IGA. Robin Bouclier. Читать онлайн. Newlib. NEWLIB.NET

Автор: Robin Bouclier
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Математика
Год издания: 0
isbn: 9781119988540
Скачать книгу
alt="images"/>

      where Qi and image are the control points of the initial Bézier curve of degree p and of the new Bézier curve of degree p + t, respectively (the numbering of control points starts at zero, here). Overall, the interesting point to note is that the degree-elevation procedure also takes the form of a linear application. A refinement matrix can be built to express the new control points of the degree-elevated B-spline entity in relation to the initial ones (Lee and Park 2002):

      [1.30] images

      [1.31] images

      where xw has one more component than x for each control point: considering Pi = (xi, yi, zi) with associated weight image The refinement strategy can also be extended to surfaces and volumes: the refinement is performed direction per direction in these situations. For more details on the refinement strategies of splines and their matrix representations, the reader is referred to Piegl and Tiller (1997), Lee and Park (2002) and Cottrell et al. (2007).

      1.2.3.2. Geometric modification

      Modification 2

Schematic illustration of example of shape modification by altering the spatial location of the control points.

      1.2.4. Spline-based finite element analysis: isogeometric principle

      1.2.4.1. General idea

      The main idea behind IGA is to use the spline-based parameterizations discussed above to build the approximation subspaces when applying Galerkin’s method for solving partial differential equations (Hughes et al. 2005; Cottrell et al. 2009). In this framework, the method can be viewed as a generalization of the FEM that considers smooth and higher-degree functions to replace typical Lagrange polynomials in the computations. Within this work, we recall that only NURBS (which constitute the most commonly used technology in CAD) and simpler B-splines are considered. In the end, this is the only major difference with the classical FEM, from an analytical point of view. IGA relies on the same variational formulations as standard FEM. When invoking the isoparametric element concept, the spline basis turns these weak forms into systems of algebraic equations. In solid mechanics, it means that the displacement field is approximated as:

      [1.32] images

      where Ri are the spline basis functions originated from the considered geometric model of the structure (see equation [1.23]). With this numerical approximation in hand, the principle of virtual work gives the following well-known linear system:

      where K is the so-called stiffness matrix, u is the displacement vector (vector collecting the degrees