The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

Abstract

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C 2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves.

The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries

“…[15][16][17]. The authors of this paper also have worked on the extension of the PIES method to solve uncertainly defined problems [18,19]. However, similarly to the classic BEM, conventional PIES produces dense and non-symmetric matrices, therefore solving of a complex or large-scale engineering problems needs a huge amount of the RAM and time-consuming computations.…”

Two Iterative Methods for Solving Linear Interval Systems

“…In this paper, uncertainty is considered as interval numbers. Some articles used this model to solve their problems [9,10].…”