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.
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