Topology Optimization Using the SIMP Method
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- 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.
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Topology Optimization Using the SIMP Method
- 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