CIVL 7/8111 Time-Dependent Problems - 2-D Diffusion and Wave Equations 1/9

Transcription

1 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 1/9 h govrig balac qatio that dcrib diffio proc i itatio ivolvig two idpdt variabl appar typically a xyt,, fxyt,, 0 t i g, t o 1 t q t x, y,0 c x, y i,, o Whr i th itrior domai, ad 1 ad form th bodary of th domai. g, t 1 x, y, t t f x, y, t 0, t q, t h phyical cotat ar ad, ad i a liar mar of th poitio o th bodary. h typ of bodary coditio pcifid o rlt from a local balac btw codctio i th itrior ad covctio ito th xtrior. 1 x, y, t t f x, y, t 0, t q, t 1 x, y, t t f x, y, t 0, t q, t g, t g, t I two dimio, th iflc of th bodary coditio i how. h lft id of th imlatio domai (x = 0) i d a a cotat orc of diffio particl, thrby cratig a tady flow ito th ara. All othr bodari ar modld by a ma bodary coditio. hi qatio i imilar i form to th lliptic bodary val problm tdid i Chaptr 3, with th vry importat additio of th tim drivativ trm i th diffrtial qatio ad th corrpodig iitial coditio. With th additio, th problm chag from a lliptic bodary val problm to a parabolic iitial bodary val problm.

2 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio /9 h baic tp of dicrtizatio, itrpolatio, lmtal formlatio, ambly, cotrait, oltio, ad comptatio of drivd variabl ar prtd i thi ctio a thy rlat to th two-dimioal parabolic iitial-bodary val problm. h Galri mthod, i coctio with th corrpodig wa formlatio to b dvlopd, will b d to grat th fiit lmt modl. Dicrtizatio - For dicrtizatio, pla rfr to th matrial i Chaptr 3. Itrpolatio - h oltio i amd to b xpribl i trm of th odally bad itrpolatio fctio i (x, y) itrodcd ad dicd i Sctio 3.. I th prt ttig, th itrpolatio fctio ar d with th midicrtizatio 1 xyt (,, ) i( t ) i( xy, ) 1 h i (x, y) ar odally bad itrpolatio fctio ad ca b liar, qadratic, or a othrwi dird. Elmtal Formlatio - h tartig poit for th lmtal formlatio i th wa formlatio of th iitialbodary val problm. h firt tp i dvlopig th wa formlatio i to mltiply th diffrtial qatio by a arbitrary tt fctio v(x, y) vaihig o 1. h rlt i th itgratd ovr th domai to obtai v f d t 0 Elmtal Formlatio - Uig th two-dimioal form of th divrgc thorm to itgrat th firt trm by part, thr rlt aftr rarragig v d v d t whr = 1 + v d vf d Elmtal Formlatio - Rcallig that v vaih o 1 ad that /x = = q - h o, it follow that v d v d t v q h d vf d Elmtal Formlatio - Sbtittig th approximatio of (x, y, t) ito th wa formlatio ad taig v =, = 1,,... yild i i i i d x x y y d h d i i i i hi qatio i th rqird wa formlatio for th twodimioal diffio problm. fd qd 1,,

3 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 3/9 Elmtal Formlatio Whr - ad - rprt th lmtal ara approximatig, ad th collctio of th lmtal dg approximatig,rpctivly. i i i i d x x y y d h d fd i i i i qd 1,, Elmtal Formlatio hi x t of liar algbraic qatio ca b writt a i 1 A B F () t 1,,..., i i i i i i Ai d hi d x x y y B d i i F f d qd Elmtal Formlatio ot that ambly i cotaid implicitly withi th formlatio. i 1 A B F () t 1,,..., i i i i i i Ai d hi d x x y y B d i i F f d qd Elmtal Formlatio I trm of th corrpodig lmtally bad itrpolatio ( x, y) h fiit lmt modl ca b xprd a A B F ' A G ag ' F= fg qg B= rg Elmtal Formlatio I trm of th corrpodig lmtally bad itrpolatio ( x, y) h fiit lmt modl ca b xprd a r f A A A da x x y y da fda a d q qd Elmtal Formlatio h iitial coditio for th ytm of firt-ordr diffrtial qatio ar obtaid from th iitial coditio prcribd for th origial iitialbodary val problm. Grally (0) i dtrmid by valatig th fctio c(x, y) at th od to obtai (0) 0 c1 c c3... c 1 c whr c i = c(x i, y i )with (x i, y i ) th coordiat of th i th od.

4 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 4/9 Cotrait - h cotrait ari from th bodary coditio pcifid o 1. Grally th val of th cotrait ar dtrmid from th q fctio with th cotraid val of at a od o 1 big ta a th val of q at that poit. h cotrait ar th forcd o th ambld qatio, rltig i th fial global cotraid t of liar firt-ordr diffrtial qatio. Soltio - h ytm of qatio i prcily th am i form ad charactr a th corrpodig qatio dvlopd for o-dimioal diffio. A aalytical mthod a wll a th mrical mthod of Elr ad improvd Elr or Cra-icolo ca b d for itgratig th abov t of qatio. 0 M K f (0) Soltio - h Elr mthod will b coditioally tabl with th critical tim tp dpdig o th maximm igval of th aociatd problm (K - M)v = 0. h improvd Elr or Cra-icolo algorithm will b coditioally tabl. Drivd variabl - h drivd variabl will b tim dpdt, ad dpdig o th particlar problm big coidrd, may d to b comptd at ach tim tp. h comptatio wold b pr lmt ad wold b carrid ot ig th tchiq dcribd i Chaptr 3. Swi Mathmaticia ad phyicit Lohard Elr dicovrd th wav qatio i thr pac dimio. Lohard Elr ( ) wa a piorig Swi mathmaticia ad phyicit. H mad importat dicovri i fild a divr a ifiitimal calcl ad graph thory. H i alo rowd for hi wor i mchaic, flid dyamic, optic, atroomy, ad mic thory h wav qatio i a importat cod-ordr liar partial diffrtial qatio for th dcriptio of wav a thy occr i phyic ch a od wav, light wav ad watr wav. It ari i fild li acotic, lctromagtic, ad flid dyamic. A oltio of th wav qatio i two dimio with a zro-diplacmt bodary coditio alog th tir otr dg.

5 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 5/9 Coidr th motio of a rctaglar mmbra (i th abc of gravity) ig th two-dimioal wav qatio. Plot of th patial part for mod ar illtratd blow. Coidr th motio of a rctaglar mmbra (i th abc of gravity) ig th two-dimioal wav qatio. h govrig qatio of motio that dcrib th propagatio of wav i itatio ivolvig two idpdt variabl appar typically a xyt,, f xyt,, 0 t i g, t o, t q, t o x, y,0 c x, y i 1,,0 dx y x y t, i Whr i th itrior domai, ad 1 ad form th bodary of th domai. g, t 1 xyt,, f xyt,, 0 t, t q, t h phyical cotat ar ad, ad i a liar mar of th poitio o th bodary. h typ of bodary coditio pcifid o rlt from a local balac btw itral ad xtral forc. 1 xyt,, f xyt,, 0 t, t q, t 1 xyt,, f xyt,, 0 t, t q, t g, t g, t

6 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 6/9 hi qatio i imilar i form to th parabolic iitialbodary val problm prtd i th prvio ctio with th vry importat chag i th tim drivativ trm from a firt to a cod drivativ, ad with th additio of a cod iitial coditio o th vlocity. With th chag, th problm chag from a parabolic iitial-bodary val problm to a hyprbolic iitial bodary val problm. h baic tp of dicrtizatio, itrpolatio, lmtal formlatio, ambly, cotrait, oltio, ad comptatio of drivd variabl ar prtd i thi ctio a thy rlat to th two-dimioal hyprbolic iitial-bodary val problm. h Galri mthod, i coctio with th corrpodig wa formlatio to b dvlopd, will b d to grat th fiit lmt modl. Dicrtizatio - Rfrrd to th matrial i Chaptr 3. Itrpolatio - h oltio i amd to b xpribl i trm of th odally bad itrpolatio fctio i (x, y) itrodcd ad dicd i Sctio 3.. I th prt ttig, th itrpolatio fctio ar d with th midicrtizatio 1 xyt (,, ) i( t ) i( xy, ) i 1 h i (x, y) ar odally bad itrpolatio fctio ad ca b liar, qadratic, or a othrwi dird. Elmtal Formlatio - h tartig poit for th lmtal formlatio i th wa formlatio of th iitialbodary val problm. h firt tp i dvlopig th wa formlatio i to mltiply th diffrtial qatio by a arbitrary tt fctio v(x, y) vaihig o 1. h rlt i th itgratd ovr th domai to obtai v f d t 0 Elmtal Formlatio - Uig th two-dimioal form of th divrgc thorm to itgrat th firt trm by part, thr rlt aftr rarragig v d v d t whr = 1 + v d vf d Elmtal Formlatio - Rcallig that v vaih o 1 ad that /x = = q - h o, it follow that v d v d t v q h d vf d hi qatio i th rqird wa formlatio for th twodimioal diffio problm.

7 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 7/9 Elmtal Formlatio - Sbtittig th approximatio of (x, y, t) ito th wa formlatio ad taig v =, = 1,,... yild i i i i d x x y y d h d i i i i Elmtal Formlatio Whr - ad - rprt th lmtal ara approximatig, ad th collctio of th lmtal dg approximatig,rpctivly. i i i i d x x y y d h d i i i i fd qd 1,, fd qd 1,, Elmtal Formlatio hi x t of liar algbraic qatio ca b writt a i 1 A B F () t 1,,..., i i i i i i Ai d hi d x x y y B d i i F f d qd Elmtal Formlatio ot that ambly i cotaid implicitly withi th formlatio. i 1 A B F () t 1,,..., i i i i i i Ai d hi d x x y y B d i i F f d qd Elmtal Formlatio I trm of th corrpodig lmtally bad itrpolatio ( x, y) h fiit lmt modl ca b xprd a A B F ' A G ag ' F= fg qg B= rg Elmtal Formlatio I trm of th corrpodig lmtally bad itrpolatio ( x, y) h fiit lmt modl ca b xprd a r f A A A da x x y y da fda a d q qd

8 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 8/9 Elmtal Formlatio h iitial coditio for th ytm of firt-ordr diffrtial qatio ar obtaid from th iitial coditio prcribd for th origial iitialbodary val problm. Grally (0) i dtrmid by valatig th fctio c(x, y) at th od to obtai (0) 0 c1 c c3... c 1 c whr c i = c(x i, y i )with (x i, y i ) th coordiat of th i th od. Elmtal Formlatio h iitial coditio for th ytm of firt-ordr diffrtial qatio ar obtaid from th iitial coditio prcribd for th origial iitialbodary val problm. Grally ů(0) i dtrmid by valatig th fctio d(x, y) at th od to obtai (0) 0 d1 d d3... d 1 d whr d i = d(x i, y i )with (x i, y i ) th coordiat of th i th od. Cotrait - h cotrait ari from th bodary coditio pcifid o 1. Grally th val of th cotrait ar dtrmid from th q fctio with th cotraid val of at a od o 1 big ta a th val of q at that poit. h cotrait ar th forcd o th ambld qatio, rltig i th fial global cotraid t of liar firt-ordr diffrtial qatio. Soltio - h ytm of qatio i prcily th am i form ad charactr a th corrpodig qatio dvlopd i Sctio 4.. for th o-dimioal wav problm. h aalytical mthod a wll a th mrical mthod ig th ctral diffrc ad wmar algorithm ca b d for itgratig th abov t of qatio. M K f (0) 0 (0) 0 Soltio - h ctral diffrc algorithm will b coditioally tabl with th critical tim tp dpdig o th maximm igval of th aociatd problm (K - M)v = 0. Drivd variabl - I a phyical itatio govrd by a wav qatio, th drivd variabl ar ally th itral forc comptd accordig to F = pr lmt for ach tim tp. h wmar algorithm will b coditioally tabl for = 0.5 ad = 0.5( + 0.5).

9 CIVL 7/8111 im-dpdt Problm - -D Diffio ad Wav Eqatio 9/9 Clor - im-dpdt problm ar ihrtly mor difficlt ad xpiv to olv tha thir corrpodig tady-tat cotrpart. h xp of gratig th global matric i highr for th tim-dpdt problm bca of th city of comptig th ma matric. h mai xtra xp, howvr, i i olvig th rltig tim-dpdt global qatio. Clor - For a aalytical approach to th oltio, additioal xp i icrrd i trm of havig to dtrmi igval ad igvctor. h actal amot of xp dpd o th pcific form of th tiff ad ma matric ad th algorithm d, bt i ay ca it i igificatly i xc of th xp of olvig th igl t of liar algbraic qatio aociatd with th tady-tat problm. Clor - For a tim domai itgratio tchiq, th additioal xp i clarly rlatd to th mbr of tim tp cary to trac ot th dird tim hitory. I additio to vral matrix mltiplicatio ad additio, ach tp ca ivolv th oltio of a t of liar algbraic qatio. I om itac thi xp ca b miimizd by ig a dcompoitio that ca b rd for th comptatio of th oltio at ach w tim. Clor - I thi rgard rcall that th Elr ad ctral diffrc algorithm rqir that th iz of th tim tp ot xcd a val proportioal to th ivr of th largt igval. For larg ytm thi critical tp iz ca b vry mall rltig i may applicatio of th algorithm to trac ot th tim hitory. Clor - h coditioally tabl Cra-icolo ad wmar algorithm, o th othr had, ca b d with arbitrary tp iz that ha b cho o a to accratly itgrat th lowr mod, with igificat improvmt i th xp rlativ to th coditioally tabl Elr ad ctral diffrc algorithm. hr ar of cor othr algorithm availabl that ar pcifically tailord to addr othr mrical i. Ed of Chaptr 4c