ANALYSIS AND NUMERICAL METHODS FOR SOME CRACK PROBLEMS

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 2, Number 2-3, Pages 155 166 c 2011 Institute for Scientific Computing and Information ANALYSIS AND NUMERICAL METHODS FOR SOME CRACK PROBLEMS XIUFANG FENG, ZHILIN LI, AND LI WANG Abstract. In this paper, finite difference schemes based on asymptotic analysis and the augmented immersed interface method are proposed for potential problems with an inclusion whose characteristic width is much smaller than the characteristic length in one and two dimensions. We call such a problem as a crack problem for simplicity. In the proposed methods, we use asymptotic analysis to approximate the problem with a single sharp interface. The jump conditions for the interface problem are derived. For one-dimensional problem, or two-dimensional problems in which the center line of the crack is parallel to one of axis, we can simply modify the finite difference scheme with added correction terms at irregular grid points. The coefficient matrix of the finite difference equations is still an M-matrix. For problems with a general thin crack, an augmented variable along the center line of the crack is introduced so that we can apply the immersed interface method to get the discretization. The augmented equation is the asymptotic jump condition. Numerical experiments including the case with large jump discontinuity in the coefficient are presented. Key words. Crack problem, open-ended interface, asymptotic analysis, jump conditions, Cartesian grid method, augmented immersed interface method. 1. Introduction Many practical application problems involved open-ended interfaces, cracks, and thin surfaces, for example, the motion of a flag or wing of a butterfly. It is challenging to solve those problems that can capture the physical behaviors near the open-ended interfaces or cracks. In this paper, we consider such a problem in the potential theory, (1) ((x) u(x)) = 0, in a domain Ω. We assume that within Ω, there is an inclusion whose characteristic width is much smaller than its characteristic length in one and two dimensions, see Figure 1 for an illustration. We call such a problem as a crack problem for short even though a real crack problem is much more complicated. Our new method is based on asymptotic analysis, augmented strategies, and modified finite difference equations. One difficulty in solving a crack problem is that the width of a crack may be so small that there is either no or very few grid points inside the crack. Nevertheless, the potentials inside and outside the crack may be significantly different. Numerical methods based on adaptive meshes can be applied to solve such problems. But the method may be complicated and can not utilize fast Poisson solvers. Our goal in this paper is to develop simple Cartesian grid methods to solve the crack problems. We were first introduced with the crack problem by some scientist from Schlumberger company in Ridgefield, Connecticut, USA. The initial idea of combining the asymptotic analysis and the augmented finite difference method was proposed by Z. Li during the scientific meeting [6]. In [17], the authors have derived the asymptotic relations of the crack problems. The results in [17] have been basis for the Received by the editors May 18, 2010. 2000 Mathematics Subject Classification. 65N06, 65N22, 65N50, 65F35. 155

156 X. FENG, Z. LI, AND L. WANG 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 Figure 1. A plot of the computed potential of a crack (a thin ellipse). The coefficient is Ω = 1 outside of the thin ellipse and f = 1/500 inside the ellipse. research in this area. Some related work can be found in [15]. The explicit-jump immersed interface method (EJIIM) [12 14, 16] has been developed for the crack problem with good numerical results. The EJIIM is an extension of the immersed interface method[1,3,8,9]. In the EJIIM approach, the solution and its up to second order derivatives are set as unknowns and coupled with the jump conditions. The discretization leads to a large system of equations and often is solved by iterative method. The study of the stability of the EJIIM is difficult. There is also a large collection of literature for crack problems using finite element formulation. In this paper we propose a finite difference scheme using simple Cartesian grids to solve the crack problem in both one and two space dimensions. We first use the asymptotic relations to approximate the problem to a two-phase problem with a line interface. Then we use the augmented immersed interface method [2,4,7,10,11] to discretize the problem. The remaining of paper is organized as follows. In the next section, we discuss the one-dimensional problem. It is easier to understand our method through onedimensional discussion even though it may not have practical value. In section 3, we present the discussion for two dimensional problems and the numerical method. Finally, some conclusions and acknowledgments are given in the last section. 2. The one dimensional algorithm and analysis We start with the one dimensional algorithm and analysis by considering the equation (u x ) x = f(x), x (a,b) where is a piecewise constant with a finite jump across the crack. We illustrate the problem in Figure 2. Within the domain (a,b), there is a crack that centered at α with the width ǫ. Thus, we assume that = o in (α ǫ, α+ǫ) is different from the in (a, α ǫ) and (α+ǫ, b). At the interface α ǫ and α+ǫ, the natural jump conditions (2) [u] = 0, [u x ] = 0, where the jump, for example, [u] is defined as (3) [u] y = lim u(x) lim u(x), x y+ x y

SAMPLE FOR HOW TO USE IJNAM.CLS 157 and so forth. In Figure 2, y is α ǫ or α+ǫ. o α ǫ α α+ǫ u u u o u + u Figure 2. A diagram of the problem in 1D. We assume that ǫ is small. The solution domain (a,b) is divided into three parts (a,α ǫ), (α ǫ,α+ǫ), and (α+ǫ,b). We denote the solution near the crack at the left side of α ǫ as u, and at the right side of α ǫ as u o respectively; and u o+ and u + in the neighborhood of α + ǫ. From the immersed interface method [9], we have the following jump conditions. = u, u + = u + (4) x = u x, u + x = u + x xx = u xx + f f ; u+ xx = u + xx + f f, where f = f(x) is defined every where except for the crack (α ǫ,α+ǫ) in which f = f (x) is defined. Our approachis to treat the thin crack as a point interface α; use the asymptotic analysisto get the jump conditions about u ±, u ± x, and u± xx ; and apply the immersed interface method to solve the problem. For this purpose, we first use the Taylor expansion at α ǫ to get (5) u + = +2ǫ x +O(ǫ2 ) u + x = x +2ǫ xx +O(ǫ 2 ). Thus the approximate jump conditions across the interface α are (6) [u] = u + u = u + u = +2ǫ x +O(ǫ2 ) u = 2ǫ u x +O(ǫ2 ), (7) [u x ] = u + x u x = = = 2ǫ = 2ǫ u + x u x ( u x +2ǫ xx +O(ǫ 2 ) ) u x xx +O(ǫ 2 ) xx f (u + f ) +O(ǫ 2 ) u = 2ǫ f +O(ǫ2 ).

158 X. FENG, Z. LI, AND L. WANG We also have (8) [u xx ] = [f]. When 0, it is called resistive crack, we have [u x ] 0 and possible large jump in u, which means that the potential may significantly changed by the presence of the crack. On the other hand, if 0, then u is almost continuous across the crack but with different rate of change. If we ignore the high order terms (O(ǫ 2 )), we can apply the immersed interface method to solve the problem with the three jump conditions. Notice that u x itself is unknown. Due to the asymptotic analysis, high order approximation is not necessary. We will seek a first order finite difference method. 2.1. The finite difference scheme for the 1D crack problem. Without loss of generality, we assume that the center of the crack is the midpoint of the interval [a, b] and, and o are two constants. We will just discuss the resistive crack case, that is, o. Let x i = a+ih, i = 0,2,,N, h = (b a)/n. Away from the crack, we apply the standard three point finite difference scheme for regular points h 2(U i 1 2U i +U i+1 ) = f i. Since we will treat the crack problem as a two phase problem u (x),u + (x), we define the solution at the crack as the limit from one particular side, say from side. We can recover the solution in the crack from the jump relations in (4). We need to modify the finite difference scheme at two grid points x N/2 and x N/2+1. At the grid point x N/2, the finite difference scheme can be written as (9) h 2(U i 1 2U i +U i+1 ) = f(x i )+ [u] h 2, where i = N/2, f(x N/2 ) means lim x xn/2,x<x N/2 f(x), see [5,9], for example, for the derivation of the finite difference scheme. We approximate [u] using (10) [u] 2ǫ u x = 2ǫ U i U i 1. h For i = N/2+1, the finite difference scheme can be written as (11) h 2(U i 1 2U i +U i+1 ) = f(x i ) [u] h 2. Note that, x i 1 = x N/2 is on the interface and is considered as from side; and x i and x i+1 are on the same side of the interface. We approximate [u] using [u] = u + u = u + = u + ( u + 2ǫu + x +O(ǫ 2 ) ) = 2ǫ u+ x 2ǫ U i+1 U i. h The different approximation of [u] is to ensure that the coefficient matrix of the finite difference equations is an M-matrix and weakly row diagonally dominant. This will guarantee the stability of the algorithm. It is obvious that the finite difference method is first order accurate. This is reasonable approach considering the asymptotic approximation that we have already made.

SAMPLE FOR HOW TO USE IJNAM.CLS 159 3.5 3 2.5 2 1.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Figure 3. Numerical solution of the 1D example with N = 40, = 1000, = 1. Remark 1. For a conductive crack, that is, we have [u] 0, and [u x ] ǫ f. The correction terms would be ±ǫ f /h which can be computed directly. 2.2. A one-dimensional example. In this example, the source term f = 0, = 1000 and = 1, the boundary condition is the following mixed boundary condition. at x = 1, u = 1 (12) u at x = 1, x = 1; The width of the crack is taken as ǫ = 10 4. In Figure 3, we plot the finite difference solution computed using N = 40, we can clearly seethe effect ofthe crack ( ǫ,ǫ). We do not know the true solution for this example. Thus we compare the computed solution with that obtained from the finest solution (N = 480). For a first order method, the ratio of the errors when the mesh size is doubled is between 2 3 and approach 3, see [5] for the proof. This is confirmed in Table 1. Table 1. Grid refinement analysis of the numerical algorithm for the 1D crack problem. The coefficient is = 1000 in the domain except for the crack ( 10 4,10 4 ) in which = 1. = 1000, = 1 N E N ratio 30 6.2500 10 2 60 2.9167 10 2 2.1428 120 1.2500 10 2 2.3334 240 4.1667 10 3 3.0000 3. The two dimensional algorithm and analysis Now we focus our attention on two-dimensional problem that is more interesting. We consider the Poisson equation (13) (u x ) x +(u y ) y = f(x,y), (x,y) [a b] [c d].

160 X. FENG, Z. LI, AND L. WANG Within the domain, there is a vertical crack centered at x = (b +a)/2; y 0 y y 1. In our numerical tests, we take y 1 = d. The width of the crack is 2ǫ(y), see Figure 4 for an illustration. For simplicity, we restrict ourselves to piecewise constant coefficient in the entire domain except for the crack in which we have =. y = y 1 o u o ǫ u u u + u y = y 0 Figure 4. A diagram of the problem in 2D. We assume that the width of the crack ǫ(y) is small. Where ǫ(y) is the signed distance fromthecenterlineofthecrackwhichisx = (b+a)/2; y 0 y y 1. 3.1. Approximating the jump conditions. As we did for the one-dimensional case, we will use the asymptotic analysis to transferthe problemtoatwo-phaseproblemwith aline interface. Todoso, weneed the interface relations for the elliptic PDE across the interfaces, the two boundaries of the crack. Since we assume that there are no sources/sinks across the crack boundary, we have the natural jump conditions [u] = 0 [u n ] = 0. Since we assume that the crack is very thin segment, the curvature is negligible. Thus, From [3,5,9], we know other interface conditions across the each side of the

SAMPLE FOR HOW TO USE IJNAM.CLS 161 crack boundary: (14) = u u + = u + ξ = u ξ u + ξ = u + ξ η = u η u + η = u + η ξξ = u ξξ + f f ηη = u ηη u + ξξ = u + ξξ + f f u + ηη = u + ηη ξη = u ξη u + ξη = u + ξη, where (ξ,η) is the local coordinates of the interface ξ = χ(η) with χ(0) = 0 and χ (0) = 0. Our approach is to treat the thin crack as a line interface ξ = 0 with width 2ǫ; use the asymptotic analysis to get the jump conditions about u ±, u ± ξ, and u± ξξ ; and apply the immersed interface method to solve the problem. For this purpose, we first use Taylor the expansion at ǫ in ξ direction to get, (15) (16) u + = +2ǫ ξ +2ǫ 2 ξξ +O(ǫ3 ) u + ξ = ξ +2ǫ ξξ +O(ǫ2 ). Where ǫ is half of the width of the crack. Thus the approximate jump conditions across the interface are (17) [u] = u + u = u + u = +2ǫ ξ +O(ǫ 2 ) u = 2ǫ u ξ +O(ǫ2 ), (18) [u ξ ] = u + ξ u ξ = = u + ξ u ξ ( ξ +2ǫ ξξ +O(ǫ2 ) = 2ǫ u ξξ +O(ǫ2 ). ) u ξ 3.2. The modified finite difference approximation for straight cracks whose center line is parallel to the y-axis. Without loss of generality, we assume that the crack is centered at the center of the domain x = x M/2. In this case, ξ = x and η = y. Again we assume that the interface x = x M/2 is in the same side as those x = x i, i < M/2. Now we need to modify the finite difference scheme at grid points (x M/2,y j ) and (x M/2+1,y j ), j = 1,2,,N 1. The finite difference equations at these grid methods arealmost exact the same as those in one-dimensional case since we know [u], [u x ]. Note that we still have [u xx ] = [f] [u yy ] = [f] for the straight cracks whose center line is parallel to the y-axis.

162 X. FENG, Z. LI, AND L. WANG 3.3. The modified finite difference approximation for arbitrary cracks. If the center line of a crack is not a straight line or parallel to the axes, then the method that we described above does not apply. There is no efficient ways to discretize u ξ or u ξξ directly. Our idea is to use the augmented immersed interface method (AIIM), see for example, [9, 10] for references. We use the resistive crack to illustrate the idea. The jump conditions are given by (17) and [u ξ ] = 0. Intheaugmentedapproach,wesetq = [u ξ ]asanaugmentedvariable. Ifweknow q, then we can solve the elliptic PDE to get a solution u(x,y;q). That is, u(x,y;q) is a function of q. In the discretization, q(s) is discretized at a set of selected points on the center line of the crack. We denote the finite difference solution U ij by U, the discrete value of q as Q. Then given Q, the finite difference solution U is the solution of the finite difference equations, (19) AU+BQ = F 1, where the matrix A corresponds to the discrete Laplacian operator, and the vector BQ corresponds to the correction terms due to the jump condition in the solution. For arbitrary Q, the interface condition (17) may not be satisfied. We should choose Q such that (17) is satisfied. In the discrete case, we use the least square interpolation to approximate the jump condition (17). Thus we can write, (20) CU+DQ = W, where C and D are two matrices, and W is a vector. The residual vector (21) R(Q) = CU+DQ W is a measurement that described how well the jump condition(17) are approximated by given Q. If we put those two matrix-vector equations (19) and (20) together, we get [ ][ ] [ ] A B U F1 (22) =. C D Q W In practice, we do not necessarily need to form the matrices A, B, C, and D. Note that the dimension of Q is O(N) (assuming M N), which is much smaller than that of U which is O(N 2 ). Eliminating U from Eq. (22), we get the Schur-complement system for Q; (23) (D CA 1 B)Q = W CA 1 F 1 def = F 2, or EQ = F 2. Note that E is not symmetric in general. We can either use a direct method, for example, the LU decomposition; or an iterative method, for example, the GMRES iteration, to solve the linear system (23) for Q. Using the GMRES method, since we do not form the matrices A, B, C, and D explicitly, one question is how to use an iterative or direct method. This has been explained in detail in the recent work [9,10]. First, we set Q = 0 and then solve the Poisson equation. The residual of the linear system (23) (or the difference between the exact and the computed boundary condition), is actually the right hand side of the Schur complement with an opposite

SAMPLE FOR HOW TO USE IJNAM.CLS 163 sign. This is because (24) (D CA 1 B)0 F 2 = F 2 from(23) = ( W CA 1 ) F 1 from(20) = W +CU(0)+D0 from(19) & (20) = R(0), which gives the right hand side of the Schur complement system with an opposite sign. Next, we explain how to find the matrix-vector multiplication given Q. This again involves only two steps: (1) solving (19) by given Q, to get U(Q); (2) interpolating U(Q) at Ω via the least squares interpolation. Once we know the matrix-vector multiplication, we can apply the GMRES or other iterative method easily. The procedure is illustrated in the following derivation: (25) (D CA 1 B ) Q = DQ CA 1 BQ from(19) = DQ C ( A 1 F 1 U(Q) ) = DQ+CU(Q) CA 1 F 1 = DQ+CU(Q) W ( CA 1 F 1 +D0 W ) = R(U(G)) R(U(0)). Thus the matrix-vector multiplication is the result of the difference of the residual of the interface condition that is to be interpolated. Alternatively, we can form the coefficient matrix E of the Schur complement by setting Q = e l, the l-th unity vector, l = 1,2,. The idea of the augmented method has been explained in [7, 9] for elliptic interface problems with piecewise constant coefficient, and Poisson equations on irregular domains. The Fortran code is available to the public either by request or by anonymous ftp. 3.4. A numerical example in two-dimensions. In this example, we set the domain as [0, 1] [0, 1], = 1000, = 1, f = 0. The boundary condition is the following: u at x = 1, n = 1, at x = 1, u n = 0; (26) u at y = 1, n = 0, x2 at y = 1, u = 4 x 2. The analytic solution is not available for this example. So we compare the computedsolutionwiththatobtainedfromthefinestmesh. InTable2,weshowtwo sets of the grid refinement analysis. The first set is M = N = 20,40,80,160 with the finest mesh being M = N = 320. The other one is M = N = 30,60,120,240 with the finest mesh being M = N = 480. 3.5. Conclusions. In this paper, we studied the potential problems with an inclusion whose characteristic width is much smaller than the characteristic length in one and two dimensions. We approximated the inclusion with a line interface and derived the jump conditions using asymptotic analysis. Two different numerical methods are proposed to deal with different inclusions. The methods are very

164 X. FENG, Z. LI, AND L. WANG Table 2. Grid refinement analysis of the numerical algorithm for the 2D crack problem. The crack width is ǫ = 10 4. = 1000, = 1 N error ratio 20 3.67971 10 2 40 2.28991 10 2 1.6070 80 1.23821 10 2 1.8468 160 5.33121 10 3 2.3396 30 2.9608 10 2 60 1.7567 10 2 1.6854 120 9.0189 10 3 1.9478 240 3.9412 10 3 2.2884 1 (a) 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 (b) The solution plot 0.5 0 0.5 1 1.5 2 1 0.5 1 0 0.5 1 1 0.5 0 0.5 Figure 5. Numerical solution of example 2 with N = 40, Number of contour line, M = 60, = 1000, = 1: (a) the contour plot; (b), the mesh plot.

SAMPLE FOR HOW TO USE IJNAM.CLS 165 useful for resistive crack problems across which the solution has a jump. For conductive crack problems, since the solution is continuous, we suggest to use a standard numerical scheme. Acknowledgments The first author was partially supported by the Key Project of Chinese Ministry of Education Grant No.209134 and the Natural Science Foundation of Ningxia Grant No. NZ1051. The first and third authors would like to thank Dr. Zhilin Li and North Carolina State University for the hospitality during the authors visit. The second author was partially supported by the US ARO grants 56349MA-MA, the AFSOR grant FA9550-09-1-0520, the US NSF grant DMS-0911434, and the US NIH grant 096195-01. The third author was partially supported by the National Natural Science Foundation of China under grant 10971102 and the Natural Science Foundation of Jiangsu Province of China under grant BK2009398, and the grant of the Ministry of Education of China 200720. References [1] Deng, S., Ito, K., and Li, Z. Three dimensional elliptic solvers for interface problems and applications. J. Comput. Phys. 184 (2003), 215 243. [2] Ito, K., Li, Z., and Lai, M.-C. An augmented method for the Navier-Stokes equations on irregular domains. J. Comput. Phys. 228 (2009), 2616 2628. [3] LeVeque, R. J., and Li, Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994), 1019 1044. [4] Li, Z.,, Lai, M.-C., He, G., and Zhao, H. Anaugmented method forfreeboundary problems with moving contact lines. Computers and Fluids 39 (2010), 1033 1040. [5] Li, Z. The Immersed Interface Method A Numerical Approach for Partial Differential Equations with Interfaces. PhD thesis, University of Washington, 1994. [6] Li, Z. Asymptotic analysis and finite difference methods for crack problems. A one-day workshop at Schlumberger, Ridgefield, Connecticut, USA, 1997. [7] Li, Z. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35 (1998), 230 254. [8] Li, Z., and Ito, K. Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23 (2001), 1225 1242. [9] Li, Z., and Ito, K. The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM Frontier Series in Applied mathematics, FR33, 2006. [10] Li, Z., Ito, K., and Lai, M.-C. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Computers and Fluids 36 (2007), 622 635. [11] Li, Z., Wan, X., Ito, K., and Lubkin, S. An augmented pressure boundary condition for a Stokes flow with a non-slip boundary condition. Communications in Computational Physics 1 (2006), 874 885. [12] Rutka, V., and Li, Z. An explicit jump immersed interface method for two-phase Navier Stokes equations with interfaces. Comput. Meth. in Appl. Mech. Eng. 197 (2008), 2317 2328. [13] Rutka, V., and Wiegmann, A. Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials. Tech. Rep. 73, Fraunhofer ITWM Kaiserslautern, 2005. [14] Wiegmann, A. The explicit jump immersed interface method and interface problems for differential equations. PhD thesis, University of Washington, 1998. [15] Wiegmann, A. Analytic solutions of a multi-interface transmission problem and crack approximation. Inverse Problems 16 (2000), 401 411. [16] Wiegmann, A., and Bube, K. The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 35 (1998), 177 200. [17] Wiegmann, A., Li, Z., and LeVeque, R. Crack jump conditions for elliptic problems. Applied Math. Letters 12 (1999), 81 88.

166 X. FENG, Z. LI, AND L. WANG School of Mathematics and Computer Sciences, Ningxia University, Yinchuan, China E-mail: xf feng@nxu.edu.cn Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA, and School of Mathematical Sciences, Nanjing Normal University, China E-mail: zhilin@math.ncsu.edu URL: http://www4.ncsu.edu/ zhilin School of Mathematical Sciences & Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, China E-mail: wangli1@njnu.edu.cn