34 Investigation of Crossing and Veering Phenomena in an Isogeometric Analysis Framework 365 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 34.4 B-spline basis functions from knot vector f0,0,0,1,2,3,4,4,5,5,5g NURBS, instead, are projection of entity from dimensiondC1 tod, where d is the number of physical spatial dimension (dD2 for a plane, dD3 for 3D) and the added dimension is a parameter called weight, and they are associated to the basis functions. The NURBS basis functions are then defined as Ri;p . / D Ni;p . /wi W . / D Ni;p . /wi Xn jD1 Nj;p . /wj (34.3) where it can be noticed that if all the weights are equals, the NURBS basis function are weighted B-Splines functions. To obtain a solid discretization, three sets of knot vectors must be defined „ D ˚ 1 2 nCpC1 , H D ˚ 1 2 nCpC1 and Z D ˚ 1 2 nCpC1 . Thanks to tensor-product structure, the solid NURBS is defined as R p;q;r i;j;k . ; ; / D Ni . /Mj . /Lk . /wi;j;k Xn b iD1 Xm b jD1 Xl b kD1 N b i . /M b j . /L b k . /w b i; b j; b k (34.4) V. ; ; / D n X iD1 m X jD1 l X kD1 Pi;j;kR p;q;r i;j;k . ; ; / (34.5) where Pi,j,k are the coordinates of the control points in the physical space and N,M,L are the elements of the B-splines generated using knot vectors „, H and Z, respectively. Taking advantage of the isoparametric concept in finite elements, where the basis functions used to geometrically define an element are also used to discretize the fields of interest (e.g. displacement field in solid mechanics) ue . ; ; / D n X aD1 de aRe a . ; ; / (34.6) where ue are the displacements in the physical directions andde a are the Degrees of Freedom to find and the elements of the solid NURBS have been attributed a unique index a. The test-rig can be represented, with remarkable simplification, using six trivariate NURBS: three bricks and three cylinders. In Fig. 34.5 are shown the different parameterizations of the bricks and the cylinders. For the first, parametric and physical dimensions are perfectly aligned, while for the latter instead of the usual circumferential vs. radial parameterization (common when modelling annular sections), the choice is the deformation of a bi-quadratic square, by moving the middle control points of the four edges. The obtained surfaces, the rectangle and the circle, are extruded adding the third parametric dimension and creating the control points for the new surface.
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