Geometric constraints for shape and topology optimization in - - PowerPoint PPT Presentation

Geometric constraints for shape and topology

• ptimization in architectural

design

Charles Dapogny1, Alexis Faure2, Georgios Michailidis2, Grégoire Allaire3, Agnes Couvelas4, Rafael Estevez2

1 CNRS & Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, Grenoble, France 2 SiMaP, Université Grenoble-Alpes, Grenoble, France 3 CMAP, UMR 7641 École Polytechnique, Palaiseau, France 4 SHAPE IKE, Kritis 13, Nea Ionia, 14231 Attiki, Greece

27th January, 2017

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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A wee bit of history (I)

• Hooke’s theorem (1675):

“As hangs the flexible chain, so but inverted will stand the rigid arch.”

• A. Gaudi designed the church of Colònia Güell (1889-1914) by using a 3d

funicular model to determine a stable arrangement of columns and vaults.

(Left) Experimental device of Gaudi, (right) tentative outline of Colònia Güell (Photo

courtesy of http://www.gaudidesigner.com). 3 / 66

A wee bit of history (II)

• Such techniques have been employed and improved by world-renowned

architects: Heinz Isler, Gustave Eiffel, Frei Otto, etc.

• Recent shape and topology optimization (S&T) techniques have been

used in the device of large-scale structures.

(Left) Front facade of the Qatar National Convention Center in Doha [Sasaki et al]. (right) Design of a 288m tall high-rise in Australia by Skidmore, Owings & Merrill.

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Shape and topology optimization for architecture

Applying shape and topology optimization techniques in the field of architectural design is tentative:

• They allow to model and optimize complex criteria of the design, related

to aesthetics, manufacturability, or mechanical performance.

• Optimal designs from the mechanical viewpoint often show ‘attractive’
• utlines, and a strongly organic nature which is praised by architects.

(Left) Soap bubble foam structure devised by Frei Otto, (right) interior view of the Manheim Garden festival (Excerpted from [La]).

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Potential limitations

• Architects generally apply construction rules based on intuition to deal

with the stringent requirements of stability, robsutness, etc. Except when aesthetics is the priority, or in the construction of exceptional structures (skyscrapers), challenging these rules is lengthy and costly, thus inefficient.

• Results of continuum-based S&T optimization techniques are difficult to

use directly, since real-life structures are often assembled from bars or beams. ⇒ A costly interpretation step of the optimized designs is necessary.

• Architectural design has a lot to do with the personal taste of the
• architect. On the contrary, most S&T optimization methods only

consider mechanical aspects: their results can be reproduced by anyone, and do not leave room for original creation!

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Objectives of the present work

• Propose a simplified theoretical and numerical S&T framework which is

mechanically relevant for conceptual architectural design.

• Introduce constraint functionals for S&T optimization problems which

allow the user to:

• Encode information about its personal taste;
• Ease the interpretation of the optimized designs as truss-like structures;
• Deal with other geometric problems plaguing the optimized structures

(visibility, elongated bars, etc.).

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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Shape optimization of linear elastic shapes (I)

A shape is a bounded domain Ω ⊂ Rd, which is

• fixed on a part ΓD of its boundary,
• submitted to surface loads g, applied on

ΓN ⊂ ∂Ω, ΓD ∩ ΓN = ∅.

• submitted to body forces f .

The displacement vector field uΩ : Ω → Rd is governed by the linear elasticity system:        −div(Ae(uΩ)) = f in Ω uΩ =

• n ΓD

Ae(uΩ)n = g

• n ΓN

Ae(uΩ)n =

• n Γ

, where e(u) = 1

2(∇uT +∇u) is the strain tensor,

and A is the Hooke’s law of the material: ∀e ∈ Sd(R), Ae = 2µe + λtr(e)I.

ΓD ΓN

• g

A ‘Cantilever’ The deformed cantilever

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Shape optimization of linear elastic shapes (II)

Goal: Optimize the compliance C(Ω) of shapes C(Ω) =

• Ω

Ae(uΩ) : e(uΩ) dx =

• Ω

f · uΩ dx +

• ΓN

g · uΩ ds, under constraints, modelled by a shape functional P(Ω), e.g. P(Ω) = Vol(Ω), the volume of shapes. Depending on the particular situation, this setting gives rise to two different kinds of optimization problems:

• Unconstrained optimization problems

min

Ω L(Ω), where L(Ω) = C(Ω) + ℓP(Ω),

where the objective criterion C(Ω) is penalized by the constraint P(Ω).

• Constrained optimization problems of the form:

min

Ω s.t. P(Ω)≤α C(Ω).

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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Differentiation with respect to the domain: Hadamard’s method

Hadamard’s boundary variation method describes variations of a reference, Lipschitz domain Ω of the form: Ω → Ωθ := (I + θ)(Ω), for ‘small’ θ ∈ W 1,∞ Rd, Rd.

Ω Ωθ θ

Definition 1.

Given a smooth domain Ω, a function F(Ω) of the domain is shape differentiable at Ω if the function W 1,∞ Rd, Rd ∋ θ → F(Ωθ) is Fréchet-differentiable at 0, i.e. the following expansion holds around 0: F(Ωθ) = F(Ω) + F ′(Ω)(θ) + o ||θ||W 1,∞(Rd,Rd) .

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Differentiation with respect to the domain: Hadamard’s method

Techniques from optimal control theory make it possible to calculate shape gradients; in the case of ‘many’ functionals of the domain J(Ω), the shape derivative has the particular structure: J′(Ω)(θ) =

• Γ

vΩ θ · n ds, where vΩ is a scalar field depending on uΩ, and possibly on an adjoint state pΩ. Example: If J(Ω) = C(Ω) =

• Ω f · uΩ dx +
• ΓN g · uΩ ds is the compliance,

vΩ = −Ae(uΩ) : e(uΩ) is the (negative) elastic energy density.

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The generic algorithm

This shape gradient provides a natural descent direction for J(Ω): for instance, defining θ as θ = −vΩn yields, for t > 0 sufficiently small (to be found numerically): J((I + tθ)(Ω)) = J(Ω) − t

• Γ

v 2

Ωds + o(t) < J(Ω)

Gradient algorithm: For n = 0, ... until convergence,

• 1. Compute the solution uΩn (and pΩn) of the elasticity system on Ωn.
• 2. Compute the shape gradient J′(Ωn) thanks to the previous formula, and

infer a descent direction θn for the cost functional.

• 3. Advect the shape Ωn according to θn, so as to get Ωn+1 := (I + θn)(Ωn).

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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The Level Set Method

A paradigm: [OSe] the motion of an evolving domain is best described in an implicit way. A domain Ω ⊂ Rd is equivalently defined by a function φ : Rd → R such that: φ(x) < 0 if x ∈ Ω ; φ(x) = 0 if x ∈ ∂Ω ; φ(x) > 0 if x ∈ cΩ

(Left) A domain Ω ⊂ R2; (right) graph of an associated level set function.

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The Level Set Method

The motion of an evolving domain Ω(t) ⊂ Rd along a velocity field v(t, x) ∈ Rd translates in terms of an associated ‘level set function’ φ(t, .) into the Level Set advection equation: ∀t, ∀x ∈ Rd, ∂φ ∂t (t, x) + v(t, x).∇φ(t, x) = 0 In many applications, the velocity v(t, x) is normal to the boundary ∂Ω(t): v(t, x) := V (t, x) ∇φ(t, x) |∇φ(t, x)|. Then the evolution equation rewrites as a Hamilton-Jacobi equation: ∀t, ∀x ∈ Rd, ∂φ ∂t (t, x) + V (t, x)|∇φ(t, x)| = 0

Ω(t) = [φ(t, .) < 0]

Ω(t + dt) = [φ(t + dt, .) < 0] v(t, x) x

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The level set method for shape optimization [AlJouToa]

• The shapes Ωn are embedded in a working

domain D equipped with a fixed mesh.

• The successive shapes Ωn are accounted for

in the level set framework, i.e. via a function φn : D → R which implicitly defines them.

• The linear elasticity equations cannot be

solved on Ωn. ⇒ Ersatz material approximation: the holes D \ Ωn are filled with a ‘very soft material’ with Hooke’s law εA, ε ≪ 1.

• This approach is very versatile and does not

require a mesh of the shapes at each iteration.

Shape accounted for with a level set description

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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The signed distance function Definition 2.

The signed distance function dΩ to a bounded domain Ω ⊂ Rd is defined by: ∀x ∈ Rd,    −d(x, ∂Ω) if x ∈ Ω if x ∈ ∂Ω d(x, ∂Ω) if x ∈ cΩ , where d(x, ∂Ω) = min

y∈∂Ω |x − y| is the usual Euclidean distance function to ∂Ω.

Graph of the signed distance function to a union of two disks (in black)

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Signed distance function and geometry (I)

Definition 3.

Let Ω ⊂ Rd be a Lipschitz, bounded open set;

• Let x ∈ Rd; the set of projections Π∂Ω(x) of x onto ∂Ω is:

Π∂Ω(x) = {y ∈ ∂Ω, d(x, ∂Ω) = |x − y|} .

• When this set is a singleton, p∂Ω(x) is the projection of x onto ∂Ω.
• The skeleton Σ of ∂Ω is:

Σ :=

• x ∈ Rd, d2

Ω is not differentiable at x

.

• For x ∈ ∂Ω, the ray emerging from x is:

ray∂Ω(x) := p−1

∂Ω(x).

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Signed distance function and geometry (II)

Ω Σ

• x

p∂Ω(x) x0

• y1

y2 z

• ray∂Ω(z)

x has a unique projection over ∂Ω, whereas x′ has two such points y1, y2.

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Signed distance function and geometry (III)

Proposition 1.

Let Ω ⊂ Rd be a Lipschitz, bounded open set;

• A point x ∈ Rd has a unique projection point p∂Ω(x) iff x /

∈ Σ. In such a case, dΩ is differentiable at x, and its gradient reads: ∇dΩ(x) = x − p∂Ω(x) dΩ(x) . In particular, |∇dΩ(x)|= 1 wherever it makes sense.

• If Ω is of class C1, this last quantity equals ∇dΩ(x) = n(p∂Ω(x)).
• If Ω is of class Ck, k ≥ 2, then dΩ is also of class Ck on a neighborhood
• f ∂Ω.

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Signed distance function and geometry (IV)

Ω Σ

x

• p∂Ω(x)

rdΩ(x) n(p∂Ω(x))

Some level sets of dΩ are depicted in color; dΩ is as smooth as the boundary ∂Ω on the shaded area (at least).

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Shape differentiability of the signed distance function (I)

Lemma 2.

Let Ω ⊂ Rd be a C1 bounded domain, and x / ∈ Σ. The function θ → dΩθ(x), from W 1,∞(Rd, Rd) into R is Gâteaux-differentiable at θ = 0, with derivative: d′

Ω(θ)(x) = −θ(p∂Ω(x)) · n(p∂Ω(x)).

Ω Ωθ

• x

p∂Ω(x) ≈ p∂Ωθ(x)

dΩ(x)

• θ(p∂Ω(x)) · n(p∂Ω(x))

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Shape differentiability of the signed distance function (II)

Remark: A more general formula holds, which encompasses the case x ∈ Σ: If x ∈ Ω, d′

Ω(θ)(x) = −

inf

y∈Π∂Ω(x) θ(y) · n(y),

If x ∈ cΩ, d′

Ω(θ)(x) = −

sup

y∈Π∂Ω(x)

θ(y) · n(y).

x

• y2

dΩ(x)

Ω Σ y1 Ωθ

θ(y2) · n(y2) θ(y1) · n(y1)

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Shape differentiability of the signed distance function (II)

• Formal clue: Taking the shape derivative in

|∇dΩ(x)|2= 1 yields: ∇d′

Ω(θ)(x) · ∇dΩ(x) = 0.

⇒ The shape derivative of dΩ is constant along the rays.

• Rigorous proof: Use of the definition:

d2

Ω(x) = min y∈∂Ω |x − y|2

in combination to a theorem for differentiating a minimum value with respect to a parameter.

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Shape differentiability of the signed distance function (III)

Lemma 3.

Let Ω be a C1 bounded domain, enclosed in a large computational domain D, and j : Rx × Rs → R be of class C1; define the functional: J(Ω) =

• D

j(x, dΩ(x)) dx. Then θ → J(Ωθ) is Gâteaux-differentiable at θ = 0 with derivative: J′(Ω)(θ) = −

• D

∂j ∂s (x, dΩ(x)) θ(p∂Ω(x)) · n(p∂Ω(x)) dx. This formula is awkward insofar it is not easily put under the form: J′(Ω)(θ) =

• Γ

v θ · n ds, and does not lend itself to the inference of a ‘natural’ descent direction for J.

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A coarea formula

Proposition 4.

Let Ω ⊂ D be a bounded domain of class C2, and let ϕ ∈ L1(D). Then,

• D

ϕ(x)dx =

• ∂Ω
• ray∂Ω(y)∩D

ϕ(z)

d−1

• i=1

(1 + dΩ(z)κi(y))dz

• dy,

where z denotes a point in the ray emerging from y ∈ ∂Ω and dz is the line integration along that ray. Hint of proof: Apply the coarea formula to the mapping: p∂Ω : D \ Σ → ∂Ω to recast the integration over D ≈ D \ Σ as an integration over ∂Ω composed with an integration over the pre-images p−1

∂Ω(x) =

ray∂Ω(x), x ∈ ∂Ω.

D Σ Ω

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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The anisotropic signed distance function (I)

Let M ∈ Sd(R) be a metric tensor, i.e. a symmetric, positive definite matrix. M accounts for another way of evaluating the distance between points: ∀x, y ∈ Rd, |x − y|2

M:= Mx, y.

Definition 4.

• The anisotropic (unsigned) distance function dM(·, K) to a compact

subset K ⊂ Rd is defined by: ∀x ∈ Rd, dM(x, ∂Ω) = inf

y∈K |x − y|M.

• The anisotropic signed distance function dM

Ω to a bounded domain

Ω ⊂ Rd is defined by: ∀x ∈ Rd, dM

Ω (x) =

   −dM(x, ∂Ω) if x ∈ Ω, if x ∈ ∂Ω, dM(x, ∂Ω) if x ∈ cΩ.

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The anisotropic signed distance function (II)

All the previous concepts extend from the isotropic to the anisotropic context, in particular as regards:

• The connections between dM

Ω and the notions of projection set ΠM ∂Ω(x), or

unique projection point pM

∂Ω(x), for a point x ∈ Rd,

• The differentiability of x → dM

Ω outside the (anisotropic) skeleton ΣM,

• The results about the shape differentiability of Ω → dM

Ω (x),

... up to giving an ‘anisotropic flavour’ to the previous formulae for the isotropic case.

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Isotropic vs. anisotropic signed distance functions

The anisotropic signed distance function dM

Ω (x) primarily evaluates the

distance from x to ∂Ω in the direction of the smallest eigenvalue of M.

• pM2

∂Ω (x)

pM1

∂Ω (x)

p∂Ω(x) x

Ω

Projections p∂Ω(x), pM1

∂Ω (x) and pM2 ∂Ω (x) of a point x ∈ Ω onto ∂Ω respectively

associated to the metrics I, M1 = 0.1 1

• and M2 =

0.01 1

• .

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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Imposing shapes to resemble a specific design

• Our goal is to impose the resemblance of the optimized design with a

user-defined target design ΩT.

• We rely on a functional used in the previous work [DeBDaFreyNa].

Pm(Ω) =

• Ω

dΩT dx. Intuition: If ΩT is connected, it is the unique (local and global) minimizer of Pm(Ω).

Theorem 5.

The function Ω → Pm(Ω) is shape differentiable and its derivative reads: P′

m(Ω) =

• Γ

dΩT θ · n ds.

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Example: optimization of the design a 2d bridge (I)

We consider the optimization of a 2d bridge under a volume constraint: min

Ω

C(Ω), s.t. Vol(Ω) ≤ VT, where C(Ω) is the compliance, and the maximum volume is VT = 0.163|D|.

250 20

(Top) Setting of the 2d bridge test case, and (bottom) resulting optimized shape.

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Example: optimization of the design a 2d bridge (II)

We would like to impose that the optimized design resembles the target shape ΩT, which has unfortunately very poor mechanical performance.

Target shape in the functional Pm(Ω) for the optimization of the 2d bridge.

We now solve the problem: min

Ω

L(Ω) s.t. Vol(Ω) ≤ VT, where L(Ω) := t C(Ω) C(ΩT) + (1 − t) Pm(Ω) Pm(ΩT), and t ∈ [0, 1].

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Example: optimization of the design a 2d bridge (III)

(a) (b) (c) (d)

Optimized 2d bridges for: (a) t = 0.35, (b) t = 0.45, (c) t = 0.55, (d) t = 0.70.

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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Fitting in a user-defined pattern

• Our aim is to influence the design of a structure Ω with a pattern.
• The pattern is supplied as the 0 level set of a function g : Rd → R.
• We rely on the constraint functional

Pp(Ω) =

• ∂Ω

|nΩ − ng|2 ds, where ng(x) =

∇g(x) |∇g(x)| is the normal vector field to the isolines of g.

Theorem 6.

The functional Ω → Pp(Ω) is shape differentiable, and: P′

p(Ω)(θ) = 2

• ∂Ω

(κ − div∂Ω(ng))θ · n ds, where div∂ΩV := div(V ) − ∇Vn, n is the tangential divergence of a smooth vector field V : ∂Ω → Rd.

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Example: optimization of the design of a 2d mast

We aim to optimize the shape of a column, while influencing its outline by the pattern function g defined by: ∀x = (x1, x2) ∈ D, g(x) = x1 − L1 10 sin 2πx2 3L2

• , L1 = 1, L2 = 2.

(Left) Boundary conditions for the column example, (right) isovalues of the pattern g.

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Example: optimization of the design of a 2d mast

We now solve the problem: min

Ω

L(Ω) s.t. Vol(Ω) ≤ VT, where L(Ω) := tC(Ω) + (1 − t)Pp(Ω)

(a) (b) (c) (d) (e) (f)

Optimized columns for: (a) t = 1.0; (b) t = 0.90; (c) t = 0.60; (d) t = 0.45; (e) t = 0.35; (f) t = 0.15.

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

43 / 66

Constraining the anisotropic maximum thickness of shapes (I)

We rely on the modelling of thickness of shapes introduced in [AlJouMi].

Definition 5.

Let Ω ⊂ Rd be a bounded domain, M be metric tensor. Then,

• Ω has maximum thickness smaller than δ > 0 if:

∀x ∈ Ω, dΩ(x) ≥ −δ/2.

• Ω has anisotropic maximum thickness smaller than δ > 0 if:

∀x ∈ Ω, dM

Ω (x) ≥ −δ/2.

Goal: Devise a constraint functional for imposing a bound δ > 0 on the vaues of the anisotropic maximum thickness of Ω.

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Constraining the anisotropic maximum thickness of shapes (II)

Recall that dM

Ω ‘sees’ the distance from x to ∂Ω in the direction of the smallest

eigenvalue of M. Example: If M = 0.1 1

• , a constraint on the anisotropic maximum

thickness of shapes concerns the horizontal length of their features.

(Left) One shape Ω (in black); (middle) isolines of the isotropic distance function dΩ; (right) isolines of the anisotropic distance function dM

Ω .

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Constraining the anisotropic maximum thickness of shapes (III)

• The maximum thickness constraint reads: ∀x ∈ Ω, dM

Ω (x) ≥ −δ/2.

• This can be equivalently formulated using an integral penalty function:

P(Ω) = 0, where P(Ω) =

• Ω
• dM

Ω − δ/2

2

− dx,

where t− = max(0, −t).

• In practice, we use the relaxed version of the constraint:

PM

a (Ω) ≤ δ

2, where PM

a (Ω) =

    1

• Ω

h(dM

Ω )dx

• Ω

h(dM

Ω ) (dM Ω )2dx

   

1 2

. and h is a regularized characteristic function:

• h(t) ≈ 1

if |t|≥ δ/2, h(t) ≈ 0

• therwise.

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Application 1: Ease the interpretation of the optimized design (I)

• We consider the optimal design of a three-dimensional bridge.

1 1 6 x y z

• We solve the simple compliance minimization problem:

min

Ω

C(Ω), s.t. Vol(Ω) ≤ VT, where VT = 0.1|D|.

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Application 1: Ease the interpretation of the optimized design (I)

The optimized design contains extended regions in the (yz)-plane, which makes difficult its interpretation as a realistic truss structure.

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Application 1: Ease the interpretation of the optimized design (III)

To alleviate this problem, we solve instead min

Ω

C(Ω), s.t. Vol(Ω) ≤ VT, PM

a (Ω) ≤ δ/2,

, where M =   m 1 1   .

Optimal bridges obtained with the coefficient (left) m = 0.1, and (right) m = 0.01.

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Application 2: Enhance the visibility of shapes (I)

• We aim to optimize the design of a three-dimensional short cantilever.
• To this end, we solve the two-load compliance minimization problem:

min

Ω

C1(Ω) + C2(Ω), s.t. Vol(Ω) ≤ VT, where VT = 0.1|D|. and Ci(Ω) is the compliance of Ω with respect to the ith load case.

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Application 2: Enhance the visibility of shapes (I)

The optimized shape presents an extended surface in the (xy)-plane, which hinders the path of light. To alleviate this problem, we add one or several constraints of the form PM

a (Ω) ≤ δ/2, where M =

  m1 m2 m3  .

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Application 2: Enhance the visibility of shapes (III)

Optimized short cantilevers using different metric tensors M: (top-left) m1 = 0.01; (top-right) m1 = 0.01, m2 = 0.01; (bottom-left) one constraint with m1 = 0.01, m2 = 1.0 and a second one with m1 = 1.0, m2 = 0.01; (bottom-right) m3 = 0.01.

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Application 3: penalization of the appearance of long bars (I)

In order to optimize a 2d MBB beam, we minimize the volume Vol(Ω) of shapes, under a compliance constraint: min

Ω

Vol(Ω), s.t. C(Ω) ≤ CT, where CT = 150.

g = 0.5 1 3

(Left): Setting of the MBB beam example, and (right) resulting optimized shape.

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Application 3: penalization of the appearance of long bars (II)

• The optimized shape shows long, horizontal bars.
• In practical realizations, these are assembled by joining two (or more)

smaller bars together, constructing strong connections between them.

Example of a junction between two HEA beams.

• Besides the increased cost of such connections, the final beam may not

behave mechanically in the predicted way. ⇒ Structural engineers generally wish to avoid such features.

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Application 3: penalization of the appearance of long bars (III)

To prevent the appearance of such bars, we solve: min

Ω Vol(Ω) s.t.

• C(Ω) ≤ CT

PM

a (Ω) ≤ δ/2. ,

with a threshold value δ = 1.0, and where the anisotropy is dictated by the metric tensor: M =

• m1

1

• .

Optimized shapes in the MBB beam example with a constraint on the anisotropic maximum thickness, and parameters (left) m1 = 0.1; (right) m1 = 0.01.

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1 Foreword: shape optimization in architecture 2 The shape and topology optimization framework

Shape optimization in linear elasticity Shape derivatives Numerical resolution with the Level Set method

3 A short detour via the signed distance function

Properties of the isotropic signed distance function Extension to the anisotropic context

4 Geometric constraints for shape optimization

Imposing shapes to resemble a specific design Imposing shapes to fit in a user-defined pattern Anisotropic maximum thickness

5 A miscellaneous, large scale illustration

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Optimization of the shape of a pylon (I)

This example is inspired by the Pylon Design Competition organized by the Royal Institute of British Architects in 2014 [RIBA].

x y z ΓD

• 57 / 66

Optimization of the shape of a pylon (II)

• x

y z

  20 −15     70 −85     70 −85     70 −85     70 −85     70 −85     70 −85  

• x

y z

  −20 −15     −70 −85     −70 −85     −70 −85     −70 −85     −70 −85     −70 −85  

• x

y z

  −25 −5     −50     −50     −50     −35 −100     −35 −100     −35 −100  

• x

y z

  25 −5     −50     −50     −50     35 −100     35 −100     35 −100  

• x

y z

  −25 −5     −50     −50     −50     −100 −35     −100 −35     −100 −35  

• x

y z

  25 −5     −50     −50     −50     100 −35     100 −35     100 −35  

Definition of the six load cases considered in the pylon design example.

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Optimization of the shape of a pylon (III)

We first solve the optimization program: min

Ω S(Ω) s.t. Vol(Ω) ≤ VT := 0.1|D|,

where S(Ω) = 6

i=1 Ci(Ω) is the sum of the compliances associated to each

independent load case.

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Optimization of a pylon: shape matching (I)

We now would like the optimal design to concentrate material around the medial axis: we impose that it resemble the target shape ΩT: To this end, we now solve the problem: min

Ω

L(Ω) s.t. Vol(Ω) ≤ VT, where L(Ω) := t C(Ω) C(ΩT) + (1 − t) Pm(Ω) Pm(ΩT),

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Optimization of a pylon: shape matching (II)

(a) (b) (c) (d)

Optimized designs under the constraint that the shape resembles the design ΩT; values of the weighting coefficient (a) t = 0.2; (b) t = 0.3; (c) t = 0.4; (d) t = 0.6.

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Optimization of a pylon: maximum thickness constraint (I)

To explore different structural systems, we add two constraints of the form PM

a (Ω) ≤ δ/2,

to the initial optimization problem; here M =   m1 m2 m3   , and successively: m1 = m3 = 0.01, m2 = 1 (constraint on the thickness in the (xz)-plane), m2 = m3 = 0.01, m1 = 1 (constraint on the thickness in the (yz)-plane).

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Optimization of a pylon: maximum thickness constraint (II)

Optimized design in the minimization of S(Ω) under a volume constraint VT = 0.1|D|, and anisotropic maximum thickness constraints with the value (top) δ = 4

• min {m1, m2, m3}∆x; (bottom) δ = 3
• min {m1, m2, m3}∆x.

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Thank you !

Thank you for your attention!

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References I

[AlJouMi] G. Allaire, F. Jouve, and G. Michailidis, Thickness control in structural optimization via a level set method, (to appear in SMO), HAL preprint: hal-00985000v1, (2014). [AlJouToa] G. Allaire, F. Jouve and A.M. Toader, Structural optimization using shape sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004) pp. 363–393. [DeBDaFreyNa] M. de Buhan, C. Dapogny, P. Frey, C. Nardoni, An

• ptimization method for elastic shape matching, C. R. Acad. Sci. Paris;
• Ser. I 354 (2016) pp. 783-787.

[DaFaMi] C. Dapogny, A. Faure, G. Michailidis, G. Allaire, A. Couvelas and R. Estevez, Geometric constraints for shape and topology

• ptimization in architectural design, submitted, (2016).

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References II

[La] A. B. Larena, Shape design methods based on the optimisation of the

• structure. historical background and application to contemporary

architecture, in Proceedings of the third international congress on construction history, 2009. [OSe] S. J. Osher and J.A. Sethian, Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J.

• Comp. Phys. 78 (1988) pp. 12-49

[RIBA] P. D. C. of the Royal Institute of British Architects, http://www.ribapylondesign.com/home.

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