An Accurate Evaluation of Integrals in Convex and Non convex Polygonal Domain by Twelve Node Quadrilateral Finite Element Method

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1 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07), pp Research Inda Publcatons 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 , Karnataka, Inda. Department of Mathematcs, Ghousa College of Engneerng, Ramanagara-5659, Karnataka, Inda. Department of Mathematcs, Cty Engneerng College, Bangalore , 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.

2 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 )

3 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

4 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 = ξ η η ξ ξ η η ξ η 3 ξ n = ξ η η ξ ξ η η ξ η 3 ξ J = ξ η ξ η η ξ ξ 4 η ξ η η ξ η 4 ξ ξ η ξ η η ξ ξ η η 4 ξ η ξ 3 m = η η ξ η ξ η ξ η 3 ξ ξ n = η η ξ η ξ η ξ η 3 ξ ξ J = η ξ η η ξ ξ 4 η ξ η η ξ η 4 ξ ξ η η 4 ξ η ξ η η ξ η ξ ξ ξ

5 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 = ξ ξ 3 η ξη ξη η ξ η n = ξ ξ 3 η ξη ξ η η ξ η Q4 Q5

6 38 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha J = ξ η η ξ ξ ξ η ηξ ξ 4 η ξ η η ξ η 4 ξ ξη ξ 3 η η 4 ξ η ξ η m = 0.385ξη ξ η ξ 3 η ξη ξ η n = ξη ξ η ξ 3 η ξη ξ η J = ξ η η ξ ξ 4 η ξ η η ξ η 4 ξ ξη η ξ ξ η ξ η η 4 ξ ξ 3 η ξ η m3 = ξη ξ η ξ 3 η ξη η ξ n3 = ξη ξ η ξ 3 η ξη ξ η J3 = ηξ η ξ η 4 ξ ξ 4 η ξ η ξη η η ξ ξ ξ η η 4 ξ ξ 3 η ξ η η ξ m4 = ξη η ξ ξ 3 η ξη η ξ n4 = η ξ η ξ J4 = ηξ ξ 4 η η ξ η ξ η ξ ξη ξη η ξ ξ 3 η ξ η 4 m5 = ξ ξ 3 n5 = η ξ ξη η ξ ξ 3 η ξη 3 J5 = ηξ ξ ξ 4 η ξ η ξ η η 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

7 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 C N= [Ref. 0] C ( x 0.5) ( y 0.5) [Ref. 0 ] C x y 0. 5 dxdy [Ref. 0 ] C 3 4x 3y dxdy [Ref. 0 ] dxdy N=0 N=0 N=

8 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 N N= = N ( x 0.5) ( y 0.5) dxdy = N= ( x ) N ( y ) = N 3 4x 3y dxdy = dxdy N=0 N= 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 [3] Md. Shafqual Islam and M. Alamgr Hossan, Numercal ntegraton over an arbtrary quadrlateral regon, Appl. Math. Computaton, Elsever (009) [4] D. A. Dunavant, Hgh degree effcent symmetrcal Gaussan Quadrature rules for trangle, Int. J. Numer. Methods Eng (985) [5] D. A. Dunavant, Hgh degree effcent symmetrcal Gaussan Quadrature rules for trangle, Int. J. Numer. Methods Eng (985) 9-48.

9 An Accurate Evaluaton of Integrals n Convex and Non convex Polygonal Doman 4 [6] G. Lague, R. Baldur, Eξtended numercal ntegraton method for trangular surfaces, Int. J. Numer. Methods Eng. (977) [7] K.T. Shvaram, Generalsed Gaussan Quadrature over a Trangle, Amercan Journal of Engneerng Research. Vol. 0, Issue-09, pp.90 93, 03. [8] S. E. Mousav, H. Ξao and N. Sukumar, Generalzed Gaussan Quadrature Rules on Arbtrary Polygons, Int. J. Numer. Meth. Engng 009, pp. 6 [9] M. A. Hossan and Md. Shafqul Islam, Generalsed Composte ntegraton rule over a polygon usng, Gaussan quadratue, Dhaka Un. J. Sc. 6() 5-9, Jan 04 [0] Chong- Jun L, Paola Lambert and Catterna Dagnno, Numercal ntegraton over polygons usng an eght- node quadrlateral splne fnte elements, Journal of Computatonal and Appled Mathematcs, 33(009),pp.79-9 [] Chong- Jun L, Catterna Dagnno, An adaptve numercal ntegraton algorthm for polygons, Journal of Appled Numercal Mathematcs, 60(00),pp [] K.T. Shvaram, Numercal Integraton of arbtrary functons over a Convex and Non convex polygonal doman by Eght noded Lnear quadrlateral Fnte Element Method, Australan Journal of Basc and Appled Scences, vol. 0, pp.04-0 (06) [3] K.T. Shvaram, Yogtha.A, Numercal Integraton of Arbtrary Functons over a Convex and non convex polygonal doman by quadrature method, Journal of Mathematcal and Computatonal Scence, vol.6, pp.77-86, (06) [4] K.T. Shvaram, Generalsed Gaussan Quadrature Rules over an arbtrary Tetrahedron n Eucldean Three- Dmensonal Space, Internatonal Journal of Appled Engneerng Research. Vol.8, pp , Number 3 (03) [5] K.T.Shvaram, H.T Prakasha, Numercal Integraton of Hghly Oscllatng Functons usng Quadrature Method, Global Journal of Pure and appled Mathematcs, Vol., pp ( 06)

10 4 K.T.Shvaram, H.R. Jyoth and A.M. Yogtha

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements

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