Meshing and refinement

Depending on the employed numerical integration scheme for the mechanical response analysis, a subdivision of NURBS patches into elements can be helpful. An efficient implementation of the element-wise Gauss integration scheme used within this work necessitates the definition of elements. The parametric space of the geometry model

(a) Physical space and control point net: The control point net is denoted by bullets con- nected by dashed lines. Red bullets denote control points which have influence on the red element. All other control points are plotted blue.

(b) Parametric space and univariate B-spline basis functions for the two parametric direc- tions ξα: All basis functions which have influ- ence on the red element are plotted bold. Ξ1= [0, 0, 0, 0, 0.5, 1, 1, 1, 1]

Ξ2=0, 0, 0, 0,1₃,2₃, 1, 1, 1, 1



Figure 4.4: Physical space and associated parametric space of a free form surface: The red element in the parametric space is mapped to the physical space with the help of the NURBS basis functions.

inherently contains a mesh of suitable elements. However, it can be necessary to refine the initial mesh, or to elevate the order of the basis functions, in order to obtain a specified level of accuracy.

Mesh generation and definition of elements

A mesh of isogeometric NURBS elements is created patch by patch from NURBS surfaces. The tensor product of the two knot vectors Ξk_αforms the parametric space of the patch k. The element edges are defined by the entries of the knot vectors. Thus, an element of the patch k is defined by

ξ1 i, ξ 1 i+1 × ξ 2 j, ξ 2 j+1  i ∈pk 1 + 1, . . . , n k 1 j ∈pk 2 + 1, . . . , n k 2 (4.39) in the parametric space. The elements in the parametric space are mapped onto the physical space with the help of the NURBS basis functions; see Figure 4.4 for a graph- ical illustration. All non-zero basis functions within an element are specified and their number is constant, see Section 4.1.4. Furthermore, inside elements defined by knot

4.2 Fundamental properties of isogeometric analysis 59

spans the B-spline basis functions are rational functions and thus can be integrated ex- actly using Gauss integration. Although it is not exact in the case of NURBS, element- wise Gauss integration performs very well; see e.g. Hughes et al. (2010) and an elabo- ration in Kiendl (2011) for more details. All this favors the idea of using the rectangles defined by knot values as elements.

Knot insertion

An enlargement of the solution space by knot insertion is the first possibility if more precise results are required. Basically, additional control points and associated basis functions are added. This corresponds to h-refinement in standard finite element analy- sis. In the case of NURBS surfaces knot insertion is performed in the parametric space by adding new values into the knot vectors, one after each other. Within every refine- ment step a new set of control points that suits to the refined knot vector while exactly maintaining the geometry has to be computed. If only one knot value is inserted, then p locally confined control points have to be redefined. Efficient algorithms for knot insertion are given in Piegl and Tiller (1997). Their computation time is negligible in comparison to the total analysis time. Multiple insertion of one knot value allows lowering the continuity at the current knot value. Thus, knot insertion allows the so- lution space to be adapted to anticipated discontinuities while the order of the basis functions is maintained. Knot insertion propagates throughout the whole patch due to the tensor-product structure of NURBS surfaces. Thus, local refinement is not possible without additional measures. A possible remedy by using local constraint conditions for B-spline surfaces was proposed in Kagan et al. (2003) and extended to NURBS in Cottrell et al. (2007). Another possibility is the usage of more flexible geometry descriptions, such as hierarchical B-splines (Vuong et al. (2011)), analysis-suitable T- splines (Scott et al. (2012)) or locally refined B-splines (Johannessen et al. (2014)). Within this thesis local refinement is not considered.

Order elevation

The accuracy of results can be increased by elevating the order p of the NURBS ba- sis functions. This slightly increases the solution space, but more important raises its approximation power. Order elevation can be carried out separately for every parametric direction. The initial continuity of splines is preserved. An elevation of order to p∗ > p in ξα-direction alters the concerned knot vector as follows. The multiplicity m of all knot values has to be elevated to m∗ = m + p∗ − p in order to keep the continuity Cp∗−m∗ = Cp−m_{constant. Within the knot spans nothing has to be} done since C∞prevails. The number of the control points as well as their positions has

to be adapted to the new knot vector. The new set of control points can be computed by solving a linear system of equations. For more details and efficient algorithms see Piegl and Tiller (1997).

p-refinement vs. k-refinement

In most practical applications of isogeometric analysis the initial geometry has to be refined for the mechanical response analysis. If besides knot insertion also order ele- vation is to be performed, two possible strategies exist.

The first one is to elevate order and continuity of the NURBS basis functions. This is referred to as k-refinement. The way of proceeding is to first order elevate the initial surface to the target order p∗. Subsequently, the required number of knots has to be inserted. Thus, at the additional knots the highest possible continuity Cp∗−1 prevails. The continuity Cp∗−m∗ stays constant at knots which are already present in the initial knot vector. k-refinement produces solution spaces with high continuity. Thus, the number of control points grows only slightly.

The second possibility is to elevate the order of the NURBS basis functions while maintaining the inter-element continuity of the initial geometry. This is achieved by carrying out knot insertion before order elevation. The continuity Cp−m_{of all knots ex-} isting in the initial knot vector is maintained while elevating the order. The continuity of newly inserted solitary knot values is Cp−1. Thus, the maximum continuity between elements is Cp−1for p-refinement. In general, the total number of control points grows significantly. If the initial order is p = 1 or the continuity is lowered manually to C0, then p-refined NURBS elements are very similar to higher order Lagrange elements. The effects of the different types of refinement are illustrated in Figure 4.5. The ini- tial geometry is given in Figure 4.5a. Pure knot insertion to obtain a mesh consisting of four elements of the initial order p = 2 yields the NURBS curve in Figure 4.5b. A p-refined mesh with four elements of order p = 6 is shown in Figure 4.5c. The k-refined counterpart is given in Figure 4.5d. The differences between p- and k- refinement are clearly visible. p-refinement produces a significantly greater number of control points than k-refinement. The control points which have influence on any ele- ment are close to this element if p-refinement is used. In contrast to that, k-refinement yields control points which lie far away from the concerned element. A more detailed discussion about refinement strategies and a graphical illustration of the resulting basis functions are given in Cottrell et al. (2009).

4.2 Fundamental properties of isogeometric analysis 61

(a) p = 2, initial curve, one non-zero ele- ments

(b) p = 2, knot insertion, four non-zero ele- ments, C1-continuity between elements

(c) p = 6, p-refinement, four non-zero ele- ments, C1-continuity between elements

(d) p = 6, k-refinement, four non-zero ele- ments, C5-continuity between elements Figure 4.5: Order elevation exemplified with a segment of a circle: The elements of the phys- ical curve are separated by strokes. The control point net is denoted by bullets connected by a dashed line. Red bullets denote control points which have influence on the element marked red. All other control points are plotted blue. Red arrows denote the nodal director vectors which have influence on the red element.