Fictitious Domain Method for the Signorini

Problem in Linear Elasticity

Karl-Heinz Hoffmann, Alexander Khludnev

no. 136

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-

gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-

sität Bonn entstanden und als Manuskript vervielfältigt worden.

Bonn, Februar 2004

FICTITIOUS DOMAIN METHOD FOR THE SIGNORINIPROBLEM IN LINEAR ELASTICITY

Karl-Heinz Hoffmann1, Alexander Khludnev2

1Caesar, Ludwig-Erhard-Allee 2, 53175 Bonn, Germany;

E-mail: hoffmann@caesar.de

2Lavrentyev Institute of Hydrodynamics of the RussianAcademy of Sciences, Novosibirsk 630090, Russia;

E-mail: khlud@hydro.nsc.ru

Fictitious domain method for the Signorini contact problemis proposed. Convergence of approximate solutions defined in awider domain is proved.

Keywords: Signorini contact problem; crack problem; ficti-tious domain. AMS Subject Classifications. 74A45

1 Introduction

In the paper, we analyze a fictitious domain method for theSignorini problem in linear elasticity theory. This method al-lows us to construct a family of auxiliary problems defined inan extended domain so that their solutions converge properlyto a solution of the Signorini problem. In fact, we propose sev-eral equivalent formulations considered in extended domains.

1

The first formulation utilizes minimization of the energy func-tional over the set of all admissible displacements. In this case,all boundary conditions that include components of the stresstensor are corollaries of the problem formulation. The secondapproach is called mixed problem formulation. It assumes thatsome restrictions of inequality type are imposed on the com-ponents of the stress tensor. Other boundary conditions thatinclude the displacement vector are corollaries of the mixedproblem formulation. We should note that each of the aboveproblems defines a family of solutions in a domain that is largethen the original one. Moreover, this extended domain containsa cut (provided we consider two dimensional cases). The thirdproblem formulation follows from the previous ones. It definesa family of solutions in a smooth auxiliary domain without anycuts. In this case, restrictions on the components of the stresstensor are imposed for some subsets of the auxiliary domain.The problems obtained in this case are analogous to contactproblems with restrictions imposed on subsets of lower dimen-sions.

Note that the above discussed boundary value problems in-cluding Signorini’s one can be classified as free boundary prob-lems. This means that a boundary condition at a given pointshould be consistent with the solution of the problem in thewhole.

It is interesting to notice that the auxiliary problems are pre-cisely crack problems with nonpenetration conditions of inequal-ity type for the crack faces, which is under active study duringlast years [5]. As for linear crack problems, we refer readers to[4], [11], [12]. In this case, boundary conditions considered onthe crack faces are linear. Applied aspects of crack theory can befound in [3], [10], [13]. As we have mentioned, one of the auxil-iary problems resembles a contact problem with inequality typerestrictions imposed on some subsets of the solution domain. Awide class of similar problems is analyzed in [6]. Fictitious do-main method for Dirichlet problems is presented in [9], see also[8]. Numerical approaches to fictitious domain method for linearelliptic problems can be found in [2].

2

2 The Signorini problem

Let Ω1 ⊂ R2 be a bounded simply connected domain withsmooth boundary Γ1, Γ1 = Γc ∪ Γ0, Γc ∩ Γ0 = ∅,measΓ0 > 0.For simplicity we assume that Γc is a smooth curve without itstip points. Denote by ν = (ν1, ν2) the internal unit normal vec-tor to Γ1. In the domain Ω1 we consider the Signorini problem.Namely, we have to find functions u = (u1, u2), σ = σij, i, j =1, 2, such that

−σij,j = fi , i = 1, 2 , in Ω1, (1)

σij = aijklεkl(u) , i, j = 1, 2, in Ω1, (2)

u = 0 on Γ0, (3)

uν ≥ 0, σν ≤ 0, στ = 0, uν · σν = 0 on Γc. (4)

Here f = (f1, f2) ∈ L2loc(R

2) is a given function, σij are the stresstensor components, εij(u) = 1

2(ui,j + uj,i) are the strain tensorcomponents, ui,j = ∂ui

∂xj, x = (x1, x2) ∈ Ω1,

σν = σijνjνi, στ = σν − σν · ν, σν = σijνj2i=1,

aijkl ∈ L∞loc(R2), aijklξklξij ≥ c|ξ|2, c > 0, ξij = ξji.

In fact, in (2) we can consider σij to be a function dependingon u, i.e. σij = σij(u). In a sequel we shall use the notationA = aijkl, hence, the Hooke law (2) can be written in theform σ(u) = Aε(u), ε(u) = εij(u)2

i,j=1. A symmetry overbelow indices, σij = σji etc., is assumed everywhere in the paper.Also the summation convention over repeated indices i, j, k, l isaccepted throughout the paper.

The problem (1)-(4) admits a variational formulation. To thisend we introduce some notations. Let H1(Ω1) be the Sobolevspace of all functions from L2(Ω1) having the first square inte-grable derivatives which belong to the space L2(Ω1). Denote

H1Γ0

(Ω1) = v = (v1, v2) ∈ H1(Ω1)| vi = 0 on Γ0, i = 1, 2

and consider a convex and closed set of admissible displacements

Kc = v = (v1, v2) ∈ H1Γ0

(Ω1) | vν ≥ 0 a.e. on Γc. (5)

3

In this case the problem (1)-(4) is equivalent to the minimizationof the energy functional

Π(v) =1

2

∫Ω1

σ(v)ε(v)−∫Ω1

fv

over the set Kc, and it can be written in the variational inequal-ity form

u ∈ Kc,

∫Ω1

σ(u)ε(v − u) ≥∫Ω1

f(v − u) ∀v ∈ Kc. (6)

Note that the variational inequality (6) has a (unique) solutionsince the functional Π is coercive and weakly lower semicontin-uous. Moreover, boundary conditions (4) have a precise inter-pretation in terms of suitable functional spaces. We omit thedetails here since the needed explanations will be given belowin a more difficult situation.

3 Auxiliary problems in a domain with a cut

It turns out that the variational inequality (6) can be consid-ered as a limit problem for a family of auxiliary boundary valueproblems defined in a wider domain as compared to the domainΩ1. Below we give necessary explanations to describe the familyof auxiliary boundary value problems. First of all we extend thedomain Ω1 to a domain Ωc, by adding the so called fictitiousdomain Ω2. Assume that Γ2 is smooth enough. Denote by Γ theexternal boundary of the domain Ωc, i.e. Γ = ∂Ωc \ (Γ+

c ∪ Γ−c ),where the faces Γ±c are defined with respect to the normal vectorν.

Denote next Σ0 = Γ1 ∩ Γ2,Σ = Σ0 \ Γ. In this case Ωc =Ω1 ∪Ω2 ∪ (Σ \ Γc). This means that the domain Ωc contains thecut Γc.

We put

aλijkl =

aijkl in Ω1

λ−1aijkl in Ω2 ,

4

with a positive parameter λ which in a sequel goes to zero.Like before we shall use the notation Aλ = aλ

ijkl. In theextended domain Ωc, we aim at solving the following bound-ary value problem. We have to find functions uλ = (uλ

1 , uλ2),

σλ = σλij, i, j = 1, 2, such that

−σλij,j = fi , i = 1, 2, in Ωc, (7)

σλij = aλ

ijklεkl(uλ) , i, j = 1, 2, in Ωc, (8)

uλ = 0 on Γ, (9)

[uλ]ν ≥ 0, [σλν ] = 0, σλ

ν ≤ 0, σλτ = 0, [uλ]ν · σλ

ν = 0 on Γc. (10)

Here [v] = v+ − v− is a jump of a function v through Γc, andv± fit to the positive and negative faces of Γ±c with respect tothe normal vector ν, σλν = σλ

ijνj2i=1, σλ

ν = σλijνjνi, σλ

τ =σλν − σλ

ν · ν. Similar to the problem (1)-(4), in (7)-(10) we canwrite σλ

ij = σλij(u

λ), i.e. σλ(uλ) = Aλε(uλ).For any λ > 0 the problem (7)-(10) has a unique solution.

This problem corresponds to a minimization of the energy func-tional

Πλ(v) =1

2

∫Ωc

σλ(v)ε(v)−∫Ωc

fv

over the set K. Here K is the set of all admissible displacements,

K = v ∈ H1Γ(Ωc) | [v]ν ≥ 0 a.e. on Γc,

H1Γ(Ωc) = v = (v1, v2) ∈ H1(Ωc) | vi = 0 on Γ , i = 1, 2.

Note that the set K is weakly closed. The solution of the prob-lem

minv∈K

Πλ(v)

exists and satisfies the variational inequality

uλ ∈ K,∫Ωc

σλ(uλ)ε(v − uλ) ≥∫Ωc

f(v − uλ) ∀v ∈ K. (11)

We should remark at this point that the boundary Γ may loosea smoothness at the points of Σ0 \Σ, i.e. at the tip points of Σ0.

5

Meanwhile, the first Korn inequality holds for functions from thespace H1

Γ(Ωc) which provides a coercivity and a weak lower semi-continuity of the functional Πλ(v) on the space H1

Γ(Ωc). Hence,for any fixed λ > 0 we can find a solution uλ from (11). Noticethat this solution is defined in the domain Ωc. Moreover, it canbe proved that as λ→ 0

uλ → u weakly in H1Γ(Ωc), uλ → 0 strongly in H1(Ω2),

(12)

uλ → u strongly in H1(Ω1).

As it turns out a restriction of the limit function u from(12) to the domain Ω1 is precisely the solution of the Signoriniproblem (1)-(4). Consequently, the Signorini problem (1)-(4)can be viewed as a limit problem for a family of problems (7)-(10).

The purpose of the arguments below is to analyze the problem(11) and, in particular, to prove the convergences (12).

Substituting v = 0, v = 2uλ in (11) as test functions we find

∫Ωc

Aλε(uλ)ε(uλ) =

∫Ωc

fuλ. (13)

Hence∫Ω1

Aε(uλ)ε(uλ) +1

λ

∫Ω2

Aε(uλ)ε(uλ) =

∫Ω1

fuλ +

∫Ω2

fuλ. (14)

As it was mentioned the first Korn inequalities hold,∫Ωi

ε(v)ε(v) ≥ ci

∫Ωi

v2

∀v = (v1, v2) ∈ H1(Ωi), v = 0 on Γ ∩ Γi, i = 1, 2,

with positive constants ci independent of v. In these cases weassume that meas(Γ ∩ Γ1) > 0. Consequently, (14) implies

‖uλ‖2H1(Ω1) +

1

λ‖uλ‖2

H1(Ω2) ≤ c3λ+ c4

6

with constants c3, c4 independent of λ. Thus, we arrive at theestimates

‖uλ‖2H1(Ω1) ≤ c5, ‖uλ‖2

H1(Ω2) ≤ c6λ, (15)

where the constants c5, c6 are independent of λ, λ ≤ λ0. Sincethe tensor aijkl is positively defined, by (13), we also have theestimate

‖uλ‖2H1

Γ(Ωc) ≤ c (16)

valid for all small positive λ.We choose a subsequence with the previous notation such

that as λ→ 0

uλ → u weakly in H1Γ(Ωc). (17)

By (15), we have

uλ → 0 strongly in H1(Ω2). (18)

In particular, the limit function u is equal to zero on Σ \ Γc.

Now we pass to the limit in the variational inequality (11).To this end we choose v ∈ K such that v ≡ 0 in Ω2 ( in thiscase vν ≥ 0 on Γ+

c ) and substitute the function v in (11). Thisimplies∫

Ω1

Aε(uλ)ε(v) ≥∫Ω1

Aε(uλ)ε(uλ) +1

λ

∫Ω2

Aε(uλ)ε(uλ)− (19)

∫Ω2

fuλ +

∫Ω1

f(v − uλ).

Passing to the lower limit in both parts of (19), by (17), (18),we derive ∫

Ω1

Aε(u)ε(v) ≥∫Ω1

Aε(u)ε(u) +

∫Ω1

f(v − u). (20)

In so doing the inequality

lim infλ→0

1

λ

∫Ω2

Aε(uλ)ε(uλ) ≥ 0

7

is taken into account. Since uλ ∈ K it follows that a restrictionof the limit function u to the domain Ω1 belongs to the set Kc,where Kc was defined in (5). As a result the inequality (20) canbe written in the form∫

Ω1

Aε(u)ε(v − u) ≥∫Ω1

f(v − u) ∀v ∈ Kc

which coincides with (6). So we have proved that a restrictionof the limit function u to the domain Ω1 is the solution of theSignorini problem (1)-(4).

It turns out that as λ→ 0 the convergence

1

λ

∫Ω2

Aε(uλ)ε(uλ) → 0 (21)

takes place. Indeed, from (14) it follows

lim supλ→0

1

λ

∫Ω2

Aε(uλ)ε(uλ) = lim supλ→0

∫Ω1

fuλ +

∫Ω2

fuλ−

∫Ω1

Aε(uλ)ε(uλ) ≤ lim supλ→0

∫Ω1

fuλ + lim supλ→0

∫Ω2

fuλ+

lim supλ→0

−∫Ω1

Aε(uλ)ε(uλ) ≤∫Ω1

fu−∫Ω1

Aε(u)ε(u).

On the other hand, relation (6) implies∫Ω1

fu =

∫Ω1

Aε(u)ε(u).

Hence the previous arguments show

0 ≤ lim infλ→0

1

λ

∫Ω2

Aε(uλ)ε(uλ) ≤ lim supλ→0

1

λ

∫Ω2

Aε(uλ)ε(uλ) ≤ 0

which proves the convergence (21).According to (21), convergence (18) can be specified, namely,

as λ→ 0

1√λuλ → 0 strongly in H1(Ω2). (22)

8

Now we shall prove more strong convergence of uλ in thedomain Ω1 as compared with (17). We prove that as λ→ 0,

uλ → u strongly in H1(Ω1). (23)

Since the weak convergence of uλ to u in H1(Ω1) is alreadyestablished it suffices to state∫

Ω1

Aε(uλ)ε(uλ) →∫Ω1

Aε(u)ε(u). (24)

From (14) it follows∫Ω1

Aε(uλ)ε(uλ) =

∫Ω2

fuλ +

∫Ω1

fuλ − 1

λ

∫Ω2

Aε(uλ)ε(uλ).

By (17), (22), the right-hand side here has a limit equal to

∫Ω1

fu,

hence

limλ→0

∫Ω1

Aε(uλ)ε(uλ) =

∫Ω1

fu.

On the other hand, as we know,∫Ω1

fu =

∫Ω1

Aε(u)ε(u),

which proves the convergence (24). As it was already noticedthe convergence (23) follows from (24).

Now we go back to the differential formulation of the problem(11), i.e. to the problem (7)-(10), and explain in what sense theboundary conditions (10) hold. First note that from (11) theequation

−divσλ(uλ) = f in Ωc (25)

follows which holds in the distributional sense. In particular,(25) implies

−divσλ(uλ) = f in Ω1 , −divσλ(uλ) = f in Ω2. (26)

9

Introduce the Hilbert spaces H1/2(Γi), i = 1, 2, equipped withthe norms, respectively,

‖v‖21/2,Γi

= ‖v‖2L2(Γi) +

∫Γi

∫Γi

|v(x)− v(y)|2

|x− y|2dxdy.

We shall use Green’s formula for the domains Ω1,Ω2 with theLipschitz boundaries Γ1,Γ2 ( see [14])

−∫Ωi

v · divσ =

∫Ωi

ε(v) · σ − 〈σn, v〉1/2 , Γi, i = 1, 2, (27)

valid for all σ = σpq, σ, divσ ∈ L2(Ωi) v = (v1, v2) ∈ H1(Ωi),where n is a unit external normal vector to Γi, and the brack-ets 〈· , ·〉1/2,Γi

denote a duality pairing between the dual spacesH−1/2(Γi) andH1/2(Γi). Observe that in this case σn ∈ H−1/2(Γi),

i = 1, 2. We introduce also the space H1/200 (Σ) with the norm

‖v‖001/2,Σ =

‖v‖21/2,Σ +

∫Σ

v2

ρ

1/2

,

where ρ(x) = dist(x, ∂Σ). By H−1/200 (Σ) we denote the space

dual of H1/200 (Σ). Denote by v an extension of any function v by

zero outside Σ, i.e.

v =

v on Σ0 on Γi \ Σ , i = 1, 2,

then v ∈ H1/2(Γi) if and only if v ∈ H1/200 (Σ) (see [5]). Also

notice that the inclusion v ∈ H−1/2(Γi) implies v ∈ H−1/200 (Σ) for

i = 1, 2. Let us take ϕ = (ϕ1, ϕ2) ∈ H10(Ω), where Ω = Ωc ∪ Γc,

and

H10(Ω) = v ∈ H1(Ω)| v = 0 on Γ.

Then we have v = uλ ± ϕ ∈ K. Substitution of v in (11) as atest function provides the equality∫

Ωc

Aλε(uλ)ε(ϕ) =

∫Ωc

fϕ.

10

Hence∫Ω1

Aε(uλ)ε(ϕ) +1

λ

∫Ω2

Aε(uλ)ε(ϕ) =

∫Ω1

fϕ+

∫Ω2

fϕ. (28)

By (26), (27), from (28) it follows

−〈σλ(uλ)ν , ϕ〉1/2,Γ1+ 〈σλ(uλ)ν , ϕ〉1/2,Γ2

= 0. (29)

In this relation the normal vector ν is defined on the boundariesΓ1,Γ2. It is assumed that on the boundary Γ2 this vector ischosen in such a way that it coincides on Γc with the vectorν introduced before. Note that on the boundaries Γi we haveϕ = (ϕ1, ϕ2) ∈ H1/2(Γi), i = 1, 2. Since ϕ = 0 on Γ ∩ Γ1 and

ϕ = 0 on Γ ∩ Γ2, we obtain ϕ ∈ H1/200 (Σ). In this case (29) can

be written as ⟨σλ(uλ)−ν − σλ(uλ)+ν , ϕ

⟩001/2,Σ = 0, (30)

where 〈· , ·〉001/2,Σ means a duality pairing between H

−1/200 (Σ) and

H1/200 (Σ). Remark that in (30) we, in fact, can take any function

ϕ = (ϕ1, ϕ2) from H1/200 (Σ). Indeed, let ϕ = (ϕ1, ϕ2) ∈ H

1/200 (Σ)

be any fixed function. Extend this function by zero to theboundaries Γ1,Γ2. In such a way we obtain the functions fromthe spaces H1/2(Γ1), H

1/2(Γ2), respectively, which can be ex-tended to Ω1,Ω2 as functions from the spaces H1(Ω1), H

1(Ω2).As a result in the domain Ω we have constructed the two-component function from the space H1

0(Ω) whose trace on Σ

coincides with the above function ϕ = (ϕ1, ϕ2) ∈ H1/200 (Σ) what

is required. Since (30) holds for all ϕ = (ϕ1, ϕ2) ∈ H1/200 (Σ) we

have

σλ(uλ)−ν = σλ(uλ)+ν (31)

in the sense of functions from the space H−1/200 (Σ). This, in

particular, implies that (31) holds in the sense of functions

from the space H−1/200 (Γc). Denoting σλ(uλ)ν = σλ(uλ)−ν and

σλν (uλ) = σλ

ij(uλ)νjνi we obtain σλ

ν (uλ) ∈ H−1/200 (Γc) due to the

smoothness of ν (see [5]). Hence the second condition of (10)

holds in the sense of the space H−1/200 (Γc).

11

Next we clarify in what sense the third condition of (10) holds.

Let ϕ ∈ H1/200 (Γc), ϕ ≥ 0. We put ψ = ϕν. Due to the smooth-

ness of ν, we have ψ = (ψ1, ψ2) ∈ H1/200 (Γc) (see [5]). Extend

this function ψ by zero to the boundary Γ1, and next extendonce again the extended function to the domain Ω1 as a func-tion from the space H1(Ω1). This provides an existence of afunction Φ = (Φ1,Φ2) ∈ H1(Ω1) such that the trace of Φ on Γc

coincides with ψ. Introduce the notation

v =

Φ in Ω1

0 in Ω2 .

Then v ∈ K. Take v = uλ + v as a test function in (11). Thisimplies the inequality∫

Ω1

σλ(uλ)ε(v) +

∫Ω2

σλ(uλ)ε(v) ≥∫Ω1

fv +

∫Ω2

fv.

The integrals over Ω2 are equal to zero by the choice of thefunction v. Applying Green’s formula (27), by (26), from thisinequality we derive⟨

σλ(uλ)n, v⟩

1/2,Γ1≥ 0 .

Consequently, by the condition v = 0 on Γ1 \ Γc, this implies

〈σλ(uλ)ν, ψ〉001/2,Γc

≤ 0.

Since

σλ(uλ)ν = σλν (uλ)ν + σλ

τ (uλ)

and (see [5])

σλν (uλ) ∈ H−1/2

00 (Γc), σλτ (uλ) = (σλ

τ (uλ)1, σλτ (uλ)2) ∈ H−1/2

00 (Γc)

we, therefore, obtain

〈σλν (uλ), ϕ〉00

1/2,Γc≤ 0 ∀ϕ ∈ H1/2

00 (Γc), ϕ ≥ 0. (32)

This inequality means σλν (uλ) ≤ 0 in the sense of function from

the space H−1/200 (Γc). So the precise interpretation of the third

12

condition of (10) is the relation (32). The fourth condition of(10) is fulfilled in the sense

〈σλτ (uλ), ϕ〉00

1/2,Γc= 0 ∀ϕ = (ϕ1, ϕ2) ∈ H1/2

00 (Γc), ϕiνi = 0 on Γc.

At last we verify in what sense the fifth condition of (10)holds. By (26), (27), from (14) it follows

−〈σλ(uλ)ν, uλ〉1/2,Γ1+ 〈σλ(uλ)ν, uλ〉1/2,Γ2

= 0

which implies

−〈σλ(uλ)+ν, (uλ)+〉001/2,Σ + 〈σλ(uλ)−ν, (uλ)−〉00

1/2,Σ = 0.

By (31), this relation can be written in the form

〈σλ(uλ)ν, [uλ]〉001/2,Σ = 0,

and consequently, we obtain

〈σλν (uλ) , [uλ]ν〉00

1/2,Γc= 0. (33)

Here the condition [uλ] = 0 on Σ\Γc is used. Thus, (33) providesa precise interpretation of the last condition of (10).

4 Auxiliary problems in a smooth domain

In this section we propose an equivalent formulation of the prob-lem (7)-(11) for which a solution is defined in the smooth domainΩ = Ωc ∪ Γc without cuts. In fact, at the first step one moreequivalent formulation (the so called mixed formulation, see [1])of the problem (7)-(11) will be given. In the mixed problemformulation, a solution is defined in the domain Ωc with the cutΓc. Hence the fictitious domain formulation is proposed in threeequivalent forms considered both in the domains Ω and Ωc.

Introduce the space of functions

Mc = σ = σij | σ, divσ ∈ L2(Ωc)

equipped with the norm

‖σ‖2Mc

= ‖σ‖2L2(Ωc) + ‖divσ‖2

L2(Ωc)

13

and consider the set of admissible stresses

Nc = σ ∈Mc | [σν] = 0 on Γc;στ = 0 , σν ≤ 0 on Γ±c .

The values σν can be defined on Γi as elements of the spacesH−1/2(Γi), i = 1, 2, for σ ∈ Mc. Hence σν = σijνjνi = σ±ν ∈H−1/200 (Σ). Here σ+

ν is obtained provided that we first considerthe restriction of σν ∈ H−1/2(Γ1) to Σ for a definition of (σν)+,and put σ+

ν = (σijνj)+νi. The values σ−ν are obtained provided

that we consider the restriction of σν ∈ H−1/2(Γ2) to Σ for adefinition of (σν)−, and next put σ−ν = (σijνj)

−νi. Boundaryconditions [σν] = 0 on Γc, στ = 0, σν ≤ 0 on Γ±c in thedefinition of the set Nc are understood as follows:

〈(σν)+ − (σν)−, ϕ〉001/2,Σ = 0 ∀ϕ = (ϕ1, ϕ2) ∈ H1/2

00 (Σ), (34)

〈στ , ϕ〉001/2,Γc

= 0 ∀ϕ = (ϕ1, ϕ2) ∈ H1/200 (Γc), ϕiνi = 0,

〈σν, ϕ〉001/2,Γc

≤ 0 ∀ϕ ∈ H1/200 (Γc), ϕ ≥ 0.

Introduce the notation B = bijkl for the inverse tensor ofA = aijkl, which means that (2) can be written in the formbijklσkl = εij(u) and put

Bλ =

B in Ω1

λB in Ω2 .

Now we are in a position to consider the so called mixedformulation of the problem (7)-(10). Write the problem (7)-(10)in the form

−divσλ = f in Ωc, (35)

Bλσλ = ε(uλ) in Ωc, (36)

uλ = 0 on Γ, (37)

[uλ]ν ≥ 0, [σλν ] = 0, σλ

ν ≤ 0, σλτ = 0, [uλ]ν · σλ

ν = 0 on Γc. (38)

The suitable space for the displacement vector in this case wouldbe L2(Ωc). In contrast to the formulation (11) the restriction ofinequality type will be imposed on the stress tensor components.

14

As for the boundary conditions which contain the displacementuλ they can be found from the variational inequality. Indeed,we have to find functions uλ = (uλ

1 , uλ2), σ

λ = σλij, i, j = 1, 2,

such that

uλ ∈ L2(Ωc), σλ ∈ Nc, (39)

−divσλ = f in Ωc, (40)∫Ωc

Bλσλ(σ − σλ) +

∫Ωc

uλ(divσ − divσλ) ≥ 0 ∀σ ∈ Nc. (41)

It is seen that the formulations (35)-(38) and (39)-(41) are for-mally equivalent. To prove this we take σ = σλ±σ, σ ∈ C∞

0 (Ωc)and substitute in (41). It immediately implies the fulfillment ofthe equation (36). Boundary conditions for uλ follow from (41)provided that the following Green’s formula is used∫Ωc

w · divσ = −∫Ωc

ε(w) · σ +

∫Γ

w · σn+

∫Γ−c

w · σν −∫Γ+

c

w · σν

(42)

valid for all smooth functions w = (w1, w2), σ = σij, i, j = 1, 2.Here n is the external unit vector to Γ. On the other hand,(39)-(41) follow from (35)-(38) after multiplication of (36) byσ − σλ and integration over Ωc with the subsequent applicationof Green’s formula (42).

It is possible to prove that there exists a solution of the prob-lem (39)-(41) for any fixed λ > 0. It can be done, for instance,by considering a regularized problem with an auxiliary parame-ter δ. We first prove an existence of a solution to the regularizedproblem for any fixed δ > 0 and next pass to the limit as δ → 0.The regularized problem is of the form

uλδ = (uλδ1 , u

λδ2 ) ∈ L2(Ωc), σ

λδ ∈ Nc, (43)

δuλδ − divσλδ = f in Ωc, (44)∫Ωc

Bλσλδ(σ − σλδ) +

∫Ωc

uλδ(divσ − divσλδ) ≥ 0 ∀σ ∈ Nc. (45)

We omit the details since they can be found in [7].

15

Now we observe that, by (40) and the boundary condition[σλν] = 0 holding on Γc in the sense (34), the equilibrium equa-tion (40) holds in the smooth domain Ω. Indeed, by (40), wehave

−divσλ = f in Ωi, i = 1, 2.

Consider the distribution divσλ+f in Ω, where σλ is arbitrarilydefined on Γc. By 〈· , ϕ〉 we denote the value of a distribution atthe point ϕ = (ϕ1, ϕ2) ∈ C∞

0 (Ω). So, let ϕ = (ϕ1, ϕ2) ∈ C∞0 (Ω)

be any fixed element. The following formulae hold,

〈divσλ + f, ϕ〉 = −∫Ω1

σλε(ϕ)−∫Ω2

σλε(ϕ) +

∫Ω

fϕ =

〈[σλν] , ϕ〉001/2,Σ +

∫Ω1

(divσλ + f)ϕ+

∫Ω2

(divσλ + f)ϕ = 0,

which proves the assertion.Now we are able to formulate the problem (39)-(41) in the

equivalent form in the smooth domain Ω. To this end, we intro-duce notations. Let

M = σ = σij | σ ∈ L2(Ω), divσ ∈ L2(Ω),

Nc = σ ∈M | σν ≤ 0 , στ = 0 on Γc.

The norm in the space M is given by the formula

‖σ‖2M = ‖σ‖2

L2(Ω) + ‖divσ‖2L2(Ω),

and the boundary conditions in the definition of Nc are fulfilledin the following sense

σν ∈ H−1/200 (Γc), στ = (στ1, στ2) ∈ H−1/2

00 (Γc),

〈σν , ϕ〉001/2,Γc

≤ 0 ∀ϕ ∈ H1/200 (Γc), ϕ ≥ 0,

〈στ , ψ〉001/2,Γc

= 0 ∀ψ = (ψ1, ψ2) ∈ H1/200 (Γc), ψiνi = 0 on Γc.

Now we can rewrite the problem (39)-(41) in the following equiv-alent form. It is necessary to find functions uλ = (uλ

1 , uλ2), σ

λ =

16

σλij, i, j = 1, 2, such that

uλ ∈ L2(Ω), σλ ∈ Nc, (46)

−divσλ = f in Ω, (47)∫Ω

Bλσλ(σ − σλ) +

∫Ω

uλ(divσ − divσλ) ≥ 0 ∀σ ∈ Nc. (48)

Solvability of the boundary problem (46)-(48) can be provedsimilar to the proof of solvability of (39)-(41) by considering aregularized auxiliary problem of the form (43)-(45) in the do-main Ω.

Thus in the smooth domain Ω we have constructed the bound-ary problem (46)-(48) which can be viewed as an approximationof the Signorini problem (1)-(4).

Similar to the problem (7)-(10) it can be shown that thesolutions uλ, σλ of the problem (46)-(48) converge properly tothe solution of the Signorini problem (1)-(4). Let us justify thisassertion. Consider any function σ such that

σ ∈ Nc, −divσ = f in Ω.

Multiply (47) by uλ and integrate over Ω. Simultaneously, wetake σ = σ in (48). Summing the relations obtained implies∫

Ω

Bλσλ · σλ ≤∫Ω

Bλσλ · σ

which proves the uniform in λ estimate

λ‖σλ‖2L2(Ω2) + ‖σλ‖2

L2(Ω1) ≤ c. (49)

Meanwhile, taking σ = σλ± σ, σ ∈ C∞0 (Ωc), from (48) it follows

that the equation

Bλσλ = ε(uλ) in Ωc (50)

holds in the distributional sense. Since uλ ∈ L2(Ω), relations(49), (50) imply uλ ∈ H1(Ωc). On the other hand, we can seefrom (48) that

uλ = 0 on Γ,

17

which together with the first Korn inequality, by (49), (50),provide the estimate

‖uλ‖H1(Ωc) ≤ c (51)

being uniform with respect to λ. Moreover, relation (50) gives

λBσλ = ε(uλ) in Ω2, (52)

hence, by (49), (50), (52), we can assume as λ→ 0

uλ → u strongly in L2(Ωc), weakly in H1(Ωc),

uλ → 0 strongly in H1(Ω2),

(53)

σλ → σ weakly in L2(Ω1),

λσλ → 0 strongly in L2(Ω2).

Taking into account the convergence (53) let us pass to the limitas λ → 0 in (46)-(48) with a fixed function σ ∈ Nc such thatσ ≡ 0 in Ω2. This implies

u = (u1, u2) ∈ L2(Ω1), σ = σij ∈ Z, (54)

−divσ = f in Ω1, (55)∫Ω1

Bσ(σ − σ) +

∫Ω1

u(divσ − divσ) ≥ 0 ∀σ ∈ Z. (56)

Here

Z = σ = σij | σ, divσ ∈ L2(Ω1), σν ≤ 0 , στ = 0 on Γc.(57)

As a result we see that the problem (54)-(56) is a limit problemfor (46)-(48). On the other hand, (54)-(56) is a mixed formula-tion of the Signorini problem (1)-(4). Indeed, rewrite (1)-(4) asfollows

−divσ = f in Ω1, (58)

Bσ = ε(u) in Ω1, (59)

u = 0 on Γ0, (60)

uν ≥ 0, σν ≤ 0, στ = 0, uν · σν = 0 on Γc. (61)

18

We choose σ ∈ Z and multiply (59) by σ−σ. Next we integratethe obtained relation over Ω. In view of (60)-(61), this provides(56).

Note that similar to the problem (46)-(48) we can pass to thelimit as λ → 0 in (39)-(41). In this case the limit problem alsocoincides with (54)-(56).

Remark. Omitting the details we formulate some state-ments for a linear problem which is obtained from the problem(1)-(4) provided that we change nonlinear conditions (4) with alinear condition σν = 0 on Γc. To be more precise, instead ofthe Signorini problem (1)-(4) we consider the following bound-ary value problem for finding u = (u1, u2), σ = σij, i, j = 1, 2,

−divσ = f in Ω1, (62)

σ = Aε(u) in Ω1, (63)

u = 0 on Γ0, (64)

σν = 0 on Γc. (65)

In such a case, instead of the problem (7)-(10) defined in theextended domain Ωc with the cut Γc, we obtain the followinglinear boundary value problem for finding the functions uλ =(uλ

1 , uλ2), σ

λ = σλij, i, j = 1, 2,

−divσλ = f in Ωc, (66)

σλ = Aλε(uλ) in Ωc, (67)

uλ = 0 on Γ, (68)

σλν = 0 on Γ±c . (69)

Variational solution of the problem (66)-(69) can be easily de-fined from the identity

uλ = (uλ1 , u

λ2) ∈ H1

Γ(Ωc),∫Ωc

σλ(uλ)ε(v) =

∫Ωc

fv ∀v = (v1, v2) ∈ H1Γ(Ωc).

Mixed formulation of the problem (66)-(69) in the domain Ωc is

19

as follows (cf. (39)-(41))

uλ = (uλ1 , u

λ2) ∈ L2(Ωc), σ

λ = σλij ∈M, (70)

−divσλ = f in Ωc, (71)∫Ωc

Bλσλ · σ +

∫Ωc

uλ · divσ = 0 ∀σ ∈M. (72)

Here

M = σ = σij | σ, divσ ∈ L2(Ωc), [σν] = 0 on Γc ,

σν = 0 , στ = 0 on Γ±c .

At last we can formulate the problem (70)-(72) in the smoothdomain Ω. Indeed, in this case we have to find functions uλ =(uλ

1 , uλ2), σ

λ = σλij, i, j = 1, 2, such that (cf. (46)-(48))

uλ ∈ L2(Ω), σλ ∈Mc, (73)

−divσλ = f in Ω, (74)∫Ω

Bλσλ · σ +

∫Ω

uλ · divσ = 0 ∀σ ∈Mc, (75)

with

Mc = σ = σij | σ, divσ ∈ L2(Ω), σν = 0 , στ = 0 on Γc.

Similar to the problem (39)-(41), we can pass to the limit as λ→0 in the problem (70)-(72) which gives the following relations forthe limit functions u = (u1, u2), σ = σij, i, j = 1, 2,

u ∈ L2(Ω1), σ ∈ Z0, (76)

−divσ = f in Ω1, (77)∫Ω1

Bσ · σ +

∫Ω1

u · divσ = 0 ∀σ ∈ Z0, (78)

where

Z0 = σ = σij | σ, divσ ∈ L2(Ω1), σν = 0 , στ = 0 on Γc.

Hence we see that the limit problem for (70)-(72) as λ→ 0 co-incides with (76)-(78). In its own turn the problem (76)-(78) is

20

the mixed formulation of the linear problem (62)-(65). Analo-gously, we can pass to the limit as λ→ 0 in (73)-(75) which alsoimplies (76)-(78).

To conclude the paper we note that the fictitious domainmethod proposed for the Signorini problem can take differentforms depending on a choice of the fictitious domain Ω2.

5 Acknowledgement

This work was done during the visit in June 2003 of the sec-ond author to Caesar (Bonn) whose support is appreciated verymuch. It was also partially supported by the Russian Fund forBasic Research (03-01-00124).

References

[1] Brezzi, F., Fortin, M. Mixed and hybrid finite elementmethods. New York, Springer (1991).

[2] Brusnikin, M.B. On effective algorithms for solving bound-ary problems in fictitious domains for multiply connectedcase. Doklady of Russian Acedemy of Sciences. 2002. v.387. N 2. pp. 151-155.

[3] Cherepanov, G.P. Mechanics of brittle fracture. McGraw-Hill (1973).

[4] Grisvard, P. Elliptic problems in nonsmooth domains.Boston-London-Melbourne, Pitman (1985).

[5] Khludnev, A.M., Kovtunenko, V.A. Analysis of cracks insolids. Southampton-Boston, WIT Press (2000).

[6] Khludnev, A.M., Sokolowski, J. Modelling and control insolid mechanics. Basel-Boston-Berlin, Birkhauser (1997).

[7] Khludnev, A.M. Smooth domain method in equilibriumproblem for a plate with a crack. Siberian Math. J. 2002.v.43. N 6. pp.1388-1400.

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[8] Konovalov, A.N. Fictitious domain method in twistingproblems. In: Numerical methods in solid mechanics.Novosibirsk. 1973. v.4. N 2. pp. 109-115.

[9] Kopchenov, V.D. Approximate solutions of the Dirichletproblem by a fictitious domain method. Differential equa-tions. 1968. v.4. N 1. pp. 151-164.

[10] Morozov, N.F. Mathematical questions of a crack theory.M., Nauka (1984).

[11] Nazarov, S.A., Plamenevski, B.A. Elliptic problems indomains with piece-wise smooth boundaries. M., Nauka(1991).

[12] Ohtsuka, K. Mathematics of brittle fracture. Theoreticalstudies on fracture mechanics in Japan. Ed. K.Ohtsuka.Hiroshima-Denki Institute of Technology. Hiroshima,1995. pp. 99-172.

[13] Parton, V.Z., Morozov, E.M. Mechanics of elastoplasticfracture. M., Nauka (1985).

[14] Temam, R. Problemes mathematiques en plasticite. Paris,Gauthier-Villars (1983).

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