Application of PML to Analysis of Nonlinear Soil-Structure-Fluid Problem Using Mixed Element

Transcription

1 In. J. of GEOMAE Jun In. J. 3 of GEOMAE ol. 4 o. Jun (l. 3 o. 8) ol o. (l. o. 8) Go. Con. Ma. & En. I:86-98(P) 86-99(O) Jaan Alaon of PML o Analy of onlnar ol-ruur-flud Probl Ung Mxd Eln Paha Rhan Hroo hor Graduad hool of Collg of n and hnology hon Unry Jaan Collg of n and hnology hon Unry Jaan ABRAC: Mxd ln ay b onnnly ud o xr non-lnar onu quaon of flud and o aod olur long. X-FEM ay b wll ud o odl donnuy of dlan bwn old and flud. Arora boundary ondon hould b a h boundar of nural odl no o rfl ougong wa. In h ar olx frquny hfd onoluon-pml whou lng of arabl dlod for xd ln and h rforan of PML ar onfrd. h forulaon of PML olly onn wh orrondng FEM or X-FEM. I an b aly xndd o any y of ln and any nonlnar onu quaon of h orrondng FEM or X-FEM. h rulng a and ffn ar for PML ar yr for lnar odl. Kyword: ol-ruur-flud nraon X-FEM Mxd forulaon Colx Frquny hfd PML. IRODUCIO ol-ruur-flud nraon ay ha gnfan ff on ron of ruur. Mxd ln ay b onnnly ud o xr non-lnar onu quaon of flud and o aod olur long. X-FEM ay b wll ud o odl donnuy of dlan bwn old and flud. In h X-FEM analy a wll a FEM and FDM analy arora boundary ondon hould b a h boundar of nural odl no o rfl ougong wa. ral hod ar rood (Wolf 988). h fr h xn h odl ung a fn ln hod or a fn dffrn hod wh aroxa nrgy ranng boundar. h ond h ubruur hod ung for xal fn ln and doan boundary ln hod. In h forr h dgr of frdo of h odl ar ofn ry larg. h lar hod ay b or ffn bu h nonlnary u b rrd whn h narby oron of ruur odld by fn ln hod.. onu quaon ar aud o b lnar a and our doan of h boundary. h hrd FEM wh PML or onolon PML(Brngr 994 Collno Bau 3 4Droar 7). PML and onoluon PML ar rod o ha ffn wa aborbng aably for lnar lao-dyna robl and h nonlnary u b rrd whn fn ln doan. In h r arhqua howr ol ay bo nonlnar o a larg xn o ha h ond and h hrd hod ay b nadqua. Conoluonal PML xndd o o wh non-lnar robl o ha nonlnar ol an b analyzd wh a ld nubr of h whou lo of auray (hor Rhan ). Bu rrd o dlan bad FEM. Hr olx frquny hfd onoluon-pml whou lng of arabl dlod for xd fn ln and for X-FEM and h rforan of PML ar onfrd. h forulaon of PML olly onn wh orrondng FEM or X-FEM. I an b aly xndd o any y of ln and any nonlnar onu quaon of h orrondng FEM or X-FEM. h rulng a and ffn ar for PML ar yr for lnar odl.. MEHOD. PML Forulaon of Mxd Eln for Flud Aung ha h ff of oy nglgbl and ha hang of dny all h quaon of oon of flud gn a: u (.) x whr aragd dny of flud dyna rur u wh oonn of dlan and x dno h oordna. Rlaonh bwn dny and dlan gn a: u (.) x whr dffrn bwn urrn dny and aragd dy. Rlaonh bwn dyna rur and xrd a follow. f( ) whr f ( ) gn a: (.3) f ( ) for (.3a) f ( ) ( ) ( )( ) (.3b) for n whh loy of ound aor rur a rur and rduon rao of ound loy afr aaon. Followng h PML rodur w nrodu olx oordna rhng funon n frquny doan a: 55

2 In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) x (.4) x= λ ()d whr x dno h oordna and x h orrondng ranford oordna and λ gn a: λ = (.5) + α +ω whr ur agnary nubr ω rular frquny and α and σ non-nga onnuou funon uh ha =and σ = a FEM-PML nrfa. A fr all quaon ar forulad n x oordna n frquny doan and hn ranford o x oordna. Equaon of oon ar gn a: u (.6) x whr rular frquny and u and ar dlan and rur alud n frquny doan rly. h rlaonh bwn dny and dlan ar gn a: u (.7) x whr rla dny alud n frquny doan. Condrng / x (/ ) / x and ullyng boh d of Eqn..6 and Eqn..7 by w g; u (.8) x u x (.9) whr dno an ngr ohr han. Inrodung wgh funon w for dlan and q for rur wa for quaon for Eqn..8 and Eqn..9 ar gn a follow. w w ud d x wn d (.) q f ( ) d (.) and /( ) Condrng h fa ha d / d d / n frquny doan orrond o d and n doan quaon n doan ar wrn a: ( ) wu ud w (.) d x wn d ( ) (.3) u u x x whr dno onoluon ngral. Dnong ( ) f ( ) f ( ) d F( ) and nrodung aroxaon F ( ) ( ) f ( ) f ( ) F( ) w g wr u ( ) d whr w U () d (.4) w r ( ) P x () d wn r ( ) P () d r ( ) r w dfn a followng: P () ( ) ( ) U() P ( ) ( ) ( ) d ( ) ( ) () u () d ( ) u() U( ) ( ) u () Lw fro Eqn..3w g: r( ) R( ) u ( ) ( ) whr r x R R () () () () and (.5) (.6) (.7) and R () [ R ( ) ()[ ( ) )] }/( () ( ) u / x Fro Eqn..7 Eqn..3aand Eqn..w g 56

3 In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) u qr ( ) r ( ) d x q R() () d (.8) Drzng doan of analy by fn ln adong Galrn forulaon and l u P and dno dlan and rur or a nodal on of a ln and nrolaon ar for dlan and rur rly arx for quaon ar oband. Fro Eqn..4 w g r du ( ) U ( ) d B R P B dp ( ) R P d ( ) P (.9) whr d ( ) d ( ) U ( ) x n n U ( ) x P ( ) r r ( ) P ( ) ( ) ( ) ( ) U( ) B R P P P P Fro Eqn..8 w g r dp ( ) (.) whr R Bdu R ( ) ( ) R d R d () () () R() R() R () and R () [ R ( ) { ( ) } /{( ) } P ( )] L H and G b dfnd a H r d G R B d. hnw g: r ( ) du G H Gu ( ) R dp ( ) dp() B dp() U G H R () d () (.) Eln ffn arx G H G ha any hourgla od and ay lad o unabl oluon. n ory rrd n non-ou flow and n flud aud o b a n h bgnnng h followng quaon ald: u u = (.) x x o n PML doan h followng quaon aud. u u (.3) x x (.4) x x whr alud of ory n frquny doan. Conrng Eqn..3 no doan and nrodung aroxaon for onoluon ngral w g ( ) r ( ) E( ) u ( ) r ( ) U ( ) x (.5) whr () ( ) () U () U ( ) u ()/ x and U () U ( ) u ()/ x. Inrodung wgh funon w w and drng wa for quaon fro Eqn..4w g w w d x x wn wn d (.6) w w d x x ranforng Eqn..6 no doan quaon and nrodung aroxaon for onoluon ngral and ubungeqn..5w g w w u u r r r ( ) r ( ) d r x x x x w w E Ed (.7) x x w w r r E() d r x x wnr wnr wn E wne d whr E () { ( ) } E( )/( ) U( ) U( ) () U U u x and () U U( ) u ( )/ x. ( ) () () ( ) ()/ 57

4 In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) Inrodung nrolaon ar h followng quaon ar oband. B ( ) Bdu r E B d E() d E B r (.8) whr x B r r B x x x By addng Eqn.. and Eqn..8w g fnal arx for quaon. ubung nrolaon funon for X-FEM w g PML for XFEM. 3 UMERICAL EXAMPLE 3. Rror Modl Rror wh dh ubd o horzonal rgd wall oon analyzd ung rood xd forulaon FE-PLM. Lngh of rror odl ar fro o 6. Boundary ondon agnd on ura boundary ar fxd boundary ou boundary wh oy and PML. Mh of odl ar hown n Fg.3.~3.3. Mh wh gn x nda PML. Ford horzonal braon of rgd wall on h lf nd of rror aud. h rul ar hown n Fg.3.4~3.9. Fxd boundary rul how rong dndny on h loaon of ura boundary. h rul wh ou boundary how l dndny on h loaon of boundary for Hz xaon bu larg dndny for 5Hz xaon. Prood PML boundary how ry ll drany bwn dffrn boundary loaon for boh Hz and 5Hz xaon. Fg.3.4 Fxd Boundary (Hz) Fg.3.5 Fxd Boundary (5Hz) Fg3.6 ou boundary(5hz) Fg.3. Rror Mh (lngh ) Fg.3. Rror Mh (lngh 3) Fg3.7ou boundary(5hz) Fg.3.3 Rror Mh(lngh 6) Fg 3.8 PML(Hz) 58

5 In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) Fg3.9 PML(5Hz) Fg 3. Mh (rror lngh=) Fg 3. Prur Drbuon(Hz) 3. Da- Rror-Foundaon Modl Da-rror-foundaon odl wa onrud ung X-FEM for angnal donnuy of dlan a old-flud nrfa and obnng rood xd forulaon FE-PML for flud wh FEM-PML for old (Fg3.).wo nd of ah ar ud (Fg.3.Fg.3.3). Donnuou dlan n angnal dron nar h o of da hown n Fg.4 ndang h ff of donnuou nrolaon funon of X-FEM. In Fg.5-7 ron of da o ung wo h odl ubd o nuodal horzonal ground oon ar oard. h rul of wo h odl ond ah ohr ndang ha odl z ha ll ff whn rood PML ud. Fg 3.3 Mh(rror lngh=3) Fg 3.4 Dlan nar o of da Rror Lngh( or 3) PML(h) PML (h) 8 PML (h) Fg 3. Da-rror-foundaon odl Fg 3.5 Ron of da o (Hz) 59

6 In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) Fg 3.6 Ron of da o (3Hz) Fg 3.7 Ron of da o (5Hz) 4 COCLUIO Conoluon PML bad on xd forulaon forulad and our od dlod for FEM and X-FEM. I wa ald o rror odl and h rforan of aborbng ougong wa wa uh br han onnonal boundary. By alyng o da-rror-foundaon odl nludng FEM and X-FEM wa donrad ha h forulaon wa qu gnral and ha ha wd la of alaon. 5 REFERECE [] Wolf J.P.(988). ol-ruur-inraon Analy n Doan Prn-Hall Englwood Clff. [] Brngr P.(994).A Prfly Mahd Layr for h Aboron of Elroagn Wa Journal of Couaonal Phy [3] CollnoF. and ogac.().alaon of h rfly ahd aborbng layr o h lnar laodyna robl n anoro hrognou da Gohy 66 : [4] DroarF.H.abd GannooulA.(7) Colx frquny hfd onoluon PMl for FD odlng of la wa Wa Moon l.44: [5] BauU and Chora A.K.(4).Prfly Mahd layr rann lodyna of unboundd doan In.J.ural Mhod Eng [6] BauU. and Chora A.K.(3) Prfly ahd layr for -haron lo dyna of unboundd doan: hory and fn-ln lnaon Cou. Mhod Al. Mh. Eng [7] H.horP.Rhan. & L J.F.() Alaon of PML for on-lnar ol-ruur Probl. 4ECEEOhrd -8CD. [8] P. RhanH.HIOJIRI Jng-Fng LI() Analyal Mhod for onlnar ol-ruur Inraon ung PMLEurodyn Lun Blgu -6CD. In. J. of GEOMAE Jun 3 ol. 4 o. (l. o. 8) M o. 67 rd Jun 3 and rwd undr GEOMAE ublaon ol. Coyrgh 3 Inrnaonal Journal of GEOMAE. All rgh rrd nludng h ang of o unl ron oband fro h oyrgh roror. Prnn duon nludng auhor lour f any wll b ublhd n h Jun 4 f h duon rd by D. 3. Corrondng Auhor: Paha Rhan 5