Aproximace a numerická realizace kontaktních

Transcription

1 Univerzita Karlova v Praze Matematicko-fyzikální fakulta DIPLOMOVÁ PRÁCE Tomáš Ligurský Aproximace a numerická realizace kontaktních úloh s daným třením a koeficientem tření, závislým na řešení v 3D. Katedra numerické matematiky Vedoucí diplomové práce: Prof. RNDr. Jaroslav Haslinger, DrSc. Studijní program: Výpočtová matematika

2 Chtěl bych srdečně poděkovat svému vedoucímu za pomoc při studiu dané problematiky i za podnětné rady a připomínky při samotné tvorbě práce. Dále bych chtěl poděkovat doc. RNDr. Radku Kučerovi, Ph.D., a ing. Oldřichu Vlachovi, Ph.D., z Vysoké školy báňské-technické univerzity Ostrava za seznámení s jimi používanou numerickou realizací kontaktních úloh, prvně jmenovanému navíc za poskytnutí svého software pro řešení těchto úloh. Prohlašuji, že jsem svou diplomovou práci napsal samostatně a výhradně s použitím citovaných pramenů. Souhlasím se zapůjčováním práce. V Praze dne 2. dubna 27 Tomáš Ligurský 2

3 Contents Introduction 5 Notation 7 2 Setting of the Problem 3 Existence Result 24 4 Finite Element Approximation 37 5 Mixed Variational Formulation 45 6 Model Examples 58 Conclusions 7 Bibliography 72 3

4 Název práce: Aproximace a numerická realizace kontaktních úloh s daným třením a koeficientem tření, závislým na řešení v 3D. Autor: Tomáš Ligurský Katedra: Katedra numerické matematiky Vedoucí diplomové práce: Prof. RNDr. Jaroslav Haslinger, DrSc. vedoucího: hasling@karlin.mff.cuni.cz Abstrakt: V práci se zabýváme trojrozměrnými kontaktními úlohami s daným třením a koeficientem tření závislým na řešení. Slabou formulaci těchto problémů danou implicitní variační nerovnicí eliptického typu převedeme na úlohu pevného bodu jistého zobrazení z prostoru stop na kontaktní části do sebe. S využitím této formulace dokážeme existenci alespoň jednoho řešení dané úlohy za předpokladu, že koeficient tření je vyjádřen kladnou, spojitou a omezenou funkcí. Za dodatečného předpokladu lipschitzovské spojitosti této funkce s malou konstantou lipschitzovskosti ukážeme dokonce jednoznačnost řešení. Úlohu diskretizujeme pomocí metody konečných prvků. V diskrétním případě provedeme podobné studium existence i jednoznačnosti řešení jako ve spojitém případě a navíc vyšetříme konvergenci řešení diskrétních modelů. Jako prostředek pro hledání pevných bodů použijeme metodu postupných aproximací. Každý její iterační krok vede na řešení kontaktní úlohy s daným třením a koeficientem, který na řešení nezávisí. Pro tuto úlohu pak uvedeme smíšenou variační formulaci, z níž odvodíme duální formulaci použitou ve výsledné numerické metodě. Ukážeme numerické výsledky několika modelových příkladů. Klíčová slova: kontaktní úloha s daným třením, jednostranný kontakt se třením, koeficient tření závislý na řešení. Title: Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution. Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: Prof. RNDr. Jaroslav Haslinger, DrSc. Supervisor s address: hasling@karlin.mff.cuni.cz Abstract: Three-dimensional contact problems with given friction and a coefficient of friction depending on the solution are studied. By means of the fixed-point approach, the existence of at least one solution is proved provided that the coefficient of friction F is represented by a continuous, positive and bounded function. Under an additional assumption, namely the Lipschitz continuity of F with a sufficiently small modulus of the Lipschitz continuity, the uniqueness of the solution is shown. The problem is discretized by the finite element method. The existence and uniqueness of the solution to the discrete problems are investigated in a similar way as it has been done in the continuous setting. Convergence of solutions to the discrete models in an appropriate sense is established. The method of successive approximations is used for finding fixed-points. Each iterative step leads to a contact problem with given friction and a coefficient of friction which does not depend on the solution. We introduce a mixed variational formulation of this problem from which the dual formulation used in computations can be derived. Numerical results of model examples are presented. Keywords: contact problem with given friction, unilateral contact with friction, solutiondependent coefficient of friction. 4

5 Introduction Contact problems form a special branch of mechanics of solids whose goal is to analyse a behaviour of loaded deformable bodies being in mutual contact. Besides unilateral conditions, one has to take into account effects of friction on contact parts. Although it is known from physical experiments that the coefficient of friction may depend on displacements themselves, mainly models of friction in which the coefficient of friction depends only on spacial variables have been used so far. This is why we shall study the case of a solutiondependent coefficient of friction, more precisely a three-dimensional Signorini problem with given friction in which a given slip bound is multiplied by a coefficient of friction depending on the norm of the tangential component of the displacement on the contact part. A weak formulation of contact problems based on this model leads to an implicit variational inequality of elliptic type. To overcome difficulties related to this problem one can derive an equivalent fixed-point formulation for a certain mapping. The first goal of this thesis was to use the fixed-point approach to prove the existence, eventually uniqueness of the solution. To propose and analyse an appropriate finite element discretization of the problem leading to an efficient numerical realization was our second goal. We extend results from [6] devoted to the analogous two-dimensional problem. The main difference between 2 and 3D case is that the tangential component of the displacements is a vector in the latter case. This fact complicates not only the theoretical analysis but mainly the numerical realization because the 3D case leads to a quadratic programming problem with simple as well as quadratic constraints instead of simple ones in 2D. The thesis consists of six chapters. In Chapter we introduce notation. In Chapter 2 we define a classical and a weak solution to the problem and establish a relation between them. In Chapter 3 the weak solution is equivalently characterized as a fixed-point of a mapping acting on the trace space defined on the contact part. Further, we prove the existence of at least one fixed-point of this mapping and we show that the fixed-point is unique provided that the coefficient of friction is Lipschitz continuous with a sufficiently small modulus of the Lipschitz continuity. Chapter 4 deals with the finite element 5

6 INTRODUCTION 6 discretization of the problem. Displacements are approximated by piecewise linear functions and the existence, uniqueness and convergence of the discrete solutions are studied. The method of successive approximations is proposed for finding fixed-points. Each step of this method is defined by a contact problem with given friction in which the coefficient of friction does not depend on the solution. Since this is a crucial step in our computations, Chapter 5 is devoted to a detailed analysis of this problem. We introduce its mixed variational formulation in terms of displacements and contact stresses. This formulation is then discretized: the displacements are approximated by piecewise linear functions again, the contact stresses by piecewise constant functions. Further, we derive the dual formulation in terms of the contact stresses. Finally, results of several numerical experiments are presented in Chapter 6.

7 Chapter Notation The Euclidean norm of a point x R 3 will be denoted by x and the scalar product of two vectors x, y R 3 by x y, in what follows. The diameter of a set T R 3 is defined by diam(t ) = sup{ x y x, y T }. Let be a bounded domain in R 3 with the Lipschitz boundary. The symbol L 2 () stands for the Lebesgue space of real square integrable functions in. This space is equipped with the norm: ( /2 v, = v dx) 2, v L 2 (). The space of all measurable real functions bounded almost everywhere in is denoted L (). The norm in this space is introduced as follows: v,, = ess sup v(x), v L (). x By C () we denote the set of real functions whose partial derivatives of all orders are continuous in and can be continuously extended up to. The symbol C () stands for the subset of C () consisting of functions with a compact support in. For every v C () and every triplet of non-negative integers α = (α, α 2, α 3 ) we denote: where D α v = α v x α x α 2 2 x α 3 3 α = α + α 2 + α 3., 7

8 CHAPTER. NOTATION 8 The same symbol D α v will be used for generalized derivatives of v. Sobolev spaces H k (), k =, 2,..., are defined by H k () = { v L 2 () D α v L 2 (), α k } with the norm ( v k, = α k and the seminorm ( v k, = α =k (D α v) 2 dx) /2, v H k (), (D α v) 2 dx) /2, v H k (). Let Γ be a non-empty and open part. The Lebesgue space of measurable square integrable functions in Γ will be denoted by L 2 (Γ). This space is endowed with the norm: ( /2 v,γ = v ds) 2, v L 2 (Γ). Γ The symbol L (Γ) stands for the space of bounded measurable functions in Γ. The norm in L (Γ) is defined as follows: v,,γ = ess sup v(x), v L (Γ). x Γ By P k, k =,,..., we shall denote the space of all polynomial functions in R 3 of degree up to k. For any subset T R 3 we set P k (T ) = { p T p P k }. Let X be one of the previous spaces. The Cartesian product (X) 3 and its elements will be denoted by bold characters. The respective norms and seminorms are introduced as follows: ( 3 /2 v, = v i,) 2, v = (v, v 2, v 3 ) L 2 (), i= ( 3 /2 v,, = v i,,) 2, v = (v, v 2, v 3 ) L (), i= ( 3 /2 v k, = v i k,) 2, v = (v, v 2, v 3 ) H k (), i= ( 3 /2 v k, = v i k,) 2, v = (v, v 2, v 3 ) H k (), i=

9 CHAPTER. NOTATION 9 ( 3 /2 v,γ = v i,γ) 2, v = (v, v 2, v 3 ) L 2 (Γ), i= ( 3 /2 v,,γ = v i,,γ) 2, v = (v, v 2, v 3 ) L (Γ). i= The symbol X sym stands for a subset of (X) 3 3 containing all symmetric (3 3)- matrix functions: X sym = { τ = (τ ij ) 3 i,j= (X) 3 3 τ ij = τ ji, i, j =, 2, 3 }. Further, ( H k () ) denotes the space of all continuous linear functionals on H k (). This so-called topological dual space of H k () is endowed with the norm: F (H k ()) = sup F (v), F ( H k () ). v H k () v k, v The divergence of a sufficiently smooth vector function v : R 3 R 3 is introduced as follows : div v(x) = v i x i (x), x R 3, while the divergence of a sufficiently smooth matrix function τ : R 3 R 3 3 is defined by div τ (x) = (τ i,i (x), τ 2i,i (x), τ 3i,i (x)), where τ ji,i (x) := τ ji x i (x), x R 3, j =, 2, 3. Let X and Y be two Banach spaces. By X c Y we denote the compact embedding of X into Y. Finally, the symbol L(X, Y ) stands for the space of all linear continuous mappings from X into Y. Here and in what follows the summation convention will be adopted.

10 Chapter 2 Setting of the Problem In this chapter we shall introduce the problem. We start with its classical formulation which consists of a system of partial differential equations and boundary conditions. Then we define a weak formulation and show its relation to the classical one. Let us consider an elastic body occupying a bounded domain R 3 with the Lipschitz boundary which is split into three open, non-empty, non-overlapping parts Γ u, Γ p and such that = Γ u Γ p. The zero displacements are prescribed on Γ u while surface tractions of density p = (p, p 2, p 3 ) L 2 (Γ p ) act on Γ p. The body is unilaterally supported by a rigid foundation S along. For the sake of simplicity of our presentation we shall suppose that is a part of a plane and S is a half-space, i.e. there is no gap p Γ u Γ p S f Figure 2.: Geometry of the model

11 CHAPTER 2. SETTING OF THE PROBLEM between S and for the undeformed configuration. Then the unit outward normal n to is a constant vector along. Besides unilateral constraints imposed on the deformation of on, we shall take into account effects of friction represented by the model with given friction in which a given slip bound g is multiplied by a coefficient of friction F which depends on the norm of the tangential component of the displacement field on. Finally, the body is subject to volume forces of density f = (f, f 2, f 3 ) L 2 (). Our aim is to find an equilibrium state of. The classical formulation of the previous problem consists in finding a displacement vector u = (u, u 2, u 3 ) which satisfies the equilibrium equations and the boundary conditions (2.) (2.5): (equilibrium equations) (kinematical boundary conditions) (static boundary conditions) (unilateral conditions) τ ij x j (u) + f i = in, i =, 2, 3 ; (2.) u i = on Γ u, i =, 2, 3 ; (2.2) T i (u) = p i on Γ p, i =, 2, 3 ; (2.3) u n, T n (u), u n T n (u) = on ; (2.4) (friction conditions) u t = = T t (u) F()g u t = T t (u) = F( u t )g u t on. (2.5) u t The symbol τ (u) = (τ ij (u)) 3 i,j= stands for a symmetric stress tensor which is related to a linearized strain tensor ε(u) = (ε ij (u)) 3 i,j= by means of linear Hooke s law: τ ij (u) = c ijkl ε kl (u), i, j =, 2, 3, where ε ij (u) = 2 ( ui + u ) j, i, j =, 2, 3, x j x i

12 CHAPTER 2. SETTING OF THE PROBLEM 2 and c ijkl L (), i, j, k, l =, 2, 3, are linear elasticity coefficients. They satisfy the following symmetry and ellipticity conditions: c ijkl = c jikl = c klij a.e. in ; (2.6) c ell > : c ijkl (x)ξ ij ξ kl c ell ξ ij ξ ij ξ ij = ξ ji R and a.a. x. (2.7) Further, u n = u n n, u t = u u n n stand for the normal, tangential component of a displacement vector u, respectively, T (u) = (T (u), T 2 (u), T 3 (u)) is a stress vector whose components are T i (u) = τ ij (u)n j and T n (u) = T n (u)n, T t (u) = T (u) T n (u)n denote the normal, tangential component of a stress vector T (u), respectively. Here u n = u n and T n (u) = (T (u)) n. Finally, F is a continuous, positive, bounded function in R + which defines the coefficient of friction depending on the magnitude u t on and g L 2 ( ), g, is a given slip bound. To give a weak formulation of our problem we introduce the following spaces: V = { v H () v = on Γ u }, V = (V ) 3 and a closed convex set of kinematically admissible displacements: K ={v V v n a.e. on }. Definition 2.. By a weak solution to a contact problem with given friction and a solution-dependent coefficient of friction F we mean any displacement field u satisfying the following implicit variational inequality of elliptic type: Find u K such that a(u, v u) + F( u t ) g ( v t u t ) ds F (v u) v K, (P) where a(u, v) = F (v) = τ ij (u)ε ij (v) dx, u, v V, f i v i dx + p i v i ds, Γ p v = (v, v 2, v 3 ) V. Before we establish a relation between the classical and the weak formulation of our problem, we introduce some notation and prove Green s type theorems.

13 CHAPTER 2. SETTING OF THE PROBLEM 3 The symbols H /2 ( ), H /2 ( ) stand for the sets of traces on of all functions belonging to H (), H (), respectively: H /2 ( ) = { ψ L 2 ( ) v H () : v = ψ on }, H /2 ( ) = { ψ L 2 ( ) v H () : v = ψ on }. Let Γ \ Γ u be a non-empty and open part. By H /2 (Γ) we denote the space of traces on Γ of all functions from V : H /2 (Γ) = { ψ L 2 (Γ) v V : v = ψ on Γ } and by H /2 (Γ) the space of functions on Γ whose extension by zero outside of Γ belongs to H /2 ( ). The space H /2 (Γ) will be equipped with the norm: ψ /2,Γ = It can be easily shown that inf v,, ψ H /2 (Γ). (2.8) v V v=ψ on Γ ψ /2,Γ = w,, where w solves the elliptic problem: Find w V such that w = ψ on Γ and ( w) v dx = v V, v = on Γ. This is the weak formulation of the following mixed boundary value problem: w = in ; w = ψ on Γ ; w = on Γ u ; w n = on \ (Γ Γ u). The relation (2.8) defines a norm also in H /2 (Γ) because H /2 (Γ) is a subspace of H /2 (Γ). It is easy to show that H /2 (Γ) is a Hilbert space with the scalar product: v w (ϕ, ψ) H /2 (Γ) = dx, ϕ, ψ H /2 (Γ), x i x i where v, w V are such that ϕ /2,Γ = v,, ψ /2,Γ = w,.

14 CHAPTER 2. SETTING OF THE PROBLEM 4 The topological dual spaces of H /2 (Γ), H /2 (Γ) will be denoted by ( H /2 (Γ) ), H /2 (Γ), respectively, while, /2,Γ stands for a duality pairing in both cases. The norms in ( H /2 (Γ) ) and H /2 (Γ) are introduced as follows: µ (H /2 (Γ)) = µ /2,Γ = µ, ψ /2,Γ sup, µ ( H /2 (Γ) ), ψ H /2 (Γ) ψ /2,Γ ψ sup ψ H /2 (Γ) ψ µ, ψ /2,Γ ψ /2,Γ, µ H /2 (Γ), respectively. The symbols H /2 + (Γ), ( H /2 (Γ) ) stand for the cones of all nonnegative elements of H /2 (Γ), ( H /2 (Γ) ) +, respectively: H /2 + (Γ) = { ψ H /2 (Γ) ψ a.e. on Γ }, ( H /2 (Γ) ) + = { µ ( H /2 (Γ) ) µ, ψ /2,Γ ψ H /2 + (Γ) }. In a similar way we define H+(Γ) /2 and H /2 + (Γ). Accordingly to our notation, H /2 (Γ) is the trace space on Γ of functions which belong to V and H /2 (Γ) is the space of all vector functions on Γ whose extension by zero vector outside of Γ belongs to H /2 ( ). The norm in H /2 (Γ) as well as in H /2 (Γ) is defined by ψ /2,Γ = inf v,, ψ H /2 (Γ). v V v=ψ on Γ It is easy to verify that ψ /2,Γ = w,, where w = (w, w 2, w 3 ) solves the elliptic problem: Find w V such that w = ψ on Γ and ( w i ) v i dx = v V, v = on Γ. The respective classical formulation says that the i-th component w i of w satisfies: w i = in ; w i = ψ i on Γ ; w i = on Γ u ; w i n = on \ (Γ Γ u),

15 CHAPTER 2. SETTING OF THE PROBLEM 5 where ψ = (ψ, ψ 2, ψ 3 ). It is readily seen that ( 3 /2 ψ /2,Γ = ψ i /2,Γ) 2. (2.9) i= The symbols ( H /2 (Γ) ), H /2 (Γ) stand for the topological dual spaces of H /2 (Γ), H /2 (Γ), respectively, and µ (H /2 (Γ)) = µ /2,Γ = Further we denote: µ, ψ /2,Γ sup, µ ( H /2 (Γ) ), ψ H /2 (Γ) ψ /2,Γ ψ sup ψ H /2 (Γ) ψ µ, ψ /2,Γ ψ /2,Γ, µ H /2 (Γ). X(Γ) = { (ϕ, ψ) L 2 (Γ) L 2 (Γ) v V : vn = ϕ, v t = ψ on Γ }. (2.) Let γ Γ : H /2 (Γ) X(Γ) be a linear operator defined by γ Γ ξ = (ξ n, ξ t ), ξ H /2 (Γ). Lemma 2.2. The operator γ Γ is an injection of H /2 (Γ) onto X(Γ). Proof. The mapping γ Γ is onto X(Γ) by the definition. Injectivity: Let (ϕ, ψ) X(Γ). Then } ξ n = ϕ on Γ ξ t = ξ ξ n n = ψ holds for every ξ H /2 (Γ) such that γ Γ ξ = (ϕ, ψ). This is satisfied if and only if ξ = ψ + ϕn on Γ. Since has only the Lipschitz boundary, it does not hold that X(Γ) = H /2 (Γ) H /2 (Γ), in general. On the other hand, the previous lemma enables us to introduce the norm in X(Γ) as follows: (ϕ, ψ) /2,Γ := γ Γ (ϕ, ψ) /2,Γ, (ϕ, ψ) X(Γ).

16 CHAPTER 2. SETTING OF THE PROBLEM 6 We denote the topological dual space of X(Γ) by X (Γ) and a duality pairing between X (Γ) and X(Γ) by [, ] /2,Γ. The space X (Γ) is endowed with the norm: µ X (Γ) = sup [µ, (ϕ, ψ)] /2,Γ, µ X (Γ). (ϕ,ψ) X(Γ) (ϕ, ψ) /2,Γ (ϕ,ψ) (,) Next we shall suppose that Γ := \ Γ u so that H /2 (Γ) = H /2 (Γ). Further let H(div, ) = { τ L 2 sym() div τ L 2 () }. Here div τ L 2 () is understood in a weak sense, i.e. there exists a function h L 2 () such that h ϕ dx = τ ij ε ij (ϕ) dx ϕ C (), and div τ := h. It is very easy to show that H(div, ) is a Hilbert space with the scalar product (τ, σ) H(div,) = τ ij σ ij dx + (div τ ) (div σ) dx, τ, σ H(div, ). Theorem 2.3. (The First Green Theorem.) There exists a unique mapping T L ( H(div, ), H /2 (Γ) ) such that τ ij ε ij (v) dx+ (div τ ) v dx = T (τ ), v /2,Γ τ H(div, ) v V, where T (τ ), v /2,Γ := T (τ ), v Γ /2,Γ. Proof. Let us set V = V C (), H /2 (Γ) = { ψ L 2 (Γ) v V : v = ψ on Γ }. Applying classical Green s formula we obtain: τ ij ε ij (v) dx + (div τ ) v dx = τ ij v i n j ds Γ τ C sym() v V. (2.)

17 CHAPTER 2. SETTING OF THE PROBLEM 7 Let τ Csym() be fixed and denote L(v) := τ ij ε ij (v) dx + (div τ ) v dx, v V. Since L depends only on the values of v on Γ, or equivalently L(v) = L(w) v, w V such that v = w on Γ, as follows from (2.), one can define L : H /2 (Γ) R by L(ψ) := L(v) = τ ij ε ij (v) dx + (div τ ) v dx ψ H /2 (Γ), where v V is an arbitrary function satisfying v = ψ on Γ. Then there exists a positive constant c such that ( ) /2 ( /2 L(ψ) τ ij τ ij dx ε ij (v)ε ij (v) dx) + div τ, v, c τ H(div,) v, ψ H /2 (Γ) v V, v = ψ on Γ, making use of the Friedrichs inequality. From the density of V in V it follows that H /2 (Γ) is dense in H /2 (Γ), as well and so that L(ψ) c τ H(div,) v, ψ H /2 (Γ) v V, v = ψ on Γ L(ψ) c τ H(div,) inf v V v=ψ on Γ v, = c τ H(div,) ψ /2,Γ (2.2) for every ψ H /2 (Γ). This proves that L is continuous in H /2 (Γ). Thus for a given τ Csym() there exists unique T (τ ) H /2 (Γ) such that T (τ ), v /2,Γ = τ ij ε ij (v) dx + (div τ ) v dx v V. (2.3) It is readily seen that T : Csym() H /2 (Γ) defined by (2.3) is a linear operator. In addition, it can be shown (see [3]) that Csym() is dense in H(div, ). Hence T L(H(div,),H /2 (Γ)) = sup T (τ ) /2,Γ c, τ C τ sym() H(div,) τ if (2.2) is taken into account. H(div, ). Thus T can be extended from C sym() on

18 CHAPTER 2. SETTING OF THE PROBLEM 8 Remark 2.4. If τ is smooth enough then T (τ ), v /2,Γ = τ ij n j v i ds v V. Γ From Lemma 2.2 we have: T (τ ), v /2,Γ = T (τ ), γ Γ γ Γv = [( ) γ /2,Γ Γ T (τ ), γγ v ] /2,Γ holds for every τ H(div, ) and every v V, where ( ) γ Γ : H /2 (Γ) X ( (Γ) ) is the adjoint mapping to γ Γ. Next, we shall write T nt(τ ) instead of γ Γ T (τ ). We arrive at the following theorem. Theorem 2.5. (The Second Green Theorem.) There exists a unique mapping T nt L(H(div, ), X (Γ)) such that τ ij ε ij (v) dx + (div τ ) v dx = [T nt (τ ), (v n, v t )] /2,Γ τ H(div, ) v V. Remark 2.6. If τ is smooth enough then [T nt (τ ), (v n, v t )] /2,Γ = (T n v n + T t v t ) ds v V, where T n = τ ij n j n i and T t = T T n n on Γ. Γ Now we are able to establish a relation between the classical and the weak formulation of (P). Let u be a solution to (P) and let ϕ C (). Inserting v = u ± ϕ K into (P) we have: τ ij (u)ε ij (ϕ) dx = f ϕ dx ϕ C (). Therefore τ (u) H(div, ) and div τ (u) = f in. Applying the First Green Theorem to (P) we obtain : T (u), v u /2,Γ + F( u t ) g ( v t u t ) ds p (v u) ds Γ p We shall write T (u) instead of T (τ (u)) here and in what follows. v K. (2.4)

19 CHAPTER 2. SETTING OF THE PROBLEM 9 Let ϕ C () be such that supp ϕ Γ p. Then v = u ± ϕ K and T (u), ϕ /2,Γ = p ϕ ds ϕ C (), supp ϕ Γ p, (2.5) Γ p which is the weak form of (2.3). Let T (u) be the restriction of T (u) on defined by T (u), v /2, := T (u), v /2,Γ p v ds v V. Γ p It is easy to show that T (u), v /2, depends only on v Γc. Indeed, let v, v V be such that v = v on. Then v v = on Γ u. Since the subset of C () of all functions ϕ with supp ϕ Γ p is dense in Ṽ = { v H () v = on Γ u }, it follows from (2.5) that T (u), v /2, = T (u), v /2,. Thus T (u) ( H /2 ( ) ). Similarly as before it holds: T (u), v /2, = T (u), γ γ Γc v /2, = [( γ ) T (u), γγc v ] /2, = [ Tnt (u), γ Γc v ] /2, v V, where γ Γc v := γ Γ v Γc and T nt (u) := ( γ ) T (u) denotes the restriction of T nt (u) on. Let us introduce the following sets: S n ( ) = { ϕ L 2 ( ) v V : v n = ϕ on }, S t ( ) = { ψ L 2 ( ) v V : v t = ψ on }. It is readily seen that functions from S t ( ) satisfy the following condition: ψ S t ( ) = ψ n = on. Since is a flat part of (by assumption), we shall prove that X( ) = S n ( ) S t ( ), S n ( ) = H /2 ( ), S t ( ) = { ψ H /2 ( ) } ψ n = a.e. on, (2.6)

20 CHAPTER 2. SETTING OF THE PROBLEM 2 where X( ) is defined by (2.) with Γ :=. This means that the normal and tangential components of displacement vectors can be treated separately. Indeed, let ϕ S n ( ) and ψ S t ( ) be given. Then one can find functions v, v 2 V such that } vn = ϕ v 2 t = v 2 vnn 2 on. = ψ Our aim is to find a function v V such that } v n = ϕ v t = v v n n = ψ on. Since n is constant along, we immediately obtain: ϕ = ( ) v n H/2 ( ), ψ = ( v 2 Γc )t H/2 ( ) so that Therefore ψ + ϕn H /2 ( ). (2.7) S n ( ) H /2 ( ), S t ( ) { ψ H /2 ( ) ψ n = a.e. on }. From (2.7) it follows that there exists a function v V such that Hence and In addition (2.6) 2,3 hold true. v = ψ + ϕn on. v n = ϕ on v t = ψ on. Let us observe that (2.6) does not hold under the assumption on Lipschitz continuity of as illustrates the following example. Let = (, 3) (, 3) (, 3), Γ u = {} (, 2) (, 2), = 2, where = (, 3) (, 3) {}, 2 = (, 3) {3} (, 3), and Γ p = \ ( Γ u ) (see Figure 2.2). Let ϕ := S n ( )

21 CHAPTER 2. SETTING OF THE PROBLEM Γ u Γ p 2 x 3 x 2.5 Γ p Γ p x Figure 2.2: Geometry and v V be such that v = (,, ) on. Then ψ := (v ) t S t ( ). If there was a function v V such that } v n = ϕ v t = v v n n = ψ on, it would satisfy: But while v = ψ + ϕn on. (2.8) ψ + ϕn = (,, ) ψ + ϕn = (,, ) on on 2 so that the second and the third component of ψ + ϕn have a jump across 2. From this it follows that which contradicts to (2.8). ψ + ϕn / H /2 ( ), Using (2.6) we can write: T (u), v /2, = [ Tn (u), v n ]/2, + [ Tt (u), v t ] /2, v V, where ( Tn (u), T t (u) ) := T nt(u) X ( ) = S n( ) S t( ). (2.9)

22 CHAPTER 2. SETTING OF THE PROBLEM 22 Here S n( ), S t( ) denote the topological dual spaces of S n ( ), S t ( ), respectively. From (2.4) and the definitions of T n (u), T t (u) we obtain: [ Tn (u), v n u n ]/2, + [ Tt (u), v t u t ] /2, + F( u t ) g ( v t u t ) ds v K. (2.2) Let us choose v V such that v n = u n ϕ where ϕ H /2 + ( ) and v t = u t on. Then v K and [ Tn (u), ϕ ] /2, ϕ H /2 + ( ). (2.2) Let v K be such that v n =, v n = 2u n, v t = u t v t = u t } on. It follows from (2.2) that [ Tn (u), u n ] /2, =. (2.22) Relations (2.2) and (2.22) represent the weak form of the unilateral boundary conditions T n (u), Tn (u)u n = on. Let v K in (2.2) be such that v n = u n, v t =, 2u t on. This choice of v leads to [ ] Tt (u), u t /2, + F( u t )g u t ds = (2.23) and (2.2), (2.22) and (2.23) imply: [ Tn (u), v n ]/2, + [ Tt (u), v t ] /2, + If v K is such that v n =, v t = ±ψ S t ( ), we get: F( u t )g v t ds v K. [ Tt (u), ψ ] /2, F( u t )g ψ ds ψ S t ( ). (2.24) Then (2.23), (2.24) are equivalent to Tt (u) F( ut )g a.e. on, (2.25) ( Tt (u) ) ut + F( u t )g u t = a.e. on, (2.26)

23 CHAPTER 2. SETTING OF THE PROBLEM 23 respectively. It is also easy to see that (2.25) and (2.26) imply (2.5) 2. Indeed, let x be such that u t (x). It holds: = ( Tt (u) ) ut + F( u t )g u t ( Tt (u) + F( ut )g ) u t at x, making use of the Cauchy-Schwartz inequality. Hence Tt (u) = F( ut )g at x (2.27) and ( Tt (u) ) ut = Tt (u) u t at x. (2.28) From (2.28) it follows that the Cauchy-Schwartz inequality is satisfied with the equality sign so that there exists a number λ R such that T t (u)(x) = λu t (x). Next, (2.27) yields: λ = F( u t )g u t and (2.28) determines the sign of λ so that T t (u) = F( u t )g u t u t at x at x.

24 Chapter 3 Existence Result In the present chapter we shall derive an equivalent fixed-point formulation of our problem. With the aid of this formulation we prove the existence of at least one solution to the problem and we give conditions guaranteeing the uniqueness of this solution. Firstly, we present several important auxiliary results which will be used in what follows. Lemma 3.. Let f C (R 3 ) be such that f L (R 3 ). Then f v H () for any v H () and (f v) x i = f y j v v j x i a.e. in, i =, 2, 3. Proof. Let v H () be arbitrary but fixed. Since C () = H (), there exists a sequence { v k} C () such that v k v in H (), k. (3.) Therefore, one can pass to a subsequence of { v k} such that v k i v i a.e. in, k, i =, 2, 3, (3.2) v k i x j v i x j a.e. in, k, i, j =, 2, 3. (3.3) First we shall show that { f v k} is a Cauchy sequence in H (). Indeed, from the mean value theorem and (3.) we get: f v k f v l 2 = (, f v k f v l) 2 dx f 2,,R 3 v k v l 2, k,l. (3.4) Here and in what follows selected subsequences will be usually denoted in the same way as the original sequence. 24

25 CHAPTER 3. EXISTENCE RESULT 25 Furthermore, let i {, 2, 3} be given. It holds: ( f v k) ( ) f v l x i x i = f v k vk j f v l vl j, y j x i y j x i, f v k vk j f v k v j y j x i y j x i + f v k v j f v v j, y j x i y j x i + f v v j f v l v j y j x i y j x i + f v l v j f v l vl j, y j x i y j x i,, =: I + I 2 + I 3 + I 4. (3.5) Similarly as in (3.4) we have: I f,,r 3 v k v k,, I 4 f,,r 3 v v l l,. (3.6) As f C (R 3 ), it follows from (3.2) that f v k f v y j y j a.e. in, k, j =, 2, 3, f v f v l y j y j a.e. in, l, j =, 2, 3, and the Lebesgue dominated convergence theorem gives: ( [( f I 2 = v k f ) ] ) 2 /2 vj k v dx, y j y j x i ( [( f I 3 = v f ) ] ) 2 /2 v l vj l dx, y j y j x i (3.7) respectively. The inequality (3.5) together with (3.6) and (3.7) yield: ( f v k) ( ) f v l x i x i, k, l., Consequently, there exists a function g H () such that f v k g in H (), k, and a subsequence of { v k} such that f v k g a.e. in, k. (3.8)

26 CHAPTER 3. EXISTENCE RESULT 26 At the same time (3.2) implies: Comparing (3.8) and (3.9) we see that f v k f v a.e. in, k. (3.9) g = f v a.e. in. Let i {, 2, 3} be given. In a similar way as before it can be shown that one can pass to a subsequence of { v k} such that As we obtain: ( f v k) x i ( f v k) x i k g = x i (f v) x i a.e. in. = f y j v k vk j x i k f y j v v j x i a.e. in, (f v) x i = f y j v v j x i a.e. in. Lemma 3.2. Let v H (). Then v H () and In addition, if v V then v V. v, v,, v, v,. Proof. Clearly, v L 2 (). Let ε > be given and denote f ε (y) = y 2 + y2 2 + y3 2 + ε 2 ε, y = (y, y 2, y 3 ) R 3. Then f ε y (y) = j y R 3 ε >, j =, 2, 3, y j y 2 + y2 2 + y3 2 + ε 2 f ε (y) y j y R3 ε >, j =, 2, 3. From the previous lemma we know that f ε v H () ε > and for every ϕ C (), i {, 2, 3} it holds: (f ε v) ϕ dx = ϕ (f ε v) dx x i x i v j v = ϕ j dx. (3.) v 2 + v2 2 + v3 2 + ε 2 x i

27 CHAPTER 3. EXISTENCE RESULT 27 Letting ε + in (3.) and using the Lebesgue dominated convergence theorem we have: v ϕ dx = ϕ ( v ) dx, x i x i where { vj v j v x ( v ) := i if v = x i if v = belongs to L 2 (), as well. Schwartz inequality implies v 2, = 3 i= 3 i= v 2, = i,j= i= { v } { v } 3 ( vj a.e. in, i =, 2, 3, Therefore v H (). Further, the Cauchy- x i ( ) 2 vj v j dx v x i ( 3 ) ( vj 2 3 ( ) ) 2 vj dx v 2 x i j= ) 2 dx = v 2,, j= 3 vi 2 dx + v 2, v 2, + v 2, = v 2,. Finally, if v = on Γ u then v = on Γ u implying v V. The same results hold for functions from the Sobolev space H ( ). We start with its definition. Since is supposed to be a flat part of, one can find an orthogonal transformation A : R 3 R 3 such that A( ) R 2 {}. Let x = (x, x 2, x 3) := A(x) = A(x, x 2, x 3 ) and G c = { (x, x 2) R 2 A (x, x 2, ) }. With any real function ψ defined on we associate a function ψ c : G c R : ψ c (x, x 2) := ψ ( A (x, x 2, ) ), (x, x 2) G c. (3.) We introduce the Sobolev space H ( ) and its norm as follows: H ( ) = { ψ L 2 ( ) ψc H (G c ) }, ψ,γc := ψ c,gc, ψ H ( ).

28 CHAPTER 3. EXISTENCE RESULT 28 In accordance with our notation, H ( ) is the Cartesian product (H ( )) 3 with the norm ( 3 ) /2 ψ,γc = ψ i 2,, ψ = (ψ, ψ 2, ψ 3 ) H ( ). It is readily seen that i= ψ,γc = ψ c,gc ψ = (ψ, ψ 2, ψ 3 ) H ( ), where the i-th component of ψ c is defined by (3.) with ψ := ψ i. Lemma 3.3. Let ψ H ( ). Then ψ H ( ) and ψ,γc ψ,γc. Proof. The assumption ψ H ( ) means that ψ c H (G c ). Arguing exactly as in Lemmas 3. and 3.2 one can verify that ψ c H (G c ) as well as ψ c,gc ψ c,gc. Consequently, ψ H ( ) and ψ,γc = ψ c,gc ψ c,gc = ψ,γc. Let γ : V L 2 ( ) denote the trace operator on : γv = v Γc, v V, and let γ n : V L 2 ( ) and γ, γ t : V L 2 ( ) be the following trace mappings: γv = (γv, γv 2, γv 3 ), γ n v = (γv) n, γ t v = (γv) t, v = (v, v 2, v 3 ) V. Next, we establish useful properties of γ, γ, γ n and γ t. Lemma 3.4. If v V, v V then γv, γ n v H /2 ( ), γv, γ t v H /2 ( ) and γv = γ v a.e. on. Proof. From (2.6) 2,3 we know that γ n v H /2 ( ), γ t v H /2 ( ) v V, respectively. The remaining assertions are straightforward.

29 CHAPTER 3. EXISTENCE RESULT 29 Lemma 3.5. If ψ H /2 ( ) then ψ, ψ n H /2 ( ) and ψ t H /2 ( ). In addition, ψ /2,Γc ψ /2,Γc ψ H /2 ( ), (3.2) ψ t /2,Γc 2 ψ /2,Γc ψ H /2 ( ). (3.3) Proof. Let ψ H /2 ( ) be given and let v V be such that ψ = γv. From Lemmas 3.2 and 3.4 we obtain: ψ = γv = γ v H /2 ( ). (3.4) As the unit outward normal vector n to is constant along, it holds that ψ n H /2 ( ) and ψ t H /2 ( ). Next, we prove (3.2). Using (3.4) we see: {v V v = ψ on } { w H () w = v in, where v V, v = ψ on Γc } and consequently This and Lemma 3.2 yield: ψ /2,Γc = inf v, inf v,. v V v V v= ψ on v=ψ on inf v, v V v= ψ on inf v, v V v=ψ on inf v, = ψ v V /2,Γc. v=ψ on Now we show (3.3). Since n is constant along, it follows from (2.9) that ( 3 ) /2 ψ t /2,Γc ψ /2,Γc + ψ n n /2,Γc = ψ /2,Γc + ψ n n i 2 /2, ψ /2,Γc + ψ n /2,Γc ψ /2,Γc + n i ψ i /2,Γc ( 3 ) /2 ( 3 ) /2 ψ /2,Γc + n 2 i ψ i 2 /2, = 2 ψ /2,Γc holds for every ψ H /2 ( ). i= i= Lemma 3.6. Let v V, { v k} V be such that Then i= v k v in H (), k. (3.5) γ t v k γ t v in H /2 ( ), k, (3.6) and consequently γt v k γt v in H /2 ( ), k. (3.7)

30 CHAPTER 3. EXISTENCE RESULT 3 Proof. Let µ ( H /2 ( ) ) be given. Using (3.3) we get: µ, γ t v /2,Γc µ (H /2 ()) γ t v /2,Γc 2 µ (H /2 ()) γv /2,Γc 2 µ (H /2 ()) v, v V, i.e. the mapping v µ, γ t v /2,Γc defines a linear continuous functional on V. From this and (3.5) we obtain (3.6). Further, weak convergence of { γ t v k} in H /2 ( ) implies its boundedness: c > : γt v k /2,Γc c k N. According to (3.2), the sequence { γt v k } is bounded in H /2 ( ), as well. Thus, there exists a subsequence { γt v l } { γt v k } and a function ϕ H /2 ( ) such that γt v l ϕ in H /2 ( ), l. It is easy to show that ϕ = γ t v a.e. on and that the whole sequence { γ t v k } tends weakly to ϕ in H /2 ( ). Alternative formulation of the problem For every ϕ H /2 + ( ) we shall define the following auxiliary problem: Find u := u(ϕ) K such that a(u, v u) + F(ϕ) g ( v t u t ) ds F (v u) v K. (P(ϕ)) The existence and uniqueness of a solution to (P(ϕ)) follows from the next theorem. Theorem 3.7. For every ϕ H /2 + ( ) there exists a unique solution u to the following minimization problem over K: } Find u := u(ϕ) K such that (P (ϕ)) J ϕ (u) J ϕ (v) v K, where J ϕ (v) = a(v, v) F (v) + j 2 ϕ(v) with j ϕ (v) = F(ϕ)g v t ds. (3.8) In addition, a function u K is a solution to (P (ϕ)) if and only if it solves (P(ϕ)).

31 CHAPTER 3. EXISTENCE RESULT 3 Proof. One can easily verify that j ϕ is a non-negative, convex, continuous functional in V for every ϕ H /2 + ( ) and a is a symmetric, continuous bilinear form on V. Korn s inequality ensures the H ()-ellipticity of a on V. The rest of the assertion follows from [4]. Let ϕ H /2 + ( ) be given and let u := u(ϕ) K be the unique solution to (P(ϕ)). As γ t u H /2 ( ), we get from Lemma 3.5 that γ t u H /2 + ( ). This makes it possible to define a mapping Ψ : H /2 + ( ) H /2 + ( ) as follows: Ψ : ϕ γ t (u(ϕ)), ϕ H /2 + ( ), (3.9) where u(ϕ) K solves (P(ϕ)). From (P) and (3.9) we arrive at the following alternative (and equivalent) definition. Definition 3.8. By a weak solution to a contact problem with given friction and a solution-dependent coefficient of friction F we mean any function u K such that γ t u is a fixed-point of Ψ in H /2 + ( ): Ψ( γ t u ) = γ t u on. To prove the existence of at least one fixed-point we examine basic properties of Ψ. Before doing this we present another auxiliary result yet. Lemma 3.9. Let v V, { v k} V be such that Then v k v in H (), k. Proof. From Korn s inequality it follows that lim inf k a( v k, v k) a(v, v). (3.2) v, := (a(v, v)) /2, v V, is a norm on V. Since every norm in a Banach space is weakly lower semicontinuous, we obtain (3.2). For every R > we set B R = { ψ H /2 + ( ) ψ /2,Γc R }.

32 CHAPTER 3. EXISTENCE RESULT 32 Lemma 3.. The mapping Ψ maps B R into itself with R := 2 F (H ()) c ell c K, where c ell > is the constant in (2.7) and c K > is the constant of Korn s inequality: c K v 2, ε ij (v)ε ij (v) dx v V. Proof. Let ϕ H /2 + ( ) be arbitrary but fixed and denote u := u(ϕ) the solution to (P(ϕ)). Inserting v := into (P(ϕ)) we obtain: a(u, u) F(ϕ)g u t ds F (u). Therefore c ell c K u 2, c ijkl ε kl (u)ε ij (u) dx + F(ϕ)g u t ds F (u) F (H ()) u, (3.2) in virtue of (2.7) and Korn s inequality. Using (3.2) and (3.3) we get: γ t u /2,Γc γ t u /2,Γc 2 γu /2,Γc 2 u,. (3.22) From (3.2) and (3.22) the assertion of the lemma follows. Lemma 3.. Let ϕ H /2 + ( ), { ϕ k} H /2 + ( ) be such that Then ϕ k ϕ in H /2 ( ), k. Ψ ( ϕ k) Ψ(ϕ) in H /2 ( ), k, i.e. the mapping Ψ is weakly continuous in H /2 + ( ). Proof. Let u k := u ( ϕ k) K be a solution to ( P ( ϕ k)), k N: a ( u k, v u k) + F ( ϕ k) g ( v t ) ( ) u k t ds F v u k v K. From (3.2) it follows that the sequence { u k} is bounded in H (): c > : u k, c k N.

33 CHAPTER 3. EXISTENCE RESULT 33 Hence there exists a subsequence { u l} { u k} and a function u V such that u l u in H (), l. (3.23) We prove that u solves (P(ϕ)). First, u K because K is a closed, convex set. Obviously it holds: From Lemma 3.9 we obtain: lim sup l Since H /2 ( ) c L 2 ( ), we have: lim a( u l, v ) = a(u, v) v K, l lim F ( v u l) = F (v u) v K. l ( a ( u l, u l)) = lim inf l a( u l, u l) a(u, u). ϕ l ϕ in L 2 ( ), l, and there exists a subsequence of { ϕ l} (denoted by the same symbol) such that ϕ l ϕ a.e. on, l. This and continuity of F yield: F ( ϕ l) F(ϕ) a.e. on, l. (3.24) Moreover, it follows from Lemma 3.6 that γt u l γt u in H /2 ( ), l, and also γt u l γt u in L 2 ( ), l. (3.25) Further F ( ϕ l) g ( v t ) u l t ds F(ϕ) g ( v t u t ) ds Γ c ( ( ) F ϕ l F(ϕ) ) g v t ds + F Γ ( ϕ l) g ( u t ) u l t ds c ( ( + )) F(ϕ) F ϕ l g u t ds =: I + I 2 + I 3. (3.26)

34 CHAPTER 3. EXISTENCE RESULT 34 From the Lebesgue dominated convergence theorem and (3.24) it follows that while (3.25) yields: I, I 3, l, (3.27) I 2 F,,R + g,γc γt u γt u l,γc l. (3.28) Finally, (3.26), (3.27) and (3.28) lead to lim F ( ϕ l) g ( v t ) u l t ds = F(ϕ) g ( v t u t ) ds. l Letting l in ( P ( ϕ l)) and using the previous results we see that a(u, v u) + F(ϕ) g ( v t u t ) ds F (v u) v K, i.e. u := u(ϕ) solves (P(ϕ)). Since (P(ϕ)) has a unique solution, the original sequence { u k} tends weakly to u in H () and making use of Lemma 3.6. γt u k γt u in H /2 ( ), k, Since H /2 ( ) is a separable Hilbert space and B R from Lemma 3. is its non-empty, closed, bounded, convex subset, the existence of at least one fixed-point of Ψ in H /2 + ( ) is ensured by the following theorem. Theorem 3.2. (The weak version of the Schauder fixed-point theorem.) Let H be a separable Hilbert space and A : K K be a weakly continuous operator, where K is a non-empty, closed, bounded, convex subset of H. Then there exists a fixed-point of A in K. For the proof we refer to [7]. On the basis of the previous results we arrive at the following existence result. Theorem 3.3. There exists a weak solution to a contact problem with given friction and a solution-dependent coefficient of friction. Next we show that Ψ is Lipschitz continuous in the L 2 ( )-norm provided that F is Lipschitz continuous in R + and g L ( ).

35 CHAPTER 3. EXISTENCE RESULT 35 Theorem 3.4. Let g L ( ), g a.e. on, and c L > be such that F(x ) F(x ) c L x x x, x R +. Then there exists a positive constant c such that Ψ(ϕ) Ψ(ϕ),Γc cc L ϕ ϕ,γc ϕ, ϕ H /2 + ( ). (3.29) Proof. Let ϕ, ϕ H /2 + ( ) be given and u, u be the respective solutions of (P(ϕ)), (P(ϕ)), i.e.: a(u, v u) + F(ϕ) g ( v t u t ) ds F (v u) v K, Γ c a(u, v u) + F(ϕ) g ( v t u t ) ds F (v u) v K. Inserting v := u into the first and v := u into the second inequality and summing them we obtain: a(u u, u u) + (F(ϕ) F(ϕ)) g ( u t u t ) ds. (3.3) From a b a b a, b R 3, a t a a R 3 we have: ( ) /2 ( ) /2 ( u t u t ) 2 ds u t u t 2 ds ( ) /2 u u 2 ds c T u u,, (3.3) where c T is the norm of the trace mapping γ : V L 2 ( ). From (2.7), Korn s inequality, (3.3) and (3.3) it follows that c ell c K u u 2, a(u u, u u) Finally, (3.3) and (3.32) yield: g,,γc F(ϕ) F(ϕ),Γc γ t u γ t u,γc c L c T g,,γc ϕ ϕ,γc u u,. (3.32) γ t u γ t u,γc c T u u, c L c 2 T g,, c ell c K ϕ ϕ,γc.

36 CHAPTER 3. EXISTENCE RESULT 36 Setting we arrive at (3.29). c := c2 T g,, c ell c K From this and the Banach fixed-point theorem we obtain the following result. Corollary 3.5. If cc L < then the mapping Ψ : H /2 + ( ) H /2 + ( ) is contractive in the L 2 ( )-norm. Consequently, Ψ has a unique fixed-point and the method of successive approximations: } ϕ H /2 + ( ) given; for k =, 2,..., set ϕ k := Ψ ( ϕ k ) (3.33) is convergent in the L 2 ( )-norm for any choice of ϕ.

37 Chapter 4 Finite Element Approximation This chapter deals with a discretization of problem (P) by the finite element method. We shall investigate the existence as well as the uniqueness of the discrete solutions in a similar way as in the continuous case. Next, we shall study convergence of the discrete solutions and as a by-product we obtain an alternative proof of the existence of a solution to (P). To avoid the use of curved elements we shall suppose that is a polyhedron. Let us consider a system of partitions {T h }, h +, of such that (i) every T T h is a closed tetrahedron; (ii) = T T h T ; (iii) T T = { T T, T T h, T T, where T is either a common vertex or a common edge or a common face of T and T ; (iv) every partition T h is compatible with the decomposition of into Γ u, Γ p and, i.e. the whole face F of any boundary tetrahedron T T h is a part of the only one of Γ u, Γ p and. Next we shall suppose that {T h }, h +, is a family of regular partitions of, i.e.: h T σ > : σ T T h h +, ρ T 37

38 CHAPTER 4. FINITE ELEMENT APPROXIMATION 38 where h T = diam(t ), ρ T = sup{diam(b) B is a ball contained in T }. In addition, we shall suppose that { T h Γ c }, h +, is a strongly regular system of triangulations of, i.e. there exist positive constants σ, ν such that where h F ρ F σ, h Γc h F ν F T h Γc h +, h F = diam(f ), ρ F = sup{diam(b) B is a circle contained in F }, h Γc = max F T h Γc h F. With any T h the following sets will be associated: V h = { v h C() vh T P (T ) T T h, v h = on Γ u }, V h = (V h ) 3, K h = { v h V h vhn (a i ) a i N h }, Λ h = { ϕ h C( ) ϕh F P (F ) F T h Γc, ϕ h (a i ) a i N h, ϕ h (x) = if x Γ u }, where N h is the set of all contact nodes, i.e. the nodes of T h lying on \ Γ u. Obviously, Λ h H /2 + ( ) h >. Moreover, taking into account that is supposed to be a part of a plane, we obtain that K h K h >. For every ϕ h Λ h we shall consider the following discrete problem: Find u h := u h (ϕ h ) K h such that a(u h, v h u h ) + F(ϕ h ) g ( v ht u ht ) ds F (v h u h ) (P(ϕ h )) h v h K h. In the same way as in the continuous case one can show that (P(ϕ h )) h has a unique solution for any ϕ h Λ h. We introduce a mapping Ψ h : Λ h Λ h by Ψ h (ϕ h ) = r h γ t (u h (ϕ h )), ϕ h Λ h, where u h (ϕ h ) K h is the solution to (P(ϕ h )) h and r h denotes the piecewise linear Lagrange interpolation operator on T h Γc, i.e. for every v C( ) the function r h v is continuous and piecewise linear on T h Γc and (r h v)(x) = v(x) for any node x of T h Γc. The mapping Ψ h can be viewed to be a discretization of Ψ defined by (3.9).

39 CHAPTER 4. FINITE ELEMENT APPROXIMATION 39 Definition 4.. We say that u h K h is a discrete solution to (P) if and only if r h u ht := r h γ t u h is a fixed-point of Ψ h in Λ h, i.e.: a(u h, v h u h ) + F(r h u ht ) g ( v ht u ht ) ds F (v h u h ) v h K h. Analogously to the continuous case we shall establish useful properties of Ψ h. To this end the following theorem will be needed. Theorem 4.2. (The inverse inequalities.) There exist positive constants c inv, c inv such that ψ h /2,Γc c inv h /2 ψ h,γc, ψ h,γc c inv h /2 ψ h /2,Γc hold for every function ψ h C( ) being piecewise linear on T h Γc, and every h +. For the proof we refer to []. Lemma 4.3. The mapping Ψ h is continuous and maps Λ h B R into Λ h B R for some R >, which does not depend on h. Proof. Let ϕ h Λ h be arbitrary but fixed and u h := u h (ϕ h ) be a solution of (P(ϕ h )) h. It holds: r h γ t u h /2,Γc r h γ t u h γ t u h /2,Γc + γ t u h /2,Γc c r h /2 γ t u h,γc + γ t u h /2,Γc c r h /2 γ t u h,γc + γ t u h /2,Γc c r c inv γ t u h /2,Γc + γ t u h /2,Γc = ( + c r c inv ) γ t u h /2,Γc, (4.) as follows from the approximation properties of r h, Lemma 3.3, (3.2) and the inverse inequality between H ( ) and H /2 ( ). The constants c r and c inv do not depend on h. Arguing exactly as in Lemma 3. one can show (see (3.2) and (3.22)) that γ t u h /2,Γc 2 u h, 2 F (H ()) c ell c K, (4.2)

40 CHAPTER 4. FINITE ELEMENT APPROXIMATION 4 where c ell, c K are the constants from (2.7) and Korn s inequality, respectively. From this and (4.) we see that Ψ h maps Λ h B R into Λ h B R with R := 2( + c r c inv ) F (H ()) c ell c K. Next we show that Ψ h is continuous in Λ h. Let ϕ k h ϕ h in H /2 ( ), ϕ k h, ϕ h Λ h, k, and let u k h := u h(ϕ k h ) K h be solutions to ( P ( )) ϕ k h, k. Since { u k h h}, { } γt u k h are bounded in H (), H /2 ( ), respectively, it can be verified in a similar way as in Lemma 3. that γ t u k h γ t u h in L 2 ( ), k, (4.3) where u h := u h (ϕ h ) solves (P(ϕ h )) h. Furthermore, we know that { r h γ t u k } h is bounded in H /2 ( ), as well. Thus there exist a subsequence { r γt h u l } { h rh γt uh k } and a function ϕ H /2 ( ) such that r γt h u l h ϕ in H /2 ( ), l. Since r h preserves monotonicity, from γt u l h γt u h γt u l h γ t u h on, l N, it follows that ( rh γt u l h γt u h ) rh γt u l h γ t u h on, l N, and consequently ( rh γt u l h γt u h ),Γc rh γt u l h γ t u h,γc, l N. Hence we can write: rh γt u l h rh γ t u h,γc rh γt u l h γ t u h,γc rh γt u l h γ t u h γt u l h γ t u h,γc + γt u l h γ t u h,γc c r h /2 γt u l h γ t u h /2,Γc + γt u l h γ t u h,γc c r h /2 γt u l h γ t u /2,Γc h + γt u l h γ t u,γc h c r c inv γt u l h γ t u h,γc + γt u l h γ t u h,γc = ( + c r c inv ) γ t u l h γ t u h,γc l (4.4)

41 CHAPTER 4. FINITE ELEMENT APPROXIMATION 4 in virtue of the approximation properties of r h, (3.2), the inverse inequality between H /2 ( ) and L 2 ( ) and (4.3). Thus ϕ = r h γ t u h and the whole sequence { r γt h u k } h tends weakly to r h γ t u h in H /2 ( ). Since the space of all piecewise linear functions on T h Γ is finite-dimensional for every h fixed, c we obtain: r γt h u k h rh γ t u h in H /2 ( ), k. The existence of a fixed-point of Ψ h in Λ h B R follows from the next theorem. Theorem 4.4. (The Brouwer fixed-point theorem.) Let K be a non-empty, closed, bounded, convex subset of a finite-dimensional normed space and A : K K be a continuous operator. Then there exists an element x K such that Ax = x. For the proof we refer to [9]. From Lemma 4.3 and Theorem 4.4 we arrive at the following result. Theorem 4.5. There exists a discrete solution to (P). Under additional assumptions on F and g one obtains the following uniqueness result. Theorem 4.6. Let g L ( ), g a.e. on, and c L > be such that F(x ) F(x ) c L x x x, x R +. Then there exists a positive constant c which does not depend on h and such that Ψ h (ϕ h ) Ψ h (ϕ h ),Γc cc L ϕ h ϕ h,γc ϕ h, ϕ h Λ h. Proof. In the same way as in Theorem 3.4 it can be shown (see (3.32)) that u h u h, c L c T g,,γc c ell c K ϕ h ϕ h,γc, (4.5) where u h, u h are the respective solutions to (P(ϕ h )) h, (P(ϕ h )) h for ϕ h, ϕ h Λ h given and c T, c ell, c K are the constants from (3.32) which do not depend on h. Moreover we know (see (4.4) and (3.3)) that r h γ t u h r h γ t u h,γc ( + c r c inv ) γ t u h γ t u h,γc c T ( + c r c inv ) u h u h,,

42 CHAPTER 4. FINITE ELEMENT APPROXIMATION 42 where c T, c r, c inv are independent of h. From this and (4.5) we see that the assertion of the theorem holds with c := c2 T ( + c rc inv ) g,,γc c ell c K. Corollary 4.7. Let h be fixed. If cc L < then the mapping Ψ h : Λ h Λ h is contractive in the L 2 ( )-norm. Consequently, Ψ h has a unique fixed-point and the method of successive approximations: } ϕ h Λ h given; is convergent for any choice of ϕ h. for k =, 2,..., set ϕ k h := Ψ h ( ϕ k h ) (4.6) Let us suppose that K C () is dense in K in the H ()-norm. Let {u h }, h +, be a sequence of discrete solutions to (P), i.e. u h K h satisfy a(u h, v h u h ) + F(r h u ht ) g ( v ht u ht ) ds F (v h u h ) v h K h. Let v K and ε > be arbitrary but fixed. From our density assumption there exists a function ω K C () such that v ω, ε/2. As ω is smooth, it is possible to define a function v h by v h := Π h ω = Π h (ω, ω 2, ω 3 ) with Π h ω = (Π h ω, Π h ω 2, Π h ω 3 ), where Π h is the piecewise linear Lagrange interpolation operator on T h. Clearly, v h K h. Moreover, from approximation properties of Π h it follows that ω v h, ε/2 if h is small enough. Thus we are able to construct a sequence {v h }, v h K h such that v h v in H (), h +. (4.7)

43 CHAPTER 4. FINITE ELEMENT APPROXIMATION 43 Since {u h } is a bounded sequence in H () and u h K h K h, one can pass to a subsequence {u h } {u h } and find a function u K such that This together with (4.7) yield: u h u in H (), h +. lim a(u h h, v h ) = a(u, v). + In a similar way as in Lemma 3. we get: lim sup( a(u h, u h )) a(u, u), h + lim F (v h h u h ) = F (v u), + γ t v h γ t v in L 2 ( ), h +, γ t u h γ t u in L 2 ( ), h +. Using the last relation, the approximation property of r h and boundedness of γ t u h in H /2 ( ) we obtain: r h u h t γ t u,γc r h u h t γ t u h,γc + γ t u h γ t u,γc c r (h ) /2 γ t u h /2,Γc + γ t u h γ t u,γc h +. (4.8) Hence, for an appropriate subsequence of {u h } denoted by the same symbol it holds: r h u h t γ t u a.e. on, h +, so that F(r h u h t ) F( γ t u ) a.e. on, h +. The previous results imply lim h + F(r h u h t ) g ( v h t u h t ) ds = F( u t ) g ( v t u t ) ds. Consequently, u K satisfies: a(u, v u) + F( u t ) g ( v t u t ) ds F (v u). Since v K was arbitrary, the function u solves (P) and γ t u is a fixed-point of Ψ. Further, boundedness of {r h u h t } in H /2 ( ) proven in Lemma 4.3 ensures that there exists a subsequence of {r h u h t }, which is weakly convergent in H /2 ( ). But from (4.8) we see that the limit point is nothing else than γ t u and that the original sequence {r h u h t } tends weakly to γ t u in H /2 ( ). The result is summarized in the following theorem.

44 CHAPTER 4. FINITE ELEMENT APPROXIMATION 44 Theorem 4.8. Let K C () be dense in K in the H ()-norm, {u h }, h +, be a sequence of discrete solutions to (P) and {ϕ h }, h +, ϕ h := r h γ t u h be a sequence of fixed-points of Ψ h in Λ h. Then one can pass to subsequences of {u h } and {ϕ h } (denoted by the same symbol) such that } u h u in H (), h +, (4.9) ϕ h ϕ in H /2 ( ), h +, where u solves (P) and ϕ = γ t u is the respective fixed-point of Ψ. addition, if (P) has a unique solution, (4.9) holds for the whole sequences. In Remark 4.9. Some cases when the density assumption of Theorem 4.8 is satisfied are studied in [5].