Equivalent Tensor Research Articles

Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit

AbstractWe show that if$A$and$H$are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of$A$to the same kind of resolution for the counit of$H$, exhibiting in this way strong links between the Hochschild homologies of$A$and$H$. This enables us to obtain a finite free resolution of the counit of$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case$E=I_n$. It follows that$\mathcal {B}(E)$is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when$E$is an antisymmetric matrix, the$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.

Slip in the 2010–2011 Canterbury earthquakes, New Zealand

The 3rd September 2010 Mw 7.1 Darfield and 21st February 2011 Mw 6.3 Christchurch (New Zealand) earthquakes occurred on previously unknown faults. We use InSAR ground displacements, SAR amplitude offsets, field mapping, aerial photographs, satellite optical imagery, a LiDAR DEM and teleseismic body‐wave modeling to constrain the pattern of faulting in these earthquakes. The InSAR measurements reveal slip on multiple strike‐slip segments and secondary reverse faults associated with the Darfield main shock. Fault orientations are consistent with those expected from the GPS‐derived strain field. The InSAR line‐of‐sight displacement field indicates the main fault rupture is about 45 km long, and is confined largely to the upper 10 km of the crust. Slip on the individual fault segments of up to 8 m at 4 km depth indicate stress drops of 6–10 MPa. In each event, rupture initiated on a reverse fault segment, before continuing onto a strike‐slip segment. The non‐double couple seismological moment tensors for each event are matched well by the sum of double couple equivalent moment tensors for fault slip determined by InSAR. The slip distributions derived from InSAR observations of both the Darfield and Christchurch events show a 15‐km‐long gap in fault slip south‐west of Christchurch, which may present a continuing seismic hazard if a further unknown fault structure of significant size should exist there.

Microstructure Design with Multiphase Materials under Mass Constraints

The microstructure design satisfying the mass constraint can reduce the structure weight more directly and effectively in comparison with the volume constraint. This paper is devoted to the topology optimization of microstructures with multiphase materials under the mass upper limitation constraint for maximizing the equivalent elastic tensors and their combinations. Firstly, the strain energy method is applied to compute the effective elastic properties of microstructures. In order to make sure that the formulation of mass constraint is linear with separable design variables, DMO (Discrete Material Optimization) model is adopted for the element density interpolation. Therefore, this optimization problem can be solved efficiently by means of mathematical programming approaches, especially the convex programming methods. Besides, the filtering technique is adopted to avoid the checkerboard pattern. There are two categories of numerical examples. In the first category, the modulus and the stiffness ratio of the solid material phase 1 are smaller than the solid material phase 2. In the second category, the modulus of the solid material phase 1 is still smaller than the solid material phase 2, but its stiffness ratio is bigger than the solid material phase 2.

Computation of Homogenized Constitutive Tensor of Elastic Solids Containing Evolving Cracks

The determination of the equivalent (homogenized) constitutive tensor is one of the most important steps in multiscale models as well as in the classical homogenization theory. In this article, a procedure for determining the homogenized instantaneous (tangent) constitutive tensor of elastic materials containing growing cracks is proposed. The primary purpose of this procedure is its use in two-way coupled multiscale finite element algorithms that can model crack formation and propagation at the local microstructure. The procedure is basically developed by relating the local displacement field to the global strain tensor at each location and using first-order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify and demonstrate the model capabilities.

Multicopy and stochastic transformation of multipartite pure states

Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable under MCSLOCC, and vice versa. Based on these environments we classify the multipartite pure states into a few inequivalent sets and orbits, between which we build the partial order to decide their transformation. In particular, we investigate the structure of SLOCC-equivalent states in terms of tensor rank, which is known as the generalized Schmidt rank. Given the tensor rank, we show that GHZ states can be used to generate all states with a smaller or equivalent tensor rank under SLOCC, and all reduced separable states with a cardinality smaller or equivalent than the tensor rank under LOCC. Using these concepts, we extended the concept of "maximally entangled state" in the multi-partite system.

Connectivity‐consistent mapping method for 2‐D discrete fracture networks

We present a new flow computation method in 2‐D discrete fracture networks (DFN) intermediary between the classical DFN flow simulation method and the projection onto continuous grids. The method divides the simulation complexity by solving for flows successively at a local mesh scale and at the global domain scale. At the local mesh scale, flows are determined by classical DFN flow simulations and approximated by an equivalent hydraulic matrix (EHM) relating heads and flow rates discretized on the mesh borders. Assembling the equivalent hydraulic matrices provides for a domain‐scale discretization of the flow equation. The equivalent hydraulic matrices transfer the connectivity and flow structure complexities from the local mesh scale to the domain scale. Compared to existing geometrical mapping or equivalent tensor methods, the EHM method broadens the simulation range of flow to all types of 2‐D fracture networks both below and above the representative elementary volume (REV). Additional computation linked to the derivation of the local mesh‐scale equivalent hydraulic matrices increases the accuracy and reliability of the method. Compared to DFN methods, the EHM method first provides a simpler domain‐scale alternative permeability model. Second, it enhances the simulation capacities to larger fracture networks where flow discretization on the DFN structure yields system sizes too large to be solved using the most advanced multigrid and multifrontal methods. We show that the EHM method continuously moves from the DFN method to the tensor representation as a function of the local mesh‐scale discretization. The balance between accuracy and model simplification can be optimally controlled by adjusting the domain‐scale and local mesh‐scale discretizations.

Theoretical model of effective stress coefficient for rock/soil-like porous materials

Physical mechanisms and influencing factors on the effective stress coefficient for rock/soil-like porous materials are investigated, based on which equivalent connectivity index is proposed. The equivalent connectivity index, relying on the meso-scale structure of porous material and the property of liquid, denotes the connectivity of pores in Representative Element Area (REA). If the conductivity of the porous material is anisotropic, the equivalent connectivity index is a second order tensor. Based on the basic theories of continuous mechanics and tensor analysis, relationship between area porosity and volumetric porosity of porous materials is deduced. Then a generalized expression, describing the relation between effective stress coefficient tensor and equivalent connectivity tensor of pores, is proposed, and the expression can be applied to isotropic media and also to anisotropic materials. Furthermore, evolution of porosity and equivalent connectivity index of the pore are studied in the strain space, and the method to determine the corresponding functions in expressions above is proposed using genetic algorithm and genetic programming. Two applications show that the results obtained by the method in this paper perfectly agree with the test data. This paper provides an important theoretical support to the coupled hydro-mechanical research.

Fabrication and characterization of anisotropic dielectrics for low-loss microwave applications

New magneto-photonic assembly designs for high-gain antennas require dielectrics with a significant anisotropy and low loss at GHz frequencies. This paper describes an approach to fabricate such dielectrics from ceramic laminates. These laminates consist of two ceramics with largely different permittivities and low dielectric losses. Alternating layers of commercially available α-Al2O3 and Nd-doped BaTiO3 were laminated using organic adhesives. Equivalent permittivity tensors and loss tangents were characterized using a resonant cavity-based approach, which was coupled with a finite-element method full-wave solver. Measured permittivity values were in good agreement with mean field predictions; a minimum loss tangent 1.1 × 10−3 was obtained when using one-component epoxy (Loctite®-3982) adhesive. Application of two-component epoxy (M-bond 610) adhesive results in a slightly higher loss but better mechanical properties and machinability. These laminates were used to demonstrate high gain in a prototype antenna with 6 misaligned anisotropic dielectric layers.

Thermal expansion behavior of composites based on non-axisymmetric ellipsoidal particles

A new model for calculating the coefficients of thermal expansion, CTE, in all three coordinate directions is developed for composites containing non-axisymmetric, ellipsoidal particles, i.e., three-dimensional ellipsoids (a1≠a2≠a3) characterized by two aspect ratios, ρα=a1/a3 and ρβ=a1/a2. The model makes use of Eshelby's equivalent tensor for the particles and utilizes the methodology from recent papers by Lee et al. The effects of the primary, ρα, and secondary, ρβ, aspect ratios and filler volume fraction (0–6%) on the various CTE for nanocomposites containing aligned inclusions are demonstrated by calculations for a matrix of nylon 6 and a filler of montmorillonite single platelets. The CTE in the longitudinal direction, α11, decreases, as both aspect ratios increase. The CTE in the transverse direction, α22, decreases as ρα increases but increases as ρβ increases. The CTE in the normal direction, α33, is greater than that of the matrix and increases as ρα increases but decreases as ρβ increases.

SUBALGEBRAS OF THE LIE ALGEBRA R6AND THEIR APPLICATIONS

As we all know, the Hamiltonian systems are the same describing forms as Newton mechanics and Lagrange mechanics. Therefore, researching for a new Hamiltonian structure of the soliton equations has important significance. In the paper, firstly, with the help of the Lie algebra R6, a few types of subalgebras are constructed, from which the corresponding equivalent tensor systems are given. For their applications, two integrable couplings hierarchies along with the multi-potential component functions generated from the soliton theory and the Virasoro symmetric algebra are obtained. Secondly, the Hamiltonian structures of the above integrable couplings are worked out, which may become another describing expression for the Newton and Lagrange mechanics. In particular, one of the integrable couplings presented above reduces to the famous AKNS hierarchy of soliton equations.

About the Symmetry of the Upscaled Equivalent Transmissivity Tensor

This paper provides a proof of the symmetry of the equivalent transmissivity tensor computed with a method based on the solution of the balance equation at the fine scale with integrated finite differences. The basic hypotheses are that the balance equation is solved on a block assigning Dirichlet boundary conditions that vary linearly with the space coordinates and the equivalent transmissivity tensor is computed as the linear tensor relating average flux and hydraulic gradient.

Modified Eshelby tensor modeling for elastic property prediction of carbon nanotube reinforced ceramic nanocomposites

A modified model using the Eshelby equivalent tensor is developed to evaluate overall elastic properties of carbon nanotube (CNT) reinforced aluminum oxide nanocomposites. This model accounts for the effect of carbon nanotube geometry and porosity on the effective elastic modulus. Experimental results are compared with the computed predictions. Computed results show higher elastic modulus values, which are attributed to the perfect bonding between the reinforcement and the matrix. It is concluded that the modified Eshelby model can predict the elastic property of CNT reinforced ceramic nanocomposites.

Geostatistical Simulation and Numerical Upscaling, to Model Ground-Water Flow in a Sandy-Gravel, Braided River, Aquifer Analogue

Abstract An integrated approach combining sedimentological, geostatistical, and numerical analyses has been applied to model groundwater flow through a glaciofluvial, braided-stream, gravel–sand aquifer of Pleistocene age, exposed in a gravel pit in the eastern Piedmont (northern Italy). The aquifer under investigation consists of a stack of braided bar and minor channel fills and scour fills, where different gravel–sand bedforms migrated. It represents an analogue of buried sediments, which host some of the most important aquifer complexes and water resources of northern Italy. The work has been performed with three main steps: (1) drawing and discretization of the geological models of two orthogonal exposures and estimate of conductivity of every facies; (2) geostatistical simulation of the 3D facies heterogeneity, based on semivariogram analysis of the geological models; (3) application of a 3D numerical model of flow to upscale the conductivity field. Semivariogram analysis developed from facies indicator functions has shown that: (a) the variability in the data set in the horizontal direction is a representation of the oriented distribution of the sedimentary facies, corresponding to the average N–S elongation of sediment units, according to the paleoflow direction; (b) the vertical correlation length approximates the average thickness of the bedsets bounded by fourth-order erosional surfaces (unit bars, scour fill, and small channel fill associations) and may represent the thickness of an individual hydrostratigraphic unit; (c) the conductivity and facies distributions are more variable in the vertical direction than in the horizontal one, with a ratio between vertical and horizontal ranges of about 1:8. The 3D conductivity distribution within the considered deposits has been determined using a sequential indicator simulation, whereas transition-probability analysis has been used to confirm the correlation model obtained with semivariogram analysis and to check the result of the simulation. The equivalent hydraulic-conductivity tensor has been computed with finite-difference modeling of groundwater flow for cubic blocks of the aquifer analogue with side length of about 13 m. At this scale the medium is almost homogeneous, but anisotropic, with the lowest values of hydraulic conductivity in the vertical direction and the greatest values along the elongation of the sedimentary units, i.e., almost parallel to average paleoflow.

Macroscopic and molecular symmetries of unconventional nematic phases

The effects of the molecular symmetry on macroscopic properties of conventional and unconventional nematic phases are investigated theoretically. These effects concern: (1) the form of the molecular orientation fluctuations, (2) the thermodynamic behavior and the list of stable ordered phases, (3) the effective symmetry of the molecules in the ordered phases. The order-parameter consists of tensors forming the main harmonics of the fluctuation distribution. In conventional models (valid for uniaxial molecules) a single tensor is sufficient while unconventional models (valid for less symmetric molecules) need several tensors with the same rank. We analyze the qualitative differences arising when the number of equivalent tensors varies. We show how to work out complete models in the general case, and to calculate the sequences of stable phases and the corresponding effective molecular symmetries. This yields, for each molecular group and each tensor rank, a complete classification and a deep insight into the structure of thermotropic nematics. This work generalizes the approach we have applied to polar nematics recently observed in polyester compounds and to unconventional uniaxial and biaxial phases of bent-core materials.

Equivalent permittivity tensor in anisotropic random media

An explicit result for the equivalent (or effective) permittivity tensor is derived for dispersions of anisotropic (biaxial) particles embedded in an isotropic matrix. The overall medium has a positional disorder but an orientational order and it exhibits a uniaxial behaviour. Each crystal is partially oriented along a given director (preferential direction) and the degree of orientation is taken into account by a suitable order parameter. The equivalent permittivity tensor of the whole heterogeneous material is described in terms of the order parameter, the volume fraction of the dispersed phase, the principal values of the permittivity tensor of the crystals and the components of the director along which the crystals are preferentially aligned.

A model for composites containing three-dimensional ellipsoidal inclusions

A model is developed for the mechanical properties of composites containing complex inclusions with no axes of symmetry, e.g. three dimensional ellipsoids (a1>a2>a3) characterized by two aspect ratios, α=a1/a3 and β=a1/a2, by using the Eshelby's equivalent tensor with a Mori–Tanaka type model.The influences of the primary and secondary aspect ratios on the effective elastic moduli of nanocomposites containing aligned isotropic inclusions are examined. The model is limited to unidirectionally aligned inclusions where both the matrix and the inclusions have linearly elastic, homogeneous properties. The longitudinal moduli (E11, E22 and E33) and the shear moduli (μ12, μ13 and μ23) are calculated. The longitudinal Young's modulus E11 increases, as the primary and secondary aspect ratios increase. However, the transverse Young's modulus E22 and shear modulus μ12 decrease, as the secondary aspect ratio increases.

Orthotropic elastic–plastic material model for paper materials

Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic–plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress–strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.

Fractional Fourier transform for elliptical Gaussian beam in spatial-frequency domain

Summary The fractional Fourier transform (FRT) is applied to elliptical Gaussian beam (EGB) in spatial-frequency domain. An equivalent tensor ABCD law for the FRT of EGB is derived in terms of the tensor method. The connection between the FRT formula and the generalized Collins formula in spatial-frequency domain is discussed. The derived formulae provide a powerful tool for analyzing and calculating the FRT of EGB in spatial-frequency domain. An application example is provided.

Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems

A new solution-adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains. Such a method is required for the numerical solution of realistic combustion problems, in which physical quantities may vary by orders of magnitude over one-tenth of a millimeter at atmospheric pressure, or over micrometers at higher pressures. The local rectangular refinement (LRR) method maintains orthogonality at grid-line intersections but lifts the tensor product restriction common to traditional grids, producing unstructured grids. Governing equations are discretized throughout the domain using newly derived forms, and Newton's method is used to solve the resulting system. On a simple test case with a known solution, the LRR method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature. For the more realistic problem of nonreacting driven square cavity flow, the LRR solution agrees very well with previously published data. When the LRR method is applied to a practical reacting flow (a rich axisymmetric laminar Bunsen flame with complex chemistry, multicomponent transport, and an optically thin radiation submodel), grid spacing highly influences the inner flame's position, which stabilizes only with adequate refinement. The vorticity–velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the LRR gridding technique. Furthermore, each LRR grid is used to form a nonuniform equivalent tensor product (ETP) grid and also, in most cases, an equispaced fully refined (FR) grid; these additional grids are supersets of the LRR grids and thus contain refinement in exactly the same regions. Performance comparisons between the LRR, ETP, and FR grids indicate that the LRR method provides substantial savings in execution time and computer memory requirements, without compromising solution accuracy.

The stress tensor for nonlocal field equations

Construction of the stress tensor for a low differential order field theory on translation-invariant space is routine. If the underlying space is rotation, or Lorentz, invariant, an equivalent symmetric tensor can be found as well. Extension to nonlocal field equations, common, for example, in the statistical mechanical theory of fluids, is not routine and is carried out in this paper.