The paper presents a non-element method of solving boundary problems defined on polygonal domains modeled by corner points. To solve these problems a parametric integral equation system (PIES) is used. The system is characterized by a separation of the approximation of boundary geometry from the approximation of boundary functions. This feature makes it possible to effectively investigate the convergence of the obtained solutions with no need of performing the approximation of boundary geometry. The testing examples included confirm high accuracy of the solutions.

The paper uses analytical modification of the classical boundary integral equations (BIEs) for the Helmholtz equation to facilitate the process of practical definition of the boundary geometry. Instead of defining the boundary by means of a boundary integral, the modification makes use of Bézier curves exclusively. As a result, a new parametric integral equation system (PIES) is obtained in which boundary geometry is taken into account in original fundamental boundary solutions. Such boundary definition makes it easy to approximate boundary functions. The proposed method to obtain numerical solution of the PIES for the Helmholtz equation is characterized by high effectiveness.

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a b s t r a c tThe paper presents the approximation strategy for solutions and derivatives of solutions of boundary value problems on the example of two-dimensional solids. The effectiveness of the proposed strategy lies in the fact that it gives possibility to calculate solutions and their derivatives continuously at all points of the boundary and the domain, irrespective of the method used to solve the boundary problem and regardless of the type of the problem. The strategy has been developed in order to: (1) improve the accuracy of solutions (displacement) and their derivatives (strains, stresses) in the vicinity of the boundary, where they are affected by errors, (2) make the ability to directly obtain the derivatives of solutions (strains, stresses) on the boundary, (3) avoid computing strongly singular integrals present in the integral identity used for obtaining solutions or their derivatives (e.g. stresses in plasticity problems). The different variants of the proposed strategy have been developed and their accuracy has been verified considering examples with analytical solutions.