Multi scale shape optimization under uncertainty · Multi scale shape optimization under...
Transcript of Multi scale shape optimization under uncertainty · Multi scale shape optimization under...
Multi scale shape optimization under uncertainty
Sergio Conti 1 Harald Held 1 Martin Pach1 Martin Rumpf 2
Rüdiger Schultz 1
1Department of MathematicsUniversity Duisburg-Essen
{conti,held,pach,schultz}@math.uni-duisburg.de
2Institute for Numerical SimulationRheinische Friedrich-Wilhelms-Universität Bonn
SPP1253 Workshop Hamburg 2008
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 1 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 2 / 33
Conceptual sketch
D
Γ0
ΓN
ΓD
O
Figure: General setting in 2D
Information Constraintdecide O −→ observe f (ω), g(ω) −→ decide u = u(O, ω)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 3 / 33
Conceptual sketch
D
Γ0
ΓN
ΓD
O
Figure: General setting in 2D
Information Constraintdecide O −→ observe f (ω), g(ω) −→ decide u = u(O, ω)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 3 / 33
Linear elasticity model
The displacement u is given by the equation system
PDE −div(Ae(u)) = f in O,u = 0 on ΓD,
(Ae(u))n = g on ΓN
Elastic body O ⊂ R3
∂O = ΓN ∪ ΓD, ΓD 6= ∅
Volume forces f in ONeumann forces g on ΓN
where e(u) = 12 (∇u +∇uT) is the linearized strain tensor
and Hooke’s law
Aξ = 2µξ + λ(trξ)Id, for any symmetric matrix ξ
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 4 / 33
Linear elasticity model
The displacement u is given by the equation system
PDE −div(Ae(u)) = f in O,u = 0 on ΓD,
(Ae(u))n = g on ΓN
Elastic body O ⊂ R3
∂O = ΓN ∪ ΓD, ΓD 6= ∅
Volume forces f in ONeumann forces g on ΓN
where e(u) = 12 (∇u +∇uT) is the linearized strain tensor
and Hooke’s law
Aξ = 2µξ + λ(trξ)Id, for any symmetric matrix ξ
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 4 / 33
Linear elasticity model
The displacement u is given by the equation system
PDE −div(Ae(u)) = f in O,u = 0 on ΓD,
(Ae(u))n = g on ΓN
Elastic body O ⊂ R3
∂O = ΓN ∪ ΓD, ΓD 6= ∅
Volume forces f in ONeumann forces g on ΓN
where e(u) = 12 (∇u +∇uT) is the linearized strain tensor
and Hooke’s law
Aξ = 2µξ + λ(trξ)Id, for any symmetric matrix ξ
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 4 / 33
Shape optimization problem
Compliance
J(O) =∫O
f · u dx +∫
ΓN
g · u ds
Least square error compared to target displacement
J(O) =(∫O|u− u0|2 dx
) 12
Shape optimization problem
minO∈Oad
J(O) + lV(O) with l ∈ R, l > 0
Oad = {O ⊂ D : ∂O Lipschitz-continuous }
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 5 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 6 / 33
Two-Stage Stochastic Linear Program
min{cTx + qTy : Tx + Wy = z(ω), y ∈ Y, x ∈ X}
Information Constraintdecide x −→ observe z(ω) −→ decide y = y(x, z(ω))
minx{cTx + min
y{qTy : Wy = z(ω)− Tx, y ∈ Y} : x ∈ X}
minx{cTx + G(x, ω) : x ∈ X}
−→ looking for a minimal member in family of random variables
{cTx + G(x, ω) : x ∈ X}
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 7 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 8 / 33
General Objective Function
J(O, u(O, ω)) =∫O
j(u) dx +∫∂O
k(u) ds + `
∫O
dx, O ∈ Uad, ` > 0
u = u(O, ω) is the solution of the PDEassume j(.) and k(.) are linear or quadratic and independent of ω
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 9 / 33
The two Stages
We now introduce random forces f (ω) and g(ω) to the shape optimization problem.First stage Non-anticipative decision on O has to be takenobserve the random forces f (ω) and g(ω) by choosing a scenarioSecond Stage The variational formulation of elasticity, given O and ω, takes therole of the second-stage problem
Information constraint heredecide O −→ observe f (ω), g(ω) −→ decide u = u(O, ω)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 10 / 33
Variational two stage formulation
variational description of linearized elasticity:
E(O, u, ω) :=12
A(O, u, u)− l(O, u, ω) with
A(O, ψ, ϑ) :=∫O
Aijkleij(ψ)ekl(ϑ)dx
l(O, ϑ, ω) :=∫O
fi(ω)ϑidx +∫∂O
gi(ω)ϑi dHd−1
Two stage shape optimization problem
minO∈Oad
{J(O, ω) : u(O, ω) = argminu E(O, u, ω)}
Oad = {O ⊂ D : ∂O Lipschitz-continuous}
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 11 / 33
Direct comparison with the linear case
shape optimization : min{J(O, ω) : u(O, ω) = argminu E(O, u, ω)}
linear program : min{j(x, ω) = cTx + min{qTy : Wy = z(ω)− T(x)}}
correspondences:O ↔ xu(O, ω) ↔ yj(x, ω) ↔ J(O, ω)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 12 / 33
optimization task
min{QE(O) := Eω(J(O, ω)) : O ∈ Oad}
load configuration:assume that ω follows a discrete distribution with scenarios ωσ and probabilities πσwith
∑Sσ=1 πσ = 1 and ’basis’ loads (f k, gm) spanning the load space:
f (ω) =K∑
k=1
αk f k , g(ω) =M∑
m=1
βm f m
by linearity :
u(O, ω) =K∑
k=1
αk ukf +
M∑m=1
βmumg
solves A(O, u(O, ωσ), ϕ) = l(O, ϕ, ωσ)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 13 / 33
Derivative of the stochcastic functional
Given u(O, ωσ) we rewrite the stochastic program
min{
QE(O) = `
∫O
dx +S∑
σ=1
πσ
∫O
j(u) dx +∫∂O
k(u) ds : O ∈ Oad
}we obtain the shape derivative
Q′E(O)(V) =S∑
σ=1
πσ J′(O, ωσ)(V)
The approach is computationally efficient if K + M � S .
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 14 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 15 / 33
Shape gradientWe consider variations Ot = (Id + t · V)(O) , t > 0 of a smooth elastic domain Ofor a smooth vector field V defined on the working domain D.The shape derivative of J(O) at O in direction V is defined as the Fréchet derivativeof the mapping t→ J(Ot), i.e.
J(Ot) = J(O) +⟨∂J∂O
,V⟩
+ o(‖V‖)
O
Ot
Ot = (Id + t V) (O)
cf. [Sokolowski, Zolesio ’92], [Delfour, Zolesio ’01]
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 16 / 33
Shape gradient
As a classical result of the shape sensitivity analysis the shape derivative takes theform
<∂J∂O
,V > =∫ΓN
(2[∂(g · u)∂n
+ hg · u + f · u]− Aε(u) : ε(u)
)V ·~n dν
+∫ΓD
(Aε(u) : ε(u)) V ·~n dν
Here h denotes the mean curvature of ∂O and ~n the outer normal.
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 17 / 33
Shape gradient in level set formulation
When the domain O is implicitly deformed by varying the level set function φ
φt = φ + tψ
the level set equation
∂tφ+ |∇φ| v · n = 0 n =∇φ|∇φ|
allows to define
<∂J∂φ, ψ >:=<
∂J∂O
,−ψ · ~n‖∇ φ‖
>
cf. [Osher, Sethian ’88],[Burger, Osher ’04]
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 18 / 33
Shape gradient in level set formulation
We take into account a regularized gradient descent, based on the metric
G(θ, ζ) =∫
Dθζ +
σ2
2∇θ · ∇ζ dx
which is related to a Gaussian filter with width σ.The shape gradient is the solution of equation
G(gradφJ, θ) = <∂J∂O
, θ > ∀θ ∈ H1,20 (D)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 19 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 20 / 33
Optimization algorithm
time continuous regularized gradient descent:
∂φ(t) = −gradφJ(φ)
with time discrete relaxation :
G(φk+1 − φk, θ) = −τ <∂J∂O
, θ > ∀θ ∈ H1,20 (D)
additional ingrediens of the algorithm :multigrid method for the primal and the dual problem (d = 3)preconditioned CG (d = 2)cascadic optimization (from coarse to fine grid resolution)morphological smoothing when switching the grid resolution(σ = 2.5h or 4.5h)
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 21 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 22 / 33
Test Setting
∂O is divided into 3 parts:ΓD: the fixed Dirichlet boundaryΓN : part of the Neumann boundary where the surface loads act on;this is also fixed and does not move during the optimization processΓ0: all other parts of the boundary; this is the only part of ∂O to be optimized
The objective function (compliance with f ≡ 0):
J(O, ω) =∫
ΓN
g(ω) · u ds + `Ri(O)
with regularization terms
R1(O) =∫∂O
ds (and volume preservation) ,
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 23 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 24 / 33
2-Stage vs. Expected Load
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 25 / 33
Initial Shape and Objective Values
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 26 / 33
Instance with 20 Scenarios
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 27 / 33
Instance with 21 Scenarios
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 28 / 33
Cantilever
optimal cantilever construction with varying number of scenarios
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25 30 35 40 45
Iterations
"Compliance""Volume"
"ObjectiveValues"
van Mises stress distribution for different scenarios
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 29 / 33
Outline
1 Introduction
2 Two-Stage Stochastic ProgrammingTwo-Stage Stochastic Linear Programming FormulationRandom Shape Optimization Problem
3 Level set methodShape gradientLevel set method
4 Numerical ResultsTest SettingExamples
5 ExtensionTopological Derivative
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 30 / 33
Different initial shapes yield different solutions
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 31 / 33
Topological Derivative
T (x) = limρ↓0
J(O \ Bρ(x))− J(O)|Bρ(x)|
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 32 / 33
Thank you !
Martin Pach (University Duisburg-Essen) Multi scale shape optimization under uncertainty SPP1253 Workshop Hamburg 2008 33 / 33