Modeling by Meshless Method LRPIM (local radial point interpolation method)

Transcription

1 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 Modelg by Mehle Mehod LRPM (local radal po erpolao mehod) Abrac: A. Mouaou a,. Bouzae b a. Deparme of phyc, Faculy of cece, Moulay mal Uvery B.P. Meke, Morocco, mouaou.phyque@gmal b. Deparme of phyc, Faculy of cece, Moulay mal Uvery B.P. Meke, Morocco, oura584@yahoo.fr Numercal oluo phycal egeerg problem eed approprae umercal approxmao mehod. Mehle mehod have araced creag aeo rece year for eekg of approxmae oluo of al boudary value problem govered by paral dffereal equao. h paper, we pree a udy of a D problem of a elac homogeou recagular plae by ug he mehod LRPM. We vegae he covergece ad accuracy of mehod LRPM ad umercal value are preeed o pecfyg he covergece doma by precg maxmum ad mmum value a a fuco of drbuo ode umber ad by ug wo radal ba fuco: Gaua (EXP) ad h plae ple (P). alo pree a comparo wh umercal reul for dffere maeral ad he radal ba fuco RBF. Fally a comparave udy of umercal reul wh aalycal oluo preeed. Keyword: Lear Elacy, recagular plae, Mehle Mehod LRPM, Radal ba fuco, uppor doma. roduco Mehle mehod ha araced more ad more aeo from reearcher rece year, ad regarded a a poeal umercal mehod compuaoal mechac. everal mehle mehod, uch a mooh parcle hydrodyamc (PH) mehod [], eleme free Galerk (EFG) mehod[3], mehle local Perov-Galerk (MLPG) mehod[4], po erpolao mehod (PM)[5-6], local po erpolao mehod (LPM) [7] ad local radal po erpolao mehod (LRPM) ha bee propoed by Lu e al. [6]. LRPM, he po erpolao ug he radal ba fuco ulzed o coruc hape fuco wh dela fuco propery. he wdely ued radal ba fuco (RBF) arer: h plae ple (P) ad Gaua (EXP) [8-9]. Local weak form are developed ug weghed redual mehod locally from he paral dffereal equao of lear elacy of D old

2 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 ad a umber of umercal example are preeed o demorae he covergece ad accuracy, valdy ad effcecy of he pree mehod. he local radal po erpolao mehod LRPM a Mehle mehod wh propere of mple mplemeao of he eeal boudary codo ad wh he lower co ha ML mehod. h paper deal wh he effec of zg parameer of ubdoma o he covergece ad accuracy of he mehod ad umercal value are preeed o pecfyg he covergece doma by precg maxmum ad mmum value a a fuco of drbuo ode umber ad by ug he radal ba fuco P (he h plae ple) ad Gaua (EXP). alo pree a comparo wh umercal reul for dffere maeral. Numercal reul agree wh he aalycal oluo of he defleco. he LRPM mehod wll be developed for olvg he problem of a h elac homogeou plae. he local weak form ad umercal mplemeao are preeed eco 3, umercal example for D problem are gve eco 4. he, he paper ed wh reul, dcuo ad fally he cocluo. RPM hape fuco mehle mehod u h (x) compoed of wo par: P j (x) Polyomal ba fuco ad (x) he radal ba h fuco RBF [-]: (x) R (x)a m u P (x)b () j he umber of feld ode he local uppor doma ad m he umber of polyomal erm. Radal ba a fuco of dace r: j ) (y y ) he above equao () ca be expreed he marx form [] : j r (x x () U he vecor of fuco value: U R Ra Pb (3) Where U u, u, u 3,..., u R he mome marx of RBF, P he mome marx of Polyomal ba fuco ad a, b he value of ukow coeffce (Radal ad Polyomal) We oe ha, o oba he uque oluo of Eq. (), he cora codo hould be appled a follow []: P j(x)a j,,..., m (4) By combg Eq. (3) ad (4) yeld a e of equao he marx form: U R P a U G a P b (5) R P he ukow vecor ca be obaed by vero of he marx G P ubuo of he vecor obaed by vero of marx G o Eq. () lead o: u h (x) Φ (x)u u (6)

3 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 3 Local weak form mehod LRPM Le u coder a wo-dmeoal problem of old mechac doma bouded by whoe rog-form of goverg equao ad he eeal boudary codo are gve by: x) b (x) (7) j,j( o j j (8) u u o u (9) Where : σ,, ] he re vecor, b b,b ] he body force vecor. [ xx yy xy [ x y (, ) deoe he vecor of u ouward ormal a a po o he aural boudare he precrbed effor, [ u, u ] he dplaceme compoe he pla ad [u, u ] o he eeal boudare. he local Perov-Galerk approache [3], oe may wre a weak form over a local quadraure doma (for ode ), whch may have a arbrary hape, ad coa he po queo, ee Fg.. he geeralzed local weak form of he dffereal Eq. (7) obaed by: x ( (x) b (x)) d () j, j Where he local doma of quadraure for ode ad he wegh or e fuco, K ( ) [4]. Geerally, mehfree mehod, he repreeao of feld ode he doma wll be aocaed o oher reparo of problem doma: fluece doma for ode erpolao, he uppor doma for accuracy. For each ode he wegh fuco doma, ad he quadraure doma for local egrao. Fgure. he local ub-doma aroud po x ad boudare Ug he dvergece heorem [4] Eq. (), we oba: j j j, j d d b d ()

4 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 Where u : he eral boudary of he quadraure doma : he par of he aural boudary ha erec wh he quadraure doma u : he par of he eeal boudary ha erec wh he quadraure doma We ca he chage he expreo of équ(): j j j j j j j, j u d d d d b d () Ug he RPM hape fuco (ee ub-eco ), we ca approxmae he ral fuco for he dplaceme a a po x ( x ) a eq.(6) h he re vecor defed by: σ ε L u (3) Where he ymmerc elacy eor of he maeral E /( ) E /( ) Eq.() ca be wre: Where,x V,y E /( ) E /( ) E / ( ) d V σd Vd Vd Vd Vbd (4) u a marx ha coa he dervave of he wegh fuco,y,x ad V he marx of wegh fuco. ubug he dffereal operaor / x L d / y o equao (3) we oba: / y / x σ B u (5) Where B wre a:,x,y,y,x. By ug he marx L, he raco of a po x ca be L σ (6) ubug Eq. (5, 6) o Eq. (4), we oba he dcree yem of lear equao for he ode. [ V Bd L BVd L BVd ] u Vd Vb d (7) he marx form of Eq. (7) ca be wre a marx form: K u f (8) Where expreo of odal marx K u K V Bd L BVd L BVd (9) u

5 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 Ad odal force vecor wh corbuo from body force appled he problem doma: f V d V b d () Where deoe he e of he ode he uppor doma of po x. wo depede lear equao ca be obaed for each ode he ere problem doma ad by aemblg all hee * equao o oba he fal global yem equao: k () * u * f * o olve he precede yem, he adard Gau quadraure formula appled wh 6 Gau po [3, 3] for calculag egral Eq (9, ) o boh boudary ad doma. he ze of quadraure doma pecfed by eg ad a regular drbuo of ode o he md-urface of plae (x, y) plae employed. 4 Numercal D elaoac example h eco abou umercal reul for a calever recagular plae ee (Fg. ). Fr, were vegaed he effec of he ze of uppor ad quadraure doma ad wa examed umercally covergece of LRPM for everal maeral; he, comparo wll be made wh he aalyc oluo for everal maeral [4]{We chooe: eel, zc, alumum ad copper wh: ( 7 E 3. N / m,. 3 ; 5 E 3. N / m,. 5 ; 7 E. N/ m, 6.34 ; E 7. N / m,. 33 ) repecvely} Dmeo of he plae are deoed: hegh 3 D m, legh L 48m, he hcke: u ad fally for Loadg: P N Fgure. alever plae ubjeced o drbued raco a he free ed. Fgure 3. Regular feld ode drbuo o he problem doma ad boudare

6 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 our umercal calculao, were codered may regular drbuo of ode : 8, 55, 9, 75 ad 89. o calculae he error eergy, a backgroud cell are requred; he, for each value of he umber of cell wa vared. o oba umercal value, he drbuo of he defleco hrough he plae, ze of uppor doma vared ad ze of he quadraure doma. he ze of uppor doma (quadraure doma rep.) are defed by: d d c r d rep.) where d c ( ( c d c rep) he odal pacg ear ode (Fg. 3) ad ( rep) he ze of he uppor doma ( local quadraure doma rep) for ode. he ze of uppor doma ( quadraure doma rep) wll be repecvely deermed x ad y dreco. For mplcy x y ( x y rep) ued for ( rep). 5 Reul ad Dcuo he adard Gaua quadraure formula wh 6 Gau po ad RPM approxmao, lear polyomal ba fuco are appled. he cubc or quadrac ple fuco ued a he e fuco he LRPM local weak-form. hroughou h eco ad for all calculao, wa fxed ad he value ( ). 5. Aaly of reul umercal of he radal ba fuco P (h plae ple) he ue of radal ba fuco RBF-P have o bee exevely uded o leraure, we gve hee paragraph umercal reul for dffere maeral. Fgure 4 how he eergy error a a fuco of, cubc ple a he e fuco wh he radal ba RBF-B ad eel ha bee choe. he reul whch are calculaed by he LRPM mehod are flueced by dffere parameer. vegaed he varao of maxmum ad mmum value of of covergece doma by creag he drbuo regular feld umber ode : = 55, 9, 75,89. ca be ee from fgure 4 ha f he value of maller ha.8, he eergy error large ad LRPM mehod o coverge. he doma of covergece reache he maxmum value a 5 for f 55. For = 9, 75 or 89 he doma covergece maller ha ha obaed wh =55. he greaer exremy value of he covergece doma ow equal o 3.66 for ( =9, 75, 89). he covergece doma oced bewee a mall value. 8 ad he greaer value 5

7 Eergy error Eergy error ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5,,8,6 55 ode 9 ode 75 ode 89 ode P-RBF, =4.,4,,,5,,5 3, 3,5 4, 4,5 5, Fgure 4. fluece of he ze of o eergy error for dffere drbuo ode umber (eel),,9,8,7,6,5,4,3,, eel Zc opper Alumum RBF-P, 55 ode, Fgure 5. Varao of he eergy error a a fuco of for dffere maeral ad ( 55 ad 3 )

8 Eergy error Eergy error ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5,,9,8,7 eel opper Zc Alum, 55 ode eel opper Zc Alum, 9 ode eel opper Zc Alum, 75 ode,6,5,4,3,,, Fgure 6.Varao of he eergy error a a fuco of for dffere maeral ad ( 55, 9, 75 ad 3 ).,,9,8 eel opper Zc Alum, 55 ode eel opper Zc Alum, 9 ode eel opper Zc Alum, 89 ode,7,6,5,4,3,,, Fgure 7.Varao of he eergy error a a fuco of for dffere maeral ad ( 55, 9, 89 ad 3) fgure 5-7 how he varao of he error eergy a a fuco of he hape parameer for dffere value of E ad e dffere maeral ad for dffere value of he umber 55, 9,75,89 ( 3 ).

9 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 We fd ha all he curve of dffere maeral have mlar pace for a fxed value he followg doma: For 55, he covergece doma : For 9, he covergece doma : For 75, he covergece doma : For 89, he covergece doma : 6. 5 he reul lluraed he fgure (5-7) o he umber of ode 55, 9, 75 ad 89 wh he radal ba RBF-P, gve a he doma of covergece large eough for he dffere ype of maeral uded. h how beer covergece ha ha gve referece [6, 5-6], he auhor gve a gle value ( 4. ). We how here ha he value ca be vared a broader egme depedg o he umber ha he doma of covergece larger ha ha gve by Lu e al., -,x -3-4,x -3 u (m) -6,x -3 Zc eel eel=5 eel=6 Zc=6 Zc=5 Aalycal oluuo : eel Aalycal oluuo: Zc 75 ode -8,x -3 -,x x (m) Fgure 8. Defleco a a fuco of x a x, for he radal ba ad aalycal oluo for wo maeral (eel ad zc, 5; 6 ad 75 ; 3 ). Fgure 8 how he varao of dplaceme a a fuco of x for x wh he hape parameer of he radal ba fuco RBF-P, for wo value of : 5 ad 6: wh wo uded maeral: eel ad Zc. We oe ha here a cocdece bewee he aalycal oluo ad ha obaed by he LRPM mehod of he radal ba RBF-P, how he covergece of LRPM mehod (ee Fgure 7). 5. Reul umercal of he radal ba fuco Gaua(EXP)

10 Eergy error Eergy error ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 We uded he effec of he parameer he radal ba fuco RBF-EXP o covergece of he LRPM mehod. fgure 9- how he varao of he error eergy a a fuco of of he radal ba RBF-EXP ad dffere umber of ode ( 55, 9,75, 89 ) ad 3. For value ragg bewee ad 6 of, alumum, zc ad copper have of curve pecfc, he he LRPM mehod o coverge, bu oly ha of eel, he mehod ha a good covergece, he doma of covergece of he eel wde (.. 3) wh repec o oher maeral. Bu for he oher maeral, he doma of coverge Fally, we ca alo ay ha he doma of he covergece broader ha ha gve he referece [, 5] whch gave he erval.3. 3 for a gle maeral. he mehod coverge whe he umber very large o a larger doma (for eel Fgure ).,,9,8,7 RBF-EXP), eel RBF-EXP), Zc RBF-EXP), opper RBF-EXP), Alumum,6,5,4,3,,,,5,,5,,5,3 c Fgure 9 Varao of he eergy error a a fuco of for dffere maeral for 55,,9,8,7,6,5,4,3 55 ode 9 ode 75 ode 89 ode eel, RBF- EXP),, legh ep.,,,,,3 Fgure Varao of he eergy error a a fuco of Fgure how he defleco reul are ploed a fuco of xa x, for dffere value of (eel)

11 U (m) ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 for he radal ba RBF-EXP, he umber of ode = 55 ad he ze of he uppor doma ( 5 ) wh he cubc ple fuco ad hape parameer: c. 3 o RBF-EXP. We codered = 55 for whch he mehod coverge ad he value of eergy error low. here a cocdece bewee he curve repreeg he radal ba RBF-EXP of he LRPM mehod ad he curve of he aalycal oluo whch correpod o he upper ed of he doma of covergece.e. bewee.8 ad 5., EXP-RBF; =.3 Aalycal oluo -,x -3-4,x -3-6,x -3-8,x -3 -,x X (m) Fgure Defleco a a fuco of x a x for he radal ba RBF-EXP ( 55 for.3 ) ad he aalycal oluo Fgure hear re ( ) a fuco of x a x L / for he radal ba RBF-EXP ad 75. We fd ha here a cocdece bewee he curve repreeg he LRPM mehod ad he aalycal oluo. he reul of he LRPM mehod wh he radal ba RBF- EXP:(. 3 wh ad ) le effecve.

12 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû RBF-EXP) Aalycal oluo 75 ode pa) x (m) 4 6 Fgure a a fuco of x a x L / for he radal RBF-EXP, ( 75 for. 3 ) ad he aalycal oluo. 6 ocluo h paper he mehle LRPM mehod employed for olvg a D elaoac problem. he goverg equao deped o he weak form ad he paro of doma. he LRPM mehod ad depedecy o zg parameer of are aocaed o dffere parameer comg ou of weak form formulao. We have vegaed for he aure of covergece doma a a fuco of ; he effec of umber ode fuco RBF., by varyg aure of maeral ad he radal ba he reul obaed for he umber of ode 55, 9, 75 ad 89 for he radal ba fuco gve a doma of covergece large eough for he dffere ype of maeral uded. h how beer covergece ha ha gve referece [6, 5-6]. we coclude ha for mall value of (55) lead o he upper exremy of covergece doma whch lmed o 5.

13 ème ogrè Fraça de Mécaque Lyo, 4 au 8 Aoû 5 Referece [] Lu GR, Lu MB, moohed parcle hydrodyamc A mehfree praccal mehod, World cefc, gapore, 3 [] Nayrole B, ouzo G, Vllo P, Geeralzg he fe eleme mehod: Dffue approxmao ad dffue eleme. ompu Mech : (99), pp [3] Belychko, Lu YY, Gu L, Eleme free Galerk mehod. J Numer Meh Eg 37: (994), pp [4] Alur N, Zhu, A ew mehle local Perov-Galerk (MLPG) approach compuaoal mechac. ompu Mech : (998), pp. 7 7 [5] Lu GR, Gu Y, A local radal po erpolao mehod (LRPM) for free vbrao aalye of -D old. J oud Vb 46(): (), pp [6] Lu GR, Ya L, Wag JG, Gu Y, Po erpolao mehod baed o local redual formulao ug radal ba fuco. ruc Eg Mech 4(6): (), pp [7] Y.. GU ad G. R. LU, A local po erpolao mehod (LPM) for ac ad dyamc aaly of h beam, ompuer Mehod Appled Mechac ad Egeerg 9: (), pp [8] M.J.D. Powell, he uform covergece of h plae ple wo dmeo, Numercal Mahemac 68 () (994), pp [9] R.L. Hardy, heory ad applcao of he mulquadrc bharmoc mehod, ompuer ad Mahemac wh Applcao 9: (99), pp [] J.G. Wag ad G.R. Lu, o he opmal hape parameer of radal ba fuco ued for -D mehle mehod, ompu Mehod Appl, Mech. Egrg. 9: (6), pp [] Lu G. R, Ya L, Wag JG, Gu Y, Po erpolao mehod baed o local redual formulao ug radal ba fuco. ruc Eg Mech 4(6): (), pp [] M.A. Golberg,.. he, H. Bowma, ome rece reul ad propoal for he ue of radal ba fuco he BEM, Egeerg Aaly wh Boudary Eleme 3: (999), pp [3] A. uarero, R. acco, F. aler, Méhode Numérque, Algorhme, aalye e applcao, prger, 7. rd [4]. P. moheko ad J. N. Gooder, heory of Elacy, 3 edo, McGraw Hll, 97. [5] J. G. Wag ad G. R. Lu, A po erpolao mehle mehod baed o radal ba fuco.. J. Numer. Meh. Eg, vol. 54 (): (a), pp [6] J. G. Wag, G. R. Lu, O he opmal hape parameer of radal ba fuco ued for -D mehle mehod. ompu Mehod Appl Mech Eg, vol. 9: (b), pp