TOPOLOGY OPTIMIZATION THEORY, METHODS AND APPLICATIONS

TOPOLOGY OPTIMIZATION THEORY, METHODS AND APPLICATIONS EIGENVALUE PROBLEMS JAKOB SØNDERGAARD JENSEN Eigenvalue problems Linear (elastic) dynamic pro...
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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



−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