Numerical Integration of Arbitrary Functions over a Convex and Non Convex Polygonal Domain by Eight Noded Linear Quadrilateral Finite Element Method

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1 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: AUSTRALIA JOURAL OF BASI AD APPLIED SIEES ISS: EISS: Joural home page: umercal Itegrato of Arbtrary Fuctos over a ovex ad o ovex Polygoal Doma by Eght oded Lear Quadrlateral Fte Elemet Method 1 A.M.Yogtha ad K.T. Shvaram 1 Assstat professor, Departmet of Mathematcs, ty Egeerg ollege, Bagalore- 6, Karataa, Ida. Assstat professor, Departmet of Mathematcs, Dayaada Sagar ollege of Egeerg, Bagalore- 78, Karataa, Ida. Address For orrespodece: K.T.Shvaram, Assstat professor, Departmet of Mathematcs, Dayaada Sagar ollege of Egeerg, Bagalore , Karataa, Ida. A R T I L E I F O Artcle hstory: Receved 11 September 016 Accepted 10 ovember 016 Publshed 8 ovember 016 Keywords: umercal Itegrato, Geeralzed Gaussa quadrature rule, ovex ad o covex polygoal doma. A B S T R A T Obectves: I ths paper, the umercal evaluato of the tegral of arbtrary fucto over a covex ad o covex polygoal doma s bee vestgated. Method: ovex ad o covex polygoal doma s dscretzed to sub polygos by eght oded lear quadrlateral Fte elemet method based o Geeralzed Gaussa quadrature rule, to evaluate the typcal tegral of arbtrary fuctos over the covex ad o covex polygoal doma. Also plotted the extracted samplg pots both covex ad o covex rego. umercal examples are gve of testg the performace of the proposed method. ITRODUTIO As a wdely used umercal method, Fte elemet method s effcet for computer aded egeerg, fte elemet aalyss (FEA) has become a tegral ad maor compoet the desg or modelg of a physcal pheomeo varous dscples. The fte elemet method has proved superor to other umercal methods due to ts better adaptablty to ay complex geometry. The fte elemet method (FEM) s a umercal procedure that ca be used to obta solutos to a large class of egeerg problems stress aalyss, heat trasfer, electromagetsm ad flud flow etc. The calculato of tegrals of arbtrary fuctos over covex ad o covex polygoal doma s oe of the most dffcult part solvg appled problems FD, electrodyamcs, quatum mechacs, heat flow across a boudary betwee materals wth dfferet coductvty etc. especally to solvg two dmesoal partal dfferetal covex ad o covex rego by explct tegrato method to extract the stffess matrx. The tegrals arsg practcal problems are ot always smple ad other quadrature scheme caot evaluate exactly. umercal Itegrato over tragle ad square rego are smple but covex ad o covex polygoal doma s challegg tas to tegrato over a arbtrary fucto. The tegrato pots have to be creased order to mprove the tegrato accuracy ad s desrable to mae these evaluatos by few gauss pots as possble. The method proposed here s termed as Geeralzed Gaussa quadrature rule. From the lterature of revew we may realze that several wors umercal tegrato usg Gaussa quadrature over varous regos have bee carred out (O.. Zeewcz, R.L. Taylor ad J. Z. Zhu,000, K. J. Bathe, 1996, Md.Shafqual Islam ad M.Alamgr Hossa, 009, R.D. oo, 1981,. T. Reddy, D. J. Ope Access Joural Publshed BY AESI Publcato 016 AESI Publsher All rghts reserved Ths wor s lcesed uder the reatve ommos Attrbuto Iteratoal Lcese ( BY). To te Ths Artcle: A.M.Yogtha ad K.T. Shvaram., umercal Itegrato of Arbtrary Fuctos over a ovex ad o ovex Polygoal Doma by Eght oded Lear Quadrlateral Fte Elemet Method. Aust. J. Basc & Appl. Sc., 10(16): , 016

2 105 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: Shppy,1981). Gaussa Legedre quadrature over two- dmesoal tragle rego gve (D.A. Duavat,1985, K.T.Shvaram,013, G.Lague, R.Baldur,1977) to costruct the umercal algorthm based o optmzato ad group theory to compute quadrature rule for umercal tegrato of bvarate polygoal over polygoal doma are preseted (S.E.Mousav, H. Xao ad.suumar, 009). umercal tegrato over polygoal rego by usg, sple fte elemet method ad cubature formula over polygos are dscussed (hog-ju L, Paola Lambert ad attera Dago,009). I ths paper, covex ad o covex polygoal doma s dscretzed to sub polygos by 8-ode quadrlateral elemets ad the we apply Geeralsed Gaussa quadrature method to evaluate the typcal tegral of arbtrary fuctos over the covex ad o covex polygoal doma. The paper s orgazed as follows. I secto. we preset the Geeralzed Gaussa Quadrature formula. I secto 3. presets the mathematcal prelmares requred for uderstadg the dervato ad dscretsed the covex ad o covex polygoal doma to sub 8- ode quadrlateral elemets ad the derve a ew Gaussa quadrature formula over a quadrlateral rego to calculate samplg pots ad weght coeffcets of varous order ad also plotted the extracted samplg pots both covex ad o covex rego. I secto 4. we compare the umercal results wth some llustratve examples. Geeralsed Gaussa Quadrature rule: Suppose the tegral of the form b ( x) x) dx w x ) a 1 x [ a, b] ad w R, 1,,3,4, Where x ad w are samplg pots ad correspodg weghts respectvely. The quadrature formula are gve Eq.(1) s sad to be Geeralzed Gaussa quadrature rule wth respect to the set of fuctos {1,l x, x, xl x, x, x l x, x, x, x l x, x l x} o [0, 1] of order = 5, 10, 15, 0, 40 are gve Table 1. (J. Ma.et al. 1996) we usg these samplg pots ad ts weghts the product of polyomal ad logarthmc fucto. 3. Formulato of tegrals over a Lear ovex Quadrlateral Elemet: The tegral of a arbtrary fucto f ( x, over a arbtrary covex quadrlateral rego s gve by I f ( x, () Let us cosder a arbtrary 8- ode quadrlateral elemet the global coordate s mapped to - square the local coordate The sopermetrc coordate trasformato from (x, plae to s gve by Where, (=1,,3,4,5,6,7,8) are the vertces of the quadrlateral elemet (x, plae ad deotes the shape fucto of ode such that (1) (3) From the Eq.(3) ad Eq.(4), we have (4)

3 106 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: Smlarly ad (a) Fg. 1: (a) ovex polygoal Doma (b) o covex polygoal Doma (b) The test the tegral doma s show Fg. 1(a) wth two quadrlateral elemets covex polygoal doma wth vertces are (0, 0.5), (0.1, 0), (0.7,0.), (1, 0.5),(0.75, 0.85) ad (0.5, 1) ad Fg. 1(b) wth fve

4 107 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: quadrlateral elemets o covex polygoal doma wth vertces are (0, 0.75), (0.5, 0.5), (0.5, 0), (0.75, 0), (1, 0.5), (0.75, 0.75), (0.75, 0.85), (0.5, 1), (7/8, 5/8) ad (1/, 5/8) 3.1. umercal Itegrato over a covex rego: Itegral form of Eq. () rewrtte as I f ( x, f ( x, Q1 Q f ( m1( 1( J1d d f ( m( ( J d d m = w w [ f ( m1(, ), 1(, )) J1 f ( m(, ), (, )) J ] 1 1 Where = = = = = Where are samplg pots ad are correspodg weghts we preset the followg algorthm to calculate samplg pots ad ts weght coeffcets as ase 1 W J1* w * w x y m1 1 ase W J * w * w x y m result = ase 1 + ase to computed the samplg pots ad correspodg weghts based o the above algorthm for order = 5, 10, 15, 0 ad plotted the dstrbuto of samplg pots covex polygoal doma of varous order

5 108 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: (a) = 15 (b) = 0 Fg. : Dstrbuto of Samplg pots ovex polygoal doma 3.. umercal Itegrato over a o covex rego: Itegral form of Eq. () rewrtte as I f ( x, f ( x, f ( x, f ( x, Q1 Q Q3 Q4 Q5 f ( x, m f ( m1( 1( J1d d f ( m( ( J d d f ( m3( 3( J3d d f ( m4( 4( J 4d d f ( m5( 5( J5d d = w w [ f ( m1(, ), 1(, )) J1 f ( m(, ), (, )) J 1 1 f ( m3(, ), 3(, )) J3 f ( m4(, ), 4(, )) J 4 f ( m5(, ), 5(, )) J5] Where = = = = = = = = = = = = = =

6 109 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: = * samplg pots ad correspodg weghts are calculated by the above equatos for order = 5, 10, 15, 0 ad plotted the dstrbuto of samplg pots o covex polygoal doma of varous order (a) = 15 (b) = 0 Fg. 3: Dstrbuto of Samplg pots o covex polygoal doma 4. umercal Results: We have compared the umercal results obtaed usg Geeralsed Gaussa quadrature method wth that of umercal results arrved (hog-ju L, Paola Lambert ad attera Dago,009) of varous order = 5, 10, 15, 0 ad are tabulated Table 1 ad Table 1: ovex rego Exact value Order omputed value 100(( x 0.5) y 0.5) ) e [Ref. 11] = ( x 0.5) ( y 0.5) [Ref. 11 ] x y [Ref. 11 ] 3 4x 3y [Ref. 11 ] =0 =0 = Table : o covex rego Exact value Order omputed value 100(( x 0.5) y 0.5) ) e [Ref. 11 ] = [Ref. 11] ( x 0.5) ( y 0.5) x y [Ref. 11 ] 3 4x 3y [Ref. 11 ] =0 =0 =

7 110 A.M.Yogtha ad K.T. Shvaram, 016 Australa Joural of Basc ad Appled Sceces, 10(16) ovember 016, Pages: ocluso: I ths paper, a ew quadrature method based o Geeralsed Gaussa quadrature rule s appled for the umercal tegrato of arbtrary fucto covex ad o covex polygoal rego s dscretsed to 8 oded quadrlateral elemet. We have compared the umercal results obtaed usg Geeralsed Gaussa quadrature method wth that of umercal results arrved (hog-ju L, Paola Lambert ad attera Dago,009) of varous order = 5, 10, 15, 0. The results obtaed are excellet agreemet wth exact values. REFEREES Bala, B., M. Selvaumar, 016. FD Aalyss of Iecto Mxer for G Eges for Optmzed Performace, Australa oural of basc ad appled sceces, pp: Bathe, K.J., Fte Elemet Procedures, Pretce Hall, Ic. Eglewood lffs,.j. hog- Ju L, Paola Lambert ad attera Dago, 009. umercal tegrato over polygos usg a eght- ode quadrlateral sple fte elemets, Joural of omputatoal ad Appled Mathematcs, 33: hog- Ju L, R.H. Wag, 006. A ew 8- ode quadrlateral sple fte elemet, Joural of omputatoal ad Appled Mathematcs, 195: oo, R.D., ocepts ad Applcatos of Fte Elemet Aalyss, ded. Joh Wley & Sos, Ic. ew Yor. Duavat, D.A., Hgh degree effcet symmetrcal Gaussa Quadrature rules for tragle, It. J. umer. Methods Eg, 1: Hossa, M.A. ad Md. Shafqul Islam, 014. Geeralsed omposte tegrato rule over a polygo usg, Gaussa quadratue, Dhaa U. J. Sc., 6(1): 5-9. Lague, G., R. Baldur, Exteded umercal tegrato method for tragular surfaces, It. J. umer. Methods Eg., 11: L,.J., P. Lambert,. Dago, 009. umercal tegrato over polygos usg a eght ode quadrlateral fte elemet, Joural of omputatoal ad Appled Mathematcs, pp: Ma, J., V. Rohl, S. Wadzura, Geeralzed Gaussa Quadrature rules for system of arbtrary fuctos, SIAM J.umer, Aal., 33(3): Maula, G., K.T. Shvaram, 016. A Fte Elemet Mesh Geerato for complex geometry, J. Math.omput. Sc., 6: Md. Shafqual Islam ad M. Alamgr Hossa, 009. umercal tegrato over a arbtrary quadrlateral rego, Appl. Math. omputato, Elsever, pp: Mousav, S.E., H. Xao ad. Suumar, 009. Geeralzed Gaussa Quadrature Rules o Arbtrary Polygos, It. J. umer. Meth. Egg, pp: 1-6. Reddy,.T., D.J. Shppy, Alteratve tegrato formulae for tragular fte elemets, It. J. umer. Methods Eg, 17: Roht, B.R., A. Athraa, 015. Desg ad Aalyss of Axal Flow ompressor to Reduce ose Usg fd, Joural of appled sceces research, 11(3): 16-. Shvaram, K.T., 013. Gauss Legedre Quadrature over a ut crcle, Iteratoal Joural of Egeerg Research & Techology, : 9. Shvaram, K.T., 013. Geeralsed Gaussa Quadrature over a Tragle, Amerca Joural of Egeerg Research, 0(09): Shvaram, K.T., 013. Geeralsed Gaussa Quadrature Rules over a arbtrary Tetrahedro Eucldea Three- Dmesoal Space, Iteratoal Joural of Appled Egeerg Research, 8: Shvaram, K.T., H.T. Praasha, 016. umercal Itegrato of Hghly oscllatg fuctos usg Quadrature Method, Global Joural of Pure ad Appled Mathematcs, 1: Sommarva, A., M. Vaello, 007. Product Gauss cubature over polygos based o Gree s Itegrato formula, BIT, umercal Mathematcs, 47: Uma Maheswar, K., S. Sathyamoorthy, 016. Forward Model Aalyss Dffuse Optcal Tomography usg RTE ad FEM, Advaces atural ad appled sceces, pp: Zeewcz, O.., R.L. Taylor ad J.Z. Zhu, 000. The Fte Elemet Method, ts bass ad fudametals, Sxth edto, Butterworth- Heema, A Imprt of Elsever.