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OPTIMAL DESIGN OF STRUCTURES
G. ALLAIRE
July 6-17th, 2015
Department of Applied Mathematics, Ecole Polytechnique
LECTURE I
AN INTRODUCTION
TO OPTIMAL DESIGN
CIMPA Summer School on Current Research in Finite Element
Methods, IIT Mumbai
G. Allaire, Ecole Polytechnique
Optimal design of structures
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A FEW DEFINITIONS
A problem of optimal design (or shape optimization) for structures is defined
by three ingredients:
☞ a model (typically a partial differential equation) to evaluate (or analyse)
the mechanical behavior of a structure,
☞ an objective function which has to be minimized or maximized, or
sometimes several objectives (also called cost functions or criteria),
☞ a set of admissible designs which precisely defines the optimization
variables, including possible constraints.
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Optimal design of structures
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Optimal design problems can roughly be classified in three categories from the
“easiest” ones to the “most difficult” ones:
☞ parametric or sizing optimization for which designs are parametrized
by a few variables (for example, thickness or member sizes), implying that
the set of admissible designs is considerably simplified,
☞ geometric or shape optimization for which all designs are obtained
from an initial guess by moving its boundary (without changing its
topology, i.e., its number of holes in 2-d),
☞ topology optimization where both the shape and the topology of the
admissible designs can vary without any explicit or implicit restrictions.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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✞
☎
Definition of topology ✆
✝
Two shapes share the same topology if there exists a continuous deformation
from one to the other.
In dimension 2 topology is characterized by the number of holes or of
connected components of the boundary.
In dimension 3 it is quite more complicated ! Not only the hole’s number
matters but also the number and intricacy of “handles” or “loops”.
(a ball 6= a ball with a hole inside 6= a torus 6= a bretzel)
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Optimal design of structures
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Example of sizing or parametric optimization
Thickness optimization of a membrane
h
Ω
➫ Ω = mean surface of a (plane) membrane
➫ h = thickness in the normal direction to the mean surface
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Optimal design of structures
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The membrane deformation is modeled by its vertical displacement
u(x) : Ω → R, solution of the following partial differential equation (p.d.e.),
the so-called membrane model,

 − div (h∇u) = f in Ω
 u=0
on ∂Ω,
with the thickness h, bounded by minimum and maximum values
0 < hmin ≤ h(x) ≤ hmax < +∞.
The thickness h is the optimization variable.
It is a sizing or parametric optimal design problem because the
computational domain Ω does not change.
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Optimal design of structures
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The set of admissible thicknesses is
Z
Uad = h(x) : Ω → R s. t. 0 < hmin ≤ h(x) ≤ hmax and
h(x) dx = h0 |Ω| ,
Ω
where h0 is an imposed average thickness.
Possible additional “feasibility” constraints: according to the
production process of membranes, the thickness h(x) can be discontinuous, or
on the contrary continuous ; its derivative h′ (x) can be uniformly bounded
(molding-type constraint) or even its second-order derivative h′′ (x), linked to
the curvature radius (milling-type constraint).
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Optimal design of structures
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The optimization criterion is linked to some mechanical property of the
membrane, evaluated through its displacement u, solution of the p.d.e.,
Z
J(h) =
j(u) dx,
Ω
where, of course, u depends on h. For example, the global rigidity of a
structure is often measured by its compliance, or work done by the load: the
smaller the work, the larger the rigidity (be careful ! compliance = - rigidity).
In such a case,
j(u) = f u.
Another example amounts to achieve (at least approximately) a target
displacement u0 (x), which means
j(u) = |u − u0 |2 .
Those two criteria are the typical examples studied in this course.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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☎
✞
Other examples of objective functions
✝
✆
☞ Introducing the stress vector σ(x) = h(x)∇u(x), we can minimize the
maximum stress norm
J(h) = sup |σ(x)|
x∈Ω
or more generally, for any p ≥ 1,
J(h) =
Z
Ω
1/p
.
|σ|p dx
☞ For a vibrating structure, introducing the first eigenfrequency ω, defined
by

 − div (h∇u) = ω 2 u in Ω
 u=0
on ∂Ω,
we consider J(h) = −ω to maximize it.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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✞
☎
Other examples of criteria (ctd.)
✝
✆
☞ Multiple loads optimization: for n given loads (fi )1≤i≤n the independent
displacements ui are solutions of

 − div (h∇u ) = f
in Ω
i
i
 ui = 0
on ∂Ω,
and we introduce an aggregated criteria
Z
n
X
ci
j(ui ) dx,
J(h) =
Ω
i=1
with given coefficients ci , or
J(h) = max
1≤i≤n
G. Allaire, Ecole Polytechnique
Z
Ω
j(ui ) dx .
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Example of geometric optimization
Optimization of a membrane’s shape
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G. Allaire, Ecole Polytechnique
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A reference domain for the membrane is denoted by Ω, with a boundary made
of three disjoint parts
∂Ω = Γ ∪ ΓN ∪ ΓD ,
where Γ is the variable part, ΓD is the Dirichlet (clamped) part and ΓN is the
Neumann part (loaded by g).
The vertical displacement u is the solution of the membrane model


−∆u = 0 in Ω




 u=0
on ΓD
∂u

on ΓN


∂n = g


 ∂u = 0
on Γ
∂n
From now on the membrane thickness is fixed, equal to 1.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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The set of admissible shapes is thus
Z
[
Uad = Ω ⊂ RN such that ΓD
dx = V0 ,
ΓN ⊂ ∂Ω and
Ω
where V0 is a given volume. The geometric shape optimization problem reads
inf J(Ω),
Ω∈Uad
with, as a criteria, the compliance
J(Ω) =
Z
gu dx,
ΓN
or a least square functional to achieve a target displacement u0 (x)
Z
|u − u0 |2 dx.
J(Ω) =
Ω
The true optimization variable is the free boundary Γ.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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Example of topology optimization
D
ΓN
ΓD
Ω
Γ
Not only the shape boundaries Γ are allowed to move but new connected
components (holes in 2-d) of Γ can appear or disappear. Topology is now
optimized too.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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G. Allaire, Ecole Polytechnique
Optimal design of structures
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✞
Shape optimization in the elasticity setting
☎
✝
✆
The model of linearized elasticity gives the displacement vector field
u(x) : Ω → RN as the solution of the system of equations


− div (A e(u)) = 0 in Ω




 u=0
on ΓD

A e(u) n = g
on ΓN




 A e(u)n = 0
on Γ
t
with e(u) = ∇u + (∇u) /2, and Aξ = 2µξ + λ(trξ) Id, where µ and λ are the
Lamé coefficients.
The domain boundary is again divided in three disjoint parts
∂Ω = Γ ∪ ΓN ∪ ΓD ,
where Γ is the free boundary, the true optimization variable.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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The set of admissible shapes is again
Z
[
Uad = Ω ⊂ RN such that ΓD
dx = V0 ,
ΓN ⊂ ∂Ω and
Ω
where V0 is a given imposed volume. The criteria is either the compliance
Z
g · u dx,
J(Ω) =
ΓN
or a least-square criteria for the target displacement u0 (x)
Z
J(Ω) =
|u − u0 |2 dx.
Ω
As before, the shape optimization problem reads
inf J(Ω).
Ω∈Uad
Three possible approaches: parametric, geometric, topology.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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✞
☎
Applications
✝
✆
See the web site http://www.cmap.polytechnique.fr/~optopo (and links
therein).
Civil engineering
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Mechanical engineering
Optimal design of structures
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Industrial examples at Airbus, Renault, Safran...
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Optimal design of structures
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G. Allaire, Ecole Polytechnique
Optimal design of structures
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An industrial example at Renault
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G. Allaire, Ecole Polytechnique
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✄
✂Commercial softwares ✁
Optistruct, Ansys DesignSpace, Genesis, MSC-Nastran, Tosca, devDept...
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Optimal design of structures
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✞
☎
Other fields related to shape optimization
✝
✆
The technical tools in this course are also useful for the following areas:
☞ Optimal control.
☞ Inverse problems.
☞ Sensitivity analysis of parameters.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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A REVIEW OF OPTIMIZATION
Let V be a Banach space, i.e., a normed vector space which is complete (any
Cauchy sequence is converging in V ).
Let K ⊂ V be a non-empty subset. Let J : V → R. We consider
inf
v∈K⊂V
J(v).
Definition. An element u is called a local minimizer of J on K if
u∈K
and ∃δ > 0 , ∀v ∈ K , kv − uk < δ =⇒ J(v) ≥ J(u) .
An element u is called a global minimizer of J on K if
u∈K
and
J(v) ≥ J(u)
∀v ∈ K .
(difference: theory ↔ global / numerics ↔ local)
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Optimal design of structures
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Optimality conditions
Theorem (Euler inequality). Let u ∈ K with K convex. We assume that
J is differentiable at u. If u is a local minimizer of J in K, then
hJ ′ (u), v − ui ≥ 0
∀v ∈ K .
If u ∈ K satisfies this inequality and if J is convex, then u is a global
minimizer of J in K.
Remarks.
☞ If u belongs to the interior of K, we deduce the Euler equation J ′ (u) = 0.
☞ The Euler inequality is usually just a necessary condition. It becomes
necessary and sufficient for convex functions.
G. Allaire, Ecole Polytechnique
Optimal design of structures
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Minimization with equality constraints
inf
v∈V, F (v)=0
J(v)
with F (v) = (F1 (v), ..., FM (v)) differentiable from V into RM .
Definition. We call Lagrangian of this problem the function
L(v, µ) = J(v) +
M
X
µi Fi (v) = J(v) + µ · F (v)
∀(v, µ) ∈ V × RM .
i=1
The new variable µ ∈ RM is called Lagrange multiplier for the constraint
F (v) = 0.
Lemma. The constrained minimization problem is equivalent to
inf
v∈V, F (v)=0
G. Allaire, Ecole Polytechnique
J(v) = inf sup L(v, µ).
v∈V µ∈RM
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☎
✞
Stationarity of the Lagrangian
✝
✆
Theorem. Assume that J and F are continuously differentiable in a
neighborhood of u ∈ V such that F (u) = 0. If u is a local minimizer and if the
′
vectors Fi (u) 1≤i≤M are linearly independent, then there exist Lagrange
multipliers λ1 , . . . , λM ∈ R such that
∂L
(u, λ) = J ′ (u) + λ · F ′ (u) = 0
∂v
G. Allaire, Ecole Polytechnique
and
∂L
(u, λ) = F (u) = 0 .
∂µ
Optimal design of structures
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Minimization with inequality constraints
inf
v∈V, F (v)≤0
J(v)
where F (v) ≤ 0 means that Fi (v) ≤ 0 for 1 ≤ i ≤ M , with F1 , . . . , FM
differentiable from V into R.
Definition. Let u be such that F (u) ≤ 0. The set
I(u) = {i ∈ {1, . . . , M } , Fi (u) = 0}
is called the set of active constraints at u. The inequality constraints are said
to be qualified at u ∈ K if the vectors (Fi′ (u))i∈I(u) are linearly independent.
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Optimal design of structures
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Definition. We call Lagrangian of the previous problem the function
L(v, µ) = J(v) +
M
X
µi Fi (v) = J(v) + µ · F (v)
∀(v, µ) ∈ V × (R+ )M .
i=1
The new non-negative variable µ ∈ (R+ )M is called Lagrange multiplier for
the constraint F (v) ≤ 0.
Lemma. The constrained minimization problem is equivalent to
inf
v∈V, F (v)≤0
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J(v) = inf
sup
v∈V µ∈(R+ )M
L(v, µ).
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✞
☎
Stationarity of the Lagrangian
✝
✆
Theorem. We assume that the constraints are qualified at u satisfying
F (u) ≤ 0. If u is a local minimizer, there exist Lagrange multipliers
λ1 , . . . , λM ≥ 0 such that
J ′ (u) +
M
X
λi Fi′ (u) = 0 ,
λi ≥ 0 ,
λi = 0 if Fi (u) < 0
∀ i ∈ {1, . . . , M } .
i=1
This condition is indeed the stationarity of the Lagrangian since
∂L
(u, λ) = J ′ (u) + λ · F ′ (u) = 0,
∂v
and the condition λ ≥ 0, F (u) ≤ 0, λ · F (u) = 0 is equivalent to the Euler
inequality for the maximization with respect to µ in the closed convex set
(R+ )M
∂L
(u, λ) · (µ − λ) = F (u) · (µ − λ) ≤ 0 ∀µ ∈ (R+ )M .
∂µ
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Optimal design of structures
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☎
✞
Interpreting the Lagrange multipliers
✝
✆
Define the Lagrangian for the minimization of J(v) under the constraint
F (v) = c
L(v, µ, c) = J(v) + µ · (F (v) − c)
We study the sensitivity of the minimal value with respect to variations of c.
Let u(c) and λ(c) be the minimizer and the optimal Lagrange multiplier. We
assume that they are differentiable with respect to c. Then
∇c J(u(c)) = −λ(c).
λ gives the derivative of the minimal value with respect to c without any
further calculation ! Indeed
∂L
(u(c), λ(c), c) = −λ(c)
∇c J(u(c)) = ∇c L(u(c), λ(c), c) =
∂c
because
∂L
∂L
(u(c), λ(c), c) = 0 ,
(u(c), λ(c), c) = 0 .
∂v
∂µ
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Optimal design of structures
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Numerical algorithms for minimization problems
A simplified classification:
☞ Stochastic algorithms: global minimization. Examples: Monte-Carlo,
simulated annealing, genetic. See the last chapter and the last course.
Inconvenient: high CPU cost.
☞ Deterministic algorithms: local minimization. Examples: gradient
methods, Newton.
Inconvenient: they require the gradient of the objective function.
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Optimal design of structures
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✞
☎
Gradient descent with an optimal step
✝
✆
The goal is to solve
inf J(v) .
v∈V
Initialization: choose u0 ∈ V . Iterations: for n ≥ 0
un+1 = un − µn J ′ (un ) ,
where µn ∈ R is chosen at each iteration such that
J(u
n+1
n
′
n
) = inf+ J u − µ J (u ) .
µ∈R
Main idea: if un+1 = un − µwn with a small µ > 0, then
J(un+1 ) = J(un ) − µhJ ′ (un ), wn i + O(µ2 ),
thus, to decrease J, the best ”first order” choice is wn proportional to J ′ (un ).
G. Allaire, Ecole Polytechnique
Optimal design of structures
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✞
☎
Projected gradient
✝
✆
Let K be a non-empty closed convex subset of V . The goal is to solve
inf J(v) .
v∈K
Initialization: choose u0 ∈ K. Iterations: for n ≥ 0
u
n+1
n
′
n
= PK u − µ J (u ) ,
where PK is the projection on K.
Remark. Another possibility is to penalize the constraints, i.e., for small
ǫ > 0 we replace
!
M
1X
inf
J(v) by inf J(v) +
[max (Fi (v), 0)]2 .
v∈V
ǫ i=1
v∈V, F (v)≤0
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Optimal design of structures
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✞
☎
Examples of projection operators PK
✝
✆
M
☞ If V = R
and K =
PK (x) = y
M
☞ If V = R
QM
with
M
[a
,
b
],
then
for
x
=
(x
,
x
,
.
.
.
,
x
)
∈
R
i
i
1
2
M
i=1
yi = min (max (ai , xi ), bi )
M
and K = {x ∈ R
PK (x) = y
with
PM
i=1
yi = xi − λ
pour 1 ≤ i ≤ M .
xi = c0 }, then
and λ =
1
M
−c0 +
M
X
xi
i=1
☞ Same if V = L2 (Ω) and K = {φ ∈ V | a(x) ≤ φ(x) ≤ b(x)} or
R
K = {φ ∈ V | Ω φ dx = c0 }.
!
.
For more general closed convex sets K, PK can be very hard to determine. In
such cases one rather uses the Uzawa algorithm which looks for a saddle point
of the Lagrangian.
G. Allaire, Ecole Polytechnique
Optimal design of structures