Maxwell Matrix Research Articles

An Artificial Anisotropy Four-Step HIE-FDTD Method With Lower Numerical Dispersion Error

In this article, by introducing artificial anisotropic (AA) parameters in the four-step hybrid implicit-explicit finite-difference time domain (HIE-FDTD) method, an AA four-step HIE-FDTD method is proposed, which can reduce the numerical dispersion error and improve the computational accuracy. Firstly, the Maxwell's matrix equation is decomposed into four sub-matrix equations by using the split-step scheme, and artificial anisotropy parameters are introduced, then the proposed AA four-step HIE-FDTD is obtained. Furthermore, the stability analysis of the AA four-step HIE-FDTD method shows that the stability condition of the proposed method is closed to the original four-step HIE-FDTD method. Next, the numerical dispersion characteristics of the proposed method are analyzed and compared with other HIE-FDTD methods. The results show that the numerical dispersion error of the proposed method is significantly reduced compared with the four-step HIE-FDTD method. Finally, the performance of the proposed method is further verified by the numerical simulation. Numerical results show that the proposed method has lower numerical dispersion error and higher computational accuracy than that of the four-step HIE-FDTD method.

Методы и алгоритмы повышения быстродействия вычислительных процессов для расчетов гидравлических сетей

The article is devoted to mathematical modeling of flow distribution in hydraulic net-works. Calculations of hydraulic networks are carried out at the stage of their design and operation. The results of numerical simulation are used to control the operation of the hy-draulic network in real time. The mathematical model of the distribution of flows in the hydraulic network is a system of nonlinear equations. The nodal pressures method used to solve the system of equations numerically is the n-dimensional Newton method. To ensure stable and fast convergence of the iterative process, it is proposed to use the initial approx-imation taking into account the network topology and parameters of its objects, use the lower relaxation factor and optimize the structure of the Maxwell matrix. The algorithms presented in the paper allow one to significantly reduce the dimension of the system of nonlinear equations being solved.

Hydraulic networks with pressure-dependent closure relations, under restrictions on the value of nodal pressures. Maxwell matrix properties and monotonicity of flow distribution problem solution

In the article, which continues the research of article [1], the results of previous article are generalized to “abstract” hydraulic networks. Additional existence theorems are proved for classical flow distribution problem (CFDP) for hydraulic networks with pressure-dependent closure relations, under restriction on nodal pressures. Hydraulic network Maxwell matrix properties are establish, related to monotonicity of CFDP solution.

TWO EFFICIENT UNCONDITIONALLY-STABLE FOUR-STAGES SPLIT-STEP FDTD METHODS WITH LOW NUMERICAL DISPERSION

Two e-cient unconditionally-stable four-stages split- step (SS) flnite-difierence time-domain (FDTD) methods based on controlling parameters are presented, which provide low numerical dispersion. Firstly, in the flrst proposed method, the Maxwell's matrix is split into four sub-matrices. Simultaneously, two controlling parameters are introduced to decrease the numerical dispersion error. Accordingly, the time step is divided into four sub-steps. The second proposed method is obtained by adjusting the sequence of the sub- matrices deduced in the flrst method. Secondly, the theoretical proofs of the unconditional stability and dispersion relations of the proposed methods are given. Furthermore, the processes of obtaining the controlling parameters for the proposed methods are shown. Thirdly, the dispersion characteristics of the proposed methods are also investigated, and numerical dispersion errors of the proposed methods can be decreased signiflcantly. Finally, to substantiate the e-ciency of the proposed methods, numerical experiments are presented.

Orientational kinetics of dipolar particles in a Maxwell fluid matrix: Inertialess limit for the rotary microrheology

We study magnetic response of an assembly of ferroparticles suspended in a viscoelastic matrix which is modeled by a Maxwell fluid with a unique stress relaxation time. The problem refers to the magnetic microrheology approach where deformational properties of a complex fluid are tested with the aid of embedded nanoparticle probes set to motion by an external ac magnetic field. A possibility is considered to simplify the description of the orientational kinetics of the system at the expense of neglecting inertia effects in particle rotary motion. It is shown that in this aspect a Maxwell matrix differs essentially from the Newtonian one. In the latter the inertialess approximation for the particles of the approximately 10nm size is valid practically unboundedly. For a viscoelastic matrix the inertialess approximation means an important restriction on the value of the stress relaxation time. Assuming weak nonequilibrium, the magneto-orientational relaxation times are found and low-frequency magnetic spectra of a viscoelastic suspension are determined in the presence of a constant (magnetizing) field.

Rotational Microrheology of Viscoelastic Fluid: Orientational Kinetics of Magnetic Particles in the Inertialess Approximation

A kinetic description is given for the suspension of dipole particles in the viscoelastic matrix simulated by the Maxwell liquid with one relaxation time. The possibility of the brief description of the orientational kinetics of the particles is considered with their inertia being disregarded. It is shown that the case of the Maxwell matrix is fundamentally different than the case of a Newtonian liquid, where the inertialess approximation for the particles with sizes of ∼10 nm has a practically unlimited domain of applicability. For viscoelastic suspension, the inertialess approximation denotes a rather strict constraint on the time of stress relaxation τ M . Times of orientational relaxation of the magnetization of a viscoelastic system and spectra of its low-frequency dynamic susceptibility in the presence of a constant biasing field are found in the approximation of the effective field (the slightly nonequilibrium state).

Transport algorithms for partially ionized and unmagnetized plasmas

A new formalism for the transport properties of partially ionized and unmagnetized plasmas is investigated from a computational point of view. Heavy particle transport expressions for shear viscosity, translational thermal conductivity, and thermal diffusion ratios are obtained from the solution of symmetric linear systems. Electron transport properties are also presented. A general Stefan–Maxwell equation and two approximate formulations deal with diffusion phenomenon. Well-posedness of the transport properties is established, provided that some conditions on the kinetic data are met. The mathematical structure of the transport matrices is readily used to build transport algorithms inspired by Ern and Giovangigli [J. Comput. Phys. 120 (1995) 105]. These algorithms rely either on a direct linear solver or on convergent iterative Krylov projection methods, such as the conjugate gradient. The Stefan–Maxwell matrix is singular and a mass conservation constraint completes the system of equations. A yet symmetric and non-singular Stefan–Maxwell matrix including the mass constraint is introduced for the direct method. A suitable projector associated with the singular form of the matrix is used for the iterative methods. An 11-species air mixture in local thermodynamic equilibrium at atmospheric pressure serves as benchmark to assess the physical model and numerical methods. Superiority of the conjugate gradient method with respect to the direct solver and approximate mixture rules found in the literature is demonstrated in terms of accuracy and computational cost.

A contact mechanism based theory of Maxwell matrix composites

Using the binder-contact laws in deriving the stress–strain relations for a composite with elastic particles/inclusions/fibers and a Maxwell type of viscoelastic matrix/binder is presented in this article. This derivation is physically based and it establishes a functional correlation of how the viscosity at the binder/matrix level is blended into that at the composite level. The derived viscoelastic stress–strain equations are multiaxial, and their equivalent models are analyzed for three scenarios. A confined creep simulation is also given with its predications being compared with the experimental results.

Analysis of error propagation in computing the Maxwell matrix of multiconductor structure from measurements

The Maxwell matrix for multiconductor transmission lines can theoretically be determined from measurements. When the measurements are performed with finite accuracy, the measurement errors propagate on the computed Maxwell matrix. Matrix relations for computation of the Maxwell matrix from measurements are presented. These relations are used in the analysis of error propagation. It is shown that the errors in determining the off-diagonal elements can be very large. A numerical example to demonstrate the role of measurement errors in determining the Maxwell matrix is supplied.

A secant-viscosity approach to the time-dependent creep of an elastic viscoplastic composite

At present no homogenization theory seems to exist to estimate the time-dependent creep behavior of a two-phase viscoplastic composite. In this paper an approach introducing a linear viscoelastic comparison composite in conjunction with the secant viscosity is proposed for this purpose. The method makes use of a Maxwell matrix in the viscoelastic composite, and sets its shear viscosity equal to the secant viscosity of the viscoplastic matrix at every stage of deformation. The property of the viscoelastic composite is in turn determined from its elastic comparison composite, and the results are cast in the rate forms to reflect the continuously changing secant viscosity of the viscoplastic matrix. This new approach is neither restricted to rigid particles nor to a viscoplastic matrix with rigid elasticity, and it is capable of accounting for the strain-hardening of the matrix phase. The theory is developed fully for the particle-reinforced composite and, when applied to a TiC/Al system, it leads to a satisfactory agreement with available experimental data. Both concepts of viscoelastic comparison composite and secant viscosity are believed to be new in a homogenization scheme, and, as the secant moduli in the rate-independent plasticity, the secant viscosity now can play a useful role in the determination of the effective behaviour of an elastic-viscoplastic composite.

Strain-Rate Sensitivity, Relaxation Behavior, and Complex Moduli of a Class of Isotropic Viscoelastic Composites

A micromechanical principle is developed to determine the strain-rate sensitivity, relaxation behavior, and complex moduli of a linear viscoelastic composite comprised of randomly oriented spheroidal inclusions. First, by taking both the matrix and inclusions as Maxwell or Voigt solids, it is found possible to construct a Maxwell or a Voigt composite when the Poisson ratios of both phases remain constant and the ratios of their shear modulus to shear viscosity (or their bulk counterparts) are equal; such a specialized composite can never be attained if either phase is purely elastic. In order to shed some light for the obtained theoretical structure, explicit results are derived next with the Maxwell matrix reinforced with spherical particles and randomly oriented disks. General calculations are performed for the glass/ED-6 system, the matrix being represented by a four-parameter model. It is found that, under the strain rates of 10−7/hr and 10−6/hr, randomly oriented disks and needles at 20 percent of concentration both give rise to a very stiff, almost linear, stress-strain behavior, whereas inclusions with an aspect ratio lying between 0.1 and 10 all lead to a softer nonlinear response. The relaxation behavior of the composite reinforced with spherical particles is found to be more pronounced than those reinforced with other inclusion shapes, with disks giving rise to the least stress relaxation. The real and imaginary parts of the overall complex moduli are also established, and found that, as the frequency increases, the real part of the complex bulk and shear moduli would approach their elastic counterparts, whereas for the imaginary part, the increase shows two maxima, and then drops to zero as the frequency continues to increase. Finally, the complex bulk modulus is examined in light of the Gibiansky and Milton bounds, and it is found that, for all inclusion shapes considered, this modulus always lies on or within the bounds.