Approximation of the derivatives of solutions in a normalized domain for 2D solids using the PIES method

Numerical methods for solving boundary value problems, despite their popularity, are also affected by defects. The most troublesome are: the necessity of discretization, the calculation of singular integrals and large computational complexity. The aim of the paper is to develop a general strategy for approximation of the derivatives of solutions. The versatility of the approach is ensured by performing approximation in a normalized domain. In order to obtain efficient and accurate tool for solving the variety of boundary problems, we should combine the mentioned strategy with the method developed by the authors, called parametric integral equations system (PIES). The effectiveness of the proposed approach lies in: lack of the discretization of both a boundary and a domain, less computational complexity, possibility of determining the derivatives of solutions in a continuous manner at any point of the boundary and the domain without the necessity of calculating singular integrals. The paper presents the different variants of the proposed strategy. Tests were performed on the examples of elastic bodies with various shapes and boundary conditions. The reliability of the proposed approach is shown in comparison to existing analytical solutions. The normalized approximation strategy for derivatives in combination with the PIES method gives accurate results, which are obtained in elementary and efficient way.