Casting constraints in structural optimization via a level-set...

Casting constraints in structural optimizationvia a level-set method

Michailidis Georgios1,2

Allaire Gregoire1

Jouve Francois3

1 CMAP, UMR 7641 Ecole Polytechnique, Palaiseau, France2 Renault, Direction de la recherche et des etudes avancees, 78280 Guyancourt, France

3 University Paris Diderot, Laboratoire J.L. Lions (UMR 7598), Paris, France

19− 24 May 2013

Allaire,Jouve,Michailidis WCSMO-10 19 − 24 May 2013 1 / 26

Table of contents

1 Shape Optimization

2 Level-Set Method

3 Casting Process

4 Molding Constraint

5 Numerical Results


Table of contents


2 Level-Set Method

3 Casting Process


5 Numerical Results


Shape Optimization

A problem of Shape Optimization in structural mechanics is mainly comprised of 3 parts:

A model (typically a PDE) that gives a mathematical description of the mechanical problem.This equation is usually called the state equation.

An objective function J to be optimized. It can be of mechanical or geometric nature (e.g.work done by the load, volume,etc).

A set of admissible shapes Uad which defines and usually constrains the set of shapesamong which we look for the optimal solution.

The final shape optimization problem reads:

minΩ∈Uad

J(Ω, u(Ω))

Main differences between different shape optimization methods:

The way the advection velocity of the boundary is computed.

The way the boundary is described.


Shape Optimization-Choice of the advection velocity

Shape sensitivity method

The variations of the shape are performed by applying a vector field θ ∈W 1,∞(Rd,Rd) on thereference smooth domain Ω such that Ωθ =

(Id+ θ

)(Ω). The shape derivative of J(Ω) at Ω can

be defined as the Frechet derivative in W 1,∞(Rd,Rd) at 0 of the applicationθ → J

((Id+ θ)(Ω)

), i.e.

J((Id+ θ)(Ω)

)= J(Ω) + J ′(Ω)(θ) + o(θ) with lim

θ→0

|o(θ)|‖θ‖

= 0 ,

where J ′(Ω) is a continuous linear form on W 1,∞(Rd,Rd).

Simplest choice

According to Hadamard’s structure theorem, the shapederivative can always be writen in the form:

J ′(Ω)(θ) =

∫Γj(s)θ(s) · n(s)ds.

Choosing θ = −tj(s)n(s), for t > 0 sufficiently small, yields:

J((Id+ θ)(Ω)

)= J(Ω)− t

∫Γj(s)2ds+ o(t) < J(Ω)

Figure: Shape variation.


Table of contents


2 Level-Set Method

3 Casting Process


5 Numerical Results


Level-Set Method

Level-set representation

We choose all admissible shapes Ω to be subsetsof a bounded working domain D ⊂ Rd. Then,the boundary of Ω is defined by means of a levelset function ψ such that

ψ(x) = 0 ⇔ x ∈ ∂Ω ∩D,ψ(x) < 0 ⇔ x ∈ Ω,

ψ(x) > 0 ⇔ x ∈(D \ Ω

).

Level-set advection

Suppose that, for every time t, the domainΩ(t) ⊂ Rd is represented by an implicit functionψ(t, .) on Rd, and is subject to an evolutiondefined by velocity v(t, x) ∈ Rd. Then

∀t, ∀x ∈ Rd,∂ψ

∂t(t, x) + v(t, x) · ∇ψ(t, x) = 0

Figure: Level-set representation (Wikipedia).


Table of contents


2 Level-Set Method

3 Casting Process


5 Numerical Results


Casting Process

A simplified sequence of steps for the construction of a cast part is the following:

1 Molds are used in order to create a cavity with the shape of the structure that we intend toconstruct.

2 The cavity is filled with molten liquid.

3 The liquid solidifies.

4 The molds are removed and the cast part is revealed.


Table of contents


2 Level-Set Method

3 Casting Process


5 Numerical Results


Molding Constraint

The molding constraint refers to the ability to remove the mold after solidification of the castpart.

Figure: Left: Castable shape; Right: Uncastable shape.


Molding Constraint

Molding condition on the advection velocity

The border of the castable structure ∂Ω can be divided into n disjoint parts Γi such thatΓi ∩ Γj = ,∪ni=1Γi = ∂Ω and Γi can be parted in the direction di.

Moldability condition : di · n(x) ≥ 0, ∀x ∈ Γi.

Proposal (Xia et al.):1 Start with a castable initial shape.2 Consider an advection velocity of the form θi(x) = λ(x)di, ∀x ∈ Γi.

Figure: Left: Castable shape; Right: Uncastable shape (Xia et al.).

(-): The major drawback of this method is that the shape can shrink but not expand in thedirection normal to di. Thus, a constraint on the minimum thickness cannot be treated ingeneral.


Molding Constraint

Generalized molding constraint

Signed-distance function

Let Ω ⊂ Rd a bounded domain. The signed distance functionto Ω is the function Rd 3 x 7→ dΩ(x) defined by :

dΩ(x) =

−d(x, ∂Ω) if x ∈ Ω0 if x ∈ ∂Ω

d(x, ∂Ω) if x ∈ cΩ

,where d(·, ∂Ω) is the usual Euclidean distance.

Constraint formulation

Predefined parting surface:dΩ (x+ ξdi) ≥ 0 ∀x ∈ Γi, ∀ξ ∈ [0, diam(D)]

Non-predefined parting surface:dΩ (x+ ξsign(n · d)d) ≥ 0 ∀x ∈ ∂Ω, ∀ξ ∈ [0, diam(D)]

where diam(D) = supx,y dist(x, y), x, y ∈ D is thediameter of the fixed domain D.

Figure: dΩ (Wikipedia).


Minimum Thickness Constraint

Offset sets

The general idea is based on the creation of offset sets of the boundary of the shape. Creating asequence of offset sets and choosing accordingly their direction, we can check the thickness of thestructure by checking the value of the signed-distance function.

Constraint formulation

We denote dmin the minimum allowed thickness anddoff the distance between the boundary and itsoffset set in the direction of the normal n(x).

Minimum thickness:

dΩ

(x− doffn (x)

)≤ 0 ∀x ∈ ∂Ω, ∀doff ∈ [0, dmin] .

Figure: Offset sets.


Molding Constraint/Shape derivative

Molding condition on the advection velocity

Starting from the general form of the shape derivative for a functional J(Ω)

J ′(Ω)(θ) =

∫∂Ω

θ(x) · n(x)V (x)dx =n∑i=1

∫Γi

θi(x) · n(x)Vi(x)dx

and considering admissible advection fields of the type θi(x) = λ(x)di, ∀x ∈ Γi, we get

J ′(Ω)(θ) =n∑i=1

∫Γi

λi(x)di(x) · n(x)Vi(x)dx,

and choosingλi(x) = −Vi(x)di(x) · n(x)

for each part Γi of the boundary ∂Ω, the shape derivative becomes

J ′(Ω)(θ) = −n∑i=1

∫Γi

(di(x) · n(x))2(Vi(x))2dx ≤ 0,

which shows that the chosen advection velocity is indeed a descent direction.




Pointwise constraint: dΩ (x+ ξdi) ≥ 0 ∀x ∈ Γi, ∀ξ ∈ [0, diam(D)].

We reformulate the pointwise constraint into a global one denoted P (Ω):

P1 (Ω) =n∑i=1

∫Γi

∫ diam(D)

0

[(dΩ (x+ ξdi))

−]2dξdx = 0,

where we have denoted: (f)− = min (f, 0).

General form:

P (Ω) =

∫∂Ω

∫ ξf

0

[(d (xm))±

]2dξdx = 0.

We use an augmented Lagrangian method to impose the constraints.Equality constraints: ci(Ω) = 0 (i = 1, ..., n).

L(Ω, `, µ) = J(Ω)−n∑i=1

ìci(Ω) +µi

2

n∑i=1

c2i (Ω),

where ` = (ì)i=1,...,n and µ = (µi)i=1,...,n are lagrange multipliers and penalty parametersfor the constraints.Update of ì: `n+1

i = `ni − µici(Ωn).The penalty parameters are augmented every 5 iterations.




After some calculus and using the fact that only the normal component of the vector field θ (x) isof interest, i.e. w (x) = θ (x) · n (x), the shape derivative of the penalty functional reads:

P ′(Ω)(θ) =∫∂Ω

∫ ξf0 w (x) [H

((dΩ (xm))±

)2+ 2

((dΩ (xm))±

)∇dΩ (xm)

∇dΩ(x)‖∇dΩ(x)‖ ]dξdx

−∫∂Ω

∫ ξf0 w

(xm|Ω

) [2(

(dΩ (xm))±)]dξdx

, where we use the notation xm = x− ξn(x) for the offset point and xm|Ω for its projection tothe boundary.

Figure: Offset point and projection onto the boundary.Allaire,Jouve,Michailidis WCSMO-10 19 − 24 May 2013 17 / 26

Table of contents


2 Level-Set Method

3 Casting Process


5 Numerical Results


Numerical Results

Mesh: 40× 40× 20 Q1 elements.

P1 : generalized casting penalty functional.

P2(dmin) : minimum thickness penalty functional.


Numerical Results

min J(Ω) =∫ΓN

g · u dx∫Ω dx = 0.2|D|


Numerical Results

Molding condition on advection velocity

Figure: Parting direction: d = (0, 0, 1); parting surface: z = 0.

min J(Ω) =∫ΓN

g · u dx∫Ω dx = 0.2|D|θ(x) = λ(x)d

Generalized casting and minimum thickness constraint

Figure: Parting direction: d = (0, 0, 1); parting surface: z = 0.

min J(Ω) =∫ΓN

g · u dx∫Ω dx = 0.2|D|P1 = 0P2(dmin = 0.4) = 0


Numerical Results


Numerical Results

Molding condition on advection velocity

Figure: Parting direction: ±d = ±(0, 0, 1); parting surface: ?.

min J(Ω) =∫ΓN

g · u dx∫Ω dx = 0.2|D|θ(x) = λ(x)d

Generalized casting and minimum thickness constraint

Figure: Parting direction: ±d = ±(0, 0, 1); parting surface: ?.

min J(Ω) =∫ΓN

g · u dx∫Ω dx = 0.2|D|P1 = 0P2(dmin = 0.3) = 0


Numerical Results


Numerical Results

Table: Compliance of the optimized structures.

ComplianceWithout molding constraint 90.14

With molding constraint 102.07and no parting surface

With molding constraint,no parting surface

and minimum thickness constraint 105.87With molding constraint 123.68

and parting surfaceWith molding, parting surface 134.68

and minimum thickness constraint


Merci!

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