r´obert path´o - math.tu-dresden.de fileoutline 1 the state problem 2 shape optimization problem 3...

Shape optimization in contact problems involving friction

Robert Patho

Charles University in Prague

Workshop Dresden-PragueDresden, 1st May 2010

Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 1 / 17

Outline

1 The state problem

2 Shape optimization problem

3 Existence of an optimal shape

4 Approximation

5 Convergence analysis


Geometrical setting

Let α ∈ C 0,1([a, b]), 0 ≤ α < γ and consider the domain Ω(α):

b

C0

γ

Ω(α)

Γ (α)

Γ

Γ

P

c

u

a

ΓP

Further denote: bΩ ≡ (a, b) × (0, γ).


Classical formulation

Signorini problem

div σ(u) + F = 0 in Ω(α),

u = 0 on Γu,

( σ(u)n ≡ ) T (u) = P on ΓP ,

( u2(·, α(·)) ≡ ) u2 α ≥ −α

T2(u) α ≥ 0(u2 α + α)T2(u) α = 0

9=; in (a, b)



Signorini problem with given friction


u = 0 on Γu,

( σ(u)n ≡ ) T (u) = P on ΓP ,

( u2(·, α(·)) ≡ ) u2 α ≥ −α

T2(u) α ≥ 0(u2 α + α)T2(u) α = 0

9=; in (a, b)

u1 = 0 ⇒ |T1(u)| ≤ F g

u1 6= 0 ⇒ T1(u) = −sgn (u1)F g

ffon Γc(α)



Signorini problem with given friction and solution-dependent coefficient of friction:


u = 0 on Γu,

( σ(u)n ≡ ) T (u) = P on ΓP ,

( u2(·, α(·)) ≡ ) u2 α ≥ −α

T2(u) α ≥ 0(u2 α + α)T2(u) α = 0

9=; in (a, b)

u1 = 0 ⇒ |T1(u)| ≤ F(0)gu1 6= 0 ⇒ T1(u) = −sgn (u1)F(|u1|)g

ffon Γc(α)



Signorini problem with given friction and solution-dependent coefficient of friction:


u = 0 on Γu,

( σ(u)n ≡ ) T (u) = P on ΓP ,

( u2(·, α(·)) ≡ ) u2 α ≥ −α

T2(u) α ≥ 0(u2 α + α)T2(u) α = 0

9=; in (a, b)

u1 = 0 ⇒ |T1(u)| ≤ F(0)gu1 6= 0 ⇒ T1(u) = −sgn (u1)F(|u1|)g

ffon Γc(α)

σij(u) = cijklεkl(u) ∀i , j = 1, 2,

cijkl = cjikl = cklij ∀i , j , k, l = 1, 2,

∃Cell > 0 : cijklξijξkl ≥ Cellξijξij ∀ξij = ξji ∈ R.


Variational formulation

Notation:

V(α) = v ∈ H1(Ω(α)) | v = 0 a.e. on Γu ,

K(α) = v ∈ V(α) | v2 α ≥ −α a.e. in (a, b) ,

a(u, v) =

Z

Ω(α)

σij(u)εij(v) dx ; L(v) =

Z

Ω(α)

Fivi dx +

Z

ΓP

Pivi ds


Variational formulation

Notation:

V(α) = v ∈ H1(Ω(α)) | v = 0 a.e. on Γu ,

K(α) = v ∈ V(α) | v2 α ≥ −α a.e. in (a, b) ,

a(u, v) =

Z

Ω(α)

σij(u)εij(v) dx ; L(v) =

Z

Ω(α)

Fivi dx +

Z

ΓP

Pivi ds

Weak formulation of the state problem:

(P(α))

8<:

Find u ∈ K(α) such that:

a(u, v − u) +

Z

Γc (α)

F(|u1|)g(|v1| − |u1|) ds ≥ L(v − u) ∀v ∈ K(α).


On the solvability of the state problem

Theorem (existence)

Let F ∈ C([0,∞)), F ≥ 0, bounded and g ∈ L2(Γc(α)), g ≥ 0. Then the problem(P(α)) has at least one solution.

Theorem (uniqueness)

If in addition g ∈ L∞(Γc(α)), g ≥ 0 and F satisfies:

|F(x) −F(y)| ≤ CL|x − y | ∀x , y ∈ [0,∞),

where

0 < CL <CellCK

C 2tr‖g‖L∞(Γc (α))

,

then problem (P(α)) has exactly one solution.


Shape optimization problem

Consider a cost functional:I : (α, y) 7→ R,

where α ∈ Uad is a suitable set, y ∈ V(α) and define the set:

G = (α, u) | α ∈ Uad , u solves (P(α)) .

The shape optimization problem then reads as:

(P)

Find (α∗, u∗) ∈ G such that:I (α∗, u∗) ≤ I (α, u) ∀(α, u) ∈ G.


Shape optimization problem

Consider a cost functional:I : (α, y) 7→ R,

where α ∈ Uad is a suitable set, y ∈ V(α) and define the set:

G = (α, u) | α ∈ Uad , u solves (P(α)) .

The shape optimization problem then reads as:

(P)

Find (α∗, u∗) ∈ G such that:I (α∗, u∗) ≤ I (α, u) ∀(α, u) ∈ G.

For which choice of Uad and I does the problem (P) have at least one solution?


Assumptions

We will consider:

Uad = α ∈ C1,1([a, b]) | 0 ≤ α(x) ≤ C0 ∀x ∈ [a, b],

|α(x) − α(y)| ≤ C1|x − y | ∀x , y ∈ [a, b],

|α′(x) − α′(y)| ≤ C2|x − y | ∀x , y ∈ [a, b],

meas Ω(α) = C3 .

Let the cost functional satisfy:

αn → α, in C 1([a, b]), αn, α ∈ Uad ,

yn y , in H1(bΩ), yn, y ∈ H

1(bΩ)

ff⇒ lim inf

n→∞

I (αn, yn|Ω(αn)) ≥ I (α, y |Ω(α)).


Existence of an optimal shape

Lemma (sequential compactness of G)

Every sequence (αn, un) ⊂ G contains a convergent subsequence (αnk, unk

) , i.e.

there exist elements α ∈ Uad and u ∈ H1(bΩ) such that:

αnk→ α in C

1([a, b]) and unk u in H

1(bΩ), k → ∞.

Moreover (α, u|Ω(α)) ∈ G.

Note: The symbol v ∈ H1(bΩ) denotes the extension of v ∈ H

1(Ω(α)) into bΩ.


Existence of an optimal shape

Lemma (sequential compactness of G)

Every sequence (αn, un) ⊂ G contains a convergent subsequence (αnk, unk

) , i.e.

there exist elements α ∈ Uad and u ∈ H1(bΩ) such that:

αnk→ α in C

1([a, b]) and unk u in H

1(bΩ), k → ∞.

Moreover (α, u|Ω(α)) ∈ G.

Note: The symbol v ∈ H1(bΩ) denotes the extension of v ∈ H

1(Ω(α)) into bΩ.

Theorem (existence)

Problem (P) has at least one solution.


Discretization of the geometry

Let a = a0 < a1 < · · · < ad = b be an equidistant partition of the interval [a, b].

Uhad = αh ∈ C

0,1([a, b]) | αh|[ai−1,ai ] ∈ P1([ai−1, ai ]) ∀i = 1, . . . , d ,

0 ≤ αh(ai ) ≤ C0 ∀i = 0, . . . , d ,

|αh(ai ) − αh(ai−1)| ≤ C1h ∀i = 1, . . . , d ,

|αh(ai+1) − 2αh(ai ) + αh(ai−1)| ≤ C2h2 ∀i = 1, . . . , d − 1,

meas Ω(αh) = C3

Let T (h, αh) be a particular triangulation of the domain Ω(αh) such that for all h > 0 T (h, αh) | αh ∈ Uh

ad is a system of topologically equivalent triangulations.


Discretized state problem

Let h > 0 be fixed. Introduce the following finite dimensional function spaces:

Vh(αh) = vh ∈ C(Ωh) | vh|T ∈ P1(T ) ∀T ∈ T (h, αh), vh = 0 on Γu ,

Kh(αh) = vh ∈ Vh(αh) | vh2(ai , αh(ai )) ≥ −αh(ai ) ∀ai ∈ Nh ,

where ai ∈ Nh ⇔ (ai , αh(ai )) ∈ Γc(αh) \ Γu.

Discretized state problem:

(Ph(αh))

8>><>>:

Find uh ∈ Kh(αh) such that:

a(uh, vh − uh) +

Z

Γc (αh)

F(rΓc

h |uh1|)g`|vh1| − |uh1|

´ds

≥ L(vh − uh) ∀vh ∈ Kh(αh),

where rΓc

h stands for the piecewise linear Lagrange interpolation operator on Γc(αh).


On the solvability of (Ph(αh))

Theorem (existence)

Problem (Ph(αh)) has at least one solution for all αh ∈ Uhad .

Theorem (uniqueness)

If in addition g ∈ L∞(Γc(αh)) and F satisfies:

|F(x) −F(y)| ≤ CL|x − y | ∀x , y ∈ [0,∞),

where

0 < CL <CellCK

(1 + C1)1/4(1 + CrCinv )C 2tr

,

then (Ph(αh)) has exactly one solution.


Discrete shape optimization problem

Denote:Gh = (αh, uh) | αh ∈ U

had , uh solves (Ph(αh)) .

Then the discrete shape optimization problem reads as:

(Ph)

Find (α∗

h , u∗

h ) ∈ Gh such that:I (α∗

h , u∗

h ) ≤ I (αh, uh) ∀(αh, uh) ∈ Gh.






(Ph)

Find (α∗

h , u∗


h , u∗

h ) ≤ I (αh, uh) ∀(αh, uh) ∈ Gh.

Let the cost functional I satisfy:

α(n)h αh, α

(n)h , αh ∈ Uh

ad ,


1(bΩ)

)⇒

lim infn→∞

I (α(n)h , yn|Ω(α

(n)h

))

≥ I (αh, y |Ω(αh)).






(Ph)

Find (α∗

h , u∗


h , u∗

h ) ≤ I (αh, uh) ∀(αh, uh) ∈ Gh.

Let the cost functional I satisfy:

α(n)h αh, α

(n)h , αh ∈ Uh

ad ,


1(bΩ)

)⇒

lim infn→∞

I (α(n)h , yn|Ω(α

(n)h

))

≥ I (αh, y |Ω(αh)).

Theorem

The problem (Ph) has at least one solution.


Convergence analysis

Lemma (density of Uhad)

For each α ∈ Uad there exists a sequence αh, αh ∈ Uhad such that αh α in [a, b],

h → 0+.





h → 0+.

Lemma

Let αh α in [a, b], h → 0+, where αh ∈ Uhad . Then α ∈ Uad .





h → 0+.

Lemma

Let αh α in [a, b], h → 0+, where αh ∈ Uhad . Then α ∈ Uad .

Lemma

Every sequence (αh, uh), (αh, uh) ∈ Gh contains a subsequence (αhj, uhj

) such that:

αhj α in [a, b], uhj

u v H1(bΩ), hj → 0+,

where α ∈ Uad and u ∈ H1(bΩ) are suitable functions. Moreover (α, u|Ω(α)) ∈ G.


Convergence analysis (cont.)

Let the cost functional be continuous in the following sense:

αh α, in [a, b], αh ∈ Uhad , α ∈ Uad ,

uh u, in H1(bΩ), uh|Ω(αh), u|Ω(α) solves (Ph(αh)), resp.(P(α)),

ff⇒

⇒ limh→0+

I (αh, uh|Ω(αh)) = I (α, u|Ω(α)).

Let us define the following set:

G := (α, u) ∈ G |∃(αh, uh), (αh, uh) ∈ Gh :

αh α in [a, b] and uh u in H1(bΩ), h → 0+



Theorem (suboptimal limit)

Every sequence (α∗

h , u∗

h ) of optimal doubles of the problems(Ph) (h → 0+) contains asubsequence so that:

α∗

hj α

∗ in [a, b] a u∗

hj u in H

1(bΩ), hj → 0+,

where α∗ ∈ Uad and u∗ = u|Ω(α∗) solves (P(α∗)), i.e. (α∗, u∗) ∈ G. Moreover:

I (α∗

, u∗) ≤ I (α, u) ∀(α, u) ∈ G.



Theorem (suboptimal limit)

Every sequence (α∗

h , u∗

h ) of optimal doubles of the problems(Ph) (h → 0+) contains asubsequence so that:

α∗

hj α

∗ in [a, b] a u∗

hj u in H

1(bΩ), hj → 0+,

where α∗ ∈ Uad and u∗ = u|Ω(α∗) solves (P(α∗)), i.e. (α∗, u∗) ∈ G. Moreover:

I (α∗

, u∗) ≤ I (α, u) ∀(α, u) ∈ G.

Corollary (optimal limit)

Let (P(α)) be uniquely solvable for each α ∈ Uad . Then G = G, in particular (α∗, u∗)from the previous theorem is optimal in the sense of the definition of (P).


Thank you for your attention!


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