NUMERICAL ANALYSIS OF SOLUTIONS OF TWO-DIMENSIONAL HEAT CONDUCTION PROBLEMS BY MESHLESS APPROACH USING FUNDAMENTAL AND GENERAL SOLUTIONS

Abstract

This article is devoted to the analysis of numerical solutions of two-dimensional heat conduction problems by meshless approach, obtained using fundamental and general solutions of the modified Helmholtz equation. The meshless method described in this article is based on a combination of the dual reciprocity method with radial basis functions and the method of particular solutions. Based on the method of particular solutions, the solution of an inhomogeneous differential equation is represented as a sum of particular and homogeneous solutions. The fundamental and general solutions are used to find the homogeneous part of the solution, and the dual reciprocity method with radial basis functions is used to obtain the particular solution. The choice of such a meshless scheme is primarily due to the fact that the method of fundamental solutions is easily programmable, has spectral convergence and allows to achieve a high order of accuracy. The fundamental solution of a differential operator is singular at the origin, which leads to the construction of a fictitious boundary outside the physical boundary of the domain of the solution of the boundary-value problem. This is done in order to circumvent the singularity of the fundamental solutions. It is important to determine the optimal location of the fictitious boundary. It may be a circle whose center coincides with the geometric center of the solution domain. With an increase in the radius of the fictitious boundary, the accuracy of the solution obtained increases, but the conditional number of the matrix of the system of linear algebraic equations deteriorates, and vice versa. The general solution of a differential operator differs from its fundamental solution in that it is non-singular everywhere. The use of the general solution makes it possible to avoid building a fictitious boundary, which in turn makes it possible to achieve a more stable solution of a boundary-value problem. Numerical analysis of the solution of boundary-value problems using fundamental and general solutions is demonstrated into two benchmark problems. Numerical solutions of boundary-value problems were obtained using fundamental and general solutions for different numbers of interpolation nodes. In the article presents table of root mean square errors of solution to benchmark problems using fundamental and general solutions for different numbers of interpolation nodes, and also shows graph of the root mean square error off the number of interpolation nodes.