Abstract
The main purpose of this study is the modification of the parametric integral equations systems (PIES) method, to include the NURBS curves into its mathematical formulas. We want to create an opportunity for defining the boundary shape using NURBS curves without the classical discretization process. We present an inclusion of the NURBS curves into PIES, and its application in process of modeling and solving the boundary value problems, on Laplace’s equation example. The correctness and accuracy of such modeling is confirmed by the examples. Obtained solutions are compared with these obtained using linear segments and B-spline curves. Additionally, we verify the correctness of modeling and modifying the shape using analytical solutions. The NURBS curves significantly expand the possibilities and increase the accuracy of the boundary shape modeling which is an important aspect of the numerical solving of boundary value problems.
Highlights
For many years researchers have used boundary integral equations (BIE) to solve boundary value problems (Banerjee and Butterfield 1981; Bonnet 1995)
It is a different way comparing to known from literature application of non-uniform rational Bspline curves (NURBS) at the stage of numerical solution of BIE
Application of NURBS curves in mathematical formulas of generalized parametric integral equations system (PIES) have a great advantage over direct application of curves to define boundary elements
Summary
For many years researchers have used boundary integral equations (BIE) to solve boundary value problems (Banerjee and Butterfield 1981; Bonnet 1995). Researchers have always been interested in application of BIE, because boundary value problems are one order reduced during solving process. It is connected with physical definition of BIE on boundary of the problem. Two-dimensional Laplace’s equation is solved as one-dimensional problem defined on boundary only There are different kinds of problems, for example, defined by Poisson equation, where integral is calculated on boundary as well as in a domain (Brebbia and Walker 1980; Brebbia et al 1984). We focus only on the problems without integration over domain
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