International audienceThe present work aims at handling uncertain loads in shape and topology optimization. More specifically, we minimize objective functions combining mean values and variances of standard cost functions and assume that uncertainties are small and generated by a finite number N of random variables. A deterministic approach that relies on a second-order Taylor expansion of the cost function has been proposed by Allaire & Dapogny. 1 That method requires a computational effort comparable to the one for an N-load problem. This work presents a general framework to handle uncertainties on arbitrary static load cases where perturbations on both surface forces and body forces are considered. We demonstrate the effectiveness of the approach in the context of level-set-based topology optimization for the robust compliance minimization on three-dimensional test cases

Bordeu, Felipe

Cortial, Julien

MANG, Chetra

Nardoni, Chiara

English

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-02471986
https://hal.archives-ouvertes.fr/hal-02471986
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A deterministic approach for shape and topology
optimization under uncertain loads
Chetra Mang, Julien Cortial, Chiara Nardoni, Felipe Bordeu
To cite this version:
Chetra Mang, Julien Cortial, Chiara Nardoni, Felipe Bordeu. A deterministic approach for shape
and topology optimization under uncertain loads. Eurogen 2019, Sep 2019, Guimarães, Portugal.
￿hal-02471986￿
EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
A deterministic approach for shape and topology optimization under uncertain
loads
Chetra Mang
Institut de Recherche Technologique SystemX (IRT SystemX)
8 Avenue de la Vauve, 91120 Palaiseau, France
Email: chetra.mang@irt-systemx.fr
Julien Cortial
Safran Tech, Modelling and Simulation
Rue des Jeunes Bois, Châteaufort, CS-80112-78777 Magny-les-Hameaux, France
Email: julien.cortial@safrangroup.com
Chiara Nardoni
Institut de Recherche Technologique SystemX (IRT SystemX)
8 Avenue de la Vauve, 91120 Palaiseau, France
Email: chiara.nardoni@irt-systemx.fr
Felipe Bordeu
Safran Tech, Modelling and Simulation
Rue des Jeunes Bois, Châteaufort, CS-80112-78777 Magny-les-Hameaux, France
Email: felipe.bordeu@safrangroup.com
Summary
The present work aims at handling uncertain loads in shape and topology optimization. More specifically, we minimize
objective functions combining mean values and variances of standard cost functions and assume that uncertainties are
small and generated by a finite number N of random variables. A deterministic approach that relies on a second-order
Taylor expansion of the cost function has been proposed by Allaire & Dapogny.1 That method requires a computational
effort comparable to the one for an N-load problem. This work presents a general framework to handle uncertainties on
arbitrary static load cases where perturbations on both surface forces and body forces are considered. We demonstrate
the effectiveness of the approach in the context of level-set-based topology optimization for the robust compliance
minimization on three-dimensional test cases.
Keywords: shape and topology optimization, level set, random uncertainties, deterministic approach.
1 Introduction
Shape and topology optimization is routinely used in
industry in the design process in mechanical parts. Most if
not all commercial grade tools assume that exact functional
specifications are available whereas in practice these can
only be determined approximately, especially during early
stages of design, when topology optimization provides the
most benefits. Moreover, the feasibility and optimality
of the solution can be very dependent on a slight change
on the actual requirements.1 Unless uncertainty has been
taken into account, the chosen design may turn out to
be suboptimal, or even worse, defective.2 Therefore, the
development of procedures that improve the robustness of
the solution should increase the added value of shape and
topology optimization in the industrial design process.
Several strategies have been proposed to account for
the effects of small uncertainties. On the one hand,
when no statistical information is available, a worst-case
design approach can be followed : The idea is simply
to minimize the maximal value of the objective function
for any possible perturbation. The computation is made
tractable by assuming that it is possible to linearize the
criterion with respect to the perturbation.3 While easy to
implement, this strategy can lead to structures with poor
nominal performances. On the other hand, when the user
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EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
has some knowledge on the statistical distribution of the
perturbation, a probabilistic description can be introduced.
Then it becomes possible to consider the mean value and
the variance of the objective fonction in the optimization
problem, leading to solutions with better performance than
in the previous case.
In this work, the latter approach is followed. The paper
is organized as follows. First, the theoretical framework,
mostly based on Allaire & Dapogny’s work,1 allows a
general treatment of uncertain loads. Then, the special case
of the robust compliance is derived and further developed.
Finally, the method is implemented in the context of
level-set-based topology optimization and illustrated on a
three-dimensional test-case.
2 Shape and topology optimisation via level set
Shape and topology optimisation
The goal of shape and topology optimization is to find a
shape that is a solution of a problem of the form:
min
Ω∈Oad
J (Ω), (1)
where J (Ω) is a cost function that depends on the
domain Ω ∈ Rd and Oad is the set of admissible shapes.
Relying on Hadamard’s notion of shape derivative, we
evaluate the cost function sensitivity with respect to a
certain class of domain perturbations in order to implement
an iterative continuous optimization algorithm. A variation
of a domain Ω is given by:
Ωθ = (I+θ)(Ω), (2)
where θ ∈ W 1,∞(Rd ,Rd) and for θ sufficiently small,
(I+θ) is diffeomorphism in Rd .
A function J (Ω) admits a shape derivative if the
mapping θ →J (Ωθ ) from W 1,∞(Rd ,Rd) to R is Fréchet
differentiable at θ = 0. The Fréchet derivative (or shape
derivative) of J (Ω) is denoted by θ →J ′(Ω)(θ) and
defined as:
J (Ωθ ) =J (Ω)+J ′(Ω)(θ)+o(θ), (3)
where
|o(θ)|
||θ ||W 1,∞(R3,R3)
θ→0−→ 0
andJ ′(Ω) is a continuous linear form on W 1,∞(Rd ,Rd).
The shape derivative depends on the normal trace of the
perturbation θ on Ω.4 In many relevant cases, it can be
written as:5
∀θ ∈W 1,∞(R3,R3), J ′(Ω)(θ) =
∫
∂Ω
vΩ(s)θnΩ ds, (4)
where nΩ is outward unit normal to the boundary ∂Ω
and vΩ(s) depends onJ (Ω).
Level set method
The level set method introduced by Osher and Sethian6 is
used for the numerical implementation of the shape and
topology optimization problem. In this framework, a shape
Ω ⊂ Rd is represented by the negative subdomain of an
auxiliary level set function φ : Rd → R :
∀x ∈ Rd ,

φ(x) < 0 if x ∈Ω
φ(x) = 0 if x ∈ ∂Ω
φ(x) > 0 if x ∈Ωc
(5)
The motion of a domain Ω(t) for t ∈ [0,T ] according
to a normal velocity field V (x, t) translates into an
Hamilton-Jacobi equation for the associated level set
function φ(t, .):
∂φ
∂ t
+V |∇φ |= 0, t ∈ (0,T ),x ∈ Rd , (6)
where t is a pseudo-time, the upper bound T is analogous to
the step size of a gradient descent and the advection field
V derives from the shape derivative DΩ of the objective
function J (Ω). Instead of taking the direction of steepest
descent V = −DΩ, a regularisation and an extension
process to compute V from DΩ is implemented.7
3 Random perturbations on mechanical loads in shape
and topology optimization
3.1 A model problem
Let us consider a linear elastic solid domain Ω ⊂ Rd .
The Dirichlet boundary condition is imposed on ΓD ⊂
∂Ω. Body and surface forces are respectively denoted by
f ∈ L2(Ω)d and g ∈ L2(ΓN)d with ΓN ⊂ ∂Ω and Γ0 =
∂Ω\(ΓN ∪ΓD). The mechanical problem obeys the system
of equations :

−div(σ(uΩ)) = f in Ω
uΩ = 0 on ΓD
σ(uΩ)n = g on ΓN
σ(uΩ)n = 0 on Γ0
(7)
We introduce small perturbations fˆ to the body forces
f = f0 + fˆ such that f0 ∈ L2(Ω)d , fˆ ∈ L2(Ω)d and gˆ to
the surface forces g = g0 + gˆ such that g0 ∈ L2(ΓN)d , gˆ ∈
L2(ΓN)d .
A second order Taylor expansion of a cost function
C (Ω, f ,g) =
∫
Ω
j( f ,uΩ)dx+
∫
ΓN
k(g,uΩ)ds (8)
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EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
around f0 ∈ L2(Ω)d and g0 ∈ L2(ΓN)d yields :
C (Ω, f0+ fˆ ,g0+ gˆ)
=
∫
Ω j( f0+ fˆ ,uΩ) dx+
∫
ΓN k(g0+ gˆ,uΩ) ds
=
∫
Ω j( f0,uΩ) dx+
∫
Ω(5 f j( f0,uΩ). fˆ
+5uΩ j( f0,uΩ).u1Ω( fˆ )) dx
+ 12
∫
Ω(52f j( f0,uΩ)( fˆ , fˆ )+25 f 5u j( f0,uΩ)( fˆ ,u1Ω( fˆ ))
+52f j( f0,uΩ)(u1Ω( fˆ ),u1Ω( fˆ ))) dx+R f ( fˆ )∫
ΓN k(g0,uΩ) ds+
∫
ΓN (5gk(g0,uΩ).gˆ
+5uΩ k(g0,uΩ).u1Ω(gˆ)) ds
+ 12
∫
ΓN (52gk(g0,uΩ)(gˆ, gˆ)+25g5u j(g0,uΩ)(gˆ,u1Ω(gˆ))
+52g j(g0,uΩ)(u1Ω(gˆ),u1Ω(gˆ))) ds+Rg(gˆ)
(9)
where
R f ( fˆ ) =
∫
Ω
∫ 1
0
∂ 3C
∂ f d
( fˆ , fˆ , fˆ ) dt dx (10)
and
Rg(gˆ) =
∫
ΓN
∫ 1
0
∂ 3C
∂gd
(gˆ, gˆ, gˆ) dt ds (11)
R( fˆ , gˆ)=R f ( fˆ )+Rg(gˆ) is the residual of the expansion.
We assume that the perturbations are random and
accordingly we set fˆ ≡ fˆ (x,ω), gˆ≡ gˆ(x,ω), for x ∈Ω and
ω ∈ O , where (O,F ,P) is a probability space.
3.2 Mean and variance of the cost function
Mean of the cost function
Let :
M (Ω) =
∫
O
C (Ω, f (.,ω),g(.,ω)) P(dω) (12)
denote the mean of the cost function
C (Ω, f (.,ω),g(.,ω))
It is assumed that fˆ ∈ L2(O,L2(Ω)d) (respectively gˆ)
is a finite sum of deterministic functions fi ∈ L2(Ω)d , i =
1, ...,N (respectively gi) weighted by independant Gaussian
random variables ξ fi (respectively ξgi ):
fˆ (x,ω) = ΣNi=1 fi(x)ξ fi(ω)χ fi (13)
gˆ(x,ω) = ΣNi=1gi(x)ξgi(ω)χgi (14)
where χ fi : Ω → {0,1} (respectively χgi ) is an
indicator function that identifies the region affected by the
perturbation fi (respectively gi ).
Then it comes:
M˜ (Ω) =
∫
Ω j( f0,uΩ)χ f0 dx
+ 12
∫
Ω(ΣNi=152f j( f0,uΩ)( fi, fi))χ fi dx
+ΣNi=1
∫
Ω(5 f 5u j( f0,uΩ)( fi,u1Ω,i))χ fi dx
+ 12
∫
Ω(ΣNi=152u j( f0,uΩ)(u1Ω,i,u1Ω,i))χ fi dx∫
ΓN k(g0,uΩ)χg0 ds
+ 12
∫
ΓN (Σ
N
i=152g k(g0,uΩ)(gi,gi))χgi ds
+ΣNi=1
∫
ΓN (5g5u k(g0,uΩ)(gi,u1Ω,i))χgi ds
+ 12
∫
ΓN (Σ
N
i=152u k(g0,uΩ)(u1Ω,i,u1Ω,i))χgi ds
(15)
such that u1Ω,i := u
1
Ω( fi,gi) is the unique solution in
H1ΓD(Ω)
d of the problem :

−div(σ(u1Ω,i)) = fiχ fi dans Ω
u1Ω,i = 0 sur ΓD
σ(u1Ω,i)n = giχgi sur ΓN
(16)
Variance of the cost function
The variance V (Ω) of the cost function C (Ω) is defined as:
V (Ω) =
∫
O
(C (Ω, f (.,ω))−M (Ω))2 P(dω) (17)
Following Allaire & Dapogny,1 we obtain the
expression:
V˜ (Ω) = ΣNi=1(a
2
Ω,i+b
2
ΓN ,i) (18)
where
aΩ,i :=
∫
Ω
(5 f j( f0,uΩ). fi+5u j( f0,uΩ).u1Ω,i)χ fi dx
and
bΓN ,i :=
∫
ΓN
(5gk(g0,uΩ).gi+5uk(g0,uΩ).u1Ω,i)χgi dx
3.3 Shape derivatives of mean and variance of the cost
function
Shape derivative of the mean of the cost function
According to Theorem 14 in Allaire & Dapogny,1 the
Fréchet derivative of the mean of the cost function is given
by :
∀θ ∈ Oad ,M˜ ′(Ω)(θ) =∫
ΓN ( j( f0,uΩ)χ f0 +
1
2Σ
N
i=152f j( f0,uΩ)( fi, fi)χ fi
+ΣNi=15 f 5u j( f0,uΩ)( fi,u1Ω,i)χ fi
+ 12Σ
N
i=152u j( f0,uΩ)(u1Ω,i,(u1Ω,i)χ fi +σ(uΩ) : e(p0Ω)χ f0
+ΣNi=1σ(u
1
Ω,i) : e(p
1
Ω,i)χ fi
− f0.p0Ωχ f0 −ΣNi=1 fi.p1Ω,iχ fi)θn ds+
∫
ΓN (
∂
∂nk(g0,uΩ)χg0
+Hk(g0,uΩ)χg0
+ 12Σ
N
i=152g k(g0,uΩ)(gi,gi)χgi
+ΣNi=15g5uk(g0,uΩ)(gi,u1Ω,i)χgi
+ 12Σ
N
i=152u k(g0,uΩ)(u1Ω,i,(u1Ω,i)χgi − ∂∂nk(g0, p0Ω)χg0
−Hk(g0, p0Ω)χg0)θn ds,
(19)
p0Ω and p
1
Ω,i for i= 1, ...,N are the (N+1) adjoint states,
defined as the unique solutions of H1ΓD(Ω)
d of the following
variational problems :
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EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
∀v ∈ H1ΓD(Ω)3,
∫
Ω(σ(p0Ω) : e(v)) dx=
−∫Ω(5u j( f0,uΩ)v) dx
− 12
∫
Ω(ΣNi=152f 5u j( f0,uΩ)( fi, fi,v)) dx
−ΣNi=1
∫
Ω(5 f 52u j( f0,uΩ)( fi,u1Ω,i,v)
+ 1253u j( f0,uΩ)(u1Ω,i,u1Ω,i,v)) dx
−∫ΓN (5uk(g0,uΩ)v) ds
− 12
∫
ΓN (Σ
N
i=152g5uk(g0,uΩ)(gi,gi,v)) ds
−ΣNi=1
∫
ΓN (5g52u k(g0,uΩ)(gi,u1Ω,i,v)
+ 1253u k(g0,uΩ)(u1Ω,i,u1Ω,i,v)) ds
(20)
∀v ∈ H1ΓD(Ω)3,
∫
Ω(σ(p1Ω,i) : e(v)) dx
=−∫Ω(5 f 5u j( f0,uΩ)( fi,v)+52u j( f0,uΩ)(u1Ω,i,v)) dx
−∫ΓN (5g5u k(g0,uΩ)(gi,v)+52uk(g0,uΩ)(u1Ω,i,v)) ds
(21)
Shape derivative of variance of the cost function
The Fréchet derivative of the variance of the cost function
is given by :
∀θ ∈ Oad , V˜ ′(Ω)(θ) =
∫
ΓN (σ(uΩ) : e(p
0
Ω)
+2ΣNi=1(aΩ,i+bΓN ,i)σ(u
1
Ω,i) : e(p
1
Ω,i))θn ds
(22)
4 Application for robust compliance
The so-called compliance function is defined as :
C (Ω) =
∫
Ω
σ : e(uΩ)dx=
∫
ΓN
g.uΩ ds+
∫
Ω
f .uΩ dx (23)
It is a special case of (8) with j( f ,uΩ) = f .uΩ and
k(g,uΩ) = g.uΩ. The expressions for the mean and the
variance of the cost function as well as the associated shape
derivatives are then :
M˜ (Ω) =
∫
Ω( f0.uΩχ f0 +Σ
N
i=1 fi.uΩ,iχ fi) dx
+
∫
ΓN (g0.uΩχg0 +Σ
N
i=1gi.uΩ,iχgi) ds
=
∫
Ω(σ(uΩ) : e(uΩ)
+ΣNi=1σ(uΩ,i) : e(uΩ,i)) dx
(24)
∀θ ∈ Oad ,M˜ ′(Ω) =∫
ΓN (σ(uΩ) : e(uΩ)+Σ
N
i=1σ(u
1
Ω,i) : e(u
1
Ω,i))nθ ds
(25)
V˜ (Ω) = ΣNi=1(a
2
Ω,i+b
2
ΓN ,i) (26)
∀θ ∈ Oad , V˜ ′(Ω) =
−4ΣNi=1(aΩ,i+bΓN ,i)
∫
ΓN σ(uΩ) : e(uΩ,i)nθ ds,
(27)
where
aΩ,i =
∫
Ω
( fi.uΩ+ f0.uΩ,i)χ fi dx
and
bΓN ,i =
∫
ΓN
(gi.uΩ+g0.uΩ,i)χgi ds
5 Numerical Example
Compliance minimization with uncertain loads
In the following example, we consider the
volume-constrained minimization of the weighted sum
J (Ω) := M˜ (Ω)+δ
√
V˜ (Ω) (28)
of the approximate mean value and standard deviation
of the compliance (23) where δ > 0.
Let us consider the test case of a three-dimensional
chair. The design domain is a subset of a 1.2×1×2 box, as
shown on Figure 1. At the bottom of the chair, four small
height cylidrical non-designed zones are clamped. The
seat and the backrest are also non-design zones, denoted
respectively by χ1 and χ2.
The vertical nominal force applied on the seat surface is
equal to f10 = −0.025e3, and the horizontal nominal force
applied on the backrest surface is equal to f20 =−0.003e1.
We introduce the perturbations fˆ11 = m1ξ1e1 on the seat
surface, and fˆ21 = m2ξ2e3 on the backrest surface, where
ξ1 and ξ2 denote two independant identically distributed
Gaussian random variables with distributionN (µ,s).
We can write the applied surface forces as :
f = ( f10+ fˆ11)χ1+( f20+ fˆ21)χ2,
In the example below, the numerical values for the mean
and the variance are set to µ = 1. and s= 0.01.
Figure 1: Geometry and boundary conditions for the 3-d
chair test case.
The optimized shapes obtained at iteration 120
for (m1,m2) ∈ {(0,0),(0.005,0.0006),(0.008,0.001)} are
displayed on Figure 2. This result clearly shows that the
optimized solution changes significantly when uncertainties
are taken into account. The convergence histories of
compliance and volume for the three cases are shown on
Figure 3.
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EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
Figure 2: Optimal shape at iteration 120 for (m1 = 0,m2 =
0) (top), (m1 = 0.005,m2 = 0.0006) (middle) and (m1 =
0.008,m2 = 0.001) (bottom).
Figure 3: Convergence histories for compliance (upper)
and volume (lower) for (m1 = 0,m2 = 0) (top), (m1 =
0.005,m2 = 0.0006) (middle) and (m1 = 0.008,m2 =
0.001) (bottom).
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EUROGEN 2019 September 12-14, 2019, Guimarães, Portugal
References
[1] Allaire, G. and Dapogny, C. A deterministic
approximation method in shape optimisation under
random uncertainties. SMAI Journal of Computational
Mathematics 1, 83–143 (2015).
[2] Pons-Prats, J., Castelltort, G. B., Ibáñez de Navarra, E.
O. n., Zárate, F., and Gómez, J. E. H. Robust shape
optimisation in aeronautics. 8th World Congress on
Structural and Multidisciplinary Optimization (2009).
[3] Allaire, G. and Dapogny, C. A linearized approach to
worst case design in parametric and geometric shape
optimisation. Mathematical Models and Methods in
Applied Sciences 24(11), 2199–2257 (2014).
[4] Henrot, A. and Pierre, M. Variation et optimisation de
formes: une analyse géométrique, volume 48. Springer
Science & Business Media, (2006).
[5] Allaire, G. Conception optimale de structures,
volume 58. Springer Berlin Heidelberg New York,
(2007).
[6] Osher, S. J. and Sethian, J. A. Front propagating
with curvature-dependent speed : Algorithms
based on Hamilton-Jacobi formulation. Journal
of Computational Physics 79, 12–49 (1988).
[7] Allaire, G., Jouve, F., and Toader, A.-M. Structural
optimization using shape sensitivity analysis and a level
set method. Journal of Computational Physics 194(1),
363–393 (2004).
6


A deterministic approach for shape and topology optimization under uncertain loads

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A deterministic approach for shape and topology optimization under uncertain loads

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