DEVELOPING INTELLIGENT SOFTWARE FOR COMPUTING PARALLELIZATION RESEARCH

One of the most important ways of improving the speed of complex task solving is employing a multiprocessor computational system. This paper describes the experience of software development for research management and solving educational problems using parallel computing technologies. The authors describe approaches to computation parallelization using a multiprocessor system with shared memory within a task of finding a numerical root of a system of linear equations with a tridiagonal coefficient matrix that appears when solving a boundary problem for a partial differential equation of parabolic type, the heat equation. Additionally, the approaches to parallelization implementation of the tridiagonal matrix method for the heat equation in the two-dimensional case within a numerical root-finding algorithm using the alternating-direction implicit method for a multiprocessor system with shared memory are described. A finite-difference method of variable directions is used to find a numerical root of a heat equation in the two-dimensional case. Sequential and parallel algorithms (two-sided Thomas algorithm and multithread horizontal block Thomas algorithm) that fit an execution on computational systems with shared memory have been used to implement a tridiagonal matrix method. Two parallel computation organization technologies for computational systems with shared memory have been used for computation parallelization: one based on OpenMP technology and one using .NET framework facilities. The parallelization process and load balance serving have been performed by means of the environment in the first case as manual operation of threads parallelization process is allowed in the latter one. As an assessment of the described approach performance, the calculation time for sequential and parallel algorithms is given depending on the task’s size and the number of threads used. Comparison of the considered parallelization algorithms and implementation technologies is performed based on the analysis of the resulting acceleration. This paper shows that total computation time is several times smaller and calculation acceleration is several times bigger when using Thread instead of OpenMP. An application has been developed that allows obtaining a visual result of modelling of process of temperature propagation in the study area using parallel calculation technologies in real time.

Prandtl and Richardson Number Effects on Mixed Convection in a Vented Enclosure on Application to the Cooling of the Fins

In the present study, a numerical investigate the transport mechanism of laminar mixed convection in a vented enclosure. The walls of the cavity were kept adiabatic except the right vertical wall which was equipped with three fins dissipating the heat at a constant temperature. The equations of considered phenomenon were established and discretized by the finite difference method. The sweeping method line-by-line and the Thomas Algorithm (TDMA) were used for the resolution of the system of discretized equations. The results obtained showed that both the variations of the Prandtl and Richardson number have important effects on the flow structure and on the heat transfer.

Impact of Heat Generation on Magneto-Nanofluid Free Convection Flow about Sphere in the Plume Region

The main aim of the current study is to analyze the physical phenomenon of free convection nanofluids heat transfer along a sphere and fluid eruption through boundary layer into a plume region above the surface of the sphere. In the current study, the effect of heat generation with the inclusion of an applied magnetic field by considering nanofluids is incorporated. The dimensioned form of formulated equations of the said phenomenon is transformed into the non-dimensional form, and then solved numerically. The developed finite difference method along with the Thomas algorithm has been utilized to approximate the given equations. The numerical simulation is carried out for the different physical parameters involved, such as magnetic field parameter, Prandtl number, thermophoresis parameter, heat generation parameter, Schmidt number, and Brownian motion parameter. Later, the quantities, such as velocity, temperature, and mass distribution, are plotted under the impacts of different values of different controlling parameters to ascertain how these quantities are affected by these pertinent parameters. Moreover, the obtained results are displayed graphically as well in tabular form. The novelty of present work is that we first secure results around different points of a sphere and then the effects of all parameters are captured above the sphere in the plume.

Numerical estimation of electrical resistivity in digital rocks using GPUs

Представлен алгоритм расчета потенциального электрического поля в образцах горных пород и предложены оценки их удельного электрического сопротивления (проводимости). Алгоритм ориентирован на расчет поля в существенно неоднородных моделях среды с частично насыщенными и полиминеральными образцами горных пород. В основе алгоритма – итерационные методы крыловского типа, в качестве предобусловливателя используется оператор, обратный к оператору Лапласа для однородной среды. Для вычисления предобусловливателя используется спектральный метод в направлениях, нормальных к основному направлению электрического тока, а серия одномерных задач решается методом прогонки. Решатель реализован с использованием графических процессоров (GPU) и позволяет обрабатывать образцы размером до 4003 вокселей на одном GPU. We present a numerical algorithm for computing the electric field in digital rock samples and estimating their electrical resistivity (conductivity). The main peculiarity of the algorithm is its applicability tostrongly heterogeneous models including partially saturated and multi-mineral rock samples. The algorithm is based on the iterative Krylov-type solver preconditioned by the inverse Laplace operator for homogeneous media. The preconditioner is computed using the spectral method in directions orthogonal to the direction of the main electric current, whereas the series of 1D problems are solved by the Thomas algorithm. We implement the algorithm using GPUs, which allows us to use a single GPU to solve the problems for samples whose size is up to 4003 voxels.

Modified method of parallel matrix sweep

The topic of this paper refers to efficient parallel solvers of block-tridiagonal linear systems of equations. Such systems occur in numerous modeling problems and require usage of high-performance multicore computation systems. One of the widely used methods for solving block-tridiagonal linear systems in parallel is the original block-tridiagonal sweep method. We consider the algorithm based on the partitioning idea. Firstly, the initial matrix is split into parts and transformations are applied to each part independently to obtain equations of a reduced block-tridiagonal system. Secondly, the reduced system is solved sequentially using the classic Thomas algorithm. Finally, all the parts are solved in parallel using the solutions of a reduced system. We propose a modification of this method. It was justified that if known stability conditions for the matrix sweep method are satisfied, then the proposed modification is stable as well.

Parallel Thomas approach development for solving tridiagonal systems in GPU programming − steady and unsteady flow simulation

The solution of tridiagonal system of equations using graphic processing units (GPU) is assessed. The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic-reduction (PCR). Lid-driven cavity problem is considered to assess these parallel approaches. This problem is also simulated using the classic Thomas algorithm that runs on a central processing unit (CPU). Runtimes and physical parameters of the mentioned GPU and CPU algorithms are compared. The results show that the speedup of CR, PCR and PTA against the CPU runtime is 4.4x,5.2x and 38.5x, respectively. Furthermore, the effect of coalesced and uncoalesced memory access to GPU global memory is examined for PTA, and a 2x-speedup is achieved for the coalesced memory access. Additionally, the PTA performance in a time dependent problem, the unsteady flow over a square, is assessed and a 9x-speedup is obtained against the CPU.

Simulation of boundary layer under laminar and turbulent modes of newtonian fluid motion in a flexible pipeline

The article deals with the modeling of boundary layer parameters for Newtonian fluids under laminar and turbulent modes of motion. Based on the system of Prandtl equations and initial boundary conditions under laminar motion, using the Gallorkin method, a tri-diagonal system of equations is formed, which connects the values of functions at the node of nets n+1 across the boundary layer. The numerical method uses the Thomas algorithm to calculate values Ujn+. The velocity value Vjn+1 is determined from the continuity equation by integration across the boundary layer. The Navier-Stokes equation in dimensionless form was used to model the turbulent boundary layer, given the velocity U is an independent variable. The differential equation system was solved using the numerical Dorodnicin method. The results of modeling the velocity distribution in the boundary layer, the thickness of the boundary layer in the section of the flexible pipeline 0.8-1.5 m from the beginning of the fluid entering the pipeline at the expense up to 0.1 kg/s are presented. Keywords: boundary layer, turbulent mode, velocity, Prandtl equation

Solution of Second Order Singular Perturbed Delay Differential Equation Using Trigonometric B-Spline

In this paper, a numerical technique is presented to approximate the solution of a singular perturbed delay differential equation. The continual emerge of singular perturbed delay differential equations in a mathematical model of real life applications trigger the researchers for the numerical treatment of these equations. The numerical technique is based on trigonometric cubic B-spline functions in which derivatives are approximated as a linear sum of basis functions. The obtained matrix system is solved by using the Thomas Algorithm. The convergence of the employed proposal is scrutinized and computational work is carried out on four examples to test the capability of the proposed scheme. The approximated solution is compared with the existing technique and to present the behavior of the obtained solution graphs are plotted.