The exact solution of an engineering problem is often impossible to be found. Consequently, the engineer is only able to approximate it. The major difference between engineers and mathematicians is that engineers don't seek for the **exact** solution, but for an **accurate** enough one to satisfy certain criteria and regulations. The real challenge is to balance between accuracy and time. The more advanced tools and techniques an engineer has in his quiver, the more optimized his projects will be.

The examined structures are designed as NURBS models in **physical space** (Cartesian system). All calculations take place in the **parameter space**,** **already known from the isoparametric concept of the **Finite Element Method** (FEM), where all** **3D** **models are represented as rectangular parallelepipeds regardless their geometry. Isogeometric analysis takes advantage of an extra space, the so-called **index space**, which plays a significant role for some kinds of SPLines, such as T-SPLines. It is worth mentioning that index space is only auxiliary for NURBS.

Control points are defined as the center of their **support**. The support of each control point is its domain of influence consisting of p+1 knot value spans for 1D, (p+1)^{2 }rectangles for 2D and (p+1)^{3 }cubes for 3D. Index space indicates the:

- interconnection between the control points
- support of a basis function
- contribution of a knot value to the basis
- overlapping between the finite elements

1D, 2D, 3D models are represented as lines, rectangles and rectangular parallelepipeds respectively, while knots are equally spaced regardless their value.

Control points are often located outside the model. This is why they are not material points and don't belong to the model in contrast to FEM’s nodes, which always belong to the mesh. Due to this, isogeometric analysis is able to utilize the existing finite element mesh of the NURBS model, something impossible in the case of the classical FEA.

W(ξ) is the Z-coordinate of the projective B-SPLine curve. Projective transformation is applied by dividing the other two coordinates of the B-SPLine curve with the Z-coordinate. NURBS shape functions are calculated as follows:

The following types of boundary conditions are taken into consideration:

- Dirichlet
- Neumann
- Robin

The equation's solution corresponds to the displacements of the control points, the so-called **pseudo-displacements**. Unlike in classical FEM, control points don't belong to the model. The displacement field is calculated from the pseudo-displacements.

The multiplication of the deformation matrix by the pseudo-displacement vector results in the **strain **vector.

Hooke’s constitutive law leads to the following equations.