- The document discusses topology optimization using the SIMP (Solid Isotropic Material with Penalization) method. - It describes how to formulate topology optimization problems using finite element analysis, including defining design variables, objective functions, and constraints. - It explains how the SIMP method works by introducing a pseudo density design variable and penalizing intermediate densities to drive solutions to 0 or 1, representing void or solid material. - Several examples are provided to illustrate different objective functions and constraints that can be used, and how to address issues like checkerboard patterns and mesh dependence.

1. Topology Optimization Using the SIMP Method
Fabian Wein
Introductary Talk @ LSE
29.10.2008
Fabian Wein Topology Optimization Using the SIMP Method

2. 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

3. Basic Optimization Problem
• Design vector x (e.g. dimensions, topology, shape, material)
• Problem
min J (x)
x
subject to
equality constraints
inequality constraints
box constraints
• Objective function J (x) → R
Fabian Wein Topology Optimization Using the SIMP Method

4. 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

5. 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

6. Linear elasticity
Hooke’s law
[σ ] = [c0 ][S]
σ (in Voigt notation: σ = [c0 ]Bu)
with
• [σ ], σ : Cauchy stress tensor
σ
• [c0 ] : tensor of elastic modului
• [S], S : linear strain tensor
• u : displacement
 ∂ ∂ ∂
T
∂x 0 0 0 ∂z ∂y
• B= 0 ∂
0 ∂
0 ∂ 
: differential operator

∂y ∂z ∂x 
∂ ∂ ∂
0 0 ∂z ∂y ∂x 0
Fabian Wein Topology Optimization Using the SIMP Method

7. Strong Formulation
PDE
Find
¯
u : Ω → R3
fulfilling
B T [c0 ]Bu = f in Ω
with the boundary conditions
u=0 on Γs
T
n [σ ] = 0
σ on ∂ ΩΓs
Fabian Wein Topology Optimization Using the SIMP Method

8. Discrete FEM Formulation
Solve
Global System
Ku = f
with
Assembly
ne
K= Ke ; Ke = [kpq ]; kpq = (B)T [c0 ]B dΩ
e=1 Ωe
Fabian Wein Topology Optimization Using the SIMP Method

9. Proportional Stiffness Model
Parametrization by design variable
• Model structure by local stiffness (full and void).
• Define local stiffness (finite) element wise: ρ = (ρ1 · · · ρne )T
• Continuous interpolation with ρmin ≤ ρe ≤ 1.
Introduce pseudo density ρ
[ce ](ρ ) = ρe [c0 ];
ρ Ke (ρ ) = ρe Ke ;
ρ K(ρ )u(ρ ) = f
ρ ρ
Fabian Wein Topology Optimization Using the SIMP Method

10. Minimal Mean Compliance
Different interpretations
• Maximize stiffness
• Minimize mean compliance
• Minimize stored mechanical energy
Minimize compliance
min J(u(ρ )) = min f T u(ρ ) = min u(ρ )T K(ρ )u(ρ )
ρ ρ ρ ρ ρ
ρ ρ ρ
Fabian Wein Topology Optimization Using the SIMP Method

11. Find Derivative
General optimization procedure
• Evaluate objective function
• Find descent direction δ (e.g.
gradient)
• Find step length along δ (line
search)
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

12. Sensitvity Analysis
• Sensitivity analysis provides analytical derivatives
• Abbreviate ∂ (·) by (·)
∂ ρe
Derive mean compliance f T u
J = f Tu + f Tu = f Tu
Find J by deriving state condition Ku = f
Solve for every u
Ku = −K u
Fabian Wein Topology Optimization Using the SIMP Method

13. Adjoint Method
The adjoint method is based on the fixed vector λ
J = f T u + λ T (Ku − f)
J = f T u + λ T (K u + Ku )
= (f T + λ T K)u + λ T K u
∂J
Solve: Kλ
λ = −f =
∂u
T
J = −u K 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

14. Naive Approach
Minimize compliance: straight forward, initial design 0.5
min f T u s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρe Ke , Ke = Ke
ρ
Fabian Wein Topology Optimization Using the SIMP Method

15. Naive Approach
Minimize compliance: straight forward, initial design 0.5
min f T u s.th.: Ku = f ρe ∈ [ρmin : 1] note: Ke = ρe Ke , Ke = Ke
ρ
The optimal topology is the trivial solution full material
Fabian Wein Topology Optimization Using the SIMP Method

16. Add Constraint
Minimize compliance: volume constraint 50%
1
min f T u s.th.: ρ ≤ V0
ρ Ω 2
Fabian Wein Topology Optimization Using the SIMP Method

17. Add Constraint
Minimize compliance: volume constraint 50%
1
min f T u s.th.: ρ ≤ V0
ρ Ω 2
“Grey” material has no physical interpretation
Fabian Wein Topology Optimization Using the SIMP Method

18. Third Try
Minimize compliance: penalize ρ by ρ p with p = 3
min f T u note: Ke = ρe Ke , Ke = 3ρe Ke
3 2
ρ
Fabian Wein Topology Optimization Using the SIMP Method

19. Third Try
Minimize compliance: penalize ρ by ρ p with p = 3
min f T u note: Ke = ρe Ke , Ke = 3ρe Ke
3 2
ρ
We have a desired 0-1 pattern but checkerboard structure
Fabian Wein Topology Optimization Using the SIMP Method

20. Forth Try
Minimize compliance: use averaged gradients
iρ 2
∑i Hi ρe 3ρe Ke
min f T u note: Ke = with Hi = rmin − dist(e, i)
ρ ∑i Hi
Fabian Wein Topology Optimization Using the SIMP Method

21. Forth Try
Minimize compliance: use averaged gradients
iρ 2
∑i Hi ρe 3ρe Ke
min f T u note: Ke = with Hi = rmin − dist(e, i)
ρ ∑i Hi
No checkerboards and no mesh dependency (view movie)
Fabian Wein Topology Optimization Using the SIMP Method

22. 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

23. Performance
Fabian Wein Topology Optimization Using the SIMP Method

24. Optimality Condition
Optimality Condition: fix-point type update scheme
 η
 max{(1 − ζ )ρek , ρmin } if ρek Bek ≤ max{(1 − η)ρek , ρmin }

η
ρek+1 = min{(1 + ζ )ρek , 1} if min{(1 + ζ )ρek , 1} ≤ ρek Bek

 η
ρek Bek else
With
• Bek = Λ−1 Ke
• 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

25. 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

26. 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

27. Synthesis of Compliant Mechanisms - aka ”no title”
Generalizing the compliance to J = lT u with l = (0 · · · 0 1 0 · · · )T .
Fabian Wein Topology Optimization Using the SIMP Method

28. Synthesis of Compliant Mechanisms - aka ”no title”
Generalizing the compliance to J = lT u 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

29. 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 + jω C − ω 2 M) u = f
T
S(ω) u = f S = S
Fabian Wein Topology Optimization Using the SIMP Method

30. Harmonic Objective Functions: J(u(ρ )) → R
ρ
Compliance
J = |uT f| J = −R(sign(J)uT S u)
J = (uT f)2 J = −2(uT f)uT S u
j
J = uT fI − uT fR
R I J = 2R(λ T S u)
λ Sλ = − ¯
λ f
2
J = uT u J = 2R(λ T S u)
¯ λ Sλ = −¯
λ u
Optimize for output
J = uT L¯ J = 2R(λ T S u) Sλ = −LT u
u λ λ ¯
• Optimize for velocity
• Optimize for coupled quantities
Fabian Wein Topology Optimization Using the SIMP Method

31. Harmonic Interpolation Functions
Classical SIMP converges faster than mass to zero

3
 ρe if ρ > 0.1 ρe
µPedersen (ρe ) = ρ µRAMP (ρe ) =
 e if ρ ≤ 0.1 1 + q(1 − ρe )
100
1e+000
SIMP
idendity
8e-001 RAMP
Contribution
6e-001
4e-001
2e-001
0e+000
0 0.2 0.4 0.6 0.8 1
Design variable
Fabian Wein Topology Optimization Using the SIMP Method

32. (Global) Dynamic Compliance
We optimize for uT u and uT L¯ with L selecting f
¯ u
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

33. Dynamic Compliance cont.
Do it again for a higher frequency (80 Hz) We optimize for uT u
¯
and uT L¯ with L selecting f
u
(a) uT u (b) uT Lu
Note: the second example did not converge and is stopped after
300 iterations. Movie
Fabian Wein Topology Optimization Using the SIMP Method

34. 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

35. 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

36. 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
0.003
0.0025
Compliance
0.002
0.0015
0.001
0.0005
Compliance
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Volume fraction
Fabian Wein Topology Optimization Using the SIMP Method

37. Hypothesis
Heretical Hypothesis
A volume constraint can remove greyness only for
solutions = global solution.
Fabian Wein Topology Optimization Using the SIMP Method

38. Initial Guess
Initial guess
• Homogeneous intermediate material
• Chosen to match volume constraint
• Mathematical feasible, physcial unfeasible
• = traditional engineering solution
• Impressive objective over iterations charts
Fabian Wein Topology Optimization Using the SIMP Method

39. 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 written
in MATLAB; 2001
• Thanks for your time!
Fabian Wein Topology Optimization Using the SIMP Method