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Space Adaptive Finite Element Methods for Dynamic SignoriniProblems

Heribert Blum · Andreas Rademacher · Andreas Schröder

the date of receipt and acceptance should be inserted later

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 presented method.

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 re-sult from the contact at the surface of the bodies under con-sideration. Typical examples for engineering processes, wherecontact problems play a dominant role, are grinding pro-cesses. The workpiece interacts with the grinding wheel onlyin a small contact zone. However, the behaviour of the grind-ing machine is strongly affected by the resulting contact

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

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

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

forces. A detailed study of this engineering problem is foundin [1]. For the reliable simulation of such a process, a preciseprediction is required for the contact forces, the contact zoneand their effects onto the whole body. Furthermore, the con-tact zone and the contact forces are strongly depending ontime. Hence, the precise consideration of these dependencesis 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 contactis contained in [21]. Convergence results for adaptive algo-rithms in the context of obstacle problems are proven in [22].

2

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 Lagrange mul-tipliers are presented in [24]. Other techniques for smooth-ing and stabilizing the computation with special finite el-ements, e.g., Mortar finite elements, are presented in [25–27]. In [28], the Newmark scheme is used with an additionalL2-projection for stabilization. Algorithms for dynamic con-tact/impact problems based on the energy- and momentumconservation are derived in [29,30]. An additive splittingof the acceleration into two parts, representing the interiorforces and the contact forces, is the basis of the methodsintroduced in [31,32]. In [33–35] algorithms based on varia-tional inequalities and optimisation algorithms are presented.Detailed surveys of this topic can be found in the mono-graphs [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 spatialerror estimator is derived serving as the basis of an adap-tive algorithm,which is explained in Section 5. In Section6, two examples illustrate the application of the presentedtechniques. The article concludes with a discussion of theresults.

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, ΓC ⊂ ∁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 applied tou. The first and sec-ond time derivatives are denoted by ˙u and u, respectively.In the following, all relations have to be understood almosteverywhere.

We choose the unconstrained trial space

V := W2,∞ (

I ;L2 (Ω))

∩L∞ (

I ;H1 (Ω ,ΓD))

for notational convenience, although the existence 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 andu(t = 0) = v0 for which

(u(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 the equiva-lent strong formulation (see [40])

u−∆u = f in Ω × I

u = 0 onΓD× I∂u∂νd

= 0 onΓN× I

u−g ≥ 0 onΓC× I∂u∂νd

≤ 0 onΓC× I

∂u∂νd

(u−g) = 0 onΓC× I ,

whereνd is the outward normal direction on the boundary.

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= uanda = u for the velocity and the acceleration, respectively.

In the Newmark method,v anda are approximated by

an =1

βk2

(

un−un−1)−1

βkvn−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 = 1

2 is required. Furthermore,the inequality 2β ≥ α ≥ 1

2 has to be valid for unconditionalstability (see [42]). For dynamic contact problems, the choiceα = β = 1

2 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 displacementby solvingthe equation(

a0,ϕ)

+(

∇u0,∇ϕ)

= ( f (0) ,ϕ) .

Here, we assume that no contact takes place in the initialconfiguration (see [42] for more details). The semi-discreteproblem then reads as follows:

Algorithm 3.1 Find u with u0 = u0, v0 = v0 and a0 = a0,such that in every time step n∈ 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. The quantities un, vn, and an are coupled bythe equations (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) withα = β = 1

2 in the inequality(3.3) leads to

(un,ϕ−un)+ 12k2 (∇un,∇(ϕ−un))

≥(

12k2 f (tn)+un−1+kvn−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+kvn−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:

Algorithm 3.2 Find (u,λ ) with u0 = u0, v0 = v0, and a0 =

a0, such that(un,λ n) ∈Vk×Λk 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 ∈ Vk, all µ ∈ Λk and all n∈ 1,2, . . . ,N. Basedon the equations (3.1) and (3.2), the functions vn and an arecalculated in a postprocessing step.

Here,Λk is the dual cone of the set

G :=

µ ∈H1/2 (ΓC)∣

∣

∣µ ≤ 0

.

The dual pairing is expressed by〈·, ·〉. The setVk := H1(Ω ,ΓD)

is the unconstrained trial space discretised in time.The equiv-alence of the two formulations is a well-known conclusionfrom the general theory of minimisation problems in Hilbertspaces presented, e.g., in [44,45].

3.2 Spatial Discretisation

A finite element approach is applied to discretise the Algo-rithm 3.2 based on the mixed formulation. We use adaptivealgorithms with dynamic meshes. Therefore, the trial spacesVn

h andΛnH may vary from time step to time step. Bilinear

basis functions on the meshTn are used for the finite ele-ment spaceVn

h . The discrete Lagrange multipliers are piece-wise constant and are contained in the setΛn

H . The indexH indicates that coarser meshes may be chosen for the La-grange multipliers. In our calculations, we useH = 2h forstability reasons.In Figure 3.1, the results for the Lagrangemultiplier with differentH are compared. The solution forH = h is obviously not stable, whereas stability is observedfor H = 2h.

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 discrete problem in space and time is

Algorithm 3.3 Find(

unh,λ

nH

)

∈ Vnh ×Λn

H with u0h = Ihu0,

v0h = Ihv0 and a0

h = Iha0, such that the system

c(unh,ϕh)+

⟨

λ nH ,γ|ΓC

(ϕh)⟩

= (Fnh ,ϕh) (3.7)

⟨

µH −λ nH,γ|ΓC

(unh)−gn⟩ ≤ 0 (3.8)

4

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

lam

bda

x(a) Lagrange multiplier withH = h

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

lam

bda

x

lambda

(b) Lagrange multiplier withH = 2h

Fig. 3.1 Results for the Lagrange multiplier with differentH.

is valid for all ϕh ∈Vnh andµH ∈Λn

H , n∈ 1,2, . . . ,N. Ad-ditionally, the equations (3.1) and (3.2) determine vn

h and anh.

Here,Fnh is given by

Fnh :=

12

k2 f (tn)+ Ihun−1h +k Ihv

n−1h .

The system (3.7-3.8) leads to the following saddle pointproblem inR

m, wherem depends on the time leveln:

Anun +Bnλ n = Fn

(

µ− λ n)T(

(Bn)T un− gn)

≤ 0,

which must hold for allµ ∈ Rm≤0. Here,An := Mn + 1

2k2Kn

is the generalised stiffness matrix,Mn ∈ Rm×m is the mass

matrix andKn ∈ Rm×m is the stiffness matrix. The matrix

Bn ∈ Rm×m represents the dual pairing in (3.8). Notice, that

all matrices may change from time step to time step.Using the Schur complement ¯un := (An)−1(

Fn−Bnλ n)

, thesaddle point problem is rewritten as the quadratic program

minλ n∈Rm

12

(

λ n)T

Qnλ n−(

λ n)T

[

(Bn)T (An)−1 F− gn]

s. t. λ n≤ 0.

The matrixQn := (Bn)T (An)−1Bn is symmetric and positivesemidefinite. Thus, a standard quadratic program with sim-ple sign constraints has to be solved in each time step. Forthe numerical solution of this quadratic program (QP) anyQP-solver can be used, which only requires a user-definedroutine for the calculation ofQnλ n.

4 Spatial Error Estimation

In this section, an error estimation is derived for the spatialerror in every time step. The estimation is easy to implementand can be evaluated fast. The temporal error is not consid-ered. The idea of the error estimation goes back to Braess[11], who presented it for static obstacle problems. This ideawas extended by Schröder [12] to static Signorini problemseven with friction by introducing a general framework for er-ror control of variational inequalities in Hilbert spaces.Here,we extend this approach to time-dependent Signorini prob-lems by applying the general framework to the semi-discreteAlgorithm 3.1. Since the functionFn is unknown in practice,we have to substituteFn by Fn

h in Algorithm 3.1 to obtainan error estimator, which can be evaluated. This leads to thefollowing saddle point problem:

Auxiliary problem 4.1 Find(

un, λ n)

∈Vk×Λk, such that

c(un,ϕ)+ 〈λ n,γ|ΓC(ϕ)〉 = (Fn

h ,ϕ) (4.1)

〈µ− λ n,γ|ΓC(un)−gn〉 ≤ 0 (4.2)

for all ϕ ∈Vk and all µ ∈Λk.

The consequences of this substitution are discussed inRemark 4.3. The basic idea for deriving the error estimationconsists in the formulation of the auxiliary problem:

Auxiliary problem 4.2 Find un⋆ ∈ Vk, such that the varia-

tional equation

c(un⋆,ϕ) = (Fn

h ,ϕ)−⟨

λ nH ,γ|ΓC

(ϕ)⟩

(4.3)

holds for allϕ ∈Vk.

Auxiliary problem 4.2 corresponds to the first line ofAuxiliary problem 4.1, but with the discrete Lagrange mul-tiplier λ n

H instead ofλ n.

5

We observe thatunh is also a discrete solution of Auxil-

iary problem 4.2. Therefore, the term‖un⋆−un

h‖ can be esti-mated by any error estimatorηn

⋆ > 0 known for variationalequations, i.e.,

‖un⋆−un

h‖2 ≤C⋆(ηn

⋆ )2.

with a constantC⋆ > 0 independent ofVnh andΛn

H .The derivation of the error estimator for the spatial error ofthe time-dependent Signorini problem can be sketched asfollows: We show (see Prop. 4.1) that

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2≤ C

(

‖un⋆−un

h‖2 +s2

0 +s21

)

with a constantC > 0 independent ofh andH. Here,‖·‖ de-notes the norm correponding to the related function spaces.We use theH1(Ω)-norm forVk and the‖ ·‖−1/2,ΓC

-norm for

functions inH−1/2(ΓC). The additional termss0 ands1 aredefined in Remark 4.1 below. Then, an error estimation isobtained by

‖u−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2≤C(ηn)2

with C > 0 and an error estimatorηn defined as

ηn := ηn⋆ +s0 +s1.

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

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2

≤ C′ ‖un⋆−un

h‖2 +C′′〈λ n−λ n

H,γ|ΓC(un

h)−gn〉.

Proof. Let ν0 > 0 be the constant of continuity ofc and letν1 be its constant of ellipticity. Then, we obtain by the ellip-ticity of c

ν1‖un−un

h‖2

≤ c(un−unh, u

n−unh)

= c(un−un⋆, u

n−unh)+c(un

⋆−unh, u

n−unh)

Equation (4.1) and the continuity ofc lead to

c(un−un⋆, u

n−unh)+c(un

⋆−unh, u

n−unh)

≤ 〈λ nH − λ n,γ|ΓC

(un−unh)〉+ ν0‖u

n⋆−un

h‖‖un−un

h‖

Using (4.3) and Young’s inequality, we get

〈λ nH − λ n,γ|ΓC

(un−unh)〉+ ν0‖u

n⋆−un

h‖‖un−un

h‖

= 〈λ nH − λ n,γ|ΓC

(un)−gn〉+ 〈λ n−λ nH ,γ|ΓC

(unh)−gn〉

+ν0‖un⋆−un

h‖‖un−un

h‖

≤ 〈λ n−λ nH,γ|ΓC

(unh)−gn〉+ ν0‖u

n⋆−un

h‖‖un−un

h‖

≤ 〈λ n−λ nH,γ|ΓC

(unh)−gn〉+ ε‖un−un

h‖2 +

ν20

4ε‖un

⋆−unh‖

2.

Therefore, it holds

‖un−unh‖

2≤ν2

0

4ε(ν1− ε)‖un

⋆−unh‖

21

+1

ν1− ε〈λ −λ n

H,γ|ΓC(un

h)−gn〉 (4.4)

with 0 < ε < ν1. In [43], it is proven that

‖λ n−λ nH‖

2≤ C‖un−un⋆‖

2

with a constantC > 0. This and inequality (4.4) yield

‖un−unh‖

2 +‖λ n−λ nH‖

2

≤ ‖un−unh‖

2 +C‖un−un⋆‖

2

≤ ‖un−unh‖

2 +C(‖un−unh‖

2 +‖un⋆−un

h‖2)

≤ C′‖un⋆−un

h‖2 +C′′〈λ −λ n

H ,γ|ΓC(un

h)−gn〉

withC′ := (1+C)/(4ε(ν1−ε))+C andC′′ := (1+C)/(ν1−

ε). ⊓⊔

As mentioned above, we are able to get rid of the term‖un

⋆− unh‖ by using an error estimatorηn

⋆ for the auxiliaryproblem. Lemma 4.1 leads to

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2

≤ C′C⋆(ηn⋆ )2 +C′′〈λ n−λ n

H ,γ|ΓC(un

h)−gn〉.

The remaining term〈λ 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 ∈ Kn :=

v∈Vk|gn− γ|ΓC

(unh)− γ|ΓC

(v)≤ 0

(4.5)

andε > 0. Then, there holds

〈λ n−λ nH,γ|ΓC

(

unh

)

−gn〉

≤ ε2

∥

∥un−unh

∥

∥

2+

(1+ε)ν20

2ε ‖d‖2

+ 12

∥

∥un⋆−un

h

∥

∥

2+ |(λ n

H ,γ|ΓC(d))|.

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

〈λ nH ,γ|ΓC

(unh)−gn〉= 0.

Furthermore, we get

〈λ n,γ|ΓC(un

h)−gn〉

= −〈λ n,gn− γ|ΓC(un

h)− γ|ΓC(d)〉− 〈λ n,γ|ΓC

(d)〉

≤ c(un,d)− (Fnh ,d),

where the defintion ofKn (4.5) and equation (4.1) have beenused. The continuity ofc and Young’s inequality lead to

c(un,d)− (Fnh ,d)

= c(un−unh,d)+c(un

h,d)− (Fnh ,d)

≤ ν0‖un−un

h‖‖d‖+c(unh,d)− (Fn

h ,d)

≤ε2‖un−un

h‖2 +

ν20

2ε‖d‖2+c(un

h,d)− (Fnh ,d).

6

The termc(unh,d)−(Fn

h ,d) is estimatedusing equation (4.3),the continuity ofc, and Young’s inequalityas follows:

c(unh,d)− (Fn

h ,d)

= c(unh−un

⋆,d)− (λ nH,γ|ΓC

(d))

≤ ν0‖un⋆−un

h‖‖d‖− (λ nH,γ|ΓC

(d))

≤12‖un

⋆−unh‖

2 +ν2

0

2‖d‖2 + |(λ n

H ,γ|ΓC(d))|.

⊓⊔

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

Proposition 4.1 There exists a constant C> 0 independentof Vn

h andΛnH , such that

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2

≤ C

(

(ηn⋆ )2 +

∥

∥

∥

(

gn− γ|ΓC(un

h))

+

∥

∥

∥

2)

+C∣

∣

∣

(

λ nH ,

(

gn− γ|ΓC(un

h))

+

)∣

∣

∣

holds. Here, f+(x) := max f (x),0 denotes the positive partof a function f .

Proof. Combining Lemma 4.1 and Lemma 4.2 yields

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2

≤ C′C⋆(ηn⋆ )2 +C′′〈λ n−λ n

H,γ|ΓC(un

h)−gn〉

≤

(

C′+12

C′′)

C⋆(ηn⋆ )2 +C′′

ε2‖un−un

h‖2

+C′′(

(1+ ε)ν20

2ε‖d‖2 + |(λ n

H ,γ|ΓC(d))|

)

.

Choosing 0< ε < 2/C′′, we get(

1−C′′ε

2

)

‖un−unh‖

2 +∥

∥

∥λ n−λ n

H

∥

∥

∥

2

≤ C(

(η⋆)2 +‖d‖2 + |(λ n

H ,γ|ΓC(d))|

)

with

C := max

(

C′+C′′

2

)

C⋆,C′′(1+ ε)ν2

0

2ε,C′′

.

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

(

gn− γ|ΓC

(

unh

))

+, which is characterised

by the minimisation problem

∥

∥dn∥

∥

21 = inf

v∈Wn‖v‖21

with

Wn :=

v∈Vn∣

∣

∣γ|ΓC

(v) =(

gn− γ|ΓC(un

h))

+

.

0.0001

0.001

0.01

0.1

1

10

10 100 1000 10000 100000

s0/s

1

dof

est. errors1s0

Fig. 4.1 Comparison of the values of the contributions to the presentederror estimator.

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

gn− γ|ΓC(un

h)− γ|ΓC

(

dn)≤ 0,

dn is an element ofKn. From the definition of the norm‖·‖1/2,ΓC

, it follows

∥

∥dn∥

∥

1 =∥

∥

∥

(

gn− γ|ΓC(un

h))

+

∥

∥

∥

1/2,ΓC

.

⊓⊔

Remark 4.1All terms in the error estimate of Proposition4.1 can be interpreted as typical sources of errors in con-tact problems. The terms1 := ‖(gn− γ|ΓC

(

unh

)

)+‖ measuresthe error of the geometrical contact condition and the terms0 := |(λ n

H ,(gn− γ|ΓC

(

unh

)

)+)| measures the violation of thecomplementarity condition.

Remark 4.2In our numerical tests, the terms1 always turnedout to be smaller than the termss0 andηn, see Figure 4.1.Since it is difficult to split this term into its elementwise con-tributions, it is neglected for the mesh refinement strategy.

In order to apply the error estimation of Proposition 4.1,we have to specify an appropriate error estimatorηn

⋆ forAuxiliary problem 4.2. In principle, each error estimator knownfrom literature of variational equations is possible to be used.See [46] or [47] for an overview. For the sake of complete-ness, a residual based error estimator for Auxiliary problem4.2 is specified:

(ηn⋆ )2 := ∑

K∈Tn

η2K

η2K := h2

K ‖r‖2L2(K) +hK ‖R‖

2L2(∂K)

7

with

r := Fnh +

12

k2unh−un

h

R :=

− 12k2

(

q−∂un

h∂νd

)

on∂Ω

− 14k2

[

∂unh

∂νd

]

else.

The quantityR represents the jump discontinuity in the ap-proximation to the normal flux on the interface. We setq= 0on ΓN andq = −λ n

H on ΓC. See [47], Section 2.2, for moredetails.

Remark 4.3We have used the discrete valueFnh instead of

Fn in Auxiliary problem 4.1, i.e., Proposition 4.1 specifiesa temporal local error estimator for the spatial discretisationerror. This technique is commonly used in the derivation oferror estimators for numerical methods for ordinary differ-ential equations, see, e.g., [48]. The presented error estima-tor expresses the spatial error distribution in the single timesteps. But it only provides information about the global errorunder the assumptionFn

h ≈ Fn, which should hold for smallk andh. 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,such an analysis does not exist and cannot be derived bystandard techniques due to the low regularity of the contin-uous solution.

Remark 4.4The 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].

5 Adaptive Algorithm

In general, adaptive algorithms for dynamic problems arebased on refinement strategies, which are known from staticproblems, see, e.g., [46,49] for a survey of adaptive algo-rithms for static problems. Commonly used adaptive algo-rithms for time dependent problems, see, e.g., [3,50], 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. The crucialpoint is that the time interval is passed only once. The toler-ance limit cannot be reached, if the solution in the previoustime step has not been calculated exactly enough. Moreover,the difference of the meshes of two succsessive time stepsmay significantly increase the error. Usually, rapid changesof the problem parameters are the reason for this behaviour.

In dynamic Signorini problems, the problem parameterschange rapidly and, thus, the above mentioned algorithms

are not appropriate. An alternative is given by algorithmsbased on the ideas in [51,52]. The adaptive solution algo-rithm is presented in Algorithm 5.1. It is split into severalcycles. The whole time interval is passed in every cycle.First, the approximative solution of the whole problem isdetermined and the error is estimated. Then, the mesh is re-fined via an appropriate spatial refinement strategy, whichcan be chosen by the user, e.g., a fixed fraction strategy, see[46,49]. Multiple hanging nodes in space and time may begenerated by this refinement process. This may cause severeproblems w.r.t. energy conservation. Therefore, the multiplehanging nodes in space are immediately removed by a spa-tial regularisation algorithm. After finishing the whole timeinterval the multiple hanging nodes in time are removed,which closely connects the meshes of different time steps.Then, the stopping criterion is checked. Possible stoppingcriterions are: The number of degrees of freedom is greaterthan a specified number, the number of cycles is greater thana given number or the estimated error is smaller than a spec-ified tolerance. A more detailed presentation of this adaptivealgorithm, its extensions and the mathematical analysis willbe given in a separate article.

Algorithm 5.1 Adaptive solving algorithm

1. Set l= 1.2. Set n= 1 and determine inital values u0

hl, v0

hl, u0

hl∈V1

hl.

3. Determine unhl, vn

hl, un

hl, λ n

Hlbased on (3.1, 3.2, 3.7, 3.8).

4. Evaluate the error estimatorηnl .

5. SetTnl+1 = T

nl and refineT

nl+1 according to the chosen

refinement strategy based onηnl .

6. Remove all multiple hanging nodes in space with Algo-rithm 5.2.

7. If n< N set n← n+1 and go to 38. Remove all double hanging nodes in time with Algorithm

5.3.9. If the stopping criterion is fulfilled, then stop, else set

l ← l +1 and go to 2.

Algorithm 5.2 Spatial regularisation algorithm

1. Mark all cells with multiple hanging nodes.2. If no cells are marked, then stop.3. Refine all marked cells.4. Go to 1

Algorithm 5.3 Temporal regularisation algorithm

1. Set n= 2.2. Mark all cells with multiple backward hanging nodes in

time inTn.

3. Refine all marked cells inTn.4. If n< N, set n← n+1 and go to 2.

8

Fig. 6.1 Geometry of the simplified Signorini example

5. Set n= N−1.6. Mark all cells with multiple forward hanging nodes in

time inTn.

7. Refine all marked cells inTn.8. If n> 1, set n← n−1 and go to 6.9. Stop.

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.

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 takesplace. In Figure 6.2 (b)-(i), themembrane gets into contact with the obstacle and the con-tact zone is adaptively refined. We observe a moving front,

(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

9

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

which is resolved by the adaptive meshes. In Figure 6.3 theestimated convergence for adaptive and uniform refinementis 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.

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.

(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

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 the

10

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

mesh 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 sophisti-cated refinement strategies can probably further enhance theefficiency. However, not every strategy known from contactfree problems seems to be suited for adaptive schemes forcontact problems. E.g., the refinement strategy presented in[52] 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.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. Away out could be the application of a space-time Galerkinmethod [53] and of DWR techniques [49].

Acknowledgments

This research work was supported by the Deutsche For-schungsgemeinschaft (DFG) within the Priority Program

1180, 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.

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