquid crystals, intermolecular forces, order parameters, anisotropic.
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Chapman, C. H., and W. S. Leaney. "Erratum: A correction to ‘A new moment-tensor decomposition for seismic events in anisotropic media’." Geophysical Journal International 199, no. 3 (October 21, 2014): 1808–10. http://dx.doi.org/10.1093/gji/ggu366.
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Соколов, А. В., А. П. Коузов, Ж. В. Булдырева, and Н. И. Егорова. "Спектральные моменты характеристик бинарных взаимодействий между линейными молекулами." Оптика и спектроскопия 129, no. 3 (2021): 253. http://dx.doi.org/10.21883/os.2021.03.50650.261-20.
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A new approach to derive symmetrized expressions of leading classical moments of spectral distributions characterizing different anisotropic terms of the interaction potential for the case of two liear molecules is presented. The results allow to calculate diffuse shapes formed by transitions between continuous eigenstates of a molecular pair and open the way to account for the nonMarkov effects (due to finite collision durations) in the rotatonal relaxation matrix of an arbitrary rank. The approach is also applied to the spectral moments of vector and tensor characteristics determining the band intensities in the collision-induced spectra of linear molecules. Generally, the use of symmetrized expressions lead to considerably faster computer codes.
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Гололобов, Петр, Peter Gololobov, Прокопий Кривошапкин, Prokopy Krivoshapkin, Гермоген Крымский, Germogen Krymsky, Владислав Григорьев, Vladislav Grigoryev, Сардаана Герасимова, and Sardaana Gerasimova. "Distribution of tensor anisotropy of cosmic rays near the neutral current sheet." Solnechno-Zemnaya Fizika 3, no. 2 (June 29, 2017): 18–21. http://dx.doi.org/10.12737/22603.
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We analyze time profiles of isotropic intensity, components of vector and tensor anisotropies of cos-mic rays (CR) when Earth crosses the neutral sheet of the interplanetary magnetic field (IMF) in solar activity cycles 23–24. The moments of the crossings are de-termined from Wilcox Observatory synoptic charts and IMF data. Periods of Forbush decreases and ground level enhancements are excluded from the analysis. The events are analyzed for the epochs of positive and negative signs of the Sun’s general magnetic field. During each epoch, the crossings from the positive sector to the negative one and vice versa are separated. In total, 213 crossing events have been selected. The first two spherical harmonics of the angular CR-distribution are obtained using the global survey method. In each case, the average number of stations is equal to 32. The analysis shows that the temporal change of the isotropic component is caused by a magnetic mirror. For the first time, the zonal harmonics are reliably distinguished, and the existence of the antisymmetric diurnal CR-variation in a low energy range, which is oriented along IMF, is recognized. We compare our results with those obtained earlier.
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Corbin, G., A. Hunt, A. Klar, F. Schneider, and C. Surulescu. "Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum." Mathematical Models and Methods in Applied Sciences 28, no. 09 (August 2018): 1771–800. http://dx.doi.org/10.1142/s0218202518400055.
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Starting from a two-scale description involving receptor binding dynamics and a kinetic transport equation for the evolution of the cell density function under velocity reorientations, we deduce macroscopic models for glioma invasion featuring partial differential equations for the mass density and momentum of a population of glioma cells migrating through the anisotropic brain tissue. The proposed first and higher-order moment closure methods enable numerical simulations of the kinetic equation. Their performance is then compared to that of the diffusion limit. The approach allows for diffusion tensor imaging (DTI)-based, patient-specific predictions of the tumor extent and its dynamic behavior.
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Ricardo, Ana M., Dimokratis G. E. Grigoriadis, and Rui M. L. Ferreira. "LES modelling of a flow within an infinite array of randomly placed cylinders: Anisotropy characterization." E3S Web of Conferences 40 (2018): 02035. http://dx.doi.org/10.1051/e3sconf/20184002035.
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The LES approach is employed to model flows within random arrays of emergent cylinders. The model is validated against laboratory data acquired with a 2D-2C Particle Image Velocimetry system. The main goals are: i) discussion of the effect of the numerical domain size and the grid resolution on the predicted flow variables; and ii) spatial characterization of the flow anisotropy. Three domains of different sizes (16 to 36 cylinders) and four grid resolutions were independently tested. A 2D methodology was proposed to characterize the flow anisotropy on the horizontal plane. The results show that the first and second order moments were not significantly affected by the size of the tested numerical domains or by the grid resolution. The comparison with laboratory data showed a fair agreement confirming that the numerical model was able to adequately reproduce all the components of the Reynolds stress tensor. The results show that turbulence is of axisymmetric expansion nature in this type of flow. Relatively to the degree of anisotropy, the highest values were found close to the cylinder, decreasing gradually downstream towards the isotropy state. However, a truly isotropic turbulence state is not reached.
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Dershowitz, Bill, Paul LaPointe, Thorsten Eiben, and Lingli Wei. "Integration of Discrete Feature Network Methods With Conventional Simulator Approaches." SPE Reservoir Evaluation & Engineering 3, no. 02 (April 1, 2000): 165–70. http://dx.doi.org/10.2118/62498-pa.
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Summary The discrete feature network (DFN) approach offers many key advantages over conventional dual porosity (DP) approaches, particularly when issues of connectivity dominate recovery and reservoir stimulation in fractured and heterogeneous reservoirs. DP models have been developed for complex multiphase and thermal effects, and have been implemented for basin scale modeling. However, DP models address only the dual porosity nature of fractured reservoirs, generally simplifying connectivity and scale-dependent heterogeneity issues which are modeled efficiently and more accurately by the DFN approach. This paper describes the development of techniques to integrate DFN and DP approaches. These techniques allow the analyst to maintain many of the advantages of the DP simulator approach without losing the realism of complex fracture system geometry and connectivity, as captured by DFN models. The techniques described are currently used within a DOE funded research project for linking a DFN and a DP thermal simulation model for the Yates field, Texas. The paper describes some of the geological and engineering aspects of the Yates field and gives two examples of how DP parameters for the thermal simulation can be derived using DFN modeling techniques. Introduction Reservoir simulation can be significantly more challenging for fractured reservoirs than it is for conventional clastic reservoirs. The dual porosity (DP) approach for the simulation of fractured reservoirs adds a second interacting continuum to reflect storage and permeability characteristics but does not adequately address connectivity issues. These effects, which play a key role in fractured reservoirs, are generally better addressed by discrete feature network (DFN) models.1 Another advantage of DFN models is that they are generally implemented as stochastic models, in which multiple realizations provide a quantitative measure for uncertainty and variability. Despite the significant simplifications made regarding the geometry of the fracture network in equivalent porous medium DP models (Fig. 1) and the recent progress made in developing powerful DFN modeling software, DP models still offer advantages regarding the level of sophistication of available multiphase flow solvers. In many cases, DP models also offer advantages regarding model size and speed. As a result, there is a need to link DP and DFN models to be able to take maximum advantage of each approach. This paper presents a series of techniques, which can be used to develop DP models that more accurately reflect the anisotropy, heterogeneity, and most important, the scale-dependent connectivity structure of fractured reservoirs. These techniques will allow the DP approach to take advantage of some of the features of the DFN approach. The approach adopted is to derive grid cell and well parameters through DFN models. The first section of this paper discusses which fracture porosity parameters can be derived for DP models from DFN models and how they are derived. The second section describes different techniques that can be used to link DP and DFN models. At the end of the paper two examples are given based on data from the Yates field, Texas. DP Input Parameter from DFN Modeling Fracture System Porosity. The fracture system porosity fF can be directly calculated as the product of the fracture intensity expressed as fracture area per unit volume (P32) and the storage aperture of the fractures (e):… Because the fracture system porosity depends on the number of fractures per unit volume, the fracture size distribution and the fracture aperture distribution, a different porosity needs to be calculated for every portion of the continuum model where these parameters vary. Using a full field DFN model, the fracture system porosity can be calculated separately for each grid cell. The primary issue in definition of fracture porosity from fracture intensity P32 is the selection of an appropriate measure for storage aperture e. Possible measures include:aperture derived from transient hydraulic response,mechanical aperture,aperture derived from fracture permeability or transmissivity ("cubic law"),aperture derived from geophysical measurements (gamma density, matrix porosity), andcorrelations to fracture size and orientation. Directional Fracture System Permeability. The permeability of the fracture system depends on the fracture intensity, the connectivity of the fracture network, and the distribution of fracture transmissivities. Approaches for calculation of approximate measures of grid cell effective directional permeability include the tensor approach of Oda,2 and the use of DFN simulations with a range of orientations for a unit gradient. Oda's2 method begins by considering the orientation of fractures in a grid cell, expressed as a unit normal vector n. Integrating the fractures over all of the unit normals N, Oda obtained the mass moment of inertia of fracture normals distributed over a unit sphere: ….For a specific grid cell with known fracture areas Ak and transmissivities Tk obtained from the DFN model, an empirical fracture tensor can be calculated by adding the individual fractures weighted by their area and transmissivity:…. Oda's permeability tensor is derived from Fij by assuming that Fij expresses fracture flow as a vector along the fracture's unit normal. Assuming that fractures are impermeable in a direction parallel to their unit normal, Fij must be rotated into the planes of permeability ….