Viktor Popov

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

This work presents a multi-domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard Boundary Element Method (BEM) is in disadvantage in comparison with classical domain schemes, such as Finite Difference (FDM) and Finite Element (FEM) methods. As in the recently developed Green Element Method (GEM), in the present approach the original domain is divided into several subdomains. In each of them the corresponding Greens integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed. In contrast with the GEM, where in each subregion the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each subregion via the Dual Reciprocity Method (DRM), using some of the most efficient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are defined in terms of isoparametric linear elements, for which the analytical integrations of the kernels of the integral representation formula are known. As in the FEM and GEM the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non-Hermitian FEM), here one is able to solve the system for the complete internal nodal variables, i.e. the field variables and their derivatives, without any additional interpolation. Finally, some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the linear and non-linear convection–diffusion equation are presented. Copyright

Mathematical modelling for predicting the growth of Pseudomonas spp. in poultry under variable temperature conditions.

A dynamic growth model under variable temperature conditions was implemented and calibrated using raw data for microbial growth of Pseudomonas spp. in poultry under aerobic conditions. The primary model was implemented using measurement data under a set of fixed temperatures. The two primary models used for predicting the growth under constant temperature conditions were: Baranyi and modified Gompertz. For the Baranyi model the maximum specific growth rate and the lag phase at constant environmental conditions are expressed in exact form and it has been shown that in limit case when maximal cells concentration is much higher than the initial concentration the maximum specific growth rate is approximately equal to the specific growth rate. The model parameters are determined in a temperature range of 2-20 degrees C. As a secondary model the square root model was used for maximum specific growth rate in both models. In both models the main assumption, that the initial physiological state of the inoculum is constant and independent of the environmental parameters, is used, and a free parameter was implemented which was determined by minimizing the mean square error (MSE) relative to the measurement data. Two temperature profiles were used for calibration of the models on the initial conditions of the cells.

DRM-MD approach for the numerical solution of gas flow in porous media, with application to landfill

Abstract In the present work two different Dual Reciprocity Method (DRM) formulations were developed, for convection–diffusion flow of a mixture of gases in a multi-layer porous media, with an application to landfills. The first method treated the whole problem domain as a single one, while the second formulation, the Dual Reciprocity Method Multi-Domain decomposition (DRM-MD), divides the initial domain into a large number of subdomains. The main advantage of the single domain formulation, in relation to domain methods, is that only surface elements are necessary, so the input data is reduced. A drawback of this approach is that it results in a fully populated matrix system, limiting it to small or medium size problems. On the other hand, the DRM-MD formulation extends the range of applications of the technique to large problems, since the final matrix of the system is sparse (band diagonal) and the matrix coefficients of geometrically similar subregions are calculated only once.

Modelling of flow-induced stresses in a Francis turbine runner

This study presents the results of large scale modelling of the water flow and the analysis flow-induced stresses in a Francis turbine runner. The modelling undergoes two stages. The first stage deals with the water flow that has been investigated by using Computational Fluid Dynamics (CFD) in order to identify the loads acting on the turbine blades. At the second stage, the finite element analysis of stresses has been performed based on the pressure distributions calculated from CFD modelling. The operational data recorded at Unit 2 of the Derbendikan power station have been used as input in the modelling. The results of calculations have revealed that the zones of high stress are situated at the trailing edge of the turbine runner, which explains observed fatigue cracks in these areas.

Modelling of laser-material interaction using semi-analytical approach

In this paper different aspects of laser-material interaction were considered. Semi-analytical method was developed and applied to analysis of spatial and temporal distribution of temperature field inside bulk materials. In particular, cases with cylindrical geometry, finite diameter and infinite length as well as cylindrical geometry, finite diameter and finite length were considered. For solving the governing partial differential equations (PDEs) the Laplace transform and the Fourier method of variables separation were used. In this way instead of the original governing PDEs, ordinary differential equations were solved. Particular solutions of the ordinary differential equations were used for evaluating the general solution, which was expressed in terms of series of particular solutions. The unknown coefficients in the series of particular solutions were determined using the boundary and initial conditions. The laser-material interaction was represented using the thermal model. These interactions for the cases of the high power laser in pulse and continuous regime were analysed. The incident intensity of laser radiation was under critical intensity.Using these methods the temperature field distribution was obtained in the Laplace transform domain. The convolution integral and the Green function were used to determine the temperature field in time domain. General semi-analytical methods and numerical solutions of appropriate transcendent equations were considered and numerical results for Al specimens were presented. The influences of laser beam parameters to the temperature field distribution and isothermal curves inside the bulk material were evaluated.

An alternative approach for calculation of the first and higher order derivatives in the DRM-MD

The partial derivatives (PDs) in the classical dual reciprocity method (DRM) are represented through PDs of the DRM approximation function. This approach is very flexible and easy to implement but may introduce singularities depending on the approximation function used, especially for higher order PDs. A new scheme is proposed that reduces the order of the PD of the DRM approximation function by one in respect to the one that appears in the classical DRM representation of PDs. This scheme can be applied for a case when the proportion of corner nodes is high, that is, in the DRM-MD approach. The new approach is tested and the results show improvements in the accuracy of at least one order of magnitude for the case when the derivatives of the approximation function introduce singularities.

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.

Three-dimensional solution for acoustic and transport problems using the radial basis integral equation method

Abstract The radial basis integral equations method (RBIEM) has been applied for solution of three-dimensional (3D) acoustic and transport problems. The acoustic problem is often described using the Helmholtz equation, while the transport problems are usually described using the Laplace equation (diffusion only), the Poisson equation (diffusion with sources/sinks) and the convection–diffusion equation. The accuracy of the numerical scheme employing the first and second order Duchon splines augmented by first and second order polynomials, respectively, was examined. The effect of the number of interpolation points used in the radial basis function approximation on the condition number of the system was investigated. Numerical results obtained for the convection–diffusion equation were compared with the solutions obtained using the multi-domain dual reciprocity boundary element method (DRM-MD). The RBIEM formulation was found to be more accurate than the DRM-MD formulation. The implementation does not involve discretization over the boundaries of the subdomains used in the RBIEM formulation when evaluating the integrals.

3D modeling of material heating with the laser beam for cylindrical geometry

In this work an analytical approach for analyzing heating of material with a laser beam is presented. A thermal model of interaction for the case of cylindrical geometry of the material and asymmetric distribution of the laser beam intensity is used and an analytical procedure is developed to analyze the temporal and the spatial distribution of the temperature field inside the bulk of material. This kind of consideration is of practical interest in cases where the excitation by the laser beam is not symmetric in respect to its position or shape, e.g., multi-mode working regimes or asymmetrical distribution of the laser beam intensity. The heating effects were considered in the temperature range up to the melting point. The thermal and the optical parameters of the material were assumed to be independent of the temperature and were given constant values in the temperature range of interest. This approach makes use of the Laplace transform, in order to eliminate dependence on time. The Fourier method of variable separation was used to obtain the temperature field distribution in the Laplace transform domain. By using the pulse response and Duhamels principle the 3D temperature field distribution in time domain is obtained. By using an appropriate set of orthogonal functions in r directions, the numerical procedure is made more effective, saving this way the CPU time. The general solutions for the temporal as well as spatial temperature field distributions are evaluated in a closed form in terms of the particular solutions of the governing partial differential equation (PDE). Because of linearity of the governing PDE, the superposition principle was used in the case of complex distributions of the laser beam intensity. The influence of different kinds of laser beam parameters to the temperature field distributions was considered.

Parametric study of acoustically-driven microbubble cavitations in a sonochemical reactor.

The bubble cavitation along a solid wall is investigated with a three-dimensional model based on the indirect boundary element method. Kinetic energy and Kelvin impulse are calculated in order to quantify the strength of cavitation. The influences of acoustic wave amplitude and frequency and liquid properties on the strength of cavitation are investigated. This study was carried out in order to better understand the relation between microscale processes and macroscale parameters in a sonochemical reactor used for impregnation of fabrics with nanoparticles.