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    Ex​amples

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Ex​amples

Thick Cylinder 3D

A typical rectangle is analysed with Geomiso.
Table 1. Geometry.
Table 2. Material.
Problem type: 3D
Linear                                
Quadratic                          
Part 1. Mesh
(a) Initial mesh.
(b) Fine mesh.
Table 3. ​​​Control points' number and polynomial degree of the basis per parametric axis.
(a) Initial mesh.
(b) Fine mesh.
Table 4. Cartesian coordinates and weights of control points.
Knot value vectors (initial mesh):
Knot value vectors (fine mesh):
Part 2. Boundary Conditions
(a) Initial mesh.
(b) Fine mesh.
Table 5. Boundary conditions. The value 1 indicates the supported degrees of freedom, while the zero the free ones.
Part 3. Loading 
(a) Initial mesh.
(b) Fine mesh.
Table 6. Loading.
Part 4. 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

(a) Initial mesh.
(b) Fine mesh.
Figure 5. Physical space.​​​

Stiffness Matrix

(a) Initial mesh.
(b) Fine mesh.
Table 7. Stiffness matrix.​​​
Figure 6. Stiffness matrix.​​​

PseudoDisplacement

(a) Initial mesh.
(b) Fine mesh.
Table 8. Pseudo-displacement.​​​

Displacement Field

(a) Initial mesh.
(b) Fine mesh.
Table 9. Displacement field (Gauss points).
Figure 7. Contour. Displacement field.​​​

Strain Field

(a) Initial mesh.
(b) Fine mesh.
Table 10. Strain field (Gauss points).
Figure 8. Contour. Strain field.​​​

Stress FIeld

(a) Initial mesh.
(b) Fine mesh.
Table 11. Stress field (Gauss points).
(a) Initial mesh.
(b) Fine mesh.
Table 12. Norm.
Figure 9. Contour. Stress field.​​​