1. IMAM
Institute of Mechanics and Advanced Materials
Sensitivity Analysis and Shape Optimisation with Isogeometric Boundary Element Method
Haojie Lian, Robert Simpson, Stéphane P.A. Bordas
Institute of Mechanics and Advanced materials, Theoretical and Computational mechanics,
Cardiff university, Cardiff, CF24 3AA, Wales, UK 1

2. Outline
Isogeometric Analysis (IGA)
Isogeometric Boundary Element Method (IGABEM)
Sensitivity analysis and shape optimisation with IGABEM
Numerical examples

3. Why isogeometric analysis
Key idea 1
The key idea of isogeometric analysis (IGA) ( Hughes et al ) is to approximate
the unknown fields with the same basis functions (NURBS, T-splines … ) as that used to generate the CAD model.
Reduce the time
No creation of analysis-suitable geometry;
Without the need of mesh generation.
Exact representation of geometry
Suitable for the problems which are
sensitive to geometric imperfections.
High order continuous field
More flexible
hpk-refinement. 3

4. NURBS curve 1. Knot vector:
a non-decreasing set of coordinates in the parametric space.
Where n is the number of basis functions,
i is the knot index and p is the curve degree,
2. Control points:
3. NURBS basis function:

5. Properties of NURBS basis functions
Partition of Unity
Non-negative
p-1 continuous derivatives
if no knot repeated
No Kronecker delta property
Tensor product property
Surface:

6. Challenges Surface representation Domain representation
domain parameterization,

7. IGABEM
Key idea 2:
Isogeometric Boundary Element Method (IGABEM) (Simpson, et al. 2011).
The NURBS basis functions are used to discretise Boundary Integral Equation
(BIE). Recently this work is extended to incorporate analysis-suitable T-splines.
Reasons:
1. Representation of boundaries;
2. Easy to represent complex geometries.

8. IGABEM formulation
Regularised form of boundary integral equation for 2D linear elasticity
where and are field point and source point respectively, and are
displacement and traction around the boundary, and are fundamental
solutions.
The geometry is discretised by:
The field is discretised by:

9. IGABEM formulation In Parametric space Integration in parent element
Matrix equation

10. Special techniques for IGABEM
1. Collocation point (Greville abscissae)
2. Boundary condition
Collocate on the prescribed boundary
3. Integration
High order Gauss integration

11. Nuclear reactor

12. Dam

13. Propeller

14. IGABEM shape optimisation
The advantages of IGABEM for shape optimisation.
More efficient
An interaction with CAD:
1. Analysis can read the CAD data directly
without any preprocessing.
2. Analysis can return the data to CAD
without any postprocessing.
More accurate
Design velocity field is exactly obtained for
gradient-based shape optimisation.

15. IGABEM sensitivity analysis
Governing equations in parametric space, which can be viewed as material coordinate system
Differentiate the equation w.r.t. design variables (implicit differentiation)
Discretise the derivatives of displacement and traction using NURBS basis
Finally

16. Sensitivity Propagation
h-refinement algorithm is also suitable for shape derivatives refinement, but need to convert NURBS in to B-splines in

17. Pressure cylinder problem
Design variable is large radius b

18. Infinite plate with a hole
Design variable is radius R

19. Cantilever Beam Design curve is AB
Minimise the area without violating von Mises stress criterion

20. Fillet Design curve is ED
Minimise the area without violating von Mises stress criterion

21. Conclusions Conclusions Future work:
An isogeometric boundary element method (IGABEM) has been introduced.
IGABEM can suppress the mesh burden and interact with CAD.
IGABEM has been applied to gradient-based shape optimisation.
IGABEM is more efficient and accurate for analysis and optimisation
Future work:
Topology optimisation with IGABEM
Easy to handle topology optimisation compared to IGAFEM
Easy to implement with the help of topology derivatives
T-spline based IGABEM for 3D shape optimisation
Local refinement
Analysis suitable and flexible to construct the complex geometry 21

23. References
TJR Hughes, JA Cottrell, and Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39-41): , 2005.
T Greville. Numerical procedures for interpolation by spline functions. Journal of the society for Industrial and Applied mathematics: Series B, Numerical Analysis, 1964.
R Johnson. Higher order B-spline collocation at the Greville abscissae. Applied Numerical Mathematics, 52:63-75, 2005.
Les Piegl and W Tiller. The NURBS book. Springer, 1995.
RN Simpson, SPA Bordas, J Trevelyan and T Rabczuk. An Isogeometric boundary element method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering (2012) 87–100.