An evolutionary algorithm for controlling numerical convergence of the radiative transfer equation with participating media using TVD interpolation schemes

Purpose This paper aims to present an evolutionary algorithm (EA) to accelerate the convergence for the radiative transfer equation (RTE) numerical solution using high-order and high-resolution schemes by the relaxation coefficients optimization. Design methodology/approach The objective function minimizes the residual value difference between iterations in each control volume until its difference is lower than the convergence criterion. The EA approach is evaluated in two configurations, a two-dimensional cavity with scattering media and absorbing media. Findings Experimental results show the capacity to obtain the numerical solution for both cases on all interpolation schemes tested by the EA approach. The EA approach reduces CPU time for the RTE numerical solution using SUPERBEE, SWEBY and MUSCL schemes until 97% and 135% in scattering and absorbing media cases, respectively. The relaxation coefficients optimized every two numerical solution iterations achieve a significant reduction of the CPU time compared to the deferred correction procedure with fixed relaxation coefficients. Originality/value The proposed EA approach for the RTE numerical solution effectively reduces the CPU time compared to the DC procedure with fixed relaxation coefficients.

Algorithm for analyzing the movement pattern of the temporomandibular joint musculararticular system based on multichannel electromyogram processing

Musculoskeletal system disorders is one of the priority directions in dentistry. They can manifest as Kosten's syndrome, snapping jaw, painful dysfunction syndrome, increased tooth abrasion, splits, breaks, pain and spasms in muscles, etc. The study set the following objectives: to develop an algorithm for analyzing the movement pattern of the muscular-articular system by developing an algorithm for recording, analyzing, filtering and processing multichannel electromyograms of the maxillofacial muscles. Analysis of the proposed algorithm for processing multichannel electromyograms showed that 7.2 % of multichannel electromyograms could not be analyzed due to patients' violations of the movement algorithm; 8.7 % of electromyogram checkpoint values were corrected. The group without dysfunctions of the temporomandibular joint is characterized by the prevalence of the relaxation coefficient of the left temporal muscle over the coefficient of the right muscle. The dysfunctioned group has the opposite result. The value of the compression ratio of the temporal muscles exceeding 2.5 is typical for the group with dysfunctions of the temporomandibular joint. The studied groups differ as much as possible when analyzing the relaxation coefficients of the temporal muscles. When analyzing this coefficient, it was possible to truly determine the presence or absence of violations in 50 %, falsely – in 16 % of cases. The coefficient of relaxation of the masticatory muscles made it possible to obtain a true state of 24 %, a false one – in 8 %. We concluded that the compression ratio is less suitable for separating patients with and without dysfunction of the temporomandibular joint.

On relaxation processes in a completely ionized plasma

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.

Investigation of the periodic axisymmetric flow of a viscoelastic fluid through a cylindrical tube

A generalized Womersley model of a nonstationary axisymmetric flow of a viscous incompressible fluid through a tube of circular cross-section to periodic pressure fluctuations at the inlet of the tube is obtained due for the case of a fluid with complicated rheology. The rheological parameters of the fluid are viscosity and four relaxation coefficients for strains and stresses of the first and second order. Such rheology is proper to the non-Newtonian viscoelastic fluids with mesostructure, namely technical and biological micro/ nanofluids. It was shown that with the increase of the relaxation coefficients of the first/second order the flow rate, the average and maximum velocities decrease/increase, accordingly. Simultaneous changes in these parameters can lead to complex changes in the velocity profile, especially for higher harmonics. The studied regularities can explain the deviations of the flow parameters of different micro/nanofluids from the values predicted by the classical Womersley solution for a homogeneous Newtonian fluid, which does not take into account viscous dissipation during the rearrangement of the fluid mesostructure.

Temperature and velocity relaxation in plasma. Spectral theory approach

The electron temperature and velocity relaxation of completely ionized plasma is studied on the basis of kinetic equation obtained from the Landau equation in a generalized Lorentz model. In this model contrary to the standard one ions form an equilibrium subsystem. Relaxation processes in the system are studied on the basis of spectral theory of the collision integral operator. This leads to an exact theory of relaxation processes of component temperatures and velocities equalizing. The relation of the developed theory with the Bogolyubov method of the reduced description of nonequilibrium systems is established, because the theory contains a proof of the relevant functional hypothesis, the idea of which is the basis of the Bogolyubov method. The temperature and velocity relaxation coefficients as eigenvalues of the collision integral operator are calculated by the method of truncated expansion of its eigenfunctions in the Sonine orthogonal polynomials. The coefficients are found in one- and two-polynomial approximation. As one can expect, convergence of this expansion is slow.

Relaxation processes in completely ionized plasma in generalized Lorentz model

On the basis of the Landau kinetic equation a generalized Lorentz model is proposed, which contrary to the standard model, considers ion system as an equilibrium one. For electron system kinetic equation of the Fokker-Planck type is obtained. In the Bogolyubov method of the reduced description, which is based on his idea of the functional hypothesis, basic equations for electron hydrodynamics construction with account for temperature and macroscopic velocity relaxation processes (kinetic modes of the system) is elaborated. The obtained equations are analyzed near the end of the relaxation processes when the theory has an additional small parameter. The main in small gradients approximation is studied in details, it corresponds to the description of relaxation processes in a spatially uniform case. The obtained equations are approximately solved by the method of truncated expansion in the Sonine polynomials. The velocity and temperature relaxation coefficients are discussed in one- and two-polynomial approximation. As a result the relaxation coefficients are calculated in one-polynomial approximation.

Stability limits of the single relaxation-time advection–diffusion lattice Boltzmann scheme

In many cases, multi-species and/or thermal flows involve large discrepancies between the different diffusion coefficients involved — momentum, heat and species diffusion. In the context of classical passive scalar lattice Boltzmann (LB) simulations, the scheme is quite sensitive to such discrepancies, as relaxation coefficients of the flow and passive scalar fields are tied together through their common lattice spacing and time-step size. This in turn leads to at least one relaxation coefficient, [Formula: see text] being either very close to 0.5 or much larger than unity which, in the case of the former (small relaxation coefficient), has been shown to cause instability. The present work first establishes the stability boundaries of the passive scalar LB method in the sense of von Neumann and as a result shows that the scheme is unconditionally stable, even for [Formula: see text], provided that the nondimensional velocity does not exceed a certain threshold. Effects of different parameters such as the distribution function and lattice speed of sound on the stability area are also investigated. It is found that the simulations diverge for small relaxation coefficients regardless of the nondimensional velocity. Numerical applications and a study of the dispersion–dissipation relations show that this behavior is due to numerical noise appearing at high wave numbers and caused by the inconsistent behavior of the dispersion relation along with low dissipation. This numerical noise, known as Gibbs oscillations, can be controlled using spatial filters. Considering that noise is limited to high wave numbers, local filters can be used to control it. In order to stabilize the scheme with minimal impact on the solution even for cases involving high wave number components, a local Total Variation Diminishing (TVD) filter is implemented as an additional step in the classical LB algorithm. Finally, numerical applications show that this filter eliminates the unwanted oscillations while closely reproducing the reference solution.

La relation d'Einstein pour le mouvement brownien des membranes

We study form fluctuations of topologically nontrivial membranes, sponges for instance, assuming that there is a principal mode described by Langevin's equation, which has a character of chaotic relaxation to the equilibrium position. The influence of the ambient liquid being taken into account by the relaxation coefficients and the source of noise. The chaotic change of the surface is characterized by the quantity Δ2 (similar to the activity of rotational Brownian motion), which satisfies Einstein's equation Δ2/t = γIT, where t is the time, I a factor depending on the form of the membrane, and γ the dissipative constant. Fluctuations are studied by using the chiral field that is obtained from the Gauss–Weingarten local frame, usual in the classical theory of surfaces. Thus, we determine the effective action for a chiral field having a supersymmetric structure, we derive the correlation functions, and develop the theory of perturbations by the curvature of the surface.[Journal translation]