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Space Adaptive Finite Element Methods for Dynamic SignoriniProblems

Heribert Blum · Andreas Rademacher · Andreas Schröder

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Abstract Space adaptive techniques for dynamic Signoriniproblems are discussed. For discretisation, the Newmarkmethod in time and low order finite elements in space areused. For the global discretisation error in space, an a pos-teriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies onan auxiliary problem, which takes the form of a variationalequation. An adaptive method based on the estimate is ap-plied to improve the finite element approximation. Numeri-cal results illustrate the performance of the presentedmethod.

Keywords Dynamic Signorini problem· A posteriori errorestimation· Mesh refinement· Finite element method

1 Introduction

Dynamic Signorini problems arise in many engineering pro-cesses, e.g., in milling and grinding processes, vehicle de-sign and ballistics. In these processes, the main effects resultfrom the contact at the surface of the bodies under consid-eration. Typical examples for engineering processes, wherecontact problems play a dominant role, are grinding pro-cesses. The workpiece interacts with the grinding wheel

Heribert BlumInstitute of Applied Mathematics, Technische UniversitätDortmund,D-44221 Dortmund, GermanyE-mail: heribert.blum@mathematik.uni-dortmund.de

Andreas RademacherInstitute of Applied Mathematics, Technische UniversitätDortmund,D-44221 Dortmund, GermanyE-mail: andreas.rademacher@mathematik.uni-dortmund.de

Andreas SchröderDepartment of Mathematics, Humboldt-Universität zu Berlin, Unterden Linden 6, D-10099 Berlin, GermanyE-mail: andreas.schroeder@mathematik.hu-berlin.de

only in a small contact zone. However, the behaviour of thegrinding machine is strongly affected by the resulting con-tact forces. A detailed study of this engineering problem isfound in [1]. For the reliable simulation of such a process, aprecise prediction is required of the contact forces, the con-tact zone and their effects onto the whole body. Furthermore,the contact zone and the contact forces are strongly depend-ing on time. Hence, the precise consideration of these de-pendences is essential in the numerical simulation.

An adequate technique, which gives rise to a flexible andefficient finite element discretisation, is based on a posteri-ori error control and resulting adaptive mesh refinement. Ingeneral, a posteriori error estimates for second order hyper-bolic problems are possible for two different discretisationapproaches. One of them uses space time Galerkin meth-ods for discretisation and applies similar techniques for errorcontrol as in the static case ([2–5]). The other one is based onfinite differences in time and finite elements in space. Here,separate error estimators are used for the space and time di-rection ([6–8]) or error estimates for the whole problem ([9,10]) are derived.In this article, finite differences in time and finite elements inspace are used to discretise the dynamic Signorini problems.Because only the data of the current time step comes intoplay, the error estimator can be evaluated efficiently. How-ever, the separation of the space and time direction compli-cates the consideration of space time effects. The aim of thisarticle is to derive an error estimator for the finite elementdiscretisation in space direction. Therefore, an error controltechnique for static contact problems is applied to the semidiscrete spatial problem. This technique goes back to Braess[11] and Schröder [12]. Other approaches to a posteriori er-ror control for static contact problems are discussed in [13–20]. In particular, an adaptive scheme for two-body contact

2

is contained in [21]. Convergence results for adaptive algo-rithms in the context of obstacle problems are proven in [22].

Adaptivity in time direction is not taken into account inthis article for notational simplicity, although it is easyto in-corporate. One can do this on the basis of error estimators,which are known from the literature of second order hyper-bolic problems [6].The temporal discretisation of dynamic contact problems isa difficult task. Several approaches based on different prob-lem formulations have been presented in literature. In [23]the penalty-method is used to solve the discrete problems.Special contact elements in combination with Lagrangemultipliers are presented in [24]. Other techniques forsmoothing and stabilizing the computation with special fi-nite elements, e.g. Mortar finite elements, are presented in[25–27]. In [28], the Newmark scheme is used with an ad-ditional L2-projection for stabilization. Algorithms for dy-namic contact/impact problems based on the energy- andmomentum conservation are derived in [29,30]. An addi-tive splitting of the acceleration into two parts, representingthe interior forces and the contact forces, is the basis of themethods introduced in [31,32]. In [33–35] algorithmsbased on variational inequalities and optimisationalgorithms are presented. Detailed surveys of this topic canbe found in the monographs [36,37].

The article is organised as follows: In the next two sec-tions, the strong and weak formulations of the dynamic sim-plified Signorini problem are introduced and the discretisa-tion of the problem is discussed. In Section 4, the spatial er-ror estimator is derived serving as basis of an adaptive algo-rithm. In Section 6, two examples illustrate the applicationof the mentioned techniques. The article concludes with adiscussion of the results.

2 Continuous Formulation

In this section, the strong and the weak formulation of thedynamic simplified Signorini problem are presented. LetΩ ⊂R

2 be the basic domain andI := [0,T ] ⊂ R a time interval.The boundary∂Ω of Ω is divided into three mutually dis-joint partsΓD, ΓC andΓN with positive measure. Homoge-neous Dirichlet and Neumann boundary conditions are pre-scribed on the closed setΓD and onΓN , respectively. Contactmay take place on the sufficiently smooth setΓC, Γ̄C ⊂ ∁ΓD.See, e.g., [38], Section 5.3 for more details. The time de-pendent rigid foundation is parameterised by a sufficientlysmooth functiong : ΓC × I → R∪{−∞} . Here, the restric-tion u ≥ g on ΓC is considered,u ≤ g can be treated analo-gously.

The initial displacementu0 is in

H1 (Ω ,ΓD) := {v ∈ H1(Ω)|γ|ΓD (v) = 0}

and the initial velocityv0 is in L2 (Ω). Here,γ denotes thetrace operator of functions inH1(Ω ,ΓD) onto the boundary∂Ω . See, e.g., [39] for more details. The gradient of the dis-placementu in space direction is denoted by∇u and∆u isthe usual Laplace operator. The first and second time deriva-tives are denoted by ˙u andü, respectively. In the following,all relations have to be understood almost everywhere.

We choose the unconstrained trial space

V := W 2,∞(

I;L2 (Ω))

∩L∞(

I;H1 (Ω ,ΓD))

for notational convenience, although the existense of a so-lution in V can not be proven, even in the contact free case[39]. The set of admissible displacements is

K :={

ϕ ∈V∣

∣γ|ΓC (ϕ) ≥ g onΓC × I}

.

The L2-scalar product is defined by(u,v) =∫

Ω uvdx foru,v ∈ L2 (Ω). The density is set equal to 1 for notationalsimplicity. Eventually, the weak formulation of the simpli-fied dynamic Signorini problem, see e.g. [40], reads

Problem 2.1 Find a functionu ∈ K with u(t = 0) = u0 andu̇(t = 0) = v0 for which

(ü(t) ,ϕ (t)−u(t))+ (∇u(t) ,∇(ϕ (t)−u(t)))≥ ( f (t) ,ϕ (t)−u(t))

holds for allϕ ∈ K and allt ∈ I.

Throughout this article, we assumef ∈ L∞(

I;L2 (Ω))

.If the solution is sufficiently smooth, we obtain theequivalent strong formulation (see [40])

ü−∆u = f in Ω × Iu = 0 onΓD × I

∂u∂ν

= 0 onΓN × I

u−g ≥ 0 onΓC × I∂u∂ν

≤ 0 onΓC × I

∂u∂ν

(u−g) = 0 onΓC × I.

3 Discretisation

We use Rothe’s method to discretise the dynamic simpli-fied Signorini problem. First, the problem is discretised intemporal direction by the Newmark method (see [41]). Theresulting spatial problems are approximately solved by loworder finite elements.

3

3.1 Temporal Discretisation

The time intervalI is split into N equidistant subintervalsIn := (tn−1,tn] of length k = tn − tn−1 with 0 =: t0 < t1 <.. . < tN−1 < tN := T . The value of a functionw at a timeinstancetn is approximated bywn. We use the notationv = u̇anda = ü for the velocity and the acceleration, respectively.

In the Newmark method,v anda are approximated by

an =1

β k2(

un −un−1)

−1

β kvn−1−

(

12β

−1

)

an−1, (3.1)

vn = vn−1+ k[

(1−α)an−1 + αan]

. (3.2)

Here,α andβ are free parameters in the interval[0,2]. Forsecond order convergence,α = 12 is required. Furthermore,the inequality 2β ≥ α ≥ 12 has to be valid for unconditionalstability (see [42]). For dynamic contact problems, the choiceα = β = 12 is recommended to guarantee conservation ofenergy and momentum (see [24,35]). For starting the New-mark method the initial accelerationa0 is needed. It can becalculated on the basis of the initial displacement and ve-locity (see [42]). The semi-discrete problem then reads asfollows:

Problem 3.1 Findu with u0 = u0, v0 = v0 anda0 = a0, suchthat in every time stepn∈ {1,2, . . . ,N}, the functionun ∈Kn

is the solution of the variational inequality

(an,ϕ −un)+ (∇un,∇(ϕ −un)) ≥ ( f (tn) ,ϕ −un) , (3.3)

for all ϕ ∈ Kn. Moreoverun, vn andan have to fulfill theequations (3.1) and (3.2).

The setKn :={

ϕ ∈ H1(Ω ,ΓD)∣

∣γ|ΓC (ϕ) ≥ gn on Ω

}

is thetime discretized set of the admissible displacements. Substi-tuting the equation (3.1) withγ = β = 12 in the inequality(3.3) leads to

(un,ϕ −un)+ 12k2 (∇un,∇(ϕ −un))

≥(1

2k2 f (tn)+ un−1+ k vn−1,ϕ −un

)

.

This can be written as

c(un,ϕ −un) ≥ (Fn,ϕ −un) , (3.4)wherec is defined by

c(ω ,ϕ) := (ω ,ϕ)+12

k2 (∇ω ,∇ϕ)

andFn as

Fn :=12

k2 f (tn)+ un−1+ k vn−1.

The bilinear formc is uniformly elliptic, continuous andsymmetric. Thus, an elliptic variational inequality has tobesolved in each time step. An efficient way for solving varia-tional inequalities is given by their mixed formulation. TheLagrange parameters may be interpreted as contact forces.The variational inequality (3.4) is equivalent to the follow-ing mixed problem:

Problem 3.2 Find(u,λ ) with u0 = u0, v0 = v0 anda0 = a0,such that(un,λ n) ∈V n ×Λ n is the solution of the system

c(un,ϕ)+〈

λ n,γ|ΓC (ϕ)〉

= (Fn,ϕ) (3.5)〈

µ −λ n,γ|ΓC (un)−gn

〉

≤ 0, (3.6)

for all ϕ ∈ V n, all µ ∈ Λ n and alln ∈ {1,2, . . . ,N}. Basedon the equations (3.1) and (3.2), the functionsvn andan arecalculated in a postprocessing step.

Here,Λ n is the dual cone of the set

G :={

µ ∈ H1/2 (ΓC)∣

∣

∣µ ≤ 0

}

.

The dual pairing is expressed by〈·, ·〉. The setV n := H1(Ω ,ΓD)is the time discretised unconstrained trial space.The equivalence of the two formulations is a well-knownconclusion from the general theory of minimisation prob-lems in Hilbert spaces presented, e.g., in [43,44].

3.2 Spatial Discretisation

A finite element approach is applied to discretise the mixedproblem 3.2. We use adaptive algorithms with dynamicmeshes. Therefore, the trial spacesV nh and Λ

nH may vary

from time step to time step. Bilinear basis functions on themeshTn are used for the finite element spaceV nh . The dis-crete Lagrange multipliers are piecewise constant and arecontained in the setΛ nH . The indexH indicates that coarsermeshes may be chosen for the Lagrange multipliers. In ourcalculations, we useH = 2h for stability reasons. A detailedstudy of the stability properties of this discretisation can befound in [12].

Because of the varying meshes, FE-functions have to betransfered to the mesh of the current time step. This processis denoted byIh and is realized by anL2-projection. Onemight also consider standard interpolation as a transfer op-erator, which needs less effort, but can lead to instabilities.The space and time discrete problem is

Problem 3.3 Find(

unh,λnH

)

∈V nh ×ΛnH with u

0h = Ihu0, v

0h =

Ihv0 anda0h = Iha0, such that the system

c(unh,ϕh)+〈

λ nH ,γ|ΓC (ϕh)〉

= (Fnh ,ϕh) (3.7)〈

µH −λ nH,γ|ΓC (unh)−g

n〉 ≤ 0 (3.8)

is valid for all ϕh ∈V nh andµH ∈ ΛnH , n ∈ {1,2, . . . ,N}. Ad-

ditionally, the equations (3.1) and (3.2) determinevnh andanh.

Here,Fnh is given by

Fnh :=12

k2 f (tn)+ Ihun−1h + k Ihv

n−1h .

4

The system (3.7-3.8) leads to the following saddle pointproblem inRm, wherem depends onn:

Anūn + Bnλ̄ n = F̄n(

µ̄ − λ̄ n)T

(

(Bn)T ūn − ḡn)

≤ 0,

which must hold for allµ̄ ∈ Rm̃≤0. Here,An := Mn + 12k

2Kn

is the generalised stiffness matrix,Mn ∈ Rm×m is the massmatrix andKn ∈ Rm×m is the stiffness matrix. The matrixBn ∈ Rm×m̃ represents the dual pairing in (3.8). Notice, thatall matrices may change from time step to time step.The saddle point problem can be rewritten as a quadraticoptimisation problem, which can be solved by substitutingūn := (An)−1

(

F̄n −Bnλ̄ n)

and applying SQP methods.More details can be found in [12].

4 Spatial Error Estimation

In this section, an error estimation is derived for the spatialerror in every time step. The estimation is easy to imple-ment and can be evaluated fast. The temporal error is notconsidered. The idea of the error estimation goes back toBraess [11], who presented it for static obstacle problems.This idea was extended by Schröder [12] to static Signoriniproblems even with friction by introducing a general frame-work for error control of variational inequalities in Hilbertspaces. In order to apply this framework here, we considerthe following saddlepoint problem:

Problem 4.1 Find(

ũn, λ̃ n)

∈V n ×Λ n, such that

c(ũn,ϕ)+ 〈λ̃ n,γ|ΓC (ϕ)〉 = (Fnh ,ϕ)

〈µ − λ̃ n,γ|ΓC (ũn)−gn〉 ≤ 0

for all ϕ ∈V n and allµ ∈ Λ n.

An essential part of the general framework in [12] is theformulation of the auxiliary problem:

Problem 4.2 Find un⋆ ∈ Vn, such that the variational equa-

tion

c(un⋆,ϕ) = (Fnh ,ϕ)−

〈

λ nH ,γ|ΓC (ϕ)〉

holds for allϕ ∈V n.

Problem 4.2 corresponds to the first line of Problem 4.1,but with the discrete Lagrangian multiplierλ nH instead of̃λ n.Applying Lemma IV.2 in [12] yields

Lemma 4.1 There are constants C′,C′′ ∈ R>0, such that

‖ũn −unh‖2 +

∥

∥

∥λ̃ n −λ nH

∥

∥

∥

2

≤ C′ ‖un⋆−unh‖

2 +C′′〈λ̃ n −λ nH,γ|ΓC (unh)−g

n〉.

Here,‖ · ‖ denotes the norm correponding to the relatedfunction spaces. We use theH1(Ω)-norm for V n. Takinginto account, that the discrete solutionunh is also a discretesolution of Problem 4.2, we are able to get rid of the term‖un⋆ − u

nh‖ by using an appropriate error estimator for the

auxiliary problem: Letηn⋆ > 0 be an error estimator of Prob-lem 4.2, i.e., there exists a constantC⋆ > 0 independent ofV nh andΛ

nH , such that

‖un⋆−unh‖

2 ≤C⋆(ηn⋆ )2.

Then, Lemma 4.1 leads to

‖ũn −unh‖2 +

∥

∥

∥λ̃ n −λ nH

∥

∥

∥

2

≤ C′C⋆(ηn⋆ )2 +C′′〈λ n −λ nH ,γ|ΓC (u

nh)−g

n〉.

The remaining term〈λ̃ n−λ nH ,γ|ΓC(

unh)

−gn〉 is estimated by

Lemma 4.2 Let ν0 > 0 be the constant of continuity of c.Furthermore, let

d ∈ K̃n :={

v ∈V n|gn − γ|ΓC (unh)− γ|ΓC(v) ≤ 0

}

and ε > 0. Then, there holds

〈λ̃ n −λ nH,γ|ΓC(

unh)

−gn〉

≤ ε2∥

∥ũn −unh∥

∥

2+

(1+ε)ν202ε ‖d‖

2

+ 12∥

∥un⋆−unh

∥

∥

2+ |(λ nH ,γ|ΓC (d))|.

Proof. Inserting 0 and 2λ nH in (3.8) yields

〈λ nH ,γ|ΓC (unh)−g

n〉 = 0.

Furthermore, we get

〈λ̃ n,γ|ΓC (unh)−g

n〉

= −〈λ̃ n,gn − γ|ΓC (unh)− γ|ΓC(d)〉− 〈λ̃

n,γ|ΓC (d)〉≤ c(ũn,d)− (Fnh ,d)

= c(ũn −unh,d)+ c(unh,d)− (F

nh ,d)

≤ ν0‖ũn −unh‖‖d‖+ c(unh,d)− (F

nh ,d)

≤ε2‖ũn −unh‖

2 +ν202ε

‖d‖2+ c(unh,d)− (Fnh ,d).

Here, we have used Young’s inequality. The termc(unh,d)−(Fnh ,d) is estimated as follows:

c(unh,d)− (Fnh ,d)

= c(unh −un⋆,d)− (λ nH ,γ|ΓC (d))

≤ ν0‖un⋆−unh‖‖d‖− (λ

nH ,γ|ΓC (d))

≤12‖un⋆−u

nh‖

2 +ν202‖d‖2 + |(λ nH,γ|ΓC (d))|.

⊓⊔

Eventually, we obtain an a posteriori error estimate bythe following proposition:

5

Proposition 4.1 There exists a constant C > 0 independentof V nh and Λ

nH , such that

‖ũn −unh‖2 +

∥

∥

∥λ̃ n −λ nH

∥

∥

∥

2

≤ C

(

(ηn⋆ )2 +

∥

∥

∥

(

gn − γ|ΓC (unh)

)

+

∥

∥

∥

2)

+C∣

∣

∣

(

λ nH ,(

gn − γ|ΓC (unh)

)

+

)∣

∣

∣

holds. Here, f+ denotes the positive part of a function f ,which means

f+(x) =

{

f (x), f (x) ≥ 0,0, f (x) < 0.

Proof. Combining Lemma 4.1 and Lemma 4.2 yields

‖ũn −unh‖2 +

∥

∥

∥λ̃ n −λ nH

∥

∥

∥

2

≤ C′C⋆(ηn⋆ )2 +C′′〈λ̃ n −λ nH,γ|ΓC (u

nh)−g

n〉

≤

(

C′ +12

C′′)

C⋆(ηn⋆ )2 +C′′

ε2‖ũn −unh‖

2

+C′′(

(1+ ε)ν202ε

‖d‖2 + |(λ nH ,γ|ΓC (d))|)

.

Choosing 0< ε < 2/C′′, we get(

1−C′′ε

2

)

‖ũn −unh‖2 +

∥

∥

∥λ̃ n −λ nH

∥

∥

∥

2

≤ C̃(

(η⋆)2 +‖d‖2 + |(λ nH ,γ|ΓC (d))|)

with

C̃ := max

{(

C′ +C′′

2

)

C⋆,C′′(1+ ε)ν20

2ε,C′′

}

.

The functiond in Lemma 4.2 can be chosen as the harmoniccontinuationd̂n of

(

gn − γ|ΓC(

unh))

+, which is characterised

by the minimisation problem∥

∥d̂n∥

∥

21 = infv∈W n

‖v‖21

with

W n :={

v ∈V n∣

∣

∣γ|ΓC (v) =

(

gn − γ|ΓC (unh)

)

+

}

.

For more details and alternatives in the choice ofd see [12].Since there holds

gn − γ|ΓC (unh)− γ|ΓC

(

d̂n)

≤ 0,

d̂n is an element ofK̃n. From the definition of the norm‖·‖1/2,ΓC , it follows∥

∥d̂n∥

∥

1 =∥

∥

∥

(

gn − γ|ΓC (unh)

)

+

∥

∥

∥

1/2,ΓC.

⊓⊔

Remark 4.1 All terms in the error estimate of Proposition4.1 can be interpreted as typical sources of errors in contactproblems. The term‖(gn − γ|ΓC

(

unh)

)+‖ measures the errorof the geometrical contact condition and the term|(λ nH ,(gn−γ|ΓC

(

unh)

)+)| measures the violation of the complementaritycondition.

Remark 4.2 In our numerical tests, the term

‖(gn − γ|ΓC (unh))+‖

always turned out to be of higher order inh, see [12]. Sinceit is difficult to split this term into its elementwise contribu-tions, it is neglected in the numerical realisation.

In order to apply the error estimation of Proposition 4.1,we have to specify an appropriate error estimatorηn⋆ forProblem 4.2. In principle, each error estimator known fromliterature of variational equations is possible to be used.See[45] or [46] for an overview. For the sake of completeness, aresidual based error estimator for Problem 4.2 is specified:

(ηn⋆ )2 := ∑

K∈Tnη2K

η2K := h2K ‖r‖

2L2(K) + hK ‖R‖

2L2(∂K)

with

r := Fnh +12

k2 △unh −unh

R :=

− 12k2(

q−∂unh∂ν

)

on ∂Ω

− 14k2[

∂unh∂ν

]

else.

The quantityR represents the jump discontinuity in the ap-proximation to the normal flux on the interface. We setq = 0on ΓN andq = −λ nH on ΓC. See [46], Section 2.2, for moredetails.

Remark 4.3 We have used the discrete valueFnh instead ofFn in Problem 4.1, i.e., Proposition 4.1 specifies a tempo-ral local error estimator for the spatial discretisation error.This technique is commonly used in the derivation of errorestimators for numerical methods for ordinary differentialequations, see, e.g., [47]. The presented error estimator ex-presses the spatial error distribution in the single time steps.But it only provides information about the global error un-der the assumptionFnh ≈ F

n, which should hold for smallkandh. An a priori error analysis of the Newmark method inthe context of dynamic contact problems is needed to makea precise statement. To the best of the authors’ knowledge,this analysis does not exist and cannot be derived by stan-dard techniques due to the low regularity of the continuoussolution.

Remark 4.4 The presented error estimate is not restricted tothe Newmark method. It can easily be used for other similartime stepping schemes. It was tested by the authors for theGeneralized-α method, the application of which to dynamiccontact problems is presented in [34,35].

6

5 Adaptive Algorithm

In general, adaptive algorithms for dynamic problems arebased on refinement strategies, which are known from staticproblems, see, e.g., [45,48] for a survey of adaptive algo-rithms for static problems. Commonly used adaptive algo-rithms for time dependent problems, see, e.g., [3,49], per-form an adaptive refinement process using a prescribed tol-erance in every time step. This refinement process is inde-pendent of previous and subsequent time steps. Here, thecrucial point is, that the time interval is passed only once.The tolerance limit cannot be reached, if the solution in theprevious time step has not been calculated exactly enough.Moreover, the difference of the meshes of two succsesivetime steps may significantly increase the error. Usually,rapid changes of the problem parameters are the reason forthis behaviour.

In dynamic Signorini problems, the problem parameterschange rapidly and, thus, the above mentioned algorithmsare not appropriate. An alternative is given by algorithmsbased on the ideas in [50,51]. The refinement procedure issplit into several cycles. The whole time interval is passedin every cycle. A cycle constists of two steps: First, the ap-proximated solution of the whole problem is determined, theerror is estimated and the mesh is refined via a usual re-finement strategy, e.g. a fixed fraction strategy, see [48,45].Multiple hanging nodes in space and time may be generatedby this refinement. In a second step, such nodes are removed.The removal of hanging nodes in time closely connects themeshes of different time steps. A detailed presentation ofthis adaptive algorithm and its extensions will be given in aseperate article.

6 Numerical Results

The error estimator and the adaptive algorithm are tested fortwo examples. The first one is a simplified Signorini prob-lem and the second one is a full 2D Signorini problem.

6.1 A simplified Signorini example

We setΩ := [0,1]2 and I := [0,1]. The initial values areu0 (x1,x2) := 0 andv0 (x1,x2) :=−sin

(

12πx1

)

. Furthermore,we setΓD := {x ∈ Ω |x1 = 0}, ΓC := {x ∈ Ω |x1 = 1} andΓN := ∂Ω\(ΓC ∩ΓD). The rigid foundation is

g := sin(πx2)−1.05.

The length of the time stepsk is chosen as 0.0025. The ini-tial mesh sizeh0 is 0.0625. Five refinement cycles are per-formed, whereas a fixed fraction strategy with constant re-finement fraction of 50% and no coarsening is used.

Fig. 6.1 Geometry of the simplified Signorini example

The geometry of the presented problem is illustrated inFigure 6.1. Meshes for different time steps are presented inFigure 6.2, the corresponding movie is shown in Animation1. The mesh in Figure 6.2 (a) corresponds to the time step,immediately before the first contact between the membraneand the rigid foundation takes places. In Figure 6.2 (b)-(i),the membrane gets into contact with the obstacle and thecontact zone is adaptively refined. We observe a movingfront, which is resolved by the adaptive meshes. In Figure6.3 the estimated convergence for adaptive and uniform re-finement is compared. The estimated error is measured by

η = max1≤n≤N

ηn,

whereηn is given in Proposition 4.1. The number of degreesof freedom is the sum of the number of degrees of freedomof all single time steps. It is obvious, that the adaptive re-finement is more efficient than the uniform refinement. Oneachieves the same accuracy with nearly a factor of 10 lessunknowns.

6.2 A Signorini example

Here, a bar of length 0.2m and height 0.05m is considered.The domain isΩ := [0,0.2]× [0,0.05] and the time inter-val is I :=

[

0,2.5 ·10−3]

. The bar is modelled using a lin-ear elastic material law in a plain strain situation withE :=73·109MPa andν := 0.33. The density isρ := 2770kg/m2.The bar is fixed at the left boundaryΓD = {x ∈ Ω |x1 = 0}.The possible contact surface is given by the set

ΓC = {x ∈ Ω |x1 ≥ 0.15∧ x2 = 0} .

There are nonhomogeneous Neumann boundary conditionsonΓN = {x ∈ Ω |x1 ≥ 0.1∧ x2 = 0.05} with

q := 3.75·107N/m2.

7

(a) Mesh atn = 20 (b) Mesh atn = 50

(c) Mesh atn = 100 (d) Mesh atn = 150

(e) Mesh atn = 200 (f) Mesh atn = 250

(g) Mesh atn = 300 (h) Mesh atn = 350

(i) Mesh atn = 400

Fig. 6.2 Meshes for different time steps of the simplified Signoriniexample

1e-06

1e-05

1e-04

0.001

100000 1e+06 1e+07 1e+08

estim

ated

err

or

number of degrees of freedom

adaptiveuniform

Fig. 6.3 Estimated error for adaptive and uniform refinement in thesimplified Signorini example

The rigid foundation is given by the set{

x ∈ R2 |0≤ x1 ≤ 0.2∧ x2 ≤−0.005}

.

The length of the time steps is 10−5 and the initial meshsizeh0 is 6.25·10−3. Again, five refinement cycles are per-formed, whereas a fixed fraction strategy with constant re-finement fraction of 50% without coarsening is used.

In Figure 6.4 meshes for different time steps are pre-sented the corresponding movie is contained in Animation2. The displacement is scaled by a factor of 5. During thecalculation, contact between the bar and the rigid founda-tion occurs several times. In the Figures 6.4 (a)-(c) and (d)-(f) a sequence is depicted, which starts before contact takesplaces and ends after contact. The influence of contact to themesh is obvious. The last sequence (g)-(i) shows the changeof the mesh during contact. The refined zone and the con-tact zone grow and shrink simultanously. The performanceof the adaptive refinement is compared with the uniform re-finement in Figure 6.5, where the same variables are used asin Figure 6.3. As in the example above, the application ofthe adaptive method is more efficient.

7 Conclusions

The presented space adaptive scheme for dynamic Signoriniproblems shows a significant improvement. More sophis-ticated refinement strategies can further enhance the effi-ciency. However, not every strategy known from non con-tact problems seems to be suited for adaptive schemes forcontact problems. E.g. the refinement strategy presented in[51] compares the refinement indicators over all time steps.The method has been tested by the authors, but the resultsare not satisfactory. The contact zone is not resolved, beforethe first contact takes place. Thus, the algorithm is not ableto detect the moment of the first contact exactly, so that theerror increases significantly.

8

(a) Mesh atn = 20 (b) Mesh atn = 25

(c) Mesh atn = 50 (d) Mesh atn = 80

(e) Mesh atn = 90 (f) Mesh atn = 100

(g) Mesh atn = 211 (h) Mesh atn = 215

(i) Mesh atn = 217

Fig. 6.4 Meshes for different time steps of the 2D Signorini example

Another method to improve the discretisation is given bytime adaptivity; error estimators for the Newmark methodas presented in [6] can be used. This technique will be con-sidered in future works.

The difficulties discussed in Remark 4.3 and the sepa-ration of the spatial and temporal discretisation complicatethe derivation of rigorous a posteriori error estimators. A

1e-04

0.001

0.01

100000 1e+06 1e+07 1e+08

estim

ated

err

or

number of degrees of freedom

adaptiveuniform

Fig. 6.5 Estimated error for adaptive and uniform refinement in theSignorini example

way out could be the application of a space-time Galerkinmethod [52] and of DWR techniques [48].

Acknowledgments

This research work was supported by the Deutsche For-schungsgemeinschaft (DFG) within the Priority Program1180, Prediction and Manipulation of Interaction betweenStructure and Process and within the Collaborative ResearchCenter 708, 3D-Surface Engineering of Tools for SheetMetal Forming - Manufacturing, Modelling, Machining.

References

1. Weinert K, Blum H, Jansen T, Rademacher A (2007) Simulationbased optimization of the NC-shape grinding process with toroidgrinding wheels. Prod Eng 1(3):245-252

2. Johnson C (1993) Discontinous Galerkin Finite Element Meth-ods for Second Order Hyperbolic Problems. Comput Methods ApplMech Eng 107:117-129

3. Li X, Wiberg N-E (1998) Implementation and Adaptivity of aSpace-Time Finite Element Method for Structural Dynamics.Com-put Methods Appl Mech Eng 156:211-229

4. Aubry D, Lucas D, Tie B (1999) Adaptive Strategy for Tran-sient/Coupled Problems. Applications to Thermoelasticity and Elas-todynamics. Comput Methods Appl Mech Eng 176:41-50

5. Bangerth W, Rannacher R (1999) Finite Element Approximation ofthe Acoustic Wave Equation: Error Control and Mesh Adaptation.East-West J Numer Math 7(4):263-282

6. Xie Y M, Zienkiewicz O C (1991) A Simple Error Estimator andAdaptive Time Stepping Procedure for Dynamic Analysis. Earth EngStruct Dyn 20:871-887

7. Li X, Wiberg N-E, Zeng L F (1992) A Posteriori Local Error Es-timation and Adaptive Time-Stepping for Newmark Integration inDynamic Analysis. Earth Eng Struct Dyn 21:555-571

8. Bornemann F A, Schemann M (1998) An Adaptive Rothe Methodfor the Wave Equation. Comput Visual Sci 1:137-144

9. Adjerid S (2002) A Posteriori Finite Element Error Estimation forSecond-Order Hyperbolic Problems. Comput Methods Appl MechEng 191:4699-4719

9

10. Bernadi C, Süli E (2005) Time and Space Adaptivity for theSecond-Order Wave Equation. Math Models Methods Appl Sci15(2):199-225

11. Braess D (2005) A posteriori error estimators for obstacle prob-lems - another look. Numer Math 101(3):415-421

12. Schröder A (2006) Fehlerkontrollierte adaptiveh- undhp-Finite-Elemente-Methoden für Kontaktprobleme mit Anwendungen inderFertigungstechnik. Bayreuther Math Schr 78

13. Johnson C (1992) Adaptive Finite Element Methods for theOb-stacle Problem. Math Models Methods Appl Sci 2:483-487

14. Ainsworth M, Oden J T, Lee C Y (1993) Local A Posteriori Er-ror Estimators for Variational Inequalities. Numer Methods PartialDiffer Equ 9:23-33

15. Hoppe R H W, Kornhuber R (1994) Adaptive Multilevel Methodsfor Obstacle Problems SIAM J Numer Anal 31:301-323

16. Chen Z, Nochetto R (2000) Residual Type A Posteriori Error Es-timates for Elliptic Obstacle Problems. Numer Math 84:527-548

17. Veeser A (2001) Efficient and Reliable A Posteriori ErrorEstima-tors for Elliptic Obstacle Problems. SIAM J Numer Anal 39:146-167

18. Nochetto R H, Siebert K G, Veeser A (2003) Pointwise A Pos-teriori Error Control for Elliptic Obstacle Problems. Numer Math95:163-195

19. Bartels S, Carstensen C (2004)Averaging Techniques yield Reli-able A Posteriori Finite Element Error Control for ObstacleProb-lems. Numer Math 99:225-249

20. Suttmeier F T (2005) On a Direct Approach to Adaptive FE-Discretisations for Elliptic Variational Inequalities. JNumer Math13:73-80

21. Wohlmuth B I (2007) An A Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes. J Sci Comput33:25-45

22. Braess D, Carstensen C, Hoppe R H W (2007) Convergence Anal-ysis of a Conforming Adaptive Finite Element Method for an Obsta-cle Problem. Numer Math 107:455-471

23. Wriggers P, Vu Van T, Stein E (1990) Finite Element Formula-tion of Large Deformation Impact-Contact Problems with Friction.Comput Struct 37(3):319-331

24. Bathe K J, Chaudhary A B (1986) A Solution Method for Staticand Dynamic Analysis of Three-Dimensional Contact Problems withFriction. Comput Struct 24(6):855-873

25. McDevitt T W, Laursen T A (2000) A Mortar-Finite Element For-mulation for Frictional Contact Problems. Int J Numer Methods Eng48(10):1525-1547

26. Puso M, Laursen T A (2002) A 3D Contact Smoothing Methodusing Gregory Patches. Int J Numer Methods Eng 54(8):1161-1194

27. Hager C, Hüeber S, Wohlmuth B I (2008) A Stable Energy Con-serving Approach for Frictional Contact Problems based on Quadra-ture Formulas.Int J Numer Methods Eng 73:205-225

28. Deuflhard P, Krause R, Ertel S (2007) A contact-stabilized New-mark method for dynamical contact problems. Int J Numer MethodsEng 73(9):1274-1290

29. Armero F, Pet̋ocz E (1998) Formulation and analysis of conserv-ing algorithms for frictionless dynamic contact/impact problems.Comput Methods Appl Mech Eng 158:269-300

30. Laursen T A, Chawla V (1997) Design of Energy Conserving Al-gorithms for Frictionless Dynamic Contact Problems. Int J NumerMethods Eng 40:863-886

31. Kane C, Repetto E A, Ortiz M, Marsden J E (1999) Finite ElementAnalysis of nonsmooth contact. Comput Methods Appl Mech Eng180:1-26

32. Pandolfi A, Kane C, Marsden J E, Ortiz M (2002) Time-discretized variational formulation of non-smooth frictional contact.Int J Numer Methods Eng 53:1801-1829

33. Talaslidis D, Panagiotopoulos P D (1982) A Linear FiniteElementApproach to the Solution of the Variational Inequalities arising inContact Problems of Structural Dynamics. Int J Numer Methods Eng18:1505-1520

34. Czekanski A, Meguid S A (2001) Analysis of Dynamic FrictionalContact Problems using Variational Inequalities. Finite Elem AnalDes 37:6861-879

35. Rademacher A (2005) Finite-Elemente-Diskretisierungen für dy-namische Probleme mit Kontakt. Diploma-Thesis, University ofDortmund. Available via http://hdl.handle.net/2003/22996

36. Laursen T A (2002) Computational Contact and Impact Mechan-ics. Springer, Berlin

37. Wriggers P (2002) Computational Contact Mechanics. John Wiley& Sons Ltd, Chichester

38. Kikuchi N, Oden J T (1988) Contact Problems in Elasticity:A Study of Variational Inequalities and Finite Element Methods.SIAM, Studies in Applied Mathematics,

39. Evans L C (1998) Partial Differential Equations, American Math-ematical Society, Providence

40. Panagiotopoulos P D (1985) Inequality Problems in Mechan-ics and Applications, Convex and Nonconvex Energy Functions.Birkhäuser, Basel

41. Newmark N M (1959) A Method of Computation for StructuralDynamics. Journal of the Engineering Mechanics Division ASCE85(EM3):67-94

42. Hughes T J R (2000) The Finite Element Method. Dover Publica-tions, Inc, Mineola

43. Cea J (1978) Lectures on optimization - theory and algorithms.Springer, Berlin

44. Ekeland I, Temam R (1976) Convex analysis and variational prob-lems, Studies in Mathematics and its Applications. North-HollandPublishing Company, Amsterdam

45. Verfürth R (1996) A Review of A Posteriori Error Estimation andAdaptive Mesh-Refinement Techniques, Wiley and Teubner, Chich-ester

46. Ainsworth M, Oden J T (2000) A Posteriori Error Estimation inFinite Element Analysis. John Wiley & Sons, Inc, New York

47. Hermann M (2004) Numerik gewöhnlicher Differentialgleichun-gen. Oldenbourg Verlag, München

48. Bangerth W, Rannacher R (2003) Adaptive Finite Element Meth-ods for Differential Equations. Birkhäuser, Basel

49. Schmidt A, Siebert K G (2004) Design of Adaptive Finite ElementSoftware. Lecture Notes in Computational Science and Engineering,Springer, Berlin

50. Bangerth W, Rannacher R (2001) Adaptive Finite Element Tech-niques for the Acoustic Wave Equation. J Comp Acoustics 9(2):575-591

51. Schmich M, Vexler B (2007) Adaptivity with dynamic meshesfor space-time finite element discretizations of parabolicequations.SIAM J Sci Comput 30(1):369-393

52. Blum H, Jansen T, Rademacher A, Weinert K (2008) Finite Ele-ments in Space and Time for Dynamic Contact Problems. Int J Nu-mer Methods Eng to appear