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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
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 2 / 17
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).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 3 / 17
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)
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 4 / 17
Classical formulation
Signorini problem with given friction
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)
u1 = 0 ⇒ |T1(u)| ≤ F g
u1 6= 0 ⇒ T1(u) = −sgn (u1)F g
ffon Γc(α)
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 4 / 17
Classical formulation
Signorini problem with given friction and solution-dependent coefficient of friction:
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)
u1 = 0 ⇒ |T1(u)| ≤ F(0)gu1 6= 0 ⇒ T1(u) = −sgn (u1)F(|u1|)g
ffon Γc(α)
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 4 / 17
Classical formulation
Signorini problem with given friction and solution-dependent coefficient of friction:
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)
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 4 / 17
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
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 5 / 17
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).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 5 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 6 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 7 / 17
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?
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 7 / 17
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 .
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 8 / 17
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)).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 8 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 9 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 9 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 10 / 17
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).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 11 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 12 / 17
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 13 / 17
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.
Let the cost functional I satisfy:
α(n)h αh, α
(n)h , αh ∈ Uh
ad ,
yn y , in H1(bΩ), yn, y ∈ H
1(bΩ)
)⇒
lim infn→∞
I (α(n)h , yn|Ω(α
(n)h
))
≥ I (αh, y |Ω(αh)).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 13 / 17
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.
Let the cost functional I satisfy:
α(n)h αh, α
(n)h , αh ∈ Uh
ad ,
yn y , in H1(bΩ), yn, y ∈ H
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 13 / 17
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+.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 14 / 17
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+.
Lemma
Let αh α in [a, b], h → 0+, where αh ∈ Uhad . Then α ∈ Uad .
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 14 / 17
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+.
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 14 / 17
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+
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Convergence analysis (cont.)
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.
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 16 / 17
Convergence analysis (cont.)
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).
Robert Patho (Charles University in Prague) Shape optimization in contact problems Workshop Dresden-Prague 16 / 17
Thank you for your attention!
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