Tensorial Relationship Research Articles

Tensor calculus in polar coordinates using Jacobi polynomials

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Feasibility of cooling positrons via conduction in conductive micro-tubes

A first order perturbation with respect to velocity has been employed to find the frictional damping force imposed on a single moving charge inside a conductive cylindrical micro-tube. The tensorial relationship between the force and velocity is derived and numerically estimated. Our results asymptotically match that of a flat geometry presented in the literature. Using the single particle analysis the cooling is formulated for an arbitrary density ensemble. It is shown that no further cooling via conduction occurs in the well-established non-neutral plasma equilibrium state. Also, the cooling rate for a weakly interacting ensemble is estimated. It is shown micro-tubes can be employed to cool down low density positron ensembles and/or to improve the beam emittance. A pack of tens of thousands of individual micro-tubes, each cooling only tens of positrons, is capable of cooling hundreds of thousands of particles in each cooling cycle. For example, with tens of particles per micro-tube in a 5 cm long micro-tube stack with the resistivity of 0.46 Ωm and the tubes of radius 50 μm, hundreds of thousands of positrons can be cooled down with a time constant of 103 μs in longitudinal and 7 μs in perpendicular direction. However, it must be noted that the cooling does not guarantee long term storage of particles in micro-tubes.

A new approach to formulate the general strength theories for anisotropic discontinuous materials. Part A: The experimental base for a new approach to formulate the general strength theories for anisotropic materials on the basis of wood

Experimental investigation carried out on wood indicates different overlapped failure mechanisms for compression and tension in the principal orthotropy plane. Different failure mechanisms result in different descriptions for compression and tensile failure stress. The consequence of these two different descriptions is the first order discontinuity (discontinuity of derivative) of failure hypersurface for normal stresses equal to zero. This discontinuity made it impossible to use failure stresses as components of the stress tensor in the whole of six-dimensional hyperspace and its sections of failure stresses in principal and structural system axes. Experimental investigation and observation of the damaged structure of wood lead to a conclusion that tensorial relationships are not satisfied in the macroscale of wood. Therefore, a new theoretical approach to describe the anisotropic materials with complicated structure on the basis of experimental data and observations of the damage processes of wood structure is proposed.

“A new approach to formulate the general strength theories for anisotropic discontinuous materials” Part B General form of polynomial to describe the strength of anisotropic discontinuous materials

A new approach to describe the failure hypersurface on the basis of assumptions presented in Part A reveals the new form of failure stress polynomial. In the presented theory new terms such as: unit tensor object, object formed on the basis of unit tensor object, the first, second and third form of the anisotropy failure function, and the first and the second type of object axis, were defined. On the basis of these terms the coefficients of their polynomials were formulated as values of the appropriate objects. The presented theory divides the six dimensional hyperspace of stresses into eight parts in which eight intersected hypersurfaces constitute the failure hypersurface. Six hypersurfaces may be assigned to two of three mutually coupled sets of elements. In general cases the theory may be used to describe the failure hypersurface for anisotropic structures where tensorial relationships do not occur. The obtained polynomial is transformed to tensor polynomial on the assumption that the failure stress tensorial relationships describe the failure hypersurface of anisotropic materials.

The direct tensor solution and higher-order acquisition schemes for generalized diffusion tensor imaging

Both in diffusion tensor imaging (DTI) and in generalized diffusion tensor imaging (GDTI) the relation between the diffusion tensor and the measured apparent diffusion coefficients is given by a tensorial equation, which needs to be inverted in order to solve the diffusion tensor. The traditional way to do this does not preserve the tensorial structure of the equation, which we consider a weakness in the method. For a physically correct measurement procedure, the condition number of the acquisition scheme, which is a determinant of the noise behavior, needs to be rotationally invariant. The method which traditionally is used to find such schemes, however, is cumbersome and mathematically unsatisfactory. This is considered a second weakness, closely connected to the first. In this paper we present an alternative inversion of the diffusion tensor equation, which does preserve the tensor form, for arbitrary order, and which is named the direct tensor solution ( DTS). The DTS is derived under the assumption that the apparent diffusion coefficient in any direction is known, i.e. in the infinite acquisition scheme. Whenever the DTS is valid for a given finite acquisition scheme and for a given order, the condition number is rotationally invariant. The DTS provides a compact, algebraic procedure to check this rotational invariance. We also present a method to construct acquisition schemes, for which the DTS is valid for the measurement of higher-order diffusion tensors. Furthermore, the DTS leads to other mathematical insights, such as tensorial relationships between diffusion tensors of different orders, and a more general understanding of the Platonic Variance Method, which we published before.

Axisymmetric Granular Collapse: A Transient 3D Flow Test of Viscoplasticity

A viscoplastic continuum theory has recently been proposed to model dense, cohesionless granular flows [P. Jop, Nature (London) 441, 727 (2006)10.1038/nature04801]. We confront this theory for the first time with a transient, three-dimensional flow situation--the simple collapse of a cylinder of granular matter onto a horizontal plane--by extracting stress and strain rate tensors directly from soft particle simulations. These simulations faithfully reproduce the different flow regimes and capture the observed scaling laws for the final deposit. Remarkably, the theoretical hypothesis that there is a simple stress-strain rate tensorial relationship does seem to hold across the whole flow even close to the rough boundary provided the flow is dense enough. These encouraging results suggest viscoplastic theory is more generally applicable to transient, multidirectional, dense flows and open the way for quantitative predictions in real applications.

Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture

The hypothesis that trabecular morphology can predict the elastic properties of cancellous bone has only partly been verified and no predictive analytical model is currently available. Such models are becoming increasingly relevant as the resolution levels of three-dimensional scanning techniques approach the size of trabeculae. This study took advantage of micro-finite-element methods and tested the aforementioned hypothesis in normal cancellous bone material collected at six anatomical locations from 56 individuals. Numerical analysis was based on high-resolution three-dimensional computer reconstructions of cancellous bone specimens from which the complete elastic characteristics and trabecular morphology, represented by three different fabric measures (the mean intercept length and two volume-based ones), were calculated. Each fabric measure was analyzed individually using the tensorial relationships derived by Cowin ( Mech Mater 4:137–147; 1985). Models for both stiffness and compliance entries were developed. The models based on stiffness entries could explain 93.4%–95.6% of the variance, whereas those based on compliance entries could explain 89.2%–89.4%. When using the former model, the MIL (mean intercept length measure) performed slightly better than the two volume-based measures, VO (volume orientation) and SVD (star volume distribution), with 23% less remaining variance. The high correlations found strongly support the hypothesis and increase the hope that, on the basis of information on trabecular morphology, it will be possible to obtain considerably better estimates of bone quality in vivo compared with the rough two-dimensional density measurements used today.

Renormalization of drift and diffusivity in random gradient flows

We investigate the relationship between the effective diffusivity and effective drift of a particle moving in a random medium. The velocity of the particle combines a white noise diffusion process with a local drift term that depends linearly on the gradient of a gaussian random field with homogeneous statistics. The theoretical analysis is confirmed by numerical simulation. For the purely isotropic case the simulation, which measures the effective drift directly in a constant gradient background field, confirms the result previously obtained theoretically, that the effective diffusivity and effective drift are renormalized by the same factor from their local values. For this isotropic case we provide an intuitive explanation, based on a {\it spatial} average of local drift, for the renormalization of the effective drift parameter relative to its local value. We also investigate situations in which the isotropy is broken by the tensorial relationship of the local drift to the gradient of the random field. We find that the numerical simulation confirms a relatively simple renormalization group calculation for the effective diffusivity and drift tensors.

Analogues of potential vorticity in electrically-conducting fluids

Abstract Exploiting the (contravariant) vectorial form for vorticity and magnetic field, and co-(variant) vectorial form for the gradient of a scalar and magnetic potential, and using the geometric Lie derivative operator, conservation of various analogues of potential vorticity are discussed for a barotropic non-dissipative electrically-conducting fluid. These analogues include the potential magnetic field, helicity, magnetic helicity, and cross helicity, together with some higher order quantities. It is noted that the volume conservation of potential vorticity continues to hold in the presence of arbitrary dissipation. However, of the analogue quantities derived for the non-dissipative system, only potential magnetic field and cross helicity have invariant integrals in the presence of dissipation. We conclude that only they are true analogues of potential vorticity. Finally, a straightforward generalisation of the method for tensorial relationships is noted.

Phasor-trajectory representation of near-resonance transients in quasilinear a.c. circuits

Some transients in linear a.c. circuits can be advantageously described by a slowly varying phasor. This is true also for a quasilinear RLC network, linearised at the point of near-resonance operation, for a small-perturbation stability analysis. The linearisation of a nonanalytical complex function indicates a tensorial relationship between the variables, but this is not uncommon in other well known electric-circuit applications such as synchronous machines and current transformers. The stability analysis is performed by calculating, step by step, the trajectory of a small initial perturbance phasor. It is illustrated by the example of a series RLC circuit and a capacitive voltage divider with a quasilinear inductive load.