Isogeometric Analysis of

Euler-Bernoulli Beam Thesis, Structural Engineering and Mechanics Student: Andrius Irkmonas Supervisor: Prof Rene de Borst Moderator: Dr Robert Simpson

Beam Properties

NURBS Properties

Set K, u, f to zeros

Quadrature

Loop over Element

Set:

Parent element – range;

Element connectivity;

Set of control points.

Compute :

NURBS basis function;

NURBS basis function derivatives.

Compute :

Stiffness matrix (K);

External force vector (f).

Solve equation

Solve system

Plot

PRE-PROCESSING

PROCESSING

POST-PROCESSING

CORE

1. Introduction

Hughes et al (2005) [1] introduced the isosgeometric analysis. The

idea come from the industry, where design to analysis took to much

time. This emerged to create more effective computational geometry

technology. Non-rational B-splines (NURBS) [2] can be used to

compute geometry for the finite element analysis (FEA). By

implementing isosgeometric analysis methodology into design and

analysis could save computational analysis time.

3. Beam analysis structure

NURBS was used in beam analysis and MatLAB code. A flow card

(see Figure 3.1 below) has been created to explain beam analysis

structure in the code. To extend the provided code for beam analysis

certain parts of the code was written. This involved finding the B-

spline basis functions, NURBS basis functions and is derivatives.

4. Results

Tests were done for a cantilever beam. For

a point load applied at the free end and

uniformly distributed load. The IGA solution

was compared with the exact solution.

Beam specifications :

• Cross-section 1 x 1 x 1

• End Load 1

• Young’s Modulus 100

• Moment of Inertia 0.0833

The refinement was required for accuracy.

4.1 Impact of refinement

It is evident that order elevation enriches the basis better than a knot

insertion. A knot insertion case, requires refinement.

4.2 Impact of loading

It is evident that loading has an effect on the IGA solution. The refinement

was required, especially for controlling distributed load.

4.3 Impact of difference in geometry and material properties

It is evident that changing geometry or material properties does not

effecting IGA and exact solution accuracy and refinement is not required.

5. Conclusion

From the results it was evident that NURBS could be used to present

geometry accurately for the finite element solution. The IGA solution could

be refined by a knot insertion or order elevation. It was also evident, that

beam had to be controlled for distributed load and refinement may

required for achieving accuracy. Also, changes in geometry and material

properties didn’t made any impact on IGA solution accuracy. It is shown,

that IGA could be applicable in a finite element framework.

References [1] T.J.R. H., J.A. C.,Y. B. Iso. An.: CAD, FE, NURBS, ex. geo.and me. ref., 4135–4195, 2005.

[2] L. A. Piegl and W. Tiller. The NURBS Book. Springer, 1996.

[3] T.J.R. H., Y. B., Iso. Ana.: Toward Integration of CAD and FEA. Wiley, 2009.

2. NURBS in

isogeometric analysis

NURBS are defined by

the set of control points, a

knot vector and a

polynomial order. A knot

vector are used to

construct basis functions

and partition of the

elements. The control

points are set up in order

to control geometry. This

can be seen in Figure 2.1

[3], where the schematic

illustration of a NURBS

patch showing the

arrangements of spaces,

the knot vectors, the

control points and the

physical mesh.

Figure 3.1: Flow card of the computation of stiffness matrix (K) and external

force vector (f) in the MAtLAB code.

Figure 4.2: Knot insertion. Figure 4.3: Order elevation.

Figure 4.4: Distributed load.

Figure 4.1: Beam specifications

Figure 4.5: Point (1) and distributed (2),(3)

loads.

Figure 4.6: Variation of Beam thickness

(1)1),(2)2). Figure 4.7: Variation of Young’s modulus

(1)100), (2)300).

Figure 2.1: A NURBS patch.

1

3

2

2

1

2

1

Loop over Element

Set:

Gauss points;

Gauss weight;

Prametric coordinare (ξ); Jacobian (𝐽ξ );

(Parent-parametric domain).

Compute :

Jacobian (𝐽ξ);

(Physical-parametric domain);

Shape function derivatives.