Example for Presentation

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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
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Outline

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

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Why isogeometric analysis
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.
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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

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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

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Challenges
Surface representation
Domain representation
domain parameterization,
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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.

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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

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IGABEM formulation
In Parametric space Integration in parent
element Matrix equation
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Special techniques for IGABEM

• 1. Collocation point (Greville abscissae)
• 2. Boundary condition
• Collocate on the prescribed boundary
• 3. Integration
• High order Gauss integration

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Nuclear reactor
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Dam
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Propeller
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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.

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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

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Sensitivity Propagation
h-refinement algorithm is also suitable for shape
derivatives refinement, but need to convert NURBS
in to B-splines in
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Pressure cylinder problem
Design variable is large radius b
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Infinite plate with a hole
Design variable is radius R
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Cantilever Beam
Design curve is AB Minimise the area without
violating von Mises stress criterion
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Fillet
Design curve is ED Minimise the area
without violating von Mises stress criterion
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Conclusions

• 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

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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)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, 5263-75, 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)
  87100.