INTEGRATION TECHNIQUES FOR TWO DIMENSIONAL DOMAINS

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1 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 INTEGRATION TECHNIQUES FOR TWO DIMENSIONAL DOMAINS Logh Peruml Lecturer, Fcult of Egeerg d Techolog, Multmed Uverst, Mlcc, Mls Astrct I ths work, three dfferet tegrto techques, whch re the umercl, sem-ltcl d ect tegrto techques re refl revewed. Numercl tegrtos re crred out usg three dfferet Qudrture rules, whch re the Clsscl Guss Qudrture, Guss Legedre d Geerlzed Guss Qudrture. Le tegrl method s used to perform sem-ltcl tegrto, whle the geerlzed equtos developed the uthor prevous works re used to crr out the ect tegrto. It s see tht Geerlzed Guss Qudrture rules outperforms other Qudrture rules durg the umercl d semltcl tegrtos. It s lso show tht the ect tegrto techque developed the uthor prevous works eld ccurte results for tegrto of moomls. Kewords: Numercl Itegrto, Sem-Altcl Itegrto, Ect Itegrto, Qudrture Rules, Clsscl Guss Qudrture, Guss Legedre, Geerlzed Guss Qudrture, Le Itegrl, d Geerlzed Equtos *** INTRODUCTION Itegrto of fucto over cert dom s frequetl ecoutered egeerg computtos. Fte Elemet Method (FEM) s oe of the computtol methods whch requre such tegrto techque to ot the stffess mtr, k ccordg to the formul elow (for sold mechcs): T k B D B d () Where Ω represets dom, B represets str dsplcemet mtr d D represets mtr propert mtr. The stffess mtr, k s lter used to determe respose of mterl to eterl lods, such s the odl dsplcemets, str d stress compoets. Aprt from sold mechcs, smlr tegrto s lso ecoutered other egeerg pplctos (whch re lzed usg FEM) such s therml prolems, flud flow prolems d etc. Thus, there hve ee m tegrto techques developed d proposed to perform the tegrto. Most commo techque used FEM to perform the tegrto s usg umercl tegrto techque. Numercl tegrto s preferred, sce t elds good results t lower computtol tme compred to ltcl method. Altcl solutos gve ccurte results, ut requre hgh computtol tme. Furthermore, ltcl solutos mght ot est for cert cses. Numercl tegrto s performed usg qudrture pots d weghts, whch re geerted specfcll for cert fuctos d/or dom. Three dfferet qudrture rules re vestgted ths work, mel Clsscl Guss Qudrture, Guss Legedre d Geerlzed Guss Qudrture. The tegrto pots d weghts re oted solvg set of fuctos wth cert tervl. These set of fuctos could e polomls, trgoometrc fuctos, d ss fuctos of prtculr fucto spce whch defe the dom []. Resultg Qudrture rules (tegrto pots d weghts) c the e used to solve tegrds of smlr tpe wth the set of fuctos chose erler. Guss Qudrture methods use set of poloml fuctos to evlute the tegrto pots d weghts, d thus t c e used to tegrte poloml fuctos ccurtel, ut performs poorl whe the fucto to e tegrted (tegrd) s dfferet from polomls such s fuctos wth frctol power, trgoometrc fuctos d etc. I order to solve for dfferet tpe of tegrds, ew set of fuctos of smlr tpe s the tegrds to e solved for eed to e selected d solved for ew Qudrture rules. Authors [] hve proposed to replce the set of fuctos used to solve for the Qudrture rules from polomls to fuctos from wder clsses, order to e le to solve tegrds from vrous fucto tpes. Bsed o ths cocept, uthors [] hve successfull preseted umercl scheme for Geerlzed Guss Qudrture whch c e used to tegrte vret of fuctos, cludg smooth fuctos d fuctos wth sgulr ed pots. Recet dvces umercl tegrto c e foud wth the cotet of polgol fte elemet method (PFEM). Those techques re [4]: Prttog of the polgol doms to severl trgles d lter perform tegrto usg umercl qudrture rules oto these trgles. Prttog the mster elemet to severl trgles d perform tegrto usg umercl qudrture rules oto these trgles the mster elemet wth soprmetrc mppg B utlzg cuture rules for rregulr polgos sed o trgles or coformg mppg. B utlzg geerlzed qudrture rules o trgles or polgos sed o smmetr groups d umercl optmzto. Volume: Issue: 7 Jul-4, 487

2 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 Severl techques to geerte ew hgher order qudrture rules for trgles c lso e see [5, 6]. Sem-ltcl tegrto ws lter troduced, to crese the ccurc of the soluto. Author [7] hs proposed ew sem-ltcl tegrto techque whch reduces doule tegrto to sgle tegrto through use of dvergece theorem. A tegrd s tegrted ltcll wth respect to vrle frst, d et tegrto wth respect to vrle s crred out umercll, usg Qudrture rule of choce. Fll, ect tegrto techques were proposed to mprove soluto ccurc. Such techque hs ee developed for qudrlterl elemets FEM. I [8], the uthor proposed ler mppg to trsform dstorted qudrlterls to the ut squre d lter tegrto s performed usg mtr covoluto. Aother developmet of ect tegrto techque over qudrlterls c e see [9]. The uthor developed ect tegrto techque for qudrlterls whch re formulted usg Wchspress shpe fuctos. Ect tegrto formuls for str sed qudrlterl elemets were lter developed []. Ths pper s rrged s follows. Secto covers mthemtcl ckgroud for the Qudrture rules used ths pper. Numercl tegrto, sem-ltcl tegrto d ect tegrto techques re epled sectos, 4 d 5, respectvel. Comprso etwee ll these tegrto techques re doe secto 6 through umercl smultos d the pper s fll cocluded secto 7.. MATHEMATICAL BACKGROUND Mthemtcl ckgrouds for the three dfferet Qudrture rules re provded the followg suhedgs.. Guss Legedre I cse of Guss Legedre (w() = ), the set of fuctos used to solve for tegrto pots d weghts re Legedre polomls []. P M / M M m m or m! m! m, whchever s teger m!! m Where represets the poloml degree. The tegrto pots re the roots for the Legedre polomls d c e determed usg umercl techque such s Newto- Rphso; () wth tl f f guess r cos 4 Where represets the terto umer, represets the poloml degree, d r represets the r th root of the poloml. Oce tegrto pots re determed, the weghts c e clculted usg the relto: P w (4) Itegrto tervl for Guss Legedre s [-, ].. 5 th Order Clsscl Guss Qudrture Clsscl Guss Qudrture (w() = ) tegrtes geerl polomls wth order - ectl, where represets tegrto order. The tegrto tervl s [, ]. The geerl polomls re of the form:... f ( ) (5) Thus, the tegrto c e represeted s: f ( ) d d ()...(6) Rewrtg the tegrto umercl form d susttutg the fucto wth geerl polomls (equto (5)) gves; f ( ) d w f ( ) w w... w... w f ( )... w f ( ) Itegrto pots d weghts w c the e oted comprg equtos (6) d (7) to get lst of equtos cosstg of vrles, w, d d solve them smulteousl. Itegrto lmts d deped o Qudrture rule of choce. For 5 th order Clsscl Guss Qudrture, = 5, = d =. Thus, equtos (6) d (7) ecome: (7) Volume: Issue: 7 Jul-4, 488

3 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: f ( ) d... 9 d (8)... 9 f ( ) d w f ( ) w 5... w... w w 5 f ( )... w 5 (9) Geerlzed Guss Qudrture f ( ) Guss Qudrture method such s Guss Legedre uses set of poloml fuctos to evlute the tegrto pots d weghts, d thus the method performs well whe the tegrd s poloml fucto. The method performs poorl whe the tegrd s dfferet from poloml fucto. J. M, V. Rokhl d S. Wdzur [] replced the set of fuctos used to solve for the Qudrture rules from polomls to fuctos from wder clss d successfull preseted umercl scheme for Geerlzed Guss Qudrture. Numer of fuctos used for geertg the Qudrture rules s, where represets tegrto order. The Qudrture rules preseted [] c e used to tegrte vret of fuctos, cludg smooth fuctos d fuctos wth sgulr ed pots. The tegrto tervl s [, ].. NUMERICAL INTEGRATION Itegrto over -dmesol dom c e represeted usg Fu s theorem []: I s( ) r( ) s( r( f (, dd or f (, dd Dom to e tegrted c e eclosed : 4 costt les costt les d fucto costt les d fuctos () It c e see the prevous su-hedgs.-. tht ech Qudrture rule s developed sed o dfferet tervls. Thus, the tegrto lmts,, r d s the equto () ove eed to e coverted ccordg to the Qudrture rule of choce. Ths c e cheved utlzg the geerlzed equtos whch hve ee developed Logh et ll [, 4]: I I s( ) r( ) s( r( f (, d d or f (, d d U U L L j f ( m u c, m v c ) m m dv du j w w m m f ( m u c, m v c ) () The U d L the equto () ove represets upper d lower lmts of the tegrto, whch c e set s,, or -, depedg o the Qudrture rule of choce. 4. SEMI-ANALYTICAL INTEGRATION Le tegrl whch ws troduced G. Dsgupt, [7] eles tegrto to e crred out wthout eed to prtto the rego. The method reduces doule tegrto to sgle tegrto through use of dvergece theorem. A tegrd s tegrted ltcll wth respect to vrle frst, d et tegrto wth respect to vrle s crred out usg oe dmesol Qudrture rule. Formul for tegrto of fucto wth respect to the secod vrle s reclled from [7]: ( ) f [ ( ), ( )] d () Sgle tegrto gve equto () s crred out umercll usg Clsscl Guss Legedre, Geerlzed Guss Qudrture d 5 th order Guss Qudrture. Equto () s coverted to umercl form through the followg relto: I f wd w f () where, represets tegrto lmts, f() represets fucto to e tegrted (tegrd), w() represets weght fuctos, w represets tegrto weghts, represets tegrto pots, =,,,, d represets tegrto order. 5. EXACT INTEGRATION The geerlzed equto () coverts rtrr tegrto lmts of,, r d s to specfed tervl [U, L] wthout volvg smolc computto (full umercl). I order to derve epressos for ect tegrto of moomls, the tegrto lmts (or the fuctos coverg the dom) should e coverted to [, ]. Altcl tegrto of moomls m c the e represeted umercll usg the relto [4]: Volume: Issue: 7 Jul-4, 489

4 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 m m dd or dd m (4) It c e see tht the method show equto (4) requres hdlg of smols to detf the vrles, d ther respectve powers, m d. The ect tegrto method s lmted to polomls fuctos. 6. NUMERICAL EXAMPLE Set of fuctos f (, re tegrted usg the tegrto schemes ove (umercl, sem-ltcl d ect tegrto techques), over smmetrcl hego wth coordtes show Fg -(). Set of fuctos selected clude poloml fucto, rtol fucto, fucto wth rtol power, turl logrthmc fucto d epoetl fucto. The smmetrcl hego s seprted to regos; R d R ccordg to the requremet of Fu s Theorem, s show Fg -(). Ech rego s eclosed two costt les d two ler fuctos. () () Fg -: Emple of dom wth ler sdes two dmesos () Smmetrcl hegol elemet wthout prttog () Prttoed hegol elemet Itegrto of fucto over the etre dom s the gve : I I R f (, dd R f (, dd f (, dd f (, dd (5) The sem-ltcl techque s performed over the dom oudr d does ot requre prttog of the dom, whle the ect tegrto techque s lmted to poloml fuctos. Results for the tegrto techques re s show Tles -6. Percetge error s clculted sed o equto (6): Altcl soluto clculted soluto % Error % (6) Altcl soluto From the results the Tles -6, t s see tht for umercl tegrto techque, the Geerlzed Guss Qudrture teds to provde covergg soluto whe tegrto order s cresed. For ll the fuctos whch hve ee tested, Geerlzed Guss Qudrture outperforms Clsscl Guss Legedre d 5 th order Guss Qudrture. Frctol fuctos could ot e solved usg Clsscl Guss Legedre d 5 th order Guss Qudrture for cert tegrto orders. Ths s due to the fct tht some of the tegrto pots or weghts of these rules cot vlue of zero (leds to sgulr pot). Aprt from tht, these Qudrture rules were ot geerted sed o frctol fuctos. As for sem-ltcl tegrto techque, the Clsscl Guss Legedre d 5 th order Guss Qudrture re le to solve frctol fuctos. Ths s cused the ltcl tegrto of the frst vrle. Accurc of the results oted from ths techque hs slghtl mproved compred to the umercl tegrto techque. However, ths techque would requre more computtol tme compred to the umercl tegrto techque, due to the smolc mpulto. Fll, the ect tegrto techque elds ccurte solutos t lower computtol tme compred to the ltcl solutos. However, ths techque s pplcle for moomls ol. The ect tegrto techque s sutle for FEM sed o stffess mtrces whch cosst of smple polomls, such s vrtul ode method d str sed elemets. Ths techque provdes ltertve for estg umercl tegrto techques, especll Guss qudrture whch requres hgh umer of tegrto pots d weghts to tegrte hgher order moomls. Volume: Issue: 7 Jul-4, 49

5 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: CONCLUSIONS Accurc of three dfferet Qudrture rules - Clsscl Guss Qudrture, Guss Legedre d Geerlzed Guss Qudrture performg umercl tegrto wth two dmesol doms hve ee successfull compred through smultos. Three dfferet tegrto techques hve ee demostrted ths work, whch re the umercl, sem-ltcl d ect tegrto techques. The tegrto lmts were coverted ccordgl through utlzto of the geerlzed equtos developed the uthor prevous work [, 4]. It s see tht Geerlzed Guss Qudrture performs etter compred to other Qudrture rules. Sem-ltcl method s recommeded for smple tegrls d whe hgh ccurc s requred, sce the method requres hgh computtol tme d soluto mght ot est for comple tegrls. The ect tegrto techque s pplcle for FEM sed o stffess mtrces whch cosst of smple polomls, such s vrtul ode method d str sed elemets. Tle -: Altcl solutos for tegrto of vrous fuctos over the smmetrcl hegol dom. Fucto Altcl soluto f (, ( / ) 5 (+) / 7 (+(l e + e ( e e) Tle -: Numercl tegrto techque - Results oted for vrous Qudrture rules d tegrto orders,. Fucto Clsscl Guss Geerlzed Guss 5 th order Guss f (, Legedre Qudrture Qudrture s used G. Dsgupt, [7] ( / ) 5 Ift (Dvso zero) Ift (Dvso zero) Ift (Dvso zero) (+) / (+(l e Volume: Issue: 7 Jul-4, 49

6 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 Fucto f (, Tle -: Numercl tegrto techque - Percetge error Geerlzed Guss Qudrture Clsscl Guss Legedre 5 th order Guss Qudrture s used G. Dsgupt, [7] ( / ) 5 N/A N/A N/A (+) / (+(l e Tle -4: Sem-ltcl tegrto techque - Results oted for vrous Qudrture rules d tegrto orders,. Fucto Clsscl Guss Legedre Geerlzed Guss 5 th order Guss Qudrture f (, Qudrture s used G. Dsgupt, [7] ( / ) (+) / (+(l e Volume: Issue: 7 Jul-4, 49

7 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 Tle -5: Sem-ltcl tegrto techque - Percetge error Fucto Clsscl Guss Geerlzed 5 th order f (, Legedre Guss Qudrture Guss Qudrture s used G. Dsgupt, [7] ( / ) (+) / (+(l e Tle -6: Ect tegrto techque - Results oted for poloml fuctos. Fucto f (, Soluto from Ect Percetge error (%) Averge mmum tme Averge mmum tme Itegrto Techque elpsed for elpsed for ect tegrto techque (secods) R. for R 8. for R R 8 R R for R.4 for R ltcl techque (secods).44 for R.45 for R.44 for R.4 for R REFERENCES []. Mousv, S. E., Xo, H. d Sukumr, N. (), Geerlzed Guss qudrture rules o rtrr polgos. It. J. Numer. Meth. Eg., Volume 8 (pp. 99 ). []. Smuel Krl, d Wllm Studde (966). Tchecheff sstems: Wth pplctos lss d sttstcs, Pure d Appled Mthemtcs, Volume XV, Iterscece Pulshers Joh Wle & Sos, New York- Lodo-Sde. []. J. M, V. Rokhl d S. Wdzur (996). Geerlzed Guss Qudrture Rules for Sstems of Artrr Fuctos, SIAM Jourl o Numercl Alss, Volume, No. (pp ). [4]. Mrkus Krus, Amrthm Rjgopl d Pul Stem, Ivestgtos o the polgol fte elemet method: Costred dptve Delu tessellto d Volume: Issue: 7 Jul-4, 49

8 IJRET: Itertol Jourl of Reserch Egeerg d Techolog eissn: 9-6 pissn: -78 coforml terpolts, Computers d Structures () 46 [5]. Frz Huss, M. S. Krm, d Rzw Ahmd Approprte Guss qudrture formule for trgles, Itertol Jourl of Appled Mthemtcs d Computto, Volume 4() 4-8. [6]. H.T.Rthod, K.V.Ngrj, B.Vektesudu d N.L.Rmesh, Guss Legedre Qudrture over Trgle, Jourl of Id Isttute of Scece, 84, pges 8-88, 4. [7]. G. Dsgupt (). Itegrto wth polgol fte elemets, Jourl of Aerospce Egeerg, Volume 6, Issue (pp. 9 8). [8]. Jck Tuml, Ect Two-Dmesol Itegrto sde Qudrlterl Boudres, Jourl of Grphcs, GPU, d Gme Tools, Volume, Issue, 6 pge 6-7. [9]. Dsgupt, G. (8). Stffess Mtrces of Isoprmetrc Four-Node Fte Elemets Ect Altcl Itegrto, J. Aerosp. Eg., (), []. D. Hmd d A. Ztr, A New Altcl Itegrto Epresso Appled To Qudrlterl Fte Elemets, Proceedgs Of The World Cogress O Egeerg (WCE), Vol II, Jul - 4, 8, Lodo, U.K. []. Kreszg, Erw (999). Advced Egeerg Mthemtcs, 8th Ed., Wle. []. Thoms, G. B., Jr. d Fe, R. L. Clculus d Altc Geometr, 8th ed. Redg, MA: Addso-Wesle, p. 99, 996. []. Logh Peruml d Thet Thet Mo, Geerlzed equtos for umercl tegrto over two dmesol doms usg Qudrture rules, Itegrto: Mthemtcl Theor d Applctos, Vol., Issue 4, 4. [4]. Logh Peruml, Ect tegrto of moomls over two dmesol regos, Itertol Coferece o Dscrete Mthemtcs d Appled Scece (ICDMAS4), M st - rd 4, Uverst of Th Chmer of Commerce (UTCC), Bgkok, Thld. BIOGRAPHIE Logh Peruml receved hs Bchelor degree Mechcl Egeerg d Mster Mechcl Egeerg from Uverst Teg Nsol, Mls. Curretl workg s lecturer wth Fcult of Egeerg d Techolog t Multmed Uverst, Mls. Hs reserch terests clude umercl methods, fte elemet method, computto d fuzz logc. He hs pulshed reserch rtcles t tol d tertol jourls, coferece proceedgs, d cotruted to chpter of ook. Volume: Issue: 7 Jul-4, 494