Explicit Decoupled Group Iterative Method for the Triangle Element Solution of 2D Helmholtz Equations

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1 Interntionl Mthemticl Forum, Vol. 12, 2017, no. 16, HIKARI Ltd, Explicit Decoupled Group Itertive Method for the Tringle Element Solution of 2D Helmholtz Equtions Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn Fculty of Science nd Nturl Resources Universiti Mlysi Sbh Kot Kinblu Sbh Mlysi Copyright c 2017 Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn. This rticle is distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct Numericl methods re used to solve the two-dimensionl Helmholtz eqution. The Explicit Decoupled Group (EDG) method is pplied to the liner system generted from the first-order tringle finite element pproximtion. Numericl experiments re conducted to verify the effectiveness of the method. Keywords: Explicit Decoupled Group; Helmholtz; Glerkin Scheme; Tringle element; hlf-sweep itertions 1 Introduction Finite element methods llow weighted residul schemes (e.g., subdomin, colloction, lest-squres, moments, nd Glerkin methods) to be used to gin pproximte solutions. Using the first-order tringle finite element pproximtion bsed on the Glerkin scheme, this pper describes n Explicit Decoupled Group (EDG) method with the GussSeidel strtegy for solving the two-dimensionl (2D) Helmholtz eqution. This pproch is compred with the Full-Sweep GussSeidel (FSGS) nd Explicit Group (EG) methods. To investigte the effectiveness of EDG, consider the 2D Helmholtz eqution

2 772 Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn () (b) Figure 1: () nd (b) show the definition of the ht function R i,j (x, y), of fullnd hlf-sweep tringle elements t the solution domin. 2 U x U αu = f (x, y), (x, y) [, b] [, b] (1) y2 with the dirichlet boundry conditions U(x, ) = g 1 (x), x b, U(x, b) = g 2 (x), x b, U(, y) = g 3 (y), x b, U(b, y) = g 4 (x), y b, Here, α is non-negtive constnt nd f (x, y) is function with sufficient smoothness. To formulte the full- nd hlf-sweep tringle element pproximtions for (1), we focus on the uniform node points shown in Figure 1. Bsed on Figure 1, the solution domin is discretized uniformly in the x nd y directions with mesh size h, defined s x = y = h = b, m = n + 1. (2) n The full- nd hlf-sweep networks of tringle finite elements re guidelines for the tringle finite element pproximtions. These pproximtions form systems of finite element pproximtions for (1). Using the sme concept of hlf-sweep itertions pplied to finite difference methods [1], finite element networks consist of severl tringle elements, ech with two solid node points of type (see Figure 1). Consequently, the full- nd hlf-sweep itertive lgorithms re pplied to such node points until the itertive convergence criterion is stisfied. Approximte solutions t the remining node points (i.e., points

3 Explicit decoupled group itertive method for the tringle element solution of type ) re clculted directly [5, 6, 7]. In the reminder of this pper, we discuss the finite element method bsed on the Glerkin scheme for discretizing (1), describe the proposed method for solving liner systems generted from tringle element pproximtions, present some numericl exmples, nd nlyze our results. 2 Hlf-Sweep Tringle Element Approximtions Using the first-order tringle finite element pproximtion, discretiztion bsed on the Glerkin scheme gives n pproximtion of (1). Considering node points of type, the generl pproximtion, in the form of n interpoltion function for n rbitrry tringle element e, is given by [2, 3] Ũ [e] (x, y) = N 1 (x, y)u 1 + N 2 (x, y)u 2 + N 3 (x, y)u 3 (3) nd the shpe functions N k (x, y), k = 1, 2, 3, cn be written s:, N k (x, y) = 1 A ( k + b k x + c k y), k = 1, 2, 3 (4) A = x 1 (y 2 y 3 ) + x 2 (y 3 y 1 ) + x 3 (y 1 y 2 ) 1 x 2 y 3 x 3 y 2 b 1 y 2 y = x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1, b 2 b 3 = y 3 y 1 y 1 y 2, c 1 c 2 c 3 = x 3 x 2 x 1 x 3 x 2 x 3 The first-order prtil derivtives of the shpe functions cn be written s: (N x k (x, y)) = y (N k (x, y)) = b k det c A k det A, }, k = 1, 2, 3 (5) Figure 2 illustrtes the ht function R r,s (x, y) in the solution domin [8, 9]. The pproximtion functions for the full- nd hlf-sweep cses over the entire domin in (1) become: Ũ(x, y) = f(x, y) = R r,s (x, y)u r,s (6) r=0 s=0 r=0 s=0 R r,s (x, y)f r,s (7)

4 774 Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn () (b) Figure 2: () nd (b) illustrte the ht function R i,j (x, y), of full- nd hlfsweep tringle elements, respectively, on the solution domin. nd Ũ(x, y) = f(x, y) = r= 0, 2, 4 s= 0, 2, 4 + m 1 m 1 r= 1, 3, 5 s= 1, 3, 5 r= 0, 2, 4 s= 0, 2, 4 + m 1 m 1 r= 1, 3, 5 s= 1, 3, 5 R r,s (x, y) f r,s (8) R r,s (x, y) f r,s R r,s (x, y) f r,s (9) R r,s (x, y) f r,s Similrly, the pproximte function f (x, y) for ll cses cn be esily defined vi the ht function. Indeed, (6) nd (8) cn be denoted s pproximte solutions for (1). To construct the full- nd hlf-sweep liner finite element pproximtions for (1), we must tke ccount of the Glerkin scheme. Consider the Glerkin residul method [10, 11, 12] to be: D R i,j (x, y)e i,j (x, y) = 0, i, j = 0, 1, 2,..., m (10), E(x, y) = 2 U + 2 U αu f(x, y) is residul function. Applying x 2 y 2 Greens theorem, (10) cn be written s

5 Explicit decoupled group itertive method for the tringle element solution ( R i,j (x, y) u λ b b, y dx + R i,j (x, y) u ( Ri,j (x, y) u x x + R i,j (x, y) y ) x dy αr i,j (x, y) U u y + αr i,j (x, y) U ) dxdy = F i,j (11) b b F i,j = R i,j (x, y) f (x, y) dxdy. Replcing (5) nd imposing the boundry conditions in (1), (11) genertes liner system for ll cses. This cn be simplified to:, K i,j,r,s = C i,j,r,s (12) b b Ci,j,r,s = (R i,j (x, y) R r,s (x, y)) dxdy. Essentilly, for the full- nd hlf-sweep cses, (12) cn be strightforwrdly expressed in stencil form s: 1. Full-sweep:[9] β 2 β 3 β 2 β 1 β 2 U i,j = β f i,j (13) β 3 β 2 F i,j = β 0 (f i 1,j + f i+1,j + f i,j 1 + f i,j+1 + f i 1,j 1 + f i+1,j+1 + 6f i,j ) β 0 = h 2 /12, β 1 = (4 + α6β 0 ), β 2 = 1 rβ 0, β 3 = rβ 0

6 776 Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn 2. Hlf-sweep:[8] γ 1 γ 3 γ 3 γ 1 γ 3 γ 3 γ 1 U i,j = γ 0 U i,j = γ 0 U i,j = γ f i,j, i = 1 f i,j, i 1, n f i,j (14) F i,j = γ 0 (f i 1,j 1 + f i+1,j 1 + f i 1,j+1 + f i+2,j + 5f i,j ), i = 1 F i,j = γ 0 (f i 1,j 1 + f i+1,j 1 + f i 1,j+1 + f i+1,j+1 + f i 2,j + f i+2,j + 6f i,j ), F i,j = γ 0 (f i 1,j 1 + f i+1,j 1 + f i 1,j+1 + f i+1,j+1 + f i 2,j + 5f i,j ), i = n γ 0 = h 2 /6, γ 1 = (4 + α6γ 0 ), γ 2 = 1 rυ 0, γ 3 = rγ 0. i 1, n In fct, the stencil forms in (13) nd (14) implicte seven node points in the pproximtion equtions. 3 Formultion of EDG Method The EDG method [1] ws introduced to solve 2D Poisson equtions using five-point rotted finite difference pproximtion. An itertive implementtion involves only red node points until convergence, with pproximte vlues of the remining node points computed by direct methods. (For discussion of this method bsed on the finite difference pproch, see [1, 6, 7]). Let four-point group be considered to form the (4x4) liner system [7] γ 1 γ γ 2 γ γ 1 γ γ 2 γ 1 U i,j U i+1,j+1 U i+1,j U i,j+1 = S 1 S 2 S 3 S 4 (15) S 1 = U i 1,j 1 + U i 1,j+1 + U i+1,j 1 F i,j, S 2 = U i,j+2 + U i+2,j + U i+2,j+2 F i+1,j+1, S 3 = U i,j 1 + U i+2,j 1 + U i+2,j+1 F i+1,j, S 4 = U i,j+1 + U i 1,j + U i 1,j+2 F i,j+1.

7 Explicit decoupled group itertive method for the tringle element solution The system in (15) cn be rewritten s two (2x2) independent liner systems. The EDG method is [ Ui,j U i+1,j+1 ] (k+1) = 1 γ 2 1 γ 2 2 [ γ1 γ 2 γ 2 γ 1 ] [ S1 S 2 ] (16) nd S 1 = U (k+1) i 1,j 1 + U (k+1) i 1,j+1 + U (k+1) i+1,j 1 F i,j, S 2 = U (k) i,j+2 + U (k) i+2,j + U (k) i+2,j+2 F i+1,j+1. [ Ui+1,j U i,j+1 ] (k+1) = 1 γ 2 1 γ 2 2 [ γ1 γ 2 γ 2 γ 1 ] [ S3 S 4 ] (17) S 3 = U (k+1) i,j 1 + U (k+1) i+2,j 1 + U (k+1) i+2,j+1 F i+1,j, i 1,j+2 F i,j+1. S 4 = U (k) i,j+1 + U (k) i 1,j + U (k) 4 Numericl Results To compre the itertive methods described in the previous section, numericl experiments were conducted on: 2 U x U αu = (cos(x + y) + cos(x y)) α cos(x) cos(y). (18) y2 The exct solution is U(x, 0) = cos x, U(x, π 2 ) = 0, U(0, y) = cos y, U(π, y) = cos y. U(x, y) = cos(x) cos(y). The effectiveness of the proposed EDG is compred with tht of EG nd FSGS. The experiments were conducted on n Intel(R) Core (TM) i7 CPU 860@2.80 GHz with 6 GB memory. The point GS method cts s the control in the comprison of numericl results. The number of itertions, execution time, nd mximum/bsolute error re used for comprison. Convergence ws determined by the tolernce ε = nd experiments were conducted for mesh sizes of 284, 308, 332, nd 356. The numericl results re presented in Tble 1.

8 778 Mohd Kmlrulzmn Md Akhir nd Jumt Sulimn Tble 1: Comprison of the number of itertions, execution time (seconds) nd mximum bsolute error for the itertive methods n Methods k t Abs.Error FSGS e EG e-7 EDG e-6 FSGS e EG e-7 EDG e-6 FSGS e EG e-7 EDG e-6 FSGS e EG e-7 EDG e-6 Tble 2: : The reduction percentges of the EDG nd EG methods compred with FSGS method. 5 Conclusions Methods k t EG EDG We hve presented n ppliction of hlf-sweep itertions with block method for solving sprse liner systems generted from the first-order tringle finite element pproximtion using the Glerkin scheme. The numericl results show tht EDG is superior to EG nd FSGS in terms of the number of itertions nd execution time. This is minly becuse of the reduced computtionl complexitythe implementtion of EDG only considers pproximtely hlf of ll interior node points in the solution domin. References [1] A.R. Abdullh, The Four Point Explicit Decoupled Group (EDG) Method: A Fst Poisson Solver, Intern. J. of Comp. Mth., 38 (1991), [2] C.A.J. Fletcher, The Glerkin method: An introduction, In Numericl Simultion of Fluid Motion, J. Noye (Ed.), North-Hollnd Publishing

9 Explicit decoupled group itertive method for the tringle element solution Compny, Amsterdm, 1978, [3] C.A.J. Fletcher, Computtionl Glerkin Methods, Springer Series in Computtionl Physics, Springer-Verlg, New York, [4] D.J. Evns, Group Explicit Methods for the Numericl Solutions of Prtil Differentil Equtions, Austrli, Gordon nd Brech Science Publishers, [5] D.M. Young, Itertive Solution of Lrge Liner Systems, London, Acdemic Press, [6] M.K.M. Akhir, M. Othmn, J. Sulimn, Z.A. Mjid nd M. Suleimn, Hlf Sweep Itertive Method for Solving Two-Dimensionl Helmholtz Equtions, Int. J. of App. Mth. nd Stt., 29 (2012), [7] M.K.M. Akhir, M. Othmn, J. Sulimn, Z.A. Mjid nd M. Suleimn, Four Point Explicit Decoupled Group Itertive Method Applied to Two- Dimensionl Helmholtz Eqution, Int. J. Mth. Anl., 20 (2012), [8] M.K.M. Akhir nd J. Sulimn, HSGS Method for the Finite Element Solution of Two-Dimensionl Helmholtz Equtions, Globl J. of Mth., 40 (2015), [9] M.K.M. Akhir nd J. Sulimn, Numericl Solutions of Helmholtz equtions the Tringle Element using Explicit Group Method, J. KALAM, (in press). [10] O.C. Zienkiewicz, Why finite elements?, ln Finite Elements in Fluids, R.H. Gllgher, J.T. Oden, C. Tylor, O.C. Zienkiewicz, (Eds), John Wiley nd Sons, London, [11] P.E. Lewis nd J.P. Wrd, The Finite Element Method: Principles nd Applictions, Addison-Wesley Publishing Compny, Wokinghm, [12] R. Vichnevetsky, Computer Methods for Prtil Differentil Equtions, Vol. I, New Jersey, Prentice-Hll, Received: July 9, 2017; Published: August 16, 2017