NUMERICAL INTEGRATION BY GAUSS-LEGENDRE QUADRATURE OVER TRIANGULAR DOMAINS

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1 NUMERICAL INTEGRATION BY GAUSS-LEGENDRE QUADRATURE OVER TRIANGULAR DOMAINS Luís Henrique Gzet de Souz Universidde Estdul Pulist Júlio de Mesquit Filho Fculdde de Engenhri de Ilh Solteir, Av. Brsil Centro nº56, Ilh Solteir, São Pulo, Brsil e-mil: João Btist Aprecido Universidde Estdul Pulist Júlio de Mesquit Filho Fculdde de Engenhri de Ilh Solteir, Av. Brsil Centro nº56, Ilh Solteir, São Pulo, Brsil e-mil: Astrct. The finite element method hs een used s powerful tool to otin pproimte solution to severl phenomen common in engineering field. When using the finite element method to discretize given prolem some integrls m rise nd its computtion must e done numericll. For two dimensionl domins s in this work it will e two dimensionl integrls. For tringulr finite elements there will e two dimensionl integrls over tringulr domins. For such domins there re not so much methods to ccomplish tht integrtion with ccurc. In this work it is proposed nother lgorithm to do numericl integrtion over two dimensionl domins shped s tringles. To ccomplish tht first liner trnsform is done to chnge the generl tringle into right tringle hving unitr sides. Afterwrds the stndrd tringle is trnsformed nonlinerl into stndrd rectngle suitle for the two dimensionl Guss-Legendre integrtion. The well known Guss-Legendre qudrture points nd weights for the two dimensionl integrtion over stndrd rectngle re then mpped ck to the stndrd tringle nd then to the generl tringle. B using supposed new trnsformtion it ws possile to otin set of Guss-Legendre smmetric points nd weights. Kewords: numericl integrtion, Guss-Legendre, finite element, tringulr element.. INTRODUCTION The Finite Element Method FEM is min numericl tool used to otin pproimte solution for mthemticl prolems tht rise from phsicl modeling, for emple, those ones ssocited to Fluid Mechnics nd Het Trnsfer. This method, computtionll developed nd coded, shows good results when pplied to solve prolems under sted stte nd unsted stte flow regimes, for liner nd non-liner equtions, nd to one, two nd three-dimensionl domins. An issue, mong others, tht need ttention in the contet of FEM is out doing the integrls tht pper during the discretiztion process of given prolem. Such integrls must e done for ech kind of element eing used. Effective numericl methods to perform tht jo re not esil found in literture. The most used tpes of elements descried in ppers re the tringulr nd qudrilterl. Hmmer 956, till we know, ws the first to ccomplish solution to the prolem of doing two-dimensionl numericl integrtion of function defined over generl tringulr domin. In the sme er Turner et l. 956 presented the first work out the Finite Element Method, followed Clough 96 nd Argris 96. Afterwrds, mn others reserchers such s Cowper 97, Lnno 977, Lurie 977, Redd nd Shipp 98 were elorting nd improving the integrtion formule. The purpose of this work is to otin method or lgorithm to do numericl integrtion over generl tringle shpes using nd dpting the ides of the Guss Legendre qudrture.. NUMERICAL METHOD FORMULATION TO DO INTEGRATION OVER A TRIANGLE Initill, to find suitle integrtion method to e used with FEM it is necessr to trnsform generl tringle into n intermedite stndrd right tringle with the following corner coordintes,;, nd,. To mp generl tringle into the stndrd one it is necessr to do: Y Finite Element X Trnsformtion, T,, Stndrd Right Tringle Element Figure. Coordinte trnsformtion from generl tringulr element into right ngle tringulr element.,

2 Such liner geometric trnsformtion cn e written s follows, ˆ ˆ, or nd. It is enough to find the coefficients in eqution, nd for tht is used the following constrints: Solving the liner sstem, we get eplicit equtions for,, nd :, 4, 4, 4c. 4d It is known from nlticl geometr tht the re, A, of generl tringle cn e otined from the coordintes of its corners. Such eqution is well-known nd is epressed A 5 Using this eqution into equtions 4-d it is possile to rewrite them s: A, A, A nd A. 6-d Using the definition of the Jcoen for the trnsformtion T, J T, 7 we otin from equtions - nd 6-d the following result

3 A A J T detj T A A A 8 Then, let us consider the integrtion over generl tringulr domin,, f,dd f, det[j, ]dd f, Add I A f, dd 9 With the previous trnsformtion ws possile to chnge the integrtion over generl tringle,, into n integrtion over stndrd right tringle,, however, it is not et useful to ppl directl the Guss-Legendre qudrture, so it is necessr to do nother trnsformtion in order to chnge the stndrd right tringle domin into stndrd squre domin. To do such trnsformtion, T, we propose the following formule v u u ; v. This trnsformtion is n etension of tht one proposed Rthod et l. 4. The difference is tht this one is smmetric, while Rthod s isn t. In Figure it is shown the trnsformtion from the tringulr domin,, to the new trnsformed squre domin, ˆ.,,, Stndrd Right Tringle Element Trnsformtion, T v,, ˆ,, u Stndrd Squre Element Figure. Coordinte sstem trnsformtion in order to trnsform the Stndrd right tringle domin into one Stndrd squre. The Jcoen of T trnsformtion is J T, u u,v v v u u v v det J u T u v, Then the integrl I in eqution 9 cn e chnged s follows I f,d ˆ fu,vdet JT dudv u v u v fu,v - dudv fu,v - dudv. ˆ In the right side of eqution the integrls intervls re [,] for oth es u or v. This fct is incomptile with the intervl definition for Legendre polnomils tht is [-,], so we do one lst trnsformtion T in order to trnslte nd scle the intervl [,] into n intervl [-,]. To ccomplish tht we use the following liner formule ξ η u ; v.

4 As this trnsformtion T is liner, see Figure, thus the Jcoen is constnt nd equl to u v u,v ξ ξ J T det J. 4 T ξ, η u v 4 η η v, ˆ, Trnsformtion, T -, η,,, u Stndrd Squre Element ξ -,-,- Figure. Liner trnsformtion T from the stndrd squre element, ˆ, into the mster element,. Thus the integrl I ˆ in eqution cn e modified to Mster Element I ˆ ˆ u v ξ η ξ η fu,v - dudv fξ,η - dξdη fξ,η - dξdη If we do the following definition ξ η gξ,η - fξ,η, then eqution 5 ecomes I gξ, ηdξdη. 7 Considering m Guss-Legendre integrtion points for the is ξ nd n integrtion points for the is η, thus the integrl I cn e pproimted s m n m n ξi η j I wi w j gξi,η j wk fξi,η j ; k i j; wk wi w j, i j k where ξ i nd w i, i,,,m re the Guss-Legendre integrtion points nd weights for the ξ-is, nd η j nd w j, j,,,n re the integrtion points nd weights for the η-is. It is interesting to rememer tht v j u η ξ ξ η i j i i j f k,k f[ - ui, - v j ] f[ -, - ] Once otined n pproimtion to the integrl I it is possile to otin through equtions, 5, 8 to otin n pproimtion for the integrl I s intended initill I m n AI AI ˆ AI A w k f k,k k.

5 For the previous eqution the Guss-Legendre points nd weights for integrtion over tringulr domin were tulted nd re shown in Tle. Tle. Guss-Legendre points nd weights for numericl integrtion. k k m n e e.97688e.7548e.7995e.5e.7995e.7548e.5e e e.58e- m n.658e.658e e e.84564e e-.8798e.6766e-.8585e e e.85657e-.75e.75e e e.78754e.789e-.6766e-.8798e.8585e e e.789e e.49649e.86966e- m n E-.6745E-.858E-.8559E E-.4586E-.6467E.4675E E E.76E-.557E E-.8559E.4586E E.75556E.756E E.94579E.5666E-.777E.7646E.9674E-.4675E-.6467E E E E.5666E E E.5877E-.688E.58546E.675E-.76E E.557E-.7646E.777E.9674E E.688E.675E E E.65E- Those results were produced using computtionl ppliction developed nd coded within FORTRAN95 compiler.. RESULTS.. Doing integrtion numericll Here re presented some emples of function integrtion over tringulr domins using eqution, developed in this work. It is done integrtion over two right tringles, one equilterl tringle, one isosceles tringle nd one sclene tringle, respectivel. All following results converged to the ect vlue, otined nlticl integrtion, for ll deciml plces shown, nd using up to four Guss-Legendre integrtion points for ech is or up to siteen points for the two directions. Right tringles with corners t,,, nd, I dd w k

6 I dd.4 5 Equilterl tringle with corners t,,, nd, I e dd Isosceles tringle with corners t -4,, -, nd -,5,- I dd 8 4 Sclene tringle with corners t -,-, -, nd 5,- I 7 8 dd dd Integrting over meshes with tringulr elements Numericl integrtion of function over meshes is crucil prt of the sptil discretiztion process of given prolem using the Finite Element Method. Such division of prolem domin into su domins, generll, is done using tringulr or qudrilterl finite elements. Now we show tht the methodolog developed in this pper works well to integrte function over tringulr meshes. Meshes used in this pper were produce using softwre developed nd coded Aprecido Mesh Unitr Squre Net is presented the mesh dt nd drwing. In the mesh dt it is shown: totl numer of elements; totl numer of nodes; node numer; -coordinte vlue; -coordinte vlue; element numer; element tpe; nd element nodes. Tle. Tringulr mesh dt. MeshD - GerMlD v.. Mesh numer of elements; Mesh numer of nodes 8 9 Node numer; Node -coordinte; Node -coordinte,e,e 5,E-,E,E,E 4,E 5,E- 5 5,E- 5,E- 6,E 5,E- 7,E,E 8 5,E-,E 9,E,E Element numer; Element tpe; Element nodes TRG 5 4 TRG 5 TRG TRG 6 5 TRG TRG TRG TRG Figure 4. Tringulr mesh drwing with 8 elements nd 9 nodes.

7 Tle. Results for integrtion over the elements of the Mesh. f, e Numer of Guss-Legendre points Elements m n m n m n 5 m n m n 8,988,49759,4964,4964,4964,988,49759,4964,4964,4964,669949,4695,4694,4694,4694 4,669949,4695,4694,4694,4694 5,669949,4695,4694,4694,4694 6,669949,4695,4694,4694,4694 7, ,57987, , , , ,57987, , , Mesh Annulr Sector This mesh hs n nnulr sector shpe with center t,, inner rdius equl to, outer rdius equl to, nti-clock strting ngle equl to º, nd finishing ngle equl to º, with tringulr elements nd 4 nodes. Figure 5 show mesh spect nd some dt. Glol mesh dt were omitted due to lck of spce. Tle 4. Results for numericl integrtion over tringulr elements of n nnulr sector shped mesh. f, Numer of Guss-Legendre points Elements m n m n m n 5 m n m n 8 984, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,7895 5,7895 5,7895 5, , , , , , , , , , , , , , , , , , , , , ,788 98, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,754

8 Figure 5. Annulr sector mesh with elements nd 4 nodes.... Mesh Qudrilterl Polgon Mesh is qudrilterl polgon with corners t -5,6; 5,-6; -8, nd,9. Such domin ws divided into 4 elements hving nodes. Mesh dt were lso omitted due to its ig etension. Following we show the mesh drwing in Figure 6. Tle 5. Numericl vlues for integrtion over the tringulr elements with the qudrilterl mesh. f, Numer of Guss-Legendre points Elements m n m n m n 5 m n m n 8-9, , , , , , , , , , , , , , , , , , , , , , , , , , ,5558 -,9875 -,9875 -,9875 -, ,7459 4, , , , , , , , , , , , , , , , , , , , , , , , ,44758,78875,78875,78875, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Integrtion over mesh elements ws ccurte nd the results shown in Tles -5 hve ver good convergence. So, the methodolog developed here hs gret potentil s n integrtion tool to e pplied to severl kind of prolems in which is necessr integrtion over tringulr domins. Meshes presented in this work re smll just for emple purpose, ut this methodolog is sclle up to meshes with millions of elements. The mesh genertor developed Aprecido 6 nd used in this work hs friendl grphic interfce tht llows comprehensive visuliztion of the mesh nd n es understnding of elements nd nodes reltionship.

9 Figure 6. Qudrilterl mesh with 4 elements nd nodes. 4. CONCLUSIONS Numericl solution of integrls is n issue tht is necessr in severl pplictions. When the integrnd function do not llow n nltic integrtion; when the domin geometr hs gret compleit, nd so son, w to ccomplish such tsk is numericl integrtion. There re in the relted literture some integrtion methods tht were developed iming to solve vriet of different kinds of integrtion. In this work we tried to develop methodolog tht hs strong theoreticl sis nd is suitle lgorithm to computtionl implementtion. Integrnds tht pper in Section. were chosen in order to let know the reder out the nltic solution of ech integrl s well to verif tht the numericl solution otined using the formultion developed in this pper provides results tht greed ver well with the nltic ones. For tht greement it ws necessr just to use up to four Guss-Legendre integrtion points for ech direction or up to siteen points for the two-dimensionl domin. Integrls pplied over meshes nd presented in Section. lso presented ver good results when done using the Guss-Legendre Qudrture methodolog developed in this work for integrtion over tringulr domins. Severl others cses were tested nd our eperience show tht is possile to otin convergence with up to eight deciml plces using just three Guss-Legendre integrtion points for ech direction or nine integrtion points for two es. Using eight Guss-Legendre points is not necessr we showed it just for testing nd documenting. High degree of Guss-Legendre integrtion is necessr just for ver comple function defined over ig domins. For simple function or for smll domins integrtion degrees of, 4 or 5, generll, re enough. This fct is enough to set vile this methodolog in pplictions to solve Fluid Mechnics nd Het Trnsfer, mong others prolems, the Method of Finite Elements. Also, this methodolog cn e used in n method in which is needed integrtion over tringulr domins. Guss-Legendre points nd weights here developed nd shown were derived directl from the clssic nd well-posed Guss-Legendre points nd weights for one dimensionl integrtion prolems, thus voiding other techniques tht m need new ones weights nd integrtion points for integrtion over two dimensionl domins. We elieve tht this technique cn e successfull etended to three or higher dimensionlit integrtions. Jing 99, Tng, Cheng nd Tsng 995, Winterscheidt nd Surn 99, Codin 998 nd severl others uthors vlidted the Finite Element Method otining good results when solving Fluid Mechnics nd Het Trnsfer prolems. Finll, this methodolog developed here hs shown to e simples, roust nd relile to ccomplish integrtions over two dimensionl tringulr domin with ccurc nd, reltive, low computtionl costs. Also, this technique is ver useful working together the Finite Element Method, providing tool to compute severl integrtions tht pper during ppliction of such method. 5. REFERENCES Aprecido, J. B., Mesh Genertor D, Personl Communiction. Argris, J. H., 96, Recent Advnces in Mtri Methods of Structurl Anlsis, Elmsford, New York, Pergmon Press. Clough, R. W., 96, The Finite Element Method in Plne Stress Anlsis. In: Conf. on Electronic Computtion, Proceedings Pittsurg, Pen: Americn Societ of Civil Engineers, pp

10 Codin, R., 988, Comprison of Some Finite Element Methods for Solving the Diffusion-Convection-Rection Eqution, Comput. Methods Appl. Mech. Engng, v. 56, pp.85-. Cowper, G. R., 97, Gussin Qudrture Formuls for Tringles, Int. J. Num. Meth. Engng, Vol. 7, pp Hmmer, P. C. nd Stroud, A. H., 956, Numericl Integrtion over Simplees, Mth. Tles Other Aids Computtion, Vol., pp.7-9. Jing, B., 99, A Lest Squres Finite Element Method for Incompressile Nvier-Stokes Prolems, Interntionl Journl for Numericl Methods in Fluids, NASA, Vol.4, pp Lnno, F. G., 977, Tringulr Finite Elements nd Numericl Integrtion, Computers Struct., Vol. 7, 6p. Lurie, D. P., 977, Automtic Numericl Integrtion over Tringle, CSIR Spec. Rep. WISK 7, Ntionl Institute for Mthemticl Sciences, Pretori. Rthod, H. T., Ngrj, K. V., Venktesudu, B. nd Rmesh, N. L., 4, Guss Legendre Qudrture over Tringle, Journl Indin Institute of Science, Bnglore, Indin, pp Redd, C. T., nd Shipp, D. J., 98, Alterntive Integrtion Formule for Tringulr Finite Elements, Int. J. Num. Meth. Engng, Vol. 7 pp. -9. Tng, L. Q., Cheng, T., Tsng, T. H., 995, Trnsient Solutions for Three Dimensionl Lid Driven Cvit Flows Lest Squres Finite Element Method, Interntionl Journl for Numericl Methods in Fluids, Vol., Leington, pp Turner, M. J., Clough, R. W., Mrtin, H. C., Topp, L. P., 956, Stiffness nd Deflection Anlsis of Comple Structures, J. Aeron. Sci., Vol., pp Winterscheidt, D., Surn, K. S., 994, p-version Lest Squres Finite Element Formultion for Two-Dimensionl, Incompressile Fluid Flow, Interntionl. Journl Numericl Methods in Fluids, Vol. 8, pp RESPONSIBILITY NOTICE The uthors re the onl responsile for the printed mteril included in this pper.