TOPOLOGY OPTIMIZATION THEORY, METHODS AND APPLICATIONS EIGENVALUE PROBLEMS JAKOB SØNDERGAARD JENSEN
Eigenvalue problems Linear (elastic) dynamic problems
Zero damping and external loads
Solution form
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Eigenvalue problems Generalized eigenvalue problem
Non-trivial (eigen-)solutions
modeshape eigenfrequency
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Free vibrations
First vibration mode: 2-node vertical bending – 3.58 Hz 4
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Free vibrations
Second vibration mode: 2-node horizontal bending – 5.86 Hz 5
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Free vibrations
Third vibration mode: 1-node torsion – 8.68 Hz 6
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Free vibrations
First rudder vibration mode ≈ 9 Hz 7
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Sensitivity analysis The design sensitivities (single eigenvalue)
After some algebra (try it yourself !)
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Optimization problems Maximizing the lowest eigenfrequency
Eigenvalue problem (subspace iterations)
Volume constraint
Box constraints 9
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Some basic papers
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Fundamental problems • Optimization problem is not well posed (less structure -> higher eigenfrequency) • Solutions: impose mass equality constraint, solve problem as a reinforcement problem or include non-structural masses
• Spurious modes in low density regions • Solution: tailored interpolation functions
• Mode switching (non-smooth) • Solutions: include more modes or use bound formulation
• Multiple eigenvalues -> incorrect sensitivities • Solution: compute correct sensitivities ! 11
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues • The rule – not the exception ! • Inherent property for 2- or 3-dimensional homogeneous and symmetric (eg square or cubic) structures • Often an outcome of the optimization procedure
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues Double eigenvalue
Normalized mode shapes
Modeshapes are orthogonal
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues Linear combination of modeshapes
Also an eigenvector
Sensitivities
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues Finding values of constants
Normalization ⇒ ⇒
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues Finding extreme values of
Leads to new (small) eigenvalueproblem
⇓
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Multiple eigenvalues Final expression for sensitivities
References
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Spurios localized modes
SIMP: p1=3, p2=1
! 19
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Modified SIMP Local ”Rayleigh quotient” – E/m
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p1=3, p2=1 p1=3, p2=6
4 3 Zero frequency!
2 1 0 20
0
0.2
0.4
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
0.6
0.8
TOPOLOGY OPTIMIZATION eigenvalue problems
1 01/07/2011
Modified SIMP SIMP: p1=3, p2=6
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
RAMP interpolation
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Alternative interpolations Discontinuos stiffness
Four-node Bezier curve
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Bezier curve interpolation Inverse ”Rayleigh quotient”
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Avoiding mode switching Weighted average (example):
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Avoiding mode switching Bound formulation
N chosen large enough to ensure smoothness (eg N=5) 26
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Example 1: separation of eigenvalues
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Separation of eigenvalues
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Separation of eigenvalues
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Optimization problem 1D structure:
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Results for 1D structure
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Optimization problem (2D) Double bound formulation (mode n and n+1):
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Material interpolation Linear mass density interpolation:
Local ”Rayleigh quotient”:
Interpolation of Ee is chosen so f penalizes intermediate densities
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Maximization of eigenvalues
Stiffness interpolation
Constants k1 and k2 determined from L 34
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Minimization of eigenvalues
Use negative L ! Maximization of gap: use both interpolations Drawback (significant): two analyses required 35
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Results: 2D
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Example 2: separation of eigenvalues – continuous spectrum
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Example 2: separation of eigenvalues – continuous spectrum
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Separation of eigenvalues – continuous spectrum • Time-harmonic motion:
• Bloch wave theory for periodic structure (unit cells with periodicity R):
Wave vector (wavelength+direction)
• Continuous eigenvalue problem (on unit cell):
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Continuous eigenvalue spectra – band diagrams Unit cells
Epoxy
Γ
X
M
Γ
Γ
X
M
Γ
Aluminum
Epoxy
Gap 2 Gap 1
Matrix Aluminum inclusions 40
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
Γ
X
M
TOPOLOGY OPTIMIZATION eigenvalue problems
Γ
01/07/2011
Gaps in the band diagram = band gaps ! 1D
2D
3D
Periodicity in different dimensions
3D crystals:
www.sandia.gov 41
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
http://ab-initio.mit.edu/photons/3d-crystal.html TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Maximizing band gaps k2
π M
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
?
−π
ω
Γ
Xπ
−π
ω2 (k )
ω1 (k ) Γ
X
M
Γ
k
Maximize gap/mid-gap ratio:
min ω j +1 (kρ, ) − max ω j k ( ρ, ) ∆ω ( ρ ) Φ (= = ρ) ωm ( ρ ) min ω j +1 (kρ, ) + max ω j k( ρ, ) / 2 42
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
k1
Phononic band gaps - elasticity
High contrast
Low contrast 43
DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Photonic band gaps - optics “TE-polarization” “TM-polarization” “TE+TM-polarization”
Rhombic cell
Square cell
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Buckling eigenvalue problems Free vibration problem
Critical load
Linear buckling problem
Buckling shape
Geometric stiffness matrix
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Topology optimization of buckling problems Some references:
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011
Take home messages Remember multiple eigenvalues – compute correct sensitivities Be aware of mode switching – make sure your objective function is differentiable (smooth) What do you aim for ? – set up a well-posed optimization problem Deal with spurious modes (easy fix: use RAMP instead of SIMP)
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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen
TOPOLOGY OPTIMIZATION eigenvalue problems
01/07/2011