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# Contents

### Isogeometric Analysis

In most occasions, the exact solution of a natural problem is neither possible nor necessary. The actual objective is to find an accurate solution that satisfies a selected convergence criterion. The ultimate challenge for an engineer is to balance between accuracy and time. Design and analysis of extraordinary geometries is a powerful asset for modern engineers, who are capable of facing surprisingly more complicated problems. Accurate geometrical representations of the natural model are designed in the familiar Cartesian system, called physical Space. Additionally, it is very helpful to envision a complex structure in an imaginary, basic space, where all geometries can be represented as lines, rectangles and cuboids. This is parameter Space. This approach is far from new; it is already known from the isoparametric concept in Finite Element Methods. The parameter space utilized in isogeometric analysis, however, holds some major differences. Furthermore, isogeometric analysis also introduces the index space. This additional space plays an important role for some kinds of SPLines, but it is only auxiliary for NURBS.
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Figure 1. B-SPLine solid.
(a) Index Space (b) Parameter Space
(c) Physical Space

# Isogeometric Analysis

Knot Value Insertion
Degree Elevation
Elevation & Insertion
​Reverse Refinement
Refinement with NURBS

# ​​Index Space

Index space is a representation of the model with respect to knot values. It is a line in 1D, containing the corresponding knot values in equally spaced positions. This space focuses upon the sequence of knot values rather than their actual numerical content.

Index space describes the contribution of each knot value to the creation of a certain B-SPLine basis function. This helps identify the level of interconnection between basis functions and the knot value support of each function.

Control points are also evaluated in the index space. In fact, control points are defined as the center of the support of knot value spans.

Expansion to 2D or 3D leads to the creation of rectangles or cuboids respectively. Due to tensor product properties, everything mentioned about 1D extends and applies to both 2D and 3D. Thus, index space provides information that can contribute to the comprehension of a complex representation.
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Figure 2. (a) Curve and (b) Surface represented in Index Space​.

# ​Parameter Space

Parameter space is a representation of the model with respect to knots. SPLine entities are always represented as orthogonal shapes in parameter space. Only lines, rectangles and cuboids exist here. In order to transform those simple patterns to virtually unlimited, complex geometries, the application of a mapping from parameter to physical space is required. Hence, parameter space is a primitive, abstract representation of physical space. The mapping between parameter space and physical space is achieved through the Jacobian Matrix and its inverse. This is something widely utilized in FEM as well.

The illustration of basis functions in the parameter space allows for a better understanding of concepts such as support, control point coordinates and the role of knots in basis function creation. Each knot marks the beginning and the end of a basis function domain. By “domain” we mean the area in which the basis function is non-zero, as all basis functions are defined throughout the parameter space, but are non-zero only in specific knot spans. Basis functions sharing the same domain are overlapping in parameter space and controlling a common part of the entity in the physical space.
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Figure 3. (a) Curve and (b) Surface represented in Parameter Space.

# ​Physical Space

Physical space is the already known cartesian space, where the real model is represented. Simple orthogonal shapes from parameter space are transformed into complex entities in the physical space. Physical coordinates of the control points play a major role in the aforementioned mapping, but an equally drastic role is set upon basis functions. In fact, for a given set of control points, only a single set of basis functions can lead to the same geometry. We will examine this thoroughly later.

Control points can often be seen outside the model in physical space in contrast to FEM’s nodes which always belong to the mesh. It is one of the reasons NURBS and SPLine entities in general can accurately represent multiple types of geometries and the understanding of this peculiarity is one of the many challenges of isogeometric analysis.
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Figure 4. (a) Curve and (b) Surface represented in Physical Space.

# ​NURBS Entities

NURBS entities are created as a linear combination of NURBS shape functions, exactly the same way as B-SPLine entities. The following is the equation for the creation of NURBS curves:
Surfaces:
and Solids:
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Figure 39. NURBS elliptical Entities.
(a) Curve, (b) Surface and (c) Solid.

# ​NURBS Examples

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Figure 40. (a) NURBS curves and (b) Shape Functions for different Weight values.
In figure 2.41, the same circle is represented by NURBS shape functions of different order. This is a closed curve, so the first and last control points coincide. Weights are shown for each control point. A NURBS circle is usually represented by four consecutive patches, bound together by a common knot value vector.
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Figure 43. NURBS Surface created from consecutive circle cross-sections.
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Figure 45. NURBS Solids.
(a) Wine Glass and (b) Abstract NURBS Solid

The glass of wine displayed in figure 2.45.a is a 3D NURBS solid. The potential of isogeometric analysis is clearly represented in this model. Observe the exact representation of conic sections and smooth surfaces, in combination with immediate mesh generation. The mesh, that is depicted in the picture, can be instantly used for analysis. An abstract form of another NURBS solid is represented in figure 2.45.b.

# ​Patches

NURBS entities are created by transforming a simple parametric shape (line, rectangle, cube) to a model in physical space (curve, surface, solid). They are used for the exact and efficient representation of complex geometrical structures. Sometimes, the mapping of a single parametric shape is not the optimum solution. A designer might need two or three parametric cubes in order to efficiently represent solids with major changes in geometrical attributes. As displayed in figure 2.46, each of these cubes, mapped to a portion of the solid in physical space, is a NURBS patch. As expected, each patch has continuity Cp-m on interior knots and C-1 on the edge.
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Figure 2.46. (a) Geometrical Representation of the famous Falkirk Wheel “abutment”.
Five separate patches are used.
(b) Each Patch portrays a cube in Parameter Space mapped as a complex shape in Physical Space.

Interconnection between patches can be roughly achieved by choosing coincident control points on the edges. Still, patch connection rarely is leak-proof. This is one of the major disadvantages of NURBS, downsized and eliminated in the next version of SPLines (T-SPLines etc.).

Sometimes, patches exist for other reasons. For example, a major change in material properties, as displayed in figure 2.47, requires a patch boundary. Interpolation through a certain control point calls for patch boundary to be established there. Even application of C0 continuity, for analysis purposes, is enabled by introduction of a patch. If the same polynomial order is used, the mapping can be unified. In these special occasions, the separate parameter spaces of the patches can be united in one parameter space, using one set of basis functions and one knot value vector. The distinction between patches can be applied by enforcing C0 continuity across the boundary. In the examples used in this book, the latter option is preferred, when possible.
Figure 47. NURBS Patches enforced in order to distinct timber from steel.
Each material requires separate Patch Stiffness Matrix evaluation,
before Global Stiffness Matrix creation.

Figure 48. Separate Knot Vectors united into one.
The Control Points at the boundary are merged. C​0 Continuity is applied.

In the geometrical models presented in this thesis, knot boundaries are drawn in blue and patch boundaries in black. This separates C-1 and C0 continuity from C1 and greater continuity.