Computational method PIES
for solving boundary value problems
The work presents the theme of computer solving two- (2D) and three-dimensional (3D) boundary value problems without classical elements, as it is used in the finite element method and the boundary element method. The discussed method PIES is based on parametric integral equations systems, which are the analytic modification of classical boundary integral equations. The mathematical formalism of these equations takes into account the shape of the boundary geometry mathematically defined by curves and surfaces used in computer graphics. Parametric integral equations systems are presented for boundary value problems modeled by Laplace’s, Poisson, Helmholtz, Stokes and Navier-Lame partial differential equations.
The modeling of 2D boundary geometries is made using Hermite’a, Béziera and B‑spline curves, whilst in the case of 3D problems Coons, Béziera and B-spline surfaces are applied. Using these curves and surfaces directly into parametric integral equation systems are modeled shapes of solved boundary value problems. In practice, modeling require only a small number of corner, boundary or control points. An effective modification of the boundary shape is made by changing position of these points and it takes place directly in parametric integral equation systems. Eventually, they will automatically adapt to the modified shape of the boundary, making easier the solution of problems that require multiple modification of the boundary shape.
The numerical solution of parametric integral equation systems is separate from the modeling of the shape and does not require the classical discretization. Verification and reliability of the method PIES are shown in many examples with analytical solutions, while its effectiveness compared to the classical numerical methods such as the finite element method and the boundary element method.
The work is addressed to those involved in computer methods to solve a variety of boundary value problems (researchers, PhD student, students).