Triangular Bézier surface patches in modeling shape of boundary geometry for potential problems in 3D

Abstract

This paper presents a new boundary shape representation for 3D boundary value problems based on parametric triangular Bézier surface patches. Formed by the surface patches, the graphical representation of the boundary is directly incorporated into the formula of parametric integral equation system (PIES). This allows us to eliminate the need for both boundary and domain discretizations. The possibility of eliminating the discretization of the boundary and the domain in PIES significantly reduces the number of input data necessary to define the boundary. In this case, the boundary is described by a small set of control points of surface patches. Three numerical examples were used to validate the solutions of PIES with analytical and numerical results available in the literature.

Numerical approximation strategy for solutions and their derivatives for two-dimensional solids

“…Using the obtained series we can determine displacements u x and u y at 121 points of the considered domain. Then, based on the analytical solutions (20), the relative errors [%] are calculated and plotted in Fig. 10.…”

“…PIESs were elaborated by one of the authors of the paper [17] and are developed through applications to solving the various kinds of boundary problems, e.g. 2D elasticity [18] or acoustics [19] and also 3D problems [20]. PIESs in these applications have confirmed their benefits including: lack of the discretization, the efficient and not cumbersome modeling of the boundary and the domain and their modification and the smaller number of data required to modeling, hence the systems of equations with smaller number of equations to solve.…”

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

“…After substituting (20), (21), and (22) in (18), we obtain the following system of the convolution integral equations…”

“…After substituting (34), (35), for (23) and their simplification in agreement with the earlier terminology [20,21], we obtain the formula termed parametric integral equation system (PIES) in the following form…”

“…However, only potential 3D problems related to the Laplace equation have been considered so far in [20,21]. Our current research focuses on the use of PIES for problems of linear elasticity, formulated by the Navier-Lamé equation.…”