Conjugate gradient-boundary element method for a Cauchy problem in the Lam6 system

Transcription

1 Conjugate gradient-boundary element method for a Cauchy problem in the Lam6 system L. Marin, D.N. Hho & D. Lesnic I Department of Applied Mathematics, University of Leeds, UK 'Hanoi Institute of Mathematics, Vietnam Abstract In this paper an iterative algorithm, based on the conjugate gradient method (CGM), for obtaining approximate solutions to the Cauchy problem in linear elasticity is analysed, using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. 1 Introduction Consider a linear elastic material which occupies an open bounded domain R C Rd, where d is the dimension o' the space in which the problem is posed, usually d E {l, 2,3), and assume that Cl is bounded by a surface I' = dr E C'. We also assume that the boundary consists of two parts, I' = I'l U r2, where rl, r2 # 0 and I'l n r2 = 0. In the absence of body forces, the equilibrium equations are given by, see e.g. Saada [l],

2 where ui, is the stress tensor and the strain tensor E;, is given by These tensors are related by the constitutive law, namely where Cijkl is the elasticity tensor and, in the case of an isotropic linear elastic material, is given by with G and v the shear modulus and Poisson ratio, respectively, and 6ij the Kronecker delta tensor. If we now substitute the constitutive law (3) into the equilibrium equations (l), and use the kinematic relations (2) together with the expression (4) for the elasticity tensor for an isotropic linear elastic material, we obtain the Lam6 system, namely, We now let n(x) be the outward normal vector at I' and t(x) be the traction vector at a point X E l? whose components are defined by Assuming that both the displacement and traction vectors can be measured on a part of the boundary l?, say r2, then this leads to the mathematical formulation of an inverse problem consisting of equations (1) or (5) and the boundary conditions where G and are prescribed vector valued functions. This problem, termed a Cauchy problem, is much more difficult to solve both analytically and numerically than the direct problem, since the solution does not satisfy the general conditions of well-posedness. Although the problem may have a unique solution, it is well known, see e.g. Hadamard [2], that this solution is unstable with respect to small perturbations in the data on r2. Thus the problem is ill-posed and we cannot use a direct approach, such as the Gauss elimination method, in order to solve the system of linear equations which arises from the discretisation of the partial differential equations (1) or (5) and the boundary conditions (7). Therefore, knowing the exact data on the

3 boundary r2, we apply a variational method to the aforementioned Cauchy problem. Since the boundary conditions on rl are unknown and have to be determined, we consider the displacement vector on the underspecified boundary as a control for a direct problem and attempt to fit the Cauchy data on the overspecified boundary r2 by minimizing a functional relating the known and calculated values of the displacement vector on r2, see e.g. Lions [3]. 2 Variational formulation Let us denote by ll2(ri), WS (ri) and W' (R) the spaces (~~(ri)) d, (~~(ri)) and (HS(R)) d, respectively, for d E {l, 2,3), i = 1,2 and some s E R, see e.g. Lions and Magenes [4]. Then the Cauchy ~roblem for the Lamk system may be recast as follows - where G E IL2(r2), 2 E ll2(r2) and U is sought in RI1l2(R). We note that t E JHI-l(r2) is sufficient for our method. Let us denote by -yi f, i = 1,2, the trace of a function f determined in R over ri, i = 1,2. First we solve the direct problem with v E L2(rl). We denote by U = u(v,'t) the solution to the problem (9) and aim to find v E IL2(rl) such that To do so, we attempt to minimize the functional with respect to v E ll2(fl). Consider the problem where p E ll2(r2).

4 232 Bourldai-y Elemenr Technology XIV Lemma 1 Let U and 1C, be the solutions of the problems (9) and (12), respectively. Then Theorem 1 Let -i E IL2(r2). If v varies in IL2(rl), then y2u(v,z) forms a dense set in IL2(rl). This theorem has important consequences, for example, we can say that the Cauchy problem (8) is solvable for almost all G,z E IL2(r2). Furthermore, we have the following result: Corollary 1 inf J(v) = 0 vcla(ra) (14) Theorem 2 The functional J(v) is Fre'chet differentiable and its first gradient has the form Furthermore, the functional J(v) is twice Fre'chet differentiable and it is strictly convex. 3 Conjugate gradient method (CGM) As we can calculate the gradient of the functional J(v) via the adjoint problem, we can now apply the conjugate gradient method (CGM) with a stopping rule, as proposed by Nemirovskii [5]. We define the linear operator and we have the following linear equation Suppose that instead of G we have only an approximation of it, say G, E IL2(r2) such that llg - G IILyr,) < - E (18) To solve our equation (8) with noisy data G,, we need to compute A+o(Aov - - U,), where A: is the adjoint of the operator A0 and C, is given by

5 Bounda~y Element Technology XIV 233 However, we observe that this is nothing else than the gradient of the functional 1 J(v) = -IIAv 2 - Gc Il~yr,) (20) Thus the CGM applied to our problem has the form of the following algorithm: Step 1. Set k = 0. Choose u(o) E L2(r2). Step 2. Solve the direct problem in order to determine the residual T ( ~ ) - - T ( ~ = ) AU(~) - G, = y2u(u(k), t) - U, Step 3. Solve the adjoint problem to determine the gradient g(k) (k) g; (X) = 71 (flij($'(oi r(k))(~))nj(~)) Calculate Pk and d(k) as follows: k=o: pk=o, d(k) = 2 l : pk= llg(k)ll~ypl) k-l) llg' lllyr1) ' Step 4. Solve the direct problem d(k) = -g(k) + ~ ~d(k-1) ( 25) in order to determine ~ od(~) given by

6 Compute a k and u(~+') as follows: Step 5. Set k = k + 1. Repeat Step 2 until a stopping criterion is prescribed. As a stopping criterion we choose the one suggested by Nemirovskii [5], namely E N : ~ ( ~ ) l 5 ~ 6~ ~ ( ~ ~ ) (29) where 6 is a constant which can be taken heuristically as suggested by Hanke and Hansen [6]. It follows from Nemirovskii's result that the above iterative procedure converges with an optimal convergence rate to the normal solution of the problem as the noise level tends to zero. A boundary element method (BEM), see e.g. Marin et al. [7], is used in order to solve the intermediate mixed well-posed boundary value problems resulting from the iterative method adopted. 4 Numerical results and discussion In order to present the performance of the numerical method proposed, we solve the Cauchy problem for examples in a smooth geometry, namely the unit disc S2 = {X = (xl, x2) I X: + X; < 1). We assume that the boundary J? = {X = (xl, 22) I X: + X; = 1) of the solution domain R is divided into two disjointed parts, namely rl = {X = (xl, 22) I X E l?, cul < cr2) and r2 = {X = (xl,x2) I X E l?, 0 < a1) U {X = (x1,x2) I X E r, a2 5 2x1, is the angular polar coordinate of X and a*, i = 1,2, are specified angles in the interval (0,27r). In order to illustrate the typical numerical results we have taken a1 = 7~/4 and a2 = 3x14 and we assume that the boundary r2 is overspecified by the prescription of both the displacement and the traction vectors while the boundary rl is underspecified with both the displacement and the traction vectors unknown. In the following test examples we consider an isotropic linear elastic medium characterized by the material constants G = 3.35 X 101 N/m2 and v = 0.34 corresponding to a copper alloy. Example 1. We consider the following analytical solution in displacements l-v (an) U' ("I = 2G(l+ v) ffo Xi i = 1,2 I in the domain fl, which corresponds to a uniform hydrostatic stress state given by

7 with a 0 = 1.5 X 1010 N/m2. Example 2. We consider the following tractions on the boundary I' where a0 = 1.5 X 10'' N/m2 and B(x) is the angular polar coordinate of X. It should be noted that the corresponding analytical expressions for the stresses uj~)(x) and displacements u!'")(r) are not available in this case for the tractions (32) and (33), but they can be obtained numerically by solving the direct problem An arbitrary vector valued function U(') E L2(rl) X L2(rl) may be specified as an initial guess for the displacement vector on PI, but in order to improve the rate of convergence of the iterative procedure we have chosen a vector valued function which ensures the continuity of the displacement vector at the endpoints of rl and which is also linear with respect to the angular polar coordinate 8. For the test examples considered, this initial guess is given by ui (0)(X) = a2 - e(x)up(xll a1 (an) a2 - a1 a 2 - a1 ui (22) (35) for i = 1,2, where ai = B(xi), xi are the endpoints of rl, and the choice of a1 = a/4 and a 2 = 37r/4 also ensures that the initial guess is not too close to the exact values uian)(x). In order to investigate the convergence of the algorithm, at every iteration we evaluate the accuracy errors where u(~) is the displacement vector on the boundary rl retrieved after k iterations, the operator A is given by equation (10) and each iteration consists of solving three direct mixed well-posed problems as described in section 3. When starting with the initial guess U(') given by equation (35) for exact input data, a convergent sequence { u(~))~>~ - of approximation functions for

8 236 Bourida~?i Elerneut Tectviology XIV Figure 1: The analytical solution U(;") (-) and the numerical solution uprn) retrieved for N = 40 (o) and N = 80 (v) constant boundary ele- ments, variable relaxation factor with amplitude A = 2.0, and input data u(an)lr2, on the underspecified boundary rl, for the Cauchy problem considered in example 2. ulr, is obtained. However, if we evaluate the error e, at every iteration using different numbers of constant boundary elements for discretising r' when input data contains noise, we note that the error e, decreases up to a specific iteration after which it starts increasing. The error E, keeps decreasing as the number of iterations Ic increases, but there is a threshold of optimality at which the iterative process should be stopped according to Nemirovskii's rule. We note that the CGM algorithm described in section 3 is convergent as we increase the number of boundary elements. However, the errors in predicting the traction vector t on the underspecified boundary are still large since we are using the analytical input data on r2 which is contaminated by numerical noise. An alternative way to generate the Cauchy data on r2 is to use the numerical solution of the direct problem and this procedure can also be used to fabricate the input data when no exact solution to the Cauchy problem is available. In this case both accuracy errors e, and E, decrease as the number of iterations k increases and the sequence {u(~))~>~ - of approximation functions for ulr, converges

9 exactly to the analytical solution u(an)lr,. However, the numerical solution for the traction vector deviates from the exact solution, especially near the ends of the underspecified boundary which is the region of high illposedeness and also where the BEM changes to mixed boundary conditions. In order to improve the results for the traction vector on the underspecified part of the boundary, we relax the marching condition (28) through the use of U(k+l) = pu(k+l) + (1 - p)u(k) (39) when passing from step 4 to step 5 of the algorithm described in section 3, with the variable relaxation factor p = p(o(x)) given by where O(x) is the angular polar coordinate of the point X E rl, cul and a2 are the angular polar coordinates of the endpoints of the boundary rl, and A E (0,2]. It should be mentioned that when using the variable relaxation factor given by equation (39), both accuracy errors e, and E, decrease continuously as the number of iterations k increases. In Figures 1 we present the numerical solutions obtained for the 22 component of the displacement vector for the Cauchy problems considered in example 2, when various numbers of boundary elements and the variable relaxation factor given by (40) with the amplitude A = 2.0 are employed. We note that the numerical solution is convergent with respect to increasing the number of boundary elements. The stability of the numerical method proposed has been investigated by perturbing the initial data ulr2 as where 6ui is a Gaussian random variable with mean zero and standard deviation U, generated by the NAG subroutine G05DDF, and p is the percentage of additive noise included in the input data ulp2 in order to simulate the inherent measurements errors. The optimal iteration numbers and the accuracy errors indicate that as p decreases then the numerical solutions approximate better the exact solutions, whilst at the same time remaining stable. 5 Conclusions In this paper we have formulated a Cauchy problem for the Lam6 system in a variational form where only weak requirements for the Cauchy data are

10 required. Consequently, the solutions of the direct problems, as well as the associated adjoint problems, are defined in a weak sense and a mathematical analysis has been undertaken. The variational approach for solving the Cauchy problem in elasticity needs the gradient of the minimization functional, which is provided by the solution of the adjoint problem. Due to the explicit representation of the gradient, the CGM was employed to solve numerically the Cauchy problem. The algorithm proposed consists of solving three direct problems at every iteration, but because of the linearity of the problem only two direct solutions are required at every iteration. In combination with Nemirovskii's stopping criterion, the CGM is known to be of optimal order when the data is sufficiently smooth. The use of a variable relaxation factor with respect to the angular polar coordinate substantially increased the accuracy of the numerical solution. Cauchy problems are inverse boundary value problems and, therefore, only the discretisation of the boundary is needed and thus the BEM is a suitable method for solving such improperly posed problems. The numerical results obtained for various numbers of constant boundary elements and various amounts of noise added into the input data were found to be in good agreement with the exact solution. References [l] A.S. Saada, Elasticity: Theory and Applications (Pergamon Press, New York 1974). [2] J. Hadamard, Lectures on Cauchy's Problem in linear partial differential equations (London, Oxford University Press 1923). [3] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer-Verlag, Berlin 1971). [4] J.L. Lions, and E. Magenes, Non-homogeneous Boundary Value Problems and Their Applications (Springer-Verlag, New York-Heidelberg 1972). [5] A.S. Nemirovskii, The Regularizing Properties of the Adjoint Gradient Method in Ill-Posed Problems, Comput. Maths. Math. Phys. 26 (1986) [6] M. Ilanke and P.C. Hansen, Regularization Methods for Large-scale Problems, Surveys Math. in Industry 3 (1993) [7] L. Marin, L. Elliott, D.B. Ingham and D. Lesnic, Boundary Element Method for the Cauchy Problem in Linear Elasticity, submitted for publication in Engineering Analysis with Boundary Elements (2000).