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T-splines are a generalization of NURBS surfaces. They retain all the properties of NURBS, but are increasingly considered as a better alternative. While the control points in NURBS lie in a rectangular grid of rows and columns, the control points in the T-splines may be incomplete. Moreover, T-splines allow for local insertion of new control points. The possibility of adding control points locally has many advantages. For the same geometric representation, T-splines require a much smaller number of control points than NURBS. Moreover, T-splines naturally eliminate the formation of gaps when connecting adjacent surfaces, which is a disadvantage of NURBS. Due to these advantages, T-splines have received increasing research attention in recent years. In the present study, we will use T-splines in combination with the Parametric Integral Equation System (PIES) to solve 3D linear elasticity problems. PIES is a computational method for finding solutions to 2D and 3D boundary value problems and is based on the formal separation of the approximation of the boundary geometry from the boundary functions. The PIES formulation is independent from the physical representation of the boundary and, for any shape, it is always defined in the parametric domain and not on the physical boundary as in the traditional Boundary Integral Equations (BIE). As a result, the boundary can be described analytically in the form of mathematical functions directly included in the mathematical formula of PIES. In the previous studies, the 2D boundary could be defined by parametric curves such as Bézier, B-spline and Hermitte, whereas in the 3D case, we can apply parametric surfaces. These were flat Coons as well as smooth Bézier surfaces. The latter have particularly positive properties with respect to their flexibility and simplicity of implementation. This correlates with the fact that parametric curves and surfaces are widespread in CAD systems, e.g. NURBS. They have been intensively studied and modified into more effective variants such as hierarchical B-splines, PB-splines (Point based splines), T-splines, LR-splines (Locally refined splines) and PHT-splines (Polynomial splines over hierarchical T-meshes). The aim of the paper is to extend the application of PIES to objects with boundaries represented by T-splines. In order to simplify the problem, we use a Bézier extraction mechanism for automatic conversion of T-splines surfaces into standard Bézier surfaces, previously used in PIES. The Bézier extraction transforms the T-splines basis functions into linear combination of standard Bernstein basis functions. Therefore, if we can transform the T-splines basis functions into Bernstein polynomials, this allows the T-spline surface to be replaced by a set of Bézier patches with the traditional parametric domain for each Bézier patch. The reliability of the proposed approach is evaluated through numerical tests based on linear elastic problems. We show how to represent the boundary by T-splines and compare the accuracy of the results with known analytical solutions. We use a commercial T-spline plugin for Rhino from Autodesk to model and manipulate T-spline surfaces. The plugin also includes the Bézier extraction module that directly converts T-spline models to the appropriate ones constructed with Bézier surfaces. The presented examples confirm the applicability of the approach with high accuracy of the solutions.

T-splines and Bézier extraction in modeling shape of boundary in Parametric Integral Equation System for 3D problems of linear elasticity