Internatonal Journal of Computatonal and Appled Mathematcs. ISSN 89-4966 Volume, Number (07), pp. 33-4 Research Inda Publcatons http://www.rpublcaton.com An Accurate Evaluaton of Integrals n Convex and Non convex Polygonal Doman by Twelve Node Quadrlateral Fnte Element Method K.T.Shvaram, H.R. Jyoth and A.M. Yogtha Department of Mathematcs, Dayananda Sagar College of Engneerng, Bangalore-560078, Karnataka, Inda. Department of Mathematcs, Ghousa College of Engneerng, Ramanagara-5659, Karnataka, Inda. Department of Mathematcs, Cty Engneerng College, Bangalore -56006, Karnataka, Inda. Abstract In ths paper, we present Gauss Legendre quadrature method to fnd numercally approxmated by ntegral of arbtrary functon over convex and non convex polygonal doman. These doman are dscretsed nto twelve noded quadrlateral element, Dervng a new Gaussan quadrature formula for generatng samplng ponts and ts weght coeffcents. Numercal results are provded to ustfy the usefulness of the proposed method. Keywords: Fnte Element Method, Gauss Legendre quadrature, Convex and Non convex polygonal doman.. INTRODUCTION The Fnte element method s a numercal method used to obtan approxmate soluton of ntegral equatons are typcally occurs n Boundary element method (BEM) and Surface fnte element method (SFM) for solvng partal dfferental equaton on surface whch requres surface ntegrals, the ntegral arsng n practcal problems are not always smple and some quadrature scheme cannot evaluate exactly. Numercal evaluaton of ntegrals over trangle and square regon are smple but convex and non convex polygonal regon are challengng task to ntegraton of arbtrary functon, the method proposed here s termed as Gauss Legendre quadrature rule.
34 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha From the lterature of revew we ponted out that numercal ntegraton of arbtrary functon over trangle and quadrlateral regon have been carred out [-7], numercal ntegraton of arbtrary functon over polygonal doman are dscussed n [ 8-0] and numercal ntegraton of arbtrary functon over polygonal doman by splne fnte element method are dscussed n [0,], numercal ntegraton of arbtrary functon over convex and non convex polygonal doman by 4 node and 8- noded quadrlateral elements by usng generalzed Gaussan quadrature rule [,3 ]. In ths paper we dvded the polygonal doman nto node quadrlateral element and then we apply Gauss Legendre quadrature rule to evaluate the numercal ntegraton of arbtrary functon over convex and non convex polygonal regon. The paper s organzed as follows. In secton. we present the mathematcal formulaton requred for understandng the dervaton and dscretsed the convex and Non convex polygonal doman nto sub - node quadrlateral elements secton 3 and 4. derve a new Gaussan quadrature formula over a convex and non convex regon to calculate samplng ponts and weght coeffcents of varous order and also plotted the extracted samplng ponts n both convex and non convex regon. In secton 5. we compare the numercal results wth some llustratve examples.. FORMULATION OF INTEGRALS OVER A LINEAR CONVEX QUADRILATERAL ELEMENT The ntegral of an arbtrary functon f ( x, over an arbtrary convex quadrlateral regon Ω s gven by I = f(x, dxdy Ω () I f ( x, dxdy () Let us consder an arbtrary - node quadrlateral element n the global coordnate s mapped nto - square n the local coordnate (ξ, η). The sopermetrc coordnate transformaton from (ξ, plane to (ξ, η) s gven by ( x y ) = (x k k= ) N y k k (ξ, η) (3) Where (x k, y k ), (k=,, 3, 4, 5, 6, 7, 8, 9, 0,, ) are the vertces of the quadrlateral element n (ξ, plane and N k (ξ, η) denotes the shape functon of node k such that N = 3 ( x)( ( 0 + 9(x + y ) N = 9 3 ( ( 3x)( x ) N3 = 9 3 ( ( + 3x)( x ) N4 = 3 ( + x)( ( 0 + 9(x + y )
An Accurate Evaluaton of Integrals n Convex and Non convex Polygonal Doman 35 N5 = 9 3 ( + x)( 3( y ) N6 = 9 3 ( + x)( + 3( y ) N7 = 3 ( + x)( + ( 0 + 9(x + y ) N8 = 9 3 ( + ( + 3x)( x ) N9 = 9 3 ( + ( 3x)( x ) N0 = 3 ( x)( + ( 0 + 9(x + y ) N = 9 3 ( x)( + 3( y ) N = 9 3 ( x)( 3( y ) J = (x, = x (ξ,η) ξ Where x = ξ and y ξ = k= x k k= y k y x η η S k ξ, x y ξ S k ξ, y η = η = k= x k k= y k S k η S k η (a) Fg. (a) Convex polygonal Doman (b) (b) Non convex polygonal Doman
36 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha we test the ntegral doman s shown n Fg. (a) wth two quadrlateral elements n convex polygonal doman wth vertces are (0, 0.5), (0., 0), (0.7,0.), (, 0.5),(0.75, 0.85) and (0.5, ) and Fg. (b) wth fve quadrlateral elements n non convex polygonal doman wth vertces are (0, 0.75), (0.5, 0.5), (0.5, 0), (0.75, 0.5), (0.75, 0), (, 0.5), (0.75, 0.75), (.0, 0.9), (0.5, ), (0.875, 0.65) and (0.5, 0.75) 3. GAUSS LEGENDRE QUADRATURE OVER A NODE CONVEX REGION Integral form of Eq. () rewrtten as I = f(x, dxdy + f(x, dxdy Q Q I = f(x(ξ, η), y(ξ, η)) J dξ dη f ( m( n( Jd d f ( m( n( J d d m n = w w [ f ( m(, ), n(, )) J f ( m(, ), n(, )) J] Where m = 0.78750 ξ + 0.4968750 η + 0.053500 η ξ 0.0487500ξ 3 0.5468750 η 3 0.0406500 η ξ 3 0.040650η 3 ξ + 0.3500000 n = 0.405650 ξ + 0.37093750 η 0.98750 η ξ + 0.476565 ξ 3 0.33593750 η 3 + 0.077343750 η ξ 3 + 0.07734375 η 3 ξ + 0.365000 J = 0.93436 ξ + 0.388989 η + 0.05834960 ξ 0.056030 η + 0.050073 η ξ 0.04805499 ξ 4 η 0.06080664 ξ η 3 0.0036939 η ξ 4 + 0.00369394 η 4 ξ 0.98347680 ξ η 0.0648005 ξ 4 0.7776977η 4 0.0008764 η 6 0.080536 ξ 3 + 0.0008764 ξ 6 0.075953 + 0.048354 η 3 + 0.09076904 η 4 ξ + 0.0305634 η ξ 3 m = 0.7578500 η 0.0656500 η ξ + 0.063850 η 3 0.773437500 ξ 3 + 0.0070350 η ξ 3 + 0.0070350 η 3 ξ + 0.73750000 + 0.4843750 ξ n = 0.449875 η 0.390650 η ξ + 0.6375000 0.6787 η 3 0.0093750 ξ 3 + 0.063850 η ξ 3 + 0.06385η 3 ξ + 0.058593750 ξ J = 0.0597480 η 0.0335459 ξ 0.6464 η + 0.09973583 η ξ 0.0453845 ξ 4 η 0.05873 ξ η 3 + 0.0033483 η ξ 4 0.0033483 η 4 ξ + 0.056398 ξ η 0.05806884 η 4 ξ 0.0086094 η ξ 3 + 0.0955734 + 0.040075683 η 3 + 0.088740654 η 4 + 0.08045043 ξ 4 + 0.000444946 η 6 + 0.00796508789 ξ 3 0.00044494 ξ 6 0.03009033 ξ
An Accurate Evaluaton of Integrals n Convex and Non convex Polygonal Doman 37 Where ξ, η are samplng ponts and w, w weght coeffcents to computed the samplng ponts and correspondng weghts of order N = 5, 0, 5 and plotted the dstrbuton of samplng ponts n convex polygonal doman of varous order N = 5 N = 0 Fg. Dstrbuton of Samplng ponts n convex polygonal doman 4. GAUSS LEGENDRE QUADRATURE OVER A NODED NON CONVEX REGION Integral form of Eq. () rewrtten as I f ( x, dxdy f ( x, dxdy f ( x, dxdy f ( x, dxdy f ( x, dxdy Q Q Q3 I = f(x(ξ, η), y(ξ, η)) J dξ dη f ( m4( n4( J 4d d f ( m5( n5( J5d d f ( m( n( Jd d f ( m( n( J d d f ( m3( n3( J3d d m n = w w [ f ( m(, ), n(, )) J f ( m(, ), n(, )) J f ( m3(, ), n3(, )) J3 f ( m4(, ), n4(, )) J4 f ( m5(, ), n5(, )) J5] Where m = 0.03556500ξ 3 + 0.0355650 ξ 3 η + 0.035565 ξη 3 0.385000ξη 0.0546875 η 3 + 0.35000 + 0.09765650ξ + 0.99687500η n = 0.08789065ξ 3 + 0.07578ξ 3 η + 0.07578500ξη 3 + 0.787500000 0.06640650ξ η 0.05734375 η 3 0.4440650 ξ + 0.464843750 η Q4 Q5
38 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha J = 0.033995ξ 4 + 0.0006798η 6 0.0547564η 4 0.00996ξ 3 0.00067ξ 6 0.0667570 + 0.0467883ξ + 0.0567η 0.0486450 ηξ + 0.0473ξ 4 η + 0.0074577ξ η 3 + 0.0085394η ξ 4 0.0085394η 4 ξ 0.07975ξη + 0.03657ξ 3 η + 0.03337097η 4 ξ 0.006866455η 3 + 0.0389099ξ + 0.0593994 η m = 0.385ξη 0.0546875ξ 3 + 0.03556η 3 + 0.03556ξ 3 η + 0.03556ξη 3 + 0.4375 + 0.996875ξ 0.0976565η n = 0.99875ξη 0.08789065ξ 3 0.08789065η 3 + 0.05734375ξ 3 η + 0.05734375ξη 3 + 0.444065ξ + 0.444065η + 0.4065 J = 0.809539795ξ 0.05039675η + 0.70507568 η ξ 0.077865600ξ 4 η 0.0595500ξ η 3 + 0.0055688η ξ 4 0.00556886η 4 ξ 0.0430750ξη + 0.0380574η 4 0.0034337ξ 3 0.0085394ξ 6 + 0.09536743 + 0.040359η 3 + 0.097434997ξ 4 + 0.00853948η 6 + 0.03657η 4 ξ + 0.0037078857ξ 3 η + 0.09699707ξ 0.0907897949η m3 = 0.0664065ξη 0.05734375ξ 3 + 0.075785η 3 + 0.075785ξ 3 η + 0.075785ξη 3 0.048885η + 0.46484375ξ + 0.84375 n3 = 0.99875ξη 0.08789065ξ 3 0.08789065η 3 + 0.05734375ξ 3 η + 0.05734375ξη 3 + 0.444065ξ + 0.444065η + 0.4065 J3 = 0.085075378ηξ + 0.0078094η ξ 4 0.007809430η 4 ξ 0.0389380ξ 4 η 0.0977600ξ η 3 0.05365ξη + 0.0009697η 6 + 0.006900787η 4 + 0.04877498ξ 4 0.00766377ξ 3 0.00096974ξ 6 + 0.04768375 + 0.00696η 3 + 0.0055688η 4 ξ + 0.00853948ξ 3 η 0.09047698ξ 0.0598364η 0.045394897η + 0.0064849853ξ m4 = 0.06640650ξη + 0.057343750η 3 0.057343750ξ 3 0.0757850ξ 3 η 0.0757850ξη 3 + 0.787500000 0.464843750η + 0.464843750ξ n4 = 0.035565η 3 0.035565ξ 3 + 0.0976565η + 0.0976565ξ + 0.6500 J4 = 0.053654ηξ + 0.0055688ξ 4 η 0.007663η 3 + 0.0085394ξ η 3 + 0.0860949 0.030899047ξ + 0.0064849853η + 0.06685485ξ 4 + 0.05365ξη 0.00556886ξη 4 0.030899047η + 0.007663ξ 3 0.0085394ξ 3 η 0.006484985ξ + 0.06685485η 4 m5 = 0.650000000 + 0.9535000ξ 0.070350000ξ 3 n5 = 0.855468750η + 0.8065000 0.0097656ξ 0.46093750ξη 0.06679680η 3 + 0.003556ξ 3 + 0.03867875ξ 3 η + 0.03867875ξη 3 J5 = 0.05347595ηξ 0.004734ξ 4 0.0447045ξ 4 η 0.00857348ξ η 3 + 0.004987304ξ 0.085339355η + 0.007553005η 3 0.009073486 samplng ponts and correspondng weghts are calculated by the above equatons for order N = 5, 0, 5 and plotted the dstrbuton of samplng ponts n Non convex polygonal doman
An Accurate Evaluaton of Integrals n Convex and Non convex Polygonal Doman 39 (a) N = 0 (b) N = 5 Fg. 3 Dstrbuton of Samplng ponts n Non convex polygonal doman 5. NUMERICAL RESULTS We have compared the numercal results obtaned usng Gauss Legendre quadrature rule wth that of numercal results arrved n [0] and [] of varous order N = 5, 0, 5 and are tabulated n Table and TABLE. Convex regon Exact value Order Computed value 00(( x 0.5) ( y 0.5) ) e dxdy 0.03445073 C N=0 0.0344585 0.034458633 [Ref. 0] 0.03445863 C ( x 0.5) ( y 0.5) 0.56855586 [Ref. 0 ] C x y 0. 5 dxdy 0.9906549435 [Ref. 0 ] C 3 4x 3y dxdy 0.5453868050054 [Ref. 0 ] dxdy N=0 N=0 N=0 0.568577 0.5685545 0.5685558 0.99065478 0.99065065 0.990654943 0.545386895 0.54538684539 0.5453868050
40 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha TABLE. 3 Non convex regon Exact value Order Relatve error Relatve error[] 00(( x 0.5) ( y 0.5) ) e dxdy 0.000000076 N N=0 0.0000000045 0.000007 = 0.030839806464 0.0000000003 N ( x 0.5) ( y 0.5) dxdy = 0.5767705385664 N=0 0.00000008 0.0000000763 0.0000000005 0.00000 ( x ) N ( y ) = 0.035433053939807 N 3 4x 3y dxdy = 0.569373639505 6 dxdy N=0 N=0 0.000000090 0.0000000085 0.0000000096 0.0000000033 0.0000000046 0.000000000 0.0000008 0.000008 5. CONCLUSIONS In ths paper, Gauss Legendre quadrature rule s appled for the numercal ntegraton of arbtrary functon over convex and non convex polygonal doman s dscretsed nto noded quadrlateral element The results obtaned are n excellent agreement wth exact values. REFERENCES [] O. C. Zenkewcz, R. L. Taylor and J. Z. Zhu, The Fnte Element Method, ts bass and fundamentals, Sξth edton, Butterworth- Henemann, An Imprnt of Elsever (000). [] C. T. Reddy, D. J. Shppy, Alternatve ntegraton formulae for trangular fnte elements, Int. J. Numer. Methods Eng 7 (98) pp.890-896 [3] Md. Shafqual Islam and M. Alamgr Hossan, Numercal ntegraton over an arbtrary quadrlateral regon, Appl. Math. Computaton, Elsever (009) 55-54. [4] D. A. Dunavant, Hgh degree effcent symmetrcal Gaussan Quadrature rules for trangle, Int. J. Numer. Methods Eng (985) 9-48. [5] D. A. Dunavant, Hgh degree effcent symmetrcal Gaussan Quadrature rules for trangle, Int. J. Numer. Methods Eng (985) 9-48.
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4 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha