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Ex​amples

Rectangle 2D (Q) - 5x5

For better presentation this example can be broken into 6 main parts.
Part 1. Geometry Definition
Problem type: 2D-Plane stress
Table 3. ​​​Control points' number and polynomial degree of the basis per parametric axis.
Table 2. Geometry.
Table 4. Cartesian coordinates and weights of control points.
Part 2. Meshing
Knot value vectors:
Part 3. Material
Table 2. Material.
Part 4. Boundary Conditions
Table 5. Boundary conditions. The value 1 indicates the supported degrees of freedom, while the zero the free ones.
Part 5. Loading Conditions
Table 5. Loading. The load is imposed on the 21th control point towards the negative of the Y axis. 
Part 6. Post-Processing

BSPLine Basis Function

A structure with rectangular shape is designed and analysed below.
The perimetric​ degrees of freedom are fixed. The load is applied to the whole surface area of the Rectangle.
The basic feature of Isogeometric Analysis is the overlapping of the basis functions, which is obvious from the stiffness matrix shape.
Figure 1. BSPLine basis function.​​​

NURBS Shape Function

Isogeometric Analysis takes the advantage of linear independency between axis ξ and η.
2D shape functions are the full tensor product of BSPLine basis functions of the axis ξ and η.
Figure 2. Shape function.​​​

Index Space

Index space is a representation of the model with respect to knot values.
Because of our dealing with 2D problem, index space consists of only rectangles. Index space is a helpful space in order to have the supervision of interconnection between basis functions and knot value support of each function.
This space focuses upon the sequence of knot values rather than their actual numerical content.
Figure 3. 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. The application of a mapping from parameter to physical space is required in order to transform those simple patterns to virtually unlimited, complex geometries,. 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 Finite Element Method as well.
Because of our having defined 3 Gauss points per knot span, arising 3 * 3 = 9 Gauss points per isogeometric element.
For this example the total number of Gauss points is equal to 81.
Figure 4. Parameter space.​​​

Physical Space

In order to create this object, the user has to define the following variables:

  • Degree of shape functions for each parametric axis.
  • Knot value vector for each parametric axis.
  • Control points (cartesian coordinates and weights).
Physical space represents the real model and is already known as cartesian space. For a given set of control points, only a single set of basis functions can lead to the same geometry.
Because of this problem control points are inside the model in physical space, but in general occasion control points can often be seen outside the model in contrast to Finite Element Method's nodes which always belong to the mesh. In physical space the discretization in finite elements and the mesh could be easily seen.
Gauss points will be used for numerical integration in order to calculate the total stiffness matrix, which describes the problem. Gauss points can be noticed in the physical space with ' * '.
Figure 5. Physical space.​​​

Stiffness Matrix

The stiffness matrix represents the system of linear equations that must be solved in order to export the geometrical accurate solution.
Elements are non zero and positive, because of the Betti - Maxwell theorem and for positive displacement, positive force response.
Non zero elements expand around the diagonal with specific attitude.
Stiffness matrix is a sparse matrix because of in the equilibrium of each node only the contracted elements take part.
Finally stiffness matrix has a strip shape.
Figure 6. Stiffness matrix.​​​

Displacement Field

The colouring depicts the intensity of each field. Red colour represents the maximum value of a specific field, while blue colour the minimum value.
Because of the nature of the problem, 2 degrees of freedom correspond to each control point. In this way the horizontal and vertical displacement contours are drawn.
Figure 7. Contour. Displacement field.​​​

Strain Field

The strain tensor of a plane stress problem consists of 3 components:
Figure 8. Contour. Strain field.​​​

Stress FIeld

The stress tensor of a plane stress problem consists of 3 components:
As regards a plane stress problem, von mises stress is calculated by the equation:
Figure 9. Contour. Stress field.​​​