Search

Search Results

• STUDY OF PARALLEL SOLUTION ORGANIZATION FOR EXTERNAL AERODYNAMICS PROBLEMS BASED ON SPLITTING SCHEMES

  V.V. Semenistyy , I. E. Gamolina

  2021-01-19

  Abstract ▼

  The aim of this work is to study the ways to organize parallel solutions of external aerodynamics
  problems. A hybrid parallel-conveyor method for numerical solution of two-dimensional
  problems is considered. It allows to simulate the flow of viscous compressible fluids around objects
  of complex shape. A parabolized system of Navier-Stokes equations is considered, for the
  numerical solution a finite-difference algorithm is chosen. Due to its features (cost-effectiveness
  and stability in the study of boundary layers of moving bodies) this algorithm was preferred. To
  implement a nonlinear finite-difference scheme, the internal iterations are used in each main section.
  The developed parallel algorithm consists constructively of nested iterative loops. The system
  of equations is solved at each internal iteration. It is organized in two stages. At the first stage the
  equations of motion are solved; at the second stage the density is determined. At each fractional
  step of the internal iteration, one-dimensional data arrays are calculated. The paper uses the
  method of splitting the operator by physical processes. For the numerical solution of the problem,
  the factorization of the stabilizing operator is carried out. The scheme of the organization of the
  process of problem solving is given in each internal iteration. The paper proposes the principle of
  organizing parallel computing. The internal parallelism of the physical problem is used here.
  To implement the parallel algorithm, a computing environment is specially selected. It contains a
  decisive field of computing devices connected by switching connections, each of computing device
  has its own RAM. Besides computing environment contains a control device. The parallel algorithm
  uses a communication topology between worker processors. Reducing the dimension of the
  problem (to 2d) allows to save time on data exchange between the processors. In this paper, time
  estimates of the effectiveness of the developed parallel algorithm for each internal iteration are
  carried out. The use of the parallel run method and the proposed principle of organizing parallel
  calculations allow to increase the effectiveness of solving problems of such class.

• COMPARATIVE ANALYSIS OF THE PARALLEL COMPUTING EFFICIENCY FOR EXPLICIT AND IMPLICIT DIFFERENCE SCHEMES FOR COMPUTATIONAL AERODYNAMICS PROBLEMS

  V.V. Semenistyy, I.E. Gamolina

  2023-02-17

  Abstract ▼

  One of the main areas of parallel computing application is solving of computational aerodynamics
  problems. The paper considers parallel modeling of gas dynamics with quasi-onedimensional
  system of equations. The system describes the gas flow through a channel with variable
  cross section using implicit and explicit difference schemes. The purpose of this work is to
  study the methods efficiency for organizing parallel computations with implicit and explicit difference
  schemes for solving internal aerodynamics problems. The article presents a comparative
  analysis of the proposed parallel models for quasi-one-dimensional equations system of gas dynamics.
  The system describes flows in a channel of variable cross section. Various parallel algorithms
  are used to solve systems of such type numerically. The method of splitting by physical processes
  is used for an implicit difference scheme. To suppress the solution oscillations a smoothing
  operator at the correction stage is introduced in calculations according to the scheme of the predictor-
  corrector type. In scheme fractional steps the parallel scalar sweep algorithm is used to
  solve tridiagonal systems with the parametric unknown choice. Besides to compare scheme mentioned
  above a parallel algorithm is constructed for McCormack's explicit scheme. Such algorithm
  is widely used in computational aerodynamics. Parallel computations are held by computing
  structures with distributed memory and by another one – with linear switching dependence between
  computing devices of the working field. The paper presents time estimations for each computing
  stage (both by implicit and explicit difference schemes). It helped to calculate the developed
  parallel algorithms efficiency. It is concluded that the acceleration factor in explicit scheme depends
  linearly on the computing devices number.

• ALGORITHM OF THE REDUCED POLYNOMIAL EQUATIONS SOLUTION USING CONTINUED FRACTIONS

  V.E. Dolgoy, I.E. Gamolina

  2023-02-17

  Abstract ▼

  The article presents an algorithm where continued fractions are used to find the zeros of nth
  degree polynomial. Our day there is a wide variety of methods and algorithms for solving such
  type problems. The proposed algorithm feature is its effective possibility using for large values of
  n. Besides this algorithm can be applied for complex roots. Any real number can be represented as
  a continued fraction: finite or infinite. The main purpose of continuous fractions is that they allow
  us to find good approximations of real numbers in the ordinary fractions form in algebraic equations
  and systems solution. The purpose of this work is algorithm with continued fractions for solving
  reduced polynomial equation that contains both real and complex roots, estimation of the
  number of arithmetic steps in its numerical solution. The analytical expressions for polynomial
  equation solutions are given in this paper. The obtained analytical expressions represent the ratio
  of the Toeplitz determinants. A distinctive feature of these determinants is the presence of coefficients
  of the solved algebraic equation as diagonal elements. A modified Rutishauser algorithm
  was used to obtain a numerical solution. Complex roots can be found using the summation algorithm
  of divergent continued fractions. As an illustration of the proposed algorithm the results of
  the numerical solution for fifth degree polynomial equation are given. The advantage of the algorithm
  is the small number of arithmetic operations required, the possibility of considering highdegree
  polynomials, and the small error of calculations.