Vectorized numerical algorithms for the solution of continuum mechanics problems

Abstract

The aim of the work is to study the possibilities provided by new information technologies, object-oriented programming tools and modern operating systems for solving boundary value problems of continuum mechanics described by partial differential equations. To discretize the basic equations, we applied the method of finite differences and finite volumes, which are widely used to solve problems in the mechanics of liquids and gases. The paper considers the implementation of the finite difference methods and the finite volume method with vectorized grid structures, including access to the inner and boundary cells of the grid, as well as the features of the implementation of algorithms at singular points of the computational domain. To solve boundary value problems described by partial differential equations, we developed an approach to the construction of vectorized algorithms and considered the features of their software implementation in the MATLAB package. Vectorization in such tasks, excluding nested loops, is ensured by appropriate data organization and the use of vectorized operations. On the one hand, the developed algorithms widely use MATLAB functions designed for processing vectors and sparse matrices, and on the other hand, they are distinguished by high efficiency and computation speed, comparable to those of programs written in C/C++. The main results imply the numerical solution of a number of problems in continuum mechanics associated with the calculation of stresses in a separate body and the calculation of the field of velocity and temperature in the flow of a viscous incompressible fluid. The features of discretization of the basic equations and the implementation of the corresponding finite-difference and finite-volume algorithms are shown. The use of the MATLAB system opens up new possibilities for the formalization and implementation of finite-difference and finite-volume methods for the numerical solution of boundary value problems in continuum mechanics. Despite the fact that the capabilities of the developed algorithms are illustrated by the example of solving fairly simple problems, they admit a relatively simple generalization to more complex problems, for example, solving the Euler equations and Navier–Stokes equations. As part of the work, computational modules were prepared with user programming tools that expand the capabilities of the package and are focused on solving problems in continuum mechanics.

Keywords

• mathematical modeling
• vectorization
• numerical algorithm
• mechanics of continuum medium