Solution of the Contact Elasticity Problem Based on an Iterative Proximal Regularization Method for the Modi ed Lagrangian Functional

Abstract

The method of successive approximation is considered for solving the contact elasticity problem which corresponds to the quasivariational Signorini problem. Auxiliary problems with given frictions arising from each external step of this method are solved by the Uzawa method with iterative proximal regularization of the modi ed Lagrangian functional. Stabilization of the sequence of auxiliary nite-element solutions of external steps of successive approximation is investigated. Numerical results are considered.

Keywords

• proximal regularization
• Lagrangian functional
• saddle point
• Uzawa method
• Delaunay triangulation
• nite element method

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Introduction ( 1 ) ( 2 )

For displacement vector v = (v1; v2), we de ne the strain tensor and the stress tensor "ij (v) = Here, i; j; k; m = 1; 2; cijkm = cjimk = ckmij , and repeated indices indicate summation.

For given functions f = (f1; f2), p = (p1; p2) and F consider the following boundary value problem (see [1]): un = 0; ij nj = pi on

; i = 1; 2 = 0 on

0 1; i = 1; 2

On the contact surface 2 of the elastic body and the absolutely rigid support, we impose the conditions: un 0; n 0; un n = 0 on 2 j j

F j nj; (F j nj j j)u = 0; u Here, n = (n1; n2) is the unit outward-pointing normal to , un = u n, u = u unn; i = ij nj , i = 1; 2; = ( 1; 2), n = ij ninj , = nn; and frictional coe cient F 0 on 2.

The main di culty in the study and construction of numerical methods for solving this nonlinear boundary value problem is that the frictional force F j n(u)j is a function of desired solution u.

De ne the sets

V = v 2 [W21( )]2 : vn

v2 = 0 on 0 ; K = fv 2 V : vn 0 on 2g : Assume that functions cijkm 2 L1( ) (i; j; k; m = 1; 2), f 2 [L2( )]2, p 2 [L2( 1 )]2 and F 2 L2( 2 ). Suppose that the solution u to the boundary value problem ( 1 ), ( 2 ) exists and belongs to the space [W22( )]2. Then u satis es the quasivariational Signorini inequality for 8 v 2 K (see [1,2]) a(u; v u) +

F j n(u)j(jv j ju j) d f (v u) d + p (v u) d ; ( 3 ) where

We use the successive approximation method for solving the quasivariational inequality (see [1,2]): 1. Set starting friction force g0 2 W21=2( 2 ), g0 0 . 2. Find uk as a solution of auxiliary variational inequality a(uk; v uk) + gk(jv j jukj) d f (v uk) d + p (v uk) d : ( 5 ) 3. Correct friction force gk+1 = F j n(uk)j.

The convergence of the successive approximation method to the solution to the quasivariational Signorini inequality ( 3 ) is still an open question. The existence of the solution is proved in the coercive case for a su ciently small coe cient F only (see [1]).

Variational inequality ( 5 ) is called the problem with given friction gk. It is equivalent to the following constrained non-di erentiable minimization problem: 8 <J (v) = :v 2 K :

Under geometric form of domain shown in g. 1, functional J (v) is not strongly convex on all space V (see [1,2]). The kernel of bilinear form a(u; v) is not trivial and consists of vector-functions = (a; 0), where a is an arbitrary number.

Z 2 a(u; v) =

Z Z 2 ij (u)"ij (v) d = cijkm"km(u)"ij (v) d :

( 4 ) Z

Z Z

Z 1 Z 1

However, if then

3 Z 1 J (v) ! +1 under kvk[W21( )]2 ! 1; v 2 K; that is, J is a coercive functional on the set K and therefore problem ( 6 ) is solvable.

Application of Dual Schemes for Solving the Auxiliary Problem with Given Friction

For problem (6), on set V

L2( 2) we de ne the classical Lagrangian functional

L(v; l) = J (v) +

lvn d : Z 2 Denote by (L2( 2 ))+ the set of nonnegative square integrable functions on 2.

De nition 1. A pair (v ; l ) 2 V (L2( 2 ))+ is called a saddle point of the Lagrangian functional L(v; l) if it satis es the two sided inequalities

L(v ; l)

L(v ; l )

L(v; l ) 8 (v; l) 2 V (L2( 2 ))+ :

It was shown in [3] that, if a solution u of auxiliary problem ( 6 ) belongs to space [W22( )]2 and mesfx 2 2 : n(u) < 0g > 0, then u is a unique solution to the problem ( 6 ) and the pair (u; n(u)) is a unique saddle point of L(v; l).

Because of the second part of saddle point is equal to n(u), we can nd accurately the frictional force on the next step of successive approximation method.

However, application of the dual Uzawa method with the classical Lagrangian functional L(v; l) does not guarantee convergence to a saddle point in the semicoercive case (see [6,7]).

To overcome this di culty the modi ed Lagrangian functional M (v; l) was considered on space V L2( K ) (see [8],[9]):

M (v; l) = J (v) + (l + rvn)+ 2

l2o d ;

De nition 2 is di erent from De nition 1. Here we use all space L2( 2 ) instead of

L2( 2)+ in the rst de nition.

However, it is known that the functionals L(v; l) and M (v; l) have the same set of saddle points (see [3], [10]).

For nding a saddle point of modi ed Lagrangian functional M (v; l) we use the method based on the combination of Uzawa method with the proximal regularization (see [4]). We note that similar method in nite-dimensional case was investigated in [5].

According to this method, the sequence f(um; lm)g is generated as follows. 1. Assign the initial approximation (u0; l0) 2 V

2. Find um+1 such that

( 7 ) ( 8 ) ( 9 ) ( 10 )

3. Calculate the next value of the dual variable by the formula

Criterion ( 7 ) implies that the exact solution um+1 is replaced by its approximation um+1, obtained by solving problem ( 8 ) numerically using the nite element method. In this case parameter m can be interpreted as the error of the numerical solution.

The regularizing term 21 kv umk[2L2( )]2 ensures that the functional to be minimized in ( 8 ) is strongly convex on V . This guarantees that the auxiliary problem ( 5 ) is uniquely solvable.

Application of the modi ed Lagrangian fucntional allows as to nd saddle point e ciently in comparison with duality methods based on classical Lagrangian functional.

It should be noted that in article [11] it was built and investigated a wide class of iterative solution methods of the semicoercive variational inequalities including the iterative proximal regularization method.

Discuss discretization of the variational problem (8).

4

Finite Element Discretization Domain is taken in the form of a trapezoid ( g. 1) with sides 2, 1, 1.5, p5=2. Under the proposed algorithm ( 7 ){( 10 ) on each step of the iteration process, we regard the minimization problem of the strongly convex functional

Mm(v) = M (v; lm) + umk[2L2( )]2 ! min; For the solving of the problem ( 11 ) nite element method have been applied. Using the Delaunay triangulation, we triangulate into a set of triangles Tk ( g. 2), so that = S1Nt Tk; where Nt is the number of triangles. Thus we have a nite element consisting of triangles, which have one point in common at the triangulation nodes.

Condensation of the mesh occurs near the contact zone 2.

Enumerate triangulation nodes from the top down, from 1 to N . For each node i it is de ned basis function 'i(x; y), for which 'i(xi; yi) = 1 and for all neighboring nodes j: 'i(xj ; yj ) = 0. For basis functions 'i we take piecewise linear functions (see [12]).

Let us introduce the following notation: h - maximum edge length of the triangulation Tk, Ph = fD1; ; DN g - triangulation node set , Ih = fM1; ; MRg - boundary node set on 2, Vh - linear span of the basis functions 'i(x; y), uh = (u1h; u2h) - piecewise interpolation of the exact solution u: uih(x; y) =

N X t(ji)'j (x; y); for i = 1; 2 and t(ji) 2 R : j=1

Note that since is a polygon, then embedding Vh

substitute the problem ( 8 ) by nite-element problem:

V is provided. Thus we can

um+1 = arg vm2iVnh fMm(v)g : kum umk[W21( )]2

Ch1=2;

C > 0 const; m = 1; 2; : : : ;

There is shown in [13] that error estimate is true for the exact solution sequence fumg: and nite-element solution sequence fumg converges on V to solution of the auxiliary problem ( 6 ) under h ! 0. ( 12 ) ( 13 )

Let us introduce the vector t = (t1; t2; : : : ; t2N ), where the rst N its components correspond to u1h(Di(xi; yi)) and last N components correspond to u2h(Di(xi; yi)). Then the minimization problem ( 11 ) reduces to nding optimal values ti. For this purpose, we use coordinate descent method.

Let sti ness matrix A and vector F be de ned as follows

8 A = <a( i; j) + :

Z 8 F = <Z (f + um) i d : i j d

Z 1 + p i d 9 = ;i;j=1;2N

; 9 = ; i=1;2N

; i = (('i; 0) ; (0; '(i N)) ; i N ; i > N : where i are vector-functions such that

Thus we have the computational formula for all nodes Di 2 Ph n Ih ts+1 = i

0 1

1 FiA : (14)

Consider next a problem for nodes Di 2 Ih. For this purpose, we pass from variables u1, u2 to un, u , where is an acute angle of the domain :

For evaluation the boundary integral on 2 (u1 = un cos + u sin ;

u2 = un sin + u cos :

1, then introduce some notations

An = Aii cos2 + Ai+Ni+N sin2 + Aii+N cos2

1 lmvn d A

Ai+Ni sin2 Ai+Ni+N cos sin ; :8<llmm((DDii))HH2 ;; DDii22ffMM21;; MR;gM;R 1g ; 8<rH ; Di 2 fM2; ; MR 1g ; :r H2 ; Di 2 fM1; MRg ; where tn = vn(Di).

To nd the optimal tn compute

A1n ts A + Xj<i Aij cos tjs+1 +

>8> i ; i lm(rDi) ; tsn+1 = :>>< A!ni ++ LB((DDii)) ; i > lm(rDi) ; (15)

A similar approach can be used for t on (i+N )th step of the Gauss-Seidel method.

Approximate

Z

R 1 X gk(Mj)jv (Mj)j + gk(Mj+1)jv (Mj+1)j jMj; Mj+1j : j=1 2 An = Aii cos sin

Aii+N sin2 + Ai+Ni cos2

Ai+Ni+N cos sin ; j<Xi+N Aij sin tjs+1 +

j6=i + j>Xi+N Ai+Nj cos tjs

A = Aii sin2

+ Ai+Ni+N cos2 + 2Aii+N cos sin ; G(Di) = <8g(Di)H ; Di 2 fM2; ; MR 1g ; :g(Di) H2 ; Di 2 fM1; MRg ;

X Aij sin tjs + j>i+N Fi sin

X j<i+N

j6=i + Fi+N cos : i = tsnAn+ Ai+Nj cos ts+1+ j We present the results of numerical computations for solving problem ( 3 ). We assume a volume load f = (f1; f2) = (0; 0), boundary loading from the left p1j 1 = 27 mPa and from the right p1j 1 = 27 mPa sides, p2j 1 = 0, frictional coe cient F = 0:3, Young's modulus E = 73000 mPa, Poisson's ratio = 0:34, constant r = 108.

Numerical solution of the considering problem is shown in g. 3. The graphs of un and j nj show that body is detached from absolutely rigid foundation at the vertex of the obtuse angle. It follows from the fact that un < 0 and j nj = 0 at this vertex.

Let us introduce the example, when body sticks together with absolutely rigid foundation. We change boundary loading from the right side p1j 1 = 21:6 mPa.

Numerical results con rmed that modi ed Lagrangian functionals e ectively remove conditions like un 0 on 2 when passing to the unconstrained minimization problem. Besides, it was revealed that successive approximation method is more e ective at larger r (r = 106; 108).

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