CVBEM for a System of Second Order Elliptic Partial Differential Equations

Transcription

1 CVBEM for a System of Second Order Elliptic Partial Differential Equations W. T. Ang and Y. S. Par Faculty of Engineering, Universiti Malaysia Sarawa, Kota Samarahan, Malaysia Abstract A boundary element method based on the Cauchy s integral formulae and the theory of complex hypersingular integrals is devised for the numerical solution of boundary value problems governed by a system of second order elliptic partial differential equations. The elliptic system has applications in physical problems involving anisotropic media. Key words: complex variable boundary element method, elliptic partial differential equations, anisotropic media. This is the preprint of the article in Engineering Analysis with Boundary Elements Vol. 1 (

2 1 INTRODUCTION Consider the system of second order elliptic partial differential equations given by X X φ a ijp =0 (i =1,,,N, (1 x j x p j=1 p=1 =1 where φ ( =1,,,N are functions of x 1 and x and a ijp (j, p =1, and i, =1,,,N are real constant coefficients which satisfy the symmetry conditions a ijp = a pij and are such that X X j=1 p=1 i=1 =1 a ijp λ ij λ p > 0 for every non-zero N real matrix [λ ij ]. Weareinterestedinsolving(1inaregionR bounded by a simple closed curve C (on the 0x 1 x plane subject to φ (x 1,x =µ (x 1,x for (x 1,x C 1 P i (x 1,x =Q i (x 1,x for (x 1,x C where µ and Q i are suitably prescribed functions of x 1 and x, C 1 and C are non-intersecting curves such that C = C 1 C and P i = X X j=1 p=1 =1 ( (3 a ijp φ x p n j (i =1,,,N (4 with n j (j =1, being components of the unit outer normal vector to R on C. The boundary value problem defined by (1 and (3 has important applications in engineering. As an example, the steady-state temperature distribution in a flat plate which is thermally anisotropic and homogeneous obeys

3 (1 with N =1. The temperature and heat flux are given by φ 1 and (P 1,P respectively, and a 1j1p are the heat conduction coefficients. The plane static deformation of a homogeneous anisotropic elastic solid is governed by (1 with N =andx 1 and x as the Cartesian coordinates. The Cartesian displacement and traction are given by (φ 1, φ and(p 1,P respectively. The coefficients a ijp are the elastic moduli of the material occupying the solid. For a specific case, the elastostatic behaviour of a transverselyisotropic material which has transverse planes perpendicular to the 0x 1 x plane and which undergoes plane deformation is governed by C φ 1 x 1 A φ x + L φ 1 x + L φ x 1 +(F + L φ x 1 x = 0, +(F + L φ 1 x 1 x = 0, (5 a special case which can be recovered from (1 if we let N =anda = A, a 1111 = C, a 11 = a 11 = F, a 11 = a 11 = a 11 = a 11 = L and the remaining a ijl be zero. Clements and Rizzo [1] provided a real boundary integral equation method (based on fundamental solutions for the numerical solution of the boundary value problem defined by (1 and (3. In the present paper, a complex variable boundary element method (CVBEM is proposed as a useful alternative numerical technique for solving the problem. The method is implemented using constant elements and used to solve some specific problems. Based on the Cauchy s integral formula, the CVBEM was originally introduced by Hromada II and Lai [] for the special case a ijp = δ jp δ i1 δ 1 [δ ij is the ronecer-delta], i.e. where the system (1 reduces to the twodimensional Laplace s equation. Further development and refinement of the method were carried out by Hromada II and his co-researchers (e.g. Hromada and Yen [3], Whitley and Hromada II [6], Hromada II and Whitley 3

4 [4], [7], and Hromada II [5]. Introducing the theory of complex Hadamard finite-part (hypersingular integrals, Linov and Mogilevsaya [8] formulated a CVBEM to solve certain elastostatic problems governed by a particular system of elliptic partial differential equations. The theory of complex hypersingular integrals is also successfully applied here to develop acvbemforsolving(1numerically subject to (3. BASIC EQUATIONS The system (1 admits solutions of the form (Clements and Rizzo [1] φ (x 1,x =Re ( N X A f (z =1, (6 where f are holomorphic functions of z = x 1 + p x in R, p are the solutions, with positive imaginary parts, of the (N-th order polynomial (characteristic equation det ³h a i11 + pa i1 + pa i1 + p a i i =0, (7 and A are solutions of the homogeneous system of equations =1 h ai11 + p a i1 + p a i1 + p a i i A =0. (8 Notice that, because of (, equation (7 admits only non-real solutions which occur in complex conjugate pairs. From (4 and (6, we find that P i (x 1,x =Re X j=1 =1 L ij f(z 0 n j, (9 4

5 where the prime denotes differentiation with respect to the relevant argument and L ij = (a ij1 + p a ij A. (10 =1 Since f are holomorphic functions of z in R, Cauchy s integral formulae give πif (ξ 1 + p ξ = πif 0 (ξ 1 + p ξ = I (x 1,x C I (x 1,x C f (x 1 + p x d (x 1 + p x (x 1 + p x ξ 1 p ξ (11 f (x 1 + p x d (x 1 + p x (x 1 + p x ξ 1 p ξ (1 for (ξ 1, ξ R. Noticethatf 0 (z denotesthefirst order derivative of f with respect to z. We assume C is assigned an anticlocwise direction. For the case where (ξ 1, ξ lies on a smooth part of C, the formulae (11 and (1 can be modified to become πif (ξ 1 + p ξ = P πif 0 (ξ 1 + p ξ = H I (x 1,x C I (x 1,x C f (x 1 + p x d (x 1 + p x (x 1 + p x ξ 1 p ξ (13 f (x 1 + p x d (x 1 + p x (x 1 + p x ξ 1 p ξ (14 where P and H denote that the integral is to be interpreted in the Cauchy principal and Hadamard finite-part sense, respectively. For further details on the theory of complex Cauchy principal and Hadamard finite-part integrals, refer to Linov and Mogilevsaya [8]. 3 CVBEM A CVBEM for the approximate solution of the boundary value problem defined by (1 and (3 may be devised using (6, (9 and (11-(14 as follows. The boundary C is discretised by putting M closely-paced well-spaced out points (x (1 1,x (1, (x ( 1,x (,, (x ( 1,x (,, and (x (M 1,x (M (in 5

6 anticlocwise direction on it. If we denote the straight line segment from (x ( 1,x ( to(x (+1 1,x (+1 byc ( [ =1,,,M], then we mae the approximation C'C (1 C ( C (M. [Notethatwetae(x (M+1 1,x (M+1 = (x (1 1,x (1.] Let us write u (x 1,x +iv (x 1,x = where u and v are real functions [u = φ ]. Inverting (15, we obtain f (z = =1 =1 A f (z, (15 N [u (x 1,x +iv (x 1,x ], (16 where [N ]istheinverseof[a ]. The existence of [N ]isguaranteedifp 1, p,,p N 1 and p N are all distinct (Clements and Rizzo [1]. Approximating u and v by u (x 1,x ' u (q v (x 1,x ' v (q for (x 1,x C (q, (17 then for (ξ 1, ξ R C the imaginary parts of both (11 and (13 give rise to the approximation v p (ξ 1, ξ ' 1 π where u (p and v (p are constants, Γ (m p (ξ 1, ξ andθ (m p (ξ 1, ξ arerealparameters defined by MX m=1 =1 n (m u Γ (m p (ξ 1, ξ +v (m Θ (m p (ξ 1, ξ o, (18 = Γ (m p (ξ 1, ξ +iθ (m p (ξ 1, ξ =1 h A p N γ (m (ξ 1, ξ +iθ (m (ξ 1, ξ i, 6

7 γ (m (m+1 (ξ 1, ξ =ln z c ln (m z c and θ (m (ξ 1, ξ = Ω (m Ω (m Ω (m (ξ 1, ξ if π Ω (m (ξ 1, ξ π (ξ 1, ξ +π if π Ω (m (ξ 1, ξ < π (ξ 1, ξ π if π < Ω (m (ξ 1, ξ π where Ω (m (ξ 1, ξ =Arg(z (m+1 c Arg(z (m c, Arg(z denotes the principal value of the argument of the complex number z, z (m p x (m and c = ξ 1 + p ξ. If the region R is convex, then θ (m θ (m (ξ 1, ξ =cos 1 z (m+1 = x (m 1 + (ξ 1, ξ can be calculated directly from c + z (m z (m+1 c (m+1 z z (m c z (m c. By letting (ξ 1, ξ =(y (p 1,y (p (x (p x (p 1,x (p+1 + x (p / (the midpoint of C (p in (18, we obtain the approximate system v (q p = 1 π MX m=1 =1 n (m u Γ (m p (y (q 1,y (q +v (m Θ (m p (y (q 1,y (q o (19 for p =1,,,N and q =1,,,M. Equations (19 constitute a system of MN linear algebraic equations in MN unnowns u (m 1,,,N; m =1,,,M. and v (m ( = Over each of the line segments C (m (m =1,,,M, either φ or P i are specified according to (3. Thus, from (3, (6, (9 and (14, we obtain either u (p = µ (y (p 1,y (p ifφ are specified over C (p, (0 or 1 π MX X m=1 =1 j=1 =1 + ³ J ij w (pm n³ Jij q (pm + K ij q (pm K ij w (pm v(m u(m o n (p j = Q i (y (p 1,y (p ifp i are specified over C (p, (1 7

8 where J ij =Re{L ij N },K ij =Im{L ij N },n (m j are the components of the unit normal vector to R on C (m, q (pm such that q (pm (z (p + iw (pm =(z (m c (p and w (pm 1 (z (m+1 c (p are real constants 1 and c (p = + z (p+1 /. We find that (0 and/or (1 gives rise to an additional MN equations in u (m and v (m. We may solve (19 [for p =1,,,N and q =1,,,M]togetherwith (0 and/or (1 as a system of MN linear algebraic equations for the MN unnowns u (m and v (m ( =1,,,N; m =1,,,M. However, from numerical experiments, we find that, depending on the discretisation of the boundary C, (19 [for p =1,,,N and q =1,,,M] together with (0 and/or (1 may be poorly conditioned and may give rise to numerical values of v that are extremely large in magnitude, ruining subsequent calculations of φ in the interior of R. Perhaps this observation is not surprising at all due to the fact that v are not uniquely determined by the boundary value problem under consideration (but u = φ are. One method which is used successfully to overcome the difficulty associated with the poorly conditioned system of linear algebraic equations is to fix thevalueofv across the segment C (M,e.gsetv (M =0, and solve (19, with p =1,,,N and q =1,,,M 1, together with (0 and/or (1 as a well conditioned system of (M 1N linear algebraic equations. Once u (m and v (m ( =1,,,N; m =1,,,Mareallnown,we can compute f approximately via f (ξ 1 + p ξ ' 1 πi MX m=1 =1 N h u (m (ξ 1, ξ +iv (m (ξ 1, ξ i ³ γ (m (ξ 1, ξ +iθ (m (ξ 1, ξ and hence φ (ξ 1, ξ [using(6]atanypoint(ξ 1, ξ in the interior of R. ( 8

9 4 SPECIFIC PROBLEMS We shall now apply the CVBEM described above to solve some specific problems. Problem 1. Tae the elliptic partial differential equation 5 φ + φ + φ =0 (3 x 1 x 1 x x which is a special case of (1 with N =1anda 1j1p =5δ j1 δ p1 + δ j1 δ p + δ j δ p1 + δ j δ p. We apply the CVBEM to solve (3 in the square region R given by subject to the conditions R = {(x 1,x :0<x 1 < 1, 0 <x < 1} φ = x 1 8x 1 4 x x =1 (x 1 +4 for 0 <x 1 < 1, x +1 φ(0,x = for 0 <x ( x +1 +4x < 1, 1 φ(x 1, 0 = (x 1 +1 for 0 <x 1 < 1, x φ(1,x = for 0 <x ( x +4x < 1. (4 It can be easily verified by substitution that the boundary value problem defined by (3 and (4 has the exact solution x 1 x +1 φ(x 1,x =. (5 (x 1 x +1 +4x To apply the CVBEM to solve the boundary value problem, we place M equally spaced out nodal points on the square boundary (the nodal points include the four vertices of the square. Once the relevant holomorphic function is completely determined on the square boundary, we compute φ(x 1,x at 9

10 various points in the interior of the square region. The numerical results obtained by using M =40andM = 10 are compared with the exact solution (5 in Table 1. It is obvious that the numerical values compare favourably with the exact ones. More importantly, there is a noticeable improvement in the numerical results as the number of nodal points increases from 40 to 10. Table 1. Comparison of numerical values of φ at various interior points (x 1,x with the exact solution. (x 1,x M =40 M =10 Exact (0.10, (0.10, (0.50, (0.90, (0.50, (0.50, (0.70, Problem. Let the elliptic partial differential equation be given by φ +4 φ =0 (6 x 1 x which is a special case of (1 with N =1anda 1j1p = δ j1 δ p1 +4δ j δ p. The boundary C is chosen to be a pentagon with vertices A (0, 0, B (0, 1, C(1, 1, D(1/, 1/ and E (1, 0. For this particular case, R is a concave region. 10

11 For a specific boundaryvalueproblem,wesolve(6inr subject to φ(0,x = cosµ 1 x φ(x 1, 1 = exp(x 1 cos for 0 <x < 1(onAB, µ 1 φ(x 1,x 1 = exp(x 1 cosµ 1 x 1 φ(x 1, 1 x 1 = exp(x 1 cosµ 1 [1 x 1] for 0 <x 1 < 1(onBC, for 1 <x 1 < 1(onCD, for 1 <x 1 < 1(onDE, φ(x 1, 0 = exp(x 1 for0<x 1 < 1(onAE. (7 It can be easily verified by direct substitution that the exact solution of the boundary value problem is φ(x 1,x =exp(x 1 cosµ 1 x. (8 We discretise the sides AB, BC and AE into 0 equal length boundary elements per side and each of DE and CD into 10 elements per side (so that M = 80 and apply the CVBEM to solve (6 in the concave region R subject to (7. Once all the relevant holomorphic functions are constructed approximately by the CVBEM, we compute φ at various interior points in the concave region. The numerical values of φ thus obtained at selected interior points are compared with the exact values from (8 in Table. The numerical and exact values show reasonable agreement with each other. 11

12 Table. Comparison of numerical values of φ at various interior points (x 1,x in the concave region with the exact solution. (x 1,x CVBEM Exact (0.5, (0.5, (0.5, (0.50, (0.50, (0.60, (0.60, Problem 3. Consider the system of partial differential equations (5 with A =16,C=18,F=7andL =4, i.e. 18 φ 1 x φ x 1 x +4 φ 1 x = 0, 16 φ x +11 φ 1 x 1 x +4 φ x 1 = 0. (9 For the system (9, the constants A, L j and N required in the CVBEM s calculations are given by " 11ip1 (18 + 4p [A ] = ip (18 + 4p 1 #, i i [N ] = 1 " i (9 + p 1 (9+p 11ip (9 + p # 1 11ν i (9 + p 1(9+p 11ip 1 (9 + p, à h ip (18 + 4p [L 1 ] = 1 1i ip [7 77 (18 + 4p ] 1!, 4i (18 7p 1 / (18 + 4p 1 4i (18 7p / (18 + 4p à 4i (18 7p 1 / (18 + 4p 1 4i (18 7p / (18 + 4p! [L ] = h ip (18 + 4p 1 1i h ip (18 + 4p 1i, 1 (30

13 where ν =(p p 1 9 ( p + p 1 andp 1 and p are the solutions of 64p p + 7 = 0 such that Im{p 1 } > 0andIm{p } > 0, i.e. p 1 ' i and p ' i. For a particular boundary value problem, we choose the region R to be given by R = {(x 1,x :0<x 1 < 1, 0 <x < 1}. We are interested in solving (9 in R subject to P (x 1, 1 = Re φ (0,x = Re ( X L [x 1 +1+p ] =1 ( X A [1 + p x ] =1 for 0 <x 1 < 1, for 0 <x < 1, ( X φ (x 1, 0 = Re A [x 1 +1] for 0 <x 1 < 1, =1 ( X φ (1,x = Re A [ + p x ] for 0 <x < 1. (31 =1 The exact solution of the boundary value problem defined by (9 and (31 is given by ( X φ (x 1,x =Re A [x 1 +1+p x ]. (3 =1 By discretising the sides of the square into 80 equal length segments, we apply the CVBEM to solve the boundary value problem and calculate φ at various points in the interior of the square domain. The numerical values of φ are compared with the exact values from (3 at various points in the interior of the square in Table 3. It is obvious that a reasonably good accuracy in the numerical results is achieved by the CVBEM. 13

14 Table 3. Comparison of numerical values of (φ 1, φ at various interior points (x 1,x in the square with the exact solution. (x 1,x CVBEM Exact (0.5, 0.5 (3.553, (3.56, (0.5, 0.50 (.614,.816 (.60,.790 (0.5, 0.75 (1.064, 4.19 (1.050, (0.50, 0.5 (5.5, (5.68, (0.50, 0.50 (4.31, (4.35, (0.50, 0.75 (.755, (.754, 5.01 (0.75, 0.5 (7.54, (7.84, (0.75, 0.50 (6.314, 3.90 (6.341, (0.75, 0.75 (4.75, (4.770, SUMMARY A simple CVBEM is presented for the numerical solution of boundary value problems involving a rather general system of second order elliptic partial differential equations. It is applied to solve some specific problems, including one involving a concave region. For each of the problems, the numerical results obtained compare favourably with the nown (exact solution. The CVBEM can be efficiently implemented on the computer. Lie other boundary element methods, it requires only the boundary of the domain of interest to be discretised. Furthermore, the coefficients of the linear algebraic equations which approximate the boundary value problems [equations (19, (0 and/or (1] are easy to compute. TheproposedCVBEMcanberefined further for better accuracy by using higher order elements in the approximation of the holomorphic functions as in Hromada II and Lai [], or by following recent developments in CVBEM, e.g. Hromada II and Whitley [4]-[7]. 14

15 References [1]Clements,D.L.&Rizzo,F.J.Amethodforthenumericalsolution of boundary value problems governed by second-order elliptic systems. Journal of the Institute of Mathematics and its Applications, 1978,, [] Hromada II, T. V. & Lai, C. The Complex Variable Boundary Element Method in Engineering Analysis. Springer-Verlag, Berlin, [3] Hromada II, T. V. & Yen, C. C. Extension of the CVBEM to higher order trial functions. Applied Mathematical Modeling, 1988, 1, [4] Hromada II, T. V. & Whitley, R. J. Expanding the CVBEM approximation in a series. Applied Numerical Mathematics, 1993, 11, [5] Hromada II, T. V. The Best Approximation Method in Computational Mechanics. Springer-Verlag, Berlin, [6] Whitley, R. J. & Hromada II, T. V. Complex logarithms, Cauchy principal values, and the complex variable boundary element method. Applied Mathematical Modeling, 1994, 18, [7] Hromada II, T. V. & Whitley, R. J. A new formulation for developing CVBEM approximation functions. Engineering Analysis with Boundary Elements, 1996, 18, [8] Linov, A. M. and Mogilevsaya, S. G., Complex hypersingular integrals and integral equations in plane elasticity, Acta Mechanica, 1994, 105,