Fabian Wein
Introduction Introduction About this document This is a fragment of a talk given interally Intended for engineers and mathematicians SIMP basics Detailed introduction (based on linear elasticity) Optimization vs. simulation Do it yourself with Sigmund s 99-line code Piezoelectric Loupspeaker Extended SIMP model Results
Optimization Introduction General optimization Objective function (scalar) Design variable Structural optimization Topology design of truss structures Shape optimization Parametrization Level set metheod (eventually topology gradient) Topology optimization Homogenization SIMP
Linear elasticity Classical SIMP Background Hooke s law [σ] = [c 0 ][S] (in Voigt notation: σ = [c 0 ]Bu) with [σ], σ : Cauchy stress tensor [c 0 ] : tensor of elastic modului [S],S : linear strain tensor u : displacement x 0 0 0 z y B = 0 y 0 z 0 x 0 0 z y x 0 T : differential operator
Strong formulation Classical SIMP Background PDE Find u : Ω R 3 fulfilling B T [c 0 ]Bu = f in Ω with the boundary conditions u = 0 n T [σ] = 0 on Γ s on ΩΓ s
Classical SIMP Discrete FEM formulation Background Solve Global System Ku = f with Assembly K = n e e=1 K e ; K e = [k pq ]; k pq = (B) T [c 0 ]B dω Ω e
Classical SIMP Proportional stiffness model Incredients Parametrization by design variable Model structure by local stiffness (full and void). Define local stiffness (finite) element wise: ρ = (ρ 1 ρ ne ) T Continuous interpolation with ρ min ρ e 1. Introduce pseudo density ρ [ c e ](ρ) = ρ e [c 0 ]; Ke (ρ) = ρ e K e ; K(ρ)u(ρ) = f
Classical SIMP Minimal mean compliance Incredients Different interpretations Maximize stiffness Minimize mean compliance Minimize stored mechanical energy Minimize compliance min ρ J(u(ρ)) = min ρ ft u(ρ) = min ρ u(ρ)t K(ρ)u(ρ)
Find derivative Classical SIMP Incredients 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
Sensitvity analysis Classical SIMP Incredients Sensitivity analysis provides analytical derivatives Abbreviate ( ) ρ e by ( ) Derive mean compliance f T u J = f T u + f T u = f T u Find J by deriving state condition Ku = f Solve for every u Ku = K u
Adjoint method Classical SIMP Incredients 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 Solve: Kλ = f = J u J = u T K u The compliance problem is self-adjoint The general adjoint problem can be efficiently solved by (incomplete) LU decomposition
Naive approach Classical SIMP Application of the SIMP method Minimize compliance: straight forward, initial design 0.5 min ρ ft u s.th.: Ku = f ρ e [ρ min : 1] note: K e = ρ e K e, K e = K e,
Naive approach Classical SIMP Application of the SIMP method Minimize compliance: straight forward, initial design 0.5 min ρ ft u s.th.: Ku = f ρ e [ρ min : 1] note: K e = ρ e K e, K e = K e, The optimal topology is the trivial solution full material
Add constraint Classical SIMP Application of the SIMP method Minimize compliance: volume constraint 50% min ρ ft u s.th.: Ω ρ 1 2 V 0
Add constraint Classical SIMP Application of the SIMP method Minimize compliance: volume constraint 50% min ρ ft u s.th.: Ω ρ 1 2 V 0 Grey material has no physical interpretation
Third try Classical SIMP Application of the SIMP method Minimize compliance: penalize ρ by ρ p with p = 3 min ρ ft u note: K e = ρ 3 e K e, K e = 3ρ 2 e K e,
Third try Classical SIMP Application of the SIMP method Minimize compliance: penalize ρ by ρ p with p = 3 min ρ ft u note: K e = ρ 3 e K e, K e = 3ρ 2 e K e, We have a desired 0-1 pattern but checkerboard structure
Forth try Classical SIMP Application of the SIMP method Minimize compliance: use averaged gradients min ρ ft u note: K e = ρ i H i i ρ e 3ρe 2 K e i H i with H i = r min dist(e,i)
Forth try Classical SIMP Application of the SIMP method Minimize compliance: use averaged gradients min ρ ft u note: K e = ρ i H i i ρ e 3ρe 2 K e i H i with H i = r min dist(e,i) No checkerboards and no mesh dependency
Classical SIMP Optimizers Comparison of different optimizers SCPIP (MMA implementation by Ch. Zillober) Optimality Condition (heuristic for SIMP) IPOPT (general second order optimizer)
Performance Classical SIMP Optimizers
Optimality Condition Classical SIMP Optimizers Optimality Condition: fix-point type update scheme With max{(1 ζ )ρ ek,ρ min } if ρ ek Be η k max{(1 η)ρ ek,ρ min } ρ ek+1 = min{(1 + ζ )ρ ek,1} if min{(1 + ζ )ρ ek,1} ρ ek Be η k ρ ek Be η k else B ek = Λ 1 K e Λ to be found by bisection Step width ζ e.g. 0.2 Damping η e.g. 0.5
Extensions to SIMP Multiple loads Complex 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
Extensions to SIMP Problem specific optimization Multiple loads Now only the left load is applied to the optimized structures Figure: The scaling of the displacement is the same
Extensions to SIMP Optimization for arbitrary nodes Synthesis of compliant mechanisms - aka no title Generalizing the compliance to J = l T u with l = (0 0 1 0 ) T.
Extensions to SIMP Optimization for arbitrary nodes Synthesis of compliant mechanisms - aka no title Generalizing the compliance to J = l T u with l = (0 0 1 0 ) T. For this case one has to apply springs to the load and output nodes
Harmonic optimization Extensions to SIMP 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 S(ω)u = f ST = S
Extensions to SIMP Harmonic optimization Harmonic objective functions: J(u(ρ)) R Compliance J = u T f J = R(sign(J)u T S u) J = (u T f) 2 J = 2(u T f)u T S u J = u T R f I u T I f R J = 2R(λ T S u) Sλ = 2 f j J = u T ū J = 2R(λ T S u) Sλ = ū Optimize for output J = u T Lū J = 2R(λ T S u) Sλ = L T ū Optimize for velocity Optimize for coupled quantities
Extensions to SIMP Harmonic tranfer functions Harmonic optimization Classical SIMP converges faster than mass to zero ρe 3 if ρ > 0.1 ρ e µ Pedersen (ρ e ) = ρ e µ RAMP (ρ e ) = if ρ 0.1 1 + q(1 ρ e ) 100 1e+000 8e-001 SIMP idendity RAMP Contribution 6e-001 4e-001 2e-001 0e+000 0 0.2 0.4 0.6 0.8 1 Design variable
Extensions to SIMP (Global) dynamic compliance Harmonic optimization We optimize for u T ū and u T Lū with L selecting f This illustrates general optimization problems One has to know what one wants One might not want what one gets
Piezoelectric Model Piezoelectricity Model We couple linear elasticity with electrostatic Material law [σ] = [c E 0 ][S] [e 0] T E, D = [e 0 ][S] + [ε S 0 ]E. E : electric field intensity in V/m D : electric displacement field C/m 2 [σ] : Cauchy stress tensor [S] : linear strain tensor [c E 0 ] : tensor of elastic moduli [c m ] : tensor of elastic moduli [ε S 0 ] : tensor of dielectric permittivities [e 0 ] : tensor of piezoelectric moduli
Strong formulation Piezoelectricity Model Loaded Electrode Grounded Electrode Piezoelectric Material Plate Strong formulation Support Find u p : Ω p R 3, u m : Ω m R 3, φ : Ω p R fulfilling ( ) B T [c E 0 ]Bu p + [e 0 ] T Bφ = 0 in Ω p, B T ( [e 0 ]Bu p [ε S 0 ] Bφ ) = 0 in Ω p, B T [c m ]Bu m = 0 in Ω m
Strong formulation Piezoelectricity Model Boundary conditions u m = 0 on Γ s, n T p [σ p ] = 0 on Ω p \ Γ g, n T m[σ m ] = 0 on Ω m \ (Γ g Γ s ), n T p [σ p ] = n T m[σ m ] on Γ g, n p = n m on Γ g, u p = u m on Γ g, φ = 0 on Γ g, φ = φ l on Γ l, n T p D = 0 on Ω p \ (Γ l Γ g )
FEM formulation Piezoelectricity Model Bilinearforms K e uu = [k uu pq ]; K e uφ = [k uφ pq ]; K e φφ = [k φφ pq ]; kpq uu = k uφ pq = k φφ pq = Ω e ( B u p ) T [c E 0 ]B u q dω, B T p [e 0 ] B q dω, Ω e BT p [ε S 0 ] B q dω. Ω e Global system K um u m K um u p 0 u m 0 K T u m u p K up u p K up φ u p = q u 0 K T u p φ K φφ φ q φ
SIMP optimization of piezoelectric devices Piezoelectric SIMP model Model Design variables pseudo density ρ pseudo polarization ρ p Extended material tensors [ c E e ] = µ c (ρ e )[c E 0 ], [ẽ e ] = µ e (ρ e )[e 0 ], [ ε S e ] = µ ε (ρ e )µ p (ρ p e )[ε S 0 ]. ρ e [ρ min ;1] ρ p e [ 1;1] Adjoint PDE for inhomogeneous forward problem No electric excitation but load from objective Gradient reduces with first order elements to J = w T e K e uuu e + w T e K e uφ φ e
SIMP optimization of piezoelectric devices Transduction basics Transduction Reciprocal theorem in elasticity t T a u b dγ = t T b u adγ Γ ta Γ tb... by knowing the body response for one load case, we can calculate the displacement at any point of the body caused by another load case.
SIMP optimization of piezoelectric devices Transduction basics Transduction Reciprocal theorem in elasticity t T a u b dγ = t T b u adγ Γ ta Γ tb... by knowing the body response for one load case, we can calculate the displacement at any point of the body caused by another load case. Extension to piezoelectricity promisses (Kögl and Silva, 2005):... the conversion of electrical into elastic energy and vice versa.
SIMP optimization of piezoelectric devices Transduction in piezoelectricity Transduction Loadcase a: f a 0 Q a = 0 Loadcase b: f b = 0 Q b 0 Loadcase c: f c 0 grounded electrodes Extension to piezoelectricity L ab = φ T b KT uφ u a + φ T a K φφ φ b L ab = û T a K û b. Difference to Kögl and Silva (2005) We have fixed supporting mechanical plate Kögl and Silva have loadcase c and volume constraint Original objective J(ρ) = w lnl ab (1 w)lnl cc, 0 w 1,
SIMP optimization of piezoelectric devices Results of mean transduction Transduction The result is the trivial result vanishing material We excite with fixed charge and fixed nodal force Loadcase a (Q a = 0): minimal stiffening (u) and maximal bending (φ) Loadcase b (f b = 0): surface charge density and E tend to infinity Displacement (m) 5.0e-005 4.5e-005 4.0e-005 3.5e-005 3.0e-005 2.5e-005 2.0e-005 1.5e-005 1.0e-005 5.0e-006 mechanical excitation electric excitation 10 20 30 40 50 60 70 80 90 100 Area void/piezo (%) Voltage (V) 1.0e+004 1.0e+003 1.0e+002 1.0e+001 1.0e+000 1.0e-001 mechanical excitation electric excitation 10 20 30 40 50 60 70 80 90 100 Area void/piezo (%) Figure: Parameter study with varying area covered by piezoelectric material
SIMP optimization of piezoelectric devices Results Optimize for maximum displacement of ground plate Thickness of layers Piezoelectric layer: 50 µm Supporting layer: 10, 20, 50, 100, 200, 500 µm No volume constraint and no filtering required!
SIMP optimization of piezoelectric devices Locking effects Results Very thin elements Shear locking occures when using linear finite elements Straight forward extension of incompatible modes by SIMP Solve local system of higher order ( K e uu K e αu K e )( uα ue K e αα α e ) = ( ) fe 0 Displacement 3.0e-004 2.5e-004 2.0e-004 1.5e-004 1.0e-004 5.0e-005 0.0e+000 linear, with locking linear, locking free quadratic 5 10 15 20 25 30 35 40 Discretization of edge Volume fraction 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 linear, with locking linear, locking free quadratic 5 10 15 20 25 30 35 40 Discretization of edge
The End 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 Thanks for your time!