Shape Optimization Directly from CAD:
an Isogeometric Boundary Element Approach
Haojie Lian
Supervisors: Professor S.P.A. Bordas Dr. P. Kerfriden
Dr. R.N. Simpson
Advanced Materials and Computational Mechanics Group The School of Engineering, Cardiff University
1. Motivation
2. CAD techniques (B-splines, NURBS, T-splines)
3. Isogeometric boundary element methods in shape optimization
4. Conclusions
Outline
Mesh Burden in Shape Optimization
Meshing
:1. Time consuming.
2. Human intervention.
3. CAD recovery from mesh.
Initial CAD model
Mesh generation
Shape sensitivity analysis Structural analysis
Optimizer for new model
Stop criteria
CAD Model recovery
End
No Yes
FEM optimization process 80% of the time!
Numerical Methods for Alleviating Mesh Burden
Meshfree Methods (Belytschko et al. 1994)
Support domain Node
XFEM (Belytschko, et al., 1999)
Isogeometric Analysis (Hughes et al., 2005)
x y
z
1.
Separate the CAD from the analysis
2.
Integrate the CAD and the analysis
Isogeometric Boundary Element Methods (IGABEM)
Use the same basis functions in CAD to
discretize Boundary Integral Equations (BIE).
(Simpson et al., 2010)
1. Seamlessly compatible with CAD due to the boundary representation.
2. No meshing and no CAD model recovery step throughout the optimization.
3. The control points can be naturally chosen as design parameters.
Initial CAD model
Mesh generation
Shape sensitivity analysis Structural analysis
Optimizer for new model
Stop criteria
CAD Model recovery
Solution directly in CAD form
No
Yes
IGABEM shape optimization flowchart
B-splines
0 0 0 0 1 2 3 4 4 4 4
Knot Parametric mesh
Knot vector
NURBS curve
NURBS surface e
e
x y
z
x
y
Control point
Curve Knot
Control polygon
− 1.0 − 0.5 0.0 0.5 1.0
˜ξ
− 0.2 0.0 0.2 0.4 0.6 0.8 1.0
y
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ξ
N3,3
Na,p(ξ)
N1,3
N2,3
N4,3
N5,3 N6,3
N7,3
B-spline Basis
B-spline basis
Fig 1. B-spline basis (left) vs quadratic polynomials (right) Linear independence.
The partition of unity.
Locally supported.
No Kronecker delta property.
NURBS
Control point
Curve Knot
Control polygon
Weight = 2 Control point
Curve Knot
Control polygon
Weight = 1
Non-Uniform B-splines (NURBS) is obtained by
where R is the NURBS basis function, N is the B-spline basis function, and w is the weight.
and
and
T-splines
T-junctions Control points
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28 29
30 31 32 33
34 35 36 37 38 39 40 41
42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57
Local refinement Water-tight geometry
Fig. 1: NURBS mesh topology
Fig. 2: T-spline mesh topology NURBS (source: Rhino3D website) T-spline (source: Rhino3D website)
B zier Extraction
Fig. 1: B zier extraction (source: Scott, 2011, IJNME )
N=CB
N: NURBS basis functions.
C: B zier extraction coefficient matrix (vary from element to element).
B: B zier basis functions (the same for all elements).
1. Easy to be incorporated into the existing FEM/BEM code.
2. Accelerate basis function evaluation time.
3. The extraction coefficients
do NOT depend on the control points.
B zier extraction formulation (Borden, 2011 and Scott, 2012):
IGABEM Formulation
Boundary Integral Equation (BIE)
source point
field point
x
1. Singular Integrals.
2. Jump terms.
The main implementations challenges:
Discretize BIE with CAD basis function R,
After discretization, matrix equations are
Regularized IGABEM Formulation
1. Regularized Boundary Integral Equation
2. Boundary condition enforcement:
No jump terms.
No strongly singular integrals.
Also available for sensitivity analysis.
Reduce to rigid body motion method.
Impose boundary conditions in "weak" sense.
IGABEM Sensitivity Formulation
Direct differentiate the above equation:
Recall the regularized BIE:
Critical points: analytical sensitivities of Green's functions .
0 10 20 30 40 50 60 70 80 90 100
−30
−20
−10 0 10 20 30 40
IGABEM
2032.5 FEM
120
IGABEM Analysis (NURBS)
0 20 40 50 60 70 80 90 100
−30
−20
−10 0 10 20 30
40 Original geometry
Control points Collocationpoints Element edges
0 1000 2000 3000 4000 5000
1.405 1.41 1.415 1.42
DOF L2norm
IGABEM Quadratic BEM
Fig. 2: Deformation
Fig. 4: Convergence study Fig. 3: Geometry description
Fig. 1: Problem definition
IGABEM Analysis (T-splines)
Fig. 1: Propeller CAD model Fig. 2: Displacement
Fig. 3: Stress Fig. 4: Stress close-up
A B C
D
NURBS curve element edge control points
0 5 10 15 20 25
−0.45
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0 0.05
Radial coordinates
Displacementsensitivitiesatboundarypoints
exact ˙ur
exact ˙uθ
IGABEM ˙ur
IGABEM ˙uθ
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
−15000
−10000
−5000 0 5000
Radial coordinates
Stresssensitivitiesatboundarypoints
exact ˙σr r
exact ˙σθθ
exact ˙σr θ
IGABEM ˙σr r
IGABEM ˙σθθ
IGABEM ˙σr θ
Sensitivity Analysis of Lam problem
Problem definition Design model
Displacement sensivities on AB Stress sensitivities on AB
NURBS curve element edge control points
−4 −3.5 −3 −2.5 −2 −1.5 −1
−0.5 0 0.5 1 1.5 2 2.5
3x 10
−3
X coordinates
Displacementsensitivitiesatboundarypoints
exact ˙ur
exact ˙uθ
IGABEM ˙ur
IGABEM ˙uθ
−4 −3.5 −3 −2.5 −2 −1.5 −1
−60
−50
−40
−30
−20
−10 0 10
X coordinates
Stresssensitivitiesatboundarypoints
exact ˙σr r
exact ˙σθθ
IGABEM ˙σr r
IGABEM ˙σθθ
IGABEM ˙σr θ
Sensitivity Analysis of Kirsch problem
Problem definition Design model
Displacement sensitivities on AB Stress sensitivities on AB
NURBS curve control points
NURBS curve element edge control points
NURBS curve element edge control points
Shape Optimization of a 2D Cantilever
Fig. 1: Problem definition Fig. 2: Design model
Fig. 4: Analysis model Fig. 3: Optimal model
NURBS curve element edge control points
NURBS curve element edge control points
A B
D C F E
NURBS curve control points
Shape Optimization of a Fillet
Fig. 1: Problem definition Fig. 2: Design model
Fig. 3: Analysis model Fig. 4: Optimal models
NURBS curve element edge control points
A B C D
E G F
H
NURBS curve element edge control points NURBS curve
control points
Shape Optimization of a Connecting Rod
Fig. 1: Problem definition Fig. 2: Analysis model
Fig. 4: Design model Fig. 3: Optimal shape
Senstivity Analysis of a Cylinder
Problem definition
Sensitivity Analysis of Cavity Problem
Displacement sensitivity comparison
Cavity problem
Shape Optimization of a 3D Beam
Initial shape of the 3D beam Optimal shape
Shape Optimization of a Hammer
Fig. 3: Initial geometry Fig. 4: Optimal geometry Fig. 1: Problem definition Fig. 2: CAD model
Shape Optimization of a T-shape component
Fig. 3: Optimal geometry
Fig. 2: Initial geometry Fig. 1: CAD model
Fig. 1: CAD model
Fig. 2: Inital geometry
Fig. 3: Optimal geometry
Shape Optimization of a Chair
Conclusions and Future Work
A shape optimization scheme using Isogeometric Boundary Element Methods (IGABEM) was presented, which possesses the following advantages:
1. Seamless integration with CAD.
2. No meshing during the steps throughout the iterative steps.
3. The CAD provides a natural parametrization for optimization.
4. NURBS for 2D and T-splines for 3D, so a water-tight geometry and local refinement is guaranteed.
Future work:
1. Acceleration algorithm, to save the memory and time.
2. Application in open domain problem optimization, such as acoustic and electromagnetics shape optimization.
