1 Evolutionary Structural Optimisation. 2 KKT Conditions for Topology Optimisation.
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Transcript of 1 Evolutionary Structural Optimisation. 2 KKT Conditions for Topology Optimisation.
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Evolutionary Structural Optimisation
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function, Lagrangian
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KKT Conditions for Topology Optimisation
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KKT Conditions (cont’d)
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∂L∂ρ e
= uT∂k
∂ρ eu+ λ 2
∂k
∂ρ eu+ λ1ve +
∂u
∂ρ e2uT k + λ 2k( )
Since λ 2 is arbitary, we select λ 2 to eliminate the ∂u
∂ρ e term, i.e. λ 2 = −2uT
∴ ∂L
∂ρ e= uT
∂k
∂ρ eu− 2uT
∂k
∂ρ eu+ λ1ve = −uT
∂k
∂ρ eu+ λ1ve = 0
Let the strain energy of a solid element (ρ e =1) be se ,
se= uek0ueand,
uT∂k
∂ρ eu =
∂ρ
∂ρ euT k0u =se
Therefore,
seλ1ve
=1
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KKT Conditions (cont’d)
Strain energy density should be constant throughout the design domain
This condition is true if strain energy density is evenly distributed in a design.
Similar to fully-stressed design.
Need to compute strain energy density
Finite Element Analysis
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seλ1ve
=1
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Evolutionary Structural Optimisation (ESO)
Fully-stressed design – von Mises stress as design sensitivity.
Total strain energy = hydrostatic + deviatoric (deviatoric component usually dominant in most continuum)
Von Mises stress represents the deviatoric component of strain energy.
Removes low stress material and adds material around high stress regions descent method
Design variables: finite elements (binary discrete)
High computational cost.
Other design requirements can been incorporated by replacing von Mises stress with other design sensitivities – 0th order method.
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ESO Algorithm
1. Define the maximum design domain, loads and boundary conditions.
2. Define evolutionary rate, ER, e.g. ER = 0.01.
3. Discretise the design domain by generating finite element mesh.
4. Finite element analysis.
5. Remove low stress elements,
6. Continue removing material until a fully stressed design is achieved
7. Examine the evolutionary history and select an optimum topology that satisfy all the design criteria.
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σ e ≤ ER × 1+σ min( )
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Cherry
Initial design domain
Fixed
Gravitational Load
ESO solution
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Michell Structure Solution
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ESO: Michell Structure
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ESO: Long Cantilevered Beam
2.5D Optimisation
Reducing thickness relative to sensitivity values rather than removing/adding the whole thickness
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Mesh Size 14436, less the 268 elements removed from mouth of spanner
Load Case 1: 2N/mm
Load Case 2: 2N/mm
Roller support
Non-Design Domain
Spanner
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Thermoelastic problems
Both temperature and mechanical loadings FE Heat Analysis to determine the temperature distribution Thermoelastic FEA to determine stress distribution due to
temperature Then ESO using these stress values
477
720
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Design Domain
P Uniform
Temperature
Plate with clamped sides and central load
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T = 7CT = 5C
T = 0C T = 3C
Group ESO
Group a set of finite elements Modification is applied to the entire set Applicable to configuration optimisation
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Example: Aircraft Spoiler
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Example: Optimum Spoiler Configuration
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Multiple Criteria
Using weighted average of sensitivities as removal/addition criteria
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AR 1.5
Mesh 45 x 30
P
P
Maximise first mode frequency & Minimise mean compliance
Optimum Solutions (70% volume)
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wstiff:wfreq = 1.0:0.0
wstiff:wfreq = 0.7:0.3 wstiff:wfreq = 0.5:0.5
wstiff:wfreq = 0.0:1.0wstiff:wfreq = 0.3:0.7
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Chequerboard Formation
Numerical instability due to discretisation. Closely linked to mesh dependency.
Piecewise linear displacement field vs. piecewise constant design update
Smooth boundary: Level-set function
Topology optimisation based on moving smooth boundary Smooth boundary is represented by level-set function Level-set function is good at merging boundaries and
guarantees realistic structures Artificially high sensitivities at nodes are reduced, and
piecewise linear update numerically more stable Manipulate implicitly through
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0,, yxyx
yxyx
yxyx
yxyx
,0,
,0,
,0,
http://en.wikipedia.org/wiki/File:Level_set_method.jpg
Topology Optimisation using Level-Set Function
Design update is achieved by moving the boundary points based on their sensitivities
Normal velocity of the boundary points are proportional to the sensitivities (ESO concept)
• Move inwards to remove material if sensitivities are low
• Move outwards to add material if sensitivities are high Move limit is usually imposed (within an element size) to
ensure stability of algorithm Holes are usually inserted where sensitivities are low (often
by using topological derivatives, proportional to strain energy)
Iteration continued until near constant strain energy/stress is reached.
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Numerical Examples
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