institute of mechanics and advanced materials sensitivity analysis and shape optimisation with...

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


• Isogeometric Analysis (IGA)

• Isogeometric Boundary Element Method (IGABEM)

• Sensitivity analysis and shape optimisation with IGABEM

• Numerical examples

Outline


Key idea 1

The key idea of isogeometric analysis (IGA) ( Hughes et al. 2005 ) 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.

Why isogeometric analysis


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:

1

C B n

i ,p ii

N .

NURBS curve

1 2 1{ , , , }n p


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:

1 1 1 1

n m n m

i ,p j ,q i ,p j ,qi j i j

N M N M

,1

( ) 1n

i pi

N

1 1

,

S B n m

i ,p j ,q i , ji j

N M


Challenges

domain parameterization,

Surface representation Domain representation


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.

IGABEM


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:

IGABEM formulation


IGABEM formulation

In Parametric space

Integration in parent element

Matrix equation


1. Collocation point (Greville abscissae)

2. Boundary condition

Collocate on the prescribed boundary

3. Integration

High order Gauss integration

Special techniques for IGABEM


Nuclear reactor


Dam


Propeller


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.

IGABEM shape optimisation


• 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

IGABEM sensitivity analysis


Sensitivity Propagation

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


Pressure cylinder problem

Design variable is large radius b


Infinite plate with a hole

Design variable is radius R


Cantilever Beam

Design curve is AB

Minimise the area without violating von Mises stress criterion


Fillet

Design curve is ED

Minimise the area without violating von Mises stress criterion


Conclusions• 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

Conclusions


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):4135-4195, 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. 209-212 (2012) 87–100.

References

institute of mechanics and advanced materials sensitivity analysis and shape optimisation with...

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