Topology Optimization Using the SIMP Method
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Topology Optimization Using the SIMP Method
Fabian Wein
Introductary Talk @ LSE29.10.2008
Fabian Wein Topology Optimization Using the SIMP Method
Optimization vs. Optimization
• Common claim
Engineers improve a system and call this ”optimizing”.But the optimum can only be found with optimization
methods.
• Modelling optimization problems is nontrivial• Design space (dimensions, topology, material, . . . )• Multiple criterions
• Different optimization methods
• Optimization results are guidelines for designers
Fabian Wein Topology Optimization Using the SIMP Method
Basic Optimization Problem
• Design vector x (e.g. dimensions, topology, shape, material)
• Problem
minx
J (x)
subject to
equality constraints
inequality constraints
box constraints
• Objective function J (x) 7→ R
Fabian Wein Topology Optimization Using the SIMP Method
Ingredients for the Optimization Problem
• Parametrization
• Iteration xk+1 = xk + td• starting point/ initial guess x0
• descent direction• step length• stopping criteria, optimality criteria
• Problems• existence• uniqueness• convergence• local optima
Fabian Wein Topology Optimization Using the SIMP Method
Optimization Approaches
• Gradient-free algorithms• stochastic algorithms (particle swarm optimization)• genetic algorithms• . . .
• Deterministic algorithms/ find descent directions• finite differences• automatic differentiation• sensitivity analysis
• Optimization domain• parameter optimization• shape optimization• topology optimization
Fabian Wein Topology Optimization Using the SIMP Method
Linear elasticity
Hooke’s law
[σσσ ] = [c0][S] (in Voigt notation: σσσ = [c0]Bu)
with
• [σσσ ],σσσ : Cauchy stress tensor
• [c0] : tensor of elastic modului
• [S],S : linear strain tensor
• u : displacement
• B =
∂
∂x 0 0 0 ∂
∂z∂
∂y
0 ∂
∂y 0 ∂
∂z 0 ∂
∂x
0 0 ∂
∂z∂
∂y∂
∂x 0
T
: differential operator
Fabian Wein Topology Optimization Using the SIMP Method
Strong Formulation
PDE
Find
u : Ω→ R3
fulfilling
BT [c0]Bu = f in Ω
with the boundary conditions
u = 0 on Γs
nT[σσσ ] = 0 on ∂ΩΓs
Fabian Wein Topology Optimization Using the SIMP Method
Discrete FEM Formulation
Solve
Global System
Ku = f
with
Assembly
K =ne∧
e=1
Ke; Ke = [kpq]; kpq =∫Ωe
(B)T [c0]BdΩ
Fabian Wein Topology Optimization Using the SIMP Method
Proportional Stiffness Model
Parametrization by design variable
• Model structure by local stiffness (full and void).
• Define local stiffness (finite) element wise: ρρρ = (ρ1 · · · ρne )T
• Continuous interpolation with ρmin ≤ ρe ≤ 1.
Introduce pseudo density ρρρ
[ce](ρρρ) = ρe [c0]; Ke(ρρρ) = ρeKe; K(ρρρ)u(ρρρ) = f
Fabian Wein Topology Optimization Using the SIMP Method
Minimal Mean Compliance
Different interpretations• Maximize stiffness• Minimize mean compliance• Minimize stored mechanical energy
Minimize compliance
minρρρ
J(u(ρρρ)) = minρρρ
fTu(ρρρ) = minρρρ
u(ρρρ)TK(ρρρ)u(ρρρ)
Fabian Wein Topology Optimization Using the SIMP Method
Find Derivative
General optimization procedure
• Evaluate objective function
• Find descent direction δδδ (e.g.gradient)
• Find step length along δδδ (linesearch)
Techniques to find descent direction
• Use gradient free methods
• Use finite differences
• Analytical first derivative
• Analytical second derivative
Fabian Wein Topology Optimization Using the SIMP Method
Sensitvity Analysis
• Sensitivity analysis provides analytical derivatives
• Abbreviate ∂(·)∂ρe
by (·)′
Derive mean compliance fTu
J ′ = f ′Tu+ fTu′ = fTu′
Find J ′ by deriving state condition Ku = f
Solve for every u′
Ku′ =−K′u
Fabian Wein Topology Optimization Using the SIMP Method
Adjoint Method
The adjoint method is based on the fixed vector λλλ
J = fTu+λλλT(Ku− f)
J ′ = fTu′+λλλT(K′u+ Ku′)
= (fT +λλλTK)u′+λλλ
TK′u
Solve: Kλλλ = −f =∂J
∂u
J ′ = −uTK′u
• The compliance problem is self-adjoint
• The general adjoint problem can be efficiently solved by(incomplete) LU decomposition
Fabian Wein Topology Optimization Using the SIMP Method
Naive Approach
Minimize compliance: straight forward, initial design 0.5
minρρρ
fTu s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρeKe, K′e = Ke
The optimal topology is the trivial solution full material
Fabian Wein Topology Optimization Using the SIMP Method
Naive Approach
Minimize compliance: straight forward, initial design 0.5
minρρρ
fTu s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρeKe, K′e = Ke
The optimal topology is the trivial solution full material
Fabian Wein Topology Optimization Using the SIMP Method
Add Constraint
Minimize compliance: volume constraint 50%
minρρρ
fTu s.th.:∫Ω
ρρρ ≤ 1
2V0
“Grey” material has no physical interpretation
Fabian Wein Topology Optimization Using the SIMP Method
Add Constraint
Minimize compliance: volume constraint 50%
minρρρ
fTu s.th.:∫Ω
ρρρ ≤ 1
2V0
“Grey” material has no physical interpretation
Fabian Wein Topology Optimization Using the SIMP Method
Third Try
Minimize compliance: penalize ρρρ by ρρρp with p = 3
minρρρ
fTu note: Ke = ρ3eKe, K
′e = 3ρ
2eKe
We have a desired 0-1 pattern but checkerboard structure
Fabian Wein Topology Optimization Using the SIMP Method
Third Try
Minimize compliance: penalize ρρρ by ρρρp with p = 3
minρρρ
fTu note: Ke = ρ3eKe, K
′e = 3ρ
2eKe
We have a desired 0-1 pattern but checkerboard structure
Fabian Wein Topology Optimization Using the SIMP Method
Forth Try
Minimize compliance: use averaged gradients
minρρρ
fTu note: K′e =
∑iHiρiρe
3ρ2eKe
∑iHiwith Hi = rmin−dist(e, i)
No checkerboards and no mesh dependency (view movie)
Fabian Wein Topology Optimization Using the SIMP Method
Forth Try
Minimize compliance: use averaged gradients
minρρρ
fTu note: K′e =
∑iHiρiρe
3ρ2eKe
∑iHiwith Hi = rmin−dist(e, i)
No checkerboards and no mesh dependency (view movie)Fabian Wein Topology Optimization Using the SIMP Method
Comparison of Different Optimizers
• SCPIP (MMA implementation by Ch. Zillober)
• Optimality Condition (heuristic for SIMP)
• IPOPT (general second order optimizer)
Fabian Wein Topology Optimization Using the SIMP Method
Performance
Fabian Wein Topology Optimization Using the SIMP Method
Optimality Condition
Optimality Condition: fix-point type update scheme
ρek+1=
max(1−ζ )ρek
,ρmin if ρekBη
ek≤max(1−η)ρek
,ρminmin(1+ζ )ρek
,1 if min(1+ζ )ρek,1 ≤ ρek
Bηek
ρekBη
ekelse
With
• Bek= Λ−1K
′e
• Λ to be found by bisection
• Step width ζ e.g. 0.2
• Damping η e.g. 0.5
Fabian Wein Topology Optimization Using the SIMP Method
Combined Load vs. Multiple Load Cases
For multiple loadcases several problems are averaged
Figure: Two loads applied simultaniously (left) and seperatly (right)
The left case is instable if the loads are not applied simultaniously
Fabian Wein Topology Optimization Using the SIMP Method
Problem Specific Optimization
Now only the left load is applied to the optimized structures
Figure: The scaling of the displacement is the same
Fabian Wein Topology Optimization Using the SIMP Method
Synthesis of Compliant Mechanisms - aka ”no title”
Generalizing the compliance to J = lTu with l = (0 · · · 0 1 0 · · ·)T.
For this case one has to apply springs to the load and output nodes
Fabian Wein Topology Optimization Using the SIMP Method
Synthesis of Compliant Mechanisms - aka ”no title”
Generalizing the compliance to J = lTu with l = (0 · · · 0 1 0 · · ·)T.
For this case one has to apply springs to the load and output nodes
Fabian Wein Topology Optimization Using the SIMP Method
Harmonic Optimization
Two common approaches
• Optimize for eigenvalues
• Perform SIMP with forced vibrations
Harmonic excitation
• Excite with a single frequency
• Gain steady-state solution in one step
• Complex numbers
Complex FEM system
(K+ jωC−ω2M)u = f
S(ω)u = f ST
= S
Fabian Wein Topology Optimization Using the SIMP Method
Harmonic Objective Functions: J(u(ρρρ))→ R
Compliance
J = |uT f| J ′ =−R(sign(J)uT S′u)
J = (uT f)2 J ′ =−2(uT f)uT S′u
J = uTRfI−uT
I fR J ′ = 2R(λλλT S′u) Sλλλ =− j
2f
J = uT u J ′ = 2R(λλλT S′u) Sλλλ =−u
Optimize for output
J = uTLu J ′ = 2R(λλλT S′u) Sλλλ =−LTu
• Optimize for velocity• Optimize for coupled quantities
Fabian Wein Topology Optimization Using the SIMP Method
Harmonic Interpolation Functions
Classical SIMP converges faster than mass to zero
µPedersen(ρe) =
ρ3e if ρ > 0.1
ρe
100if ρ ≤ 0.1
µRAMP(ρe) =ρe
1+q(1−ρe)
0e+000
2e-001
4e-001
6e-001
8e-001
1e+000
0 0.2 0.4 0.6 0.8 1
Con
trib
utio
n
Design variable
SIMPidendity
RAMP
Fabian Wein Topology Optimization Using the SIMP Method
(Global) Dynamic Compliance
We optimize for uT u and uTLu with L selecting f
This illustrates general optimization problems
• One has to know what one wants
• One might not want what one gets
Fabian Wein Topology Optimization Using the SIMP Method
Dynamic Compliance cont.
Do it again for a higher frequency (80 Hz) We optimize for uT uand uTLu with L selecting f
(a) uTu (b) uTLu
Note: the second example did not converge and is stopped after300 iterations. Movie
Fabian Wein Topology Optimization Using the SIMP Method
Thee Dimensions
(a) Single load (b) Surface load
Problems
• Requires iterative solvers (ILUPACK)
• Difficult to visualize (greyness!)
• Neither GiD not GMV are optimal
Fabian Wein Topology Optimization Using the SIMP Method
Volume Constraints
Volume constraint
• Avoids trivial solution (full or void)
• Removes greyness by penalization
Choice of volume constraint
• Engineering requirements (weight, price)
• Well optimization behaviour
• Nice pictures
Fabian Wein Topology Optimization Using the SIMP Method
Multicriterial optimization
Constrained Optimization
• Mathematically objective function and constraints are similar
• Consider the volume constraint as design variable
Multicriterial optimization
• Pareto efficiency:No criterion can be improved without worse another one.
• Solution is a Pareto front
• Requires user choice
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Com
plia
nce
Volume fraction
Compliance
Fabian Wein Topology Optimization Using the SIMP Method
Hypothesis
Heretical Hypothesis
A volume constraint can remove greyness only forsolutions 6= global solution.
Fabian Wein Topology Optimization Using the SIMP Method
Initial Guess
Initial guess
• Homogeneous intermediate material
• Chosen to match volume constraint
• Mathematical feasible, physcial unfeasible
• 6= traditional engineering solution
• Impressive objective over iterations charts
Fabian Wein Topology Optimization Using the SIMP Method
The End
Last comments
• The optimal solution lays inside the PDE (plus adjoint RHS)
• Optimization helps to understand systems better
• Optimization is the next step after simulation
• Ole Sigmund: A 99 Line Topology Optimization Code writtenin MATLAB; 2001
• Thanks for your time!
Fabian Wein Topology Optimization Using the SIMP Method