Chape 5 inie Vlue Meh f Nn-hgnal Mehe flui echanic pble f inee Enginee he gee f he pble can n be epeene b a Caeian eh. Inea i i cn f he bunaie be cue in pace. The Caeian finie lue eh ecibe in Chape 2 an 4 u be exene a nn-hgnal eh all nn-ecangula geeie be accuael elle. A ucue eh i plgicall ecangula, bu a be efe in pace uch ha i i n lnge Caeian. Such a eh a ill be hgnal, ih he ineecin f he eh line being a, bu uch a eh can be ifficul geneae, an a cpleel geneal eh ill be nn-hgnal ih n eicin n he angle f he ineecin f he eh line. In hi chape he finie lue iceiain f he anp equain n a nn-hgnal eh i icue in fie pa. i hee i a ecipin f he gee f he eh, hich i flle b he iceiain f he Diffuin e f he anp equain. The aecie e ill hen be iceie an he buna cniin ecibe. inall he luin f he Naie Ske equain n a nn-hgnal eh ill be ecibe, uing he SIMPLE chee icue in Chape 4. The eh ue iceie he equain i iila ha ecibe b Jne[2] an Peić[129, 43]. 5.1 The Gee f a Nn-hgnal Meh The calculain f he geeic ppeie f a nn-hgnal eh i nn-iial, an i eee icuin befe iceiing a PDE upn he eh. Bh an hee ieninal ehe ae icue, he iplificain aiing f he ue f a -ieninal eh aaning hei epaae icuin. A nn-hgnal egula -ieninal eh i hn in igue 5.1, ih he eh being hn in bh cpuainal an phical pace. In cpuainal pace he cinae pace i epeene b he axe, hil in phical pace he axi e i ue, ih hee being a ne ne apping beeen he cinae e. a finie lue iceiain he eh i efine b he cell eex pin. hee pin e ih fin he cell cene, he cene f he cell face, he aea ec f he cell face, an he cell lue. he ce ue in hi u he cell cene an he cene f he cell face ee appxiae b 115
, N 0 0 / / - $ ' ' & & % $ ' ' & & % T $ ' ' & & % $ ' ' & & %, 0 0 / / - CHAPTE 5. NN-THGNAL MESHES 11 ( ( ( ( ) ) ) ) * * * * + + + + "!!!! # # # #!!!! # # # #,.-!!! 2! # # # # ( $ % ( "!!!! # # # # ( ( ) $ % ) ) ) * $ % $ %,1- * * * + + + + igue 5.1 T Dieninal gi in cpuainal pace (p) an phical pace (b). he aeage f he cell eex alue. hil hi en gie he geeic ceni f he cell i a cniee a eanable appxiain. ieninal cell he cell face aea ec can be calculae a he c puc f he cell face an he ec nal in he iecin f a hi (ficiiu) axi. Thu f he ea face f he cell in igue 5.2 he cell face ec i 354894; =>4 (5.1) an he face aea i A354 CBEDG In cpnen f hi bece H JI 3 H LK C3 (5.2) M (5.3) The lue f he cell, N, can be eail calculae a half he agniue f he c puc f he cell iagnal, he lue f he cell in igue 5.2 being P QS UT 4; V>4LY B 4;.Z[4J\ I i puen ue he agniue f he lue peen he inaeen calculain f negaie lue. a hee ieninal eh he cell face aea ae fun a half he c puc f he face iagnal, f he cell face in igue 5.3 he face aea i gien b Q T^] BE_ (5.4) (5.5)
N a CHAPTE 5. NN-THGNAL MESHES 11 ne n P A e e x igue 5.2 The gee f a -ieninal cell. b a A n x n z x igue 5.3 The gee f a hee-ieninal cell, an he calculain f he aea f ne face f he cell. The lue f he cell can be calculae b inegaing he face aea e he uface f he cell uing he Gau iegence hee. The lue f he cell i hu hee b. ` 4 (5.) i he ec aea f he face b fun ih Equain (5.5), an 4; i he ceni f he face 5.2 The Diceiain f he Diffuin Te n a Nn-hgnal Meh T iceie he iffuin equain n a nn-hgnal eh e nee fin an appxiain f he iffuie flux ac a face f a eh lue in a anne iila ha ue in Secin 2.2. u exaple e ill ake he eaen face f a cell, bu he pce can be applie an face f he lue. The gee f he face in phical pace i hn in igue 5.4 alng ih he gee in cpuainal pace. ick la he iffuie flux i gien b hee b i he iecin nal he uface. If ccfe b (5.) i he aea ec f he face, an H gih ;h i
e e k CHAPTE 5. NN-THGNAL MESHES 118 NE N η A 12 ξ A 11 E P SE S x igue 5.4 The aea ec f he eaen face f a -ieninal eh. he agniue f he face aea, hen he al flux ac he face i appxiae b The aea H H cjfe b (5.8) i eail calculae f he eh gee a a ecibe in he peiu ecin,. e can eail ap- c an i a gien ppe, all ha eain i appxiae he eiaie kml pxiae he eiaie in he cpuainal axe uing cene iffeence appxiain. Unfunael f a nn-hgnal eh he face nal i n neceail in he iecin f an f he eh axe, bu i can be appxiae uing he eiaie in he eh axe b ue f he chain ule fe b b fe n b A a enine, he eiaie f e in cpuainal pace can be appxiae b cene iffeence appxiain, hich f he ea face f he cell a be ien a e Z fe eqp e fe 1 J T e fe ex = e T ez 3 ning ha f a uni cell in cpuainal pace, A3 b eup fe e ez p e{ } ~M 2J e p e{ p ƒ The eh anfain e can be calculae f a puc f he cell face uni nal an he cnaaian bai ec ~ 4 ~ Q 4 ~ ` 4 (5.9) (5.10) (5.11)
N 4 4 4 4 4 4 4 4 4 N N N CHAPTE 5. NN-THGNAL MESHES 119 ih an ih he uni nal being gien b b b b V V V h h (5.12) (5.13) e 2 1 e e 2 e 1 ~ igue 5.5 The caian ~ Q ~ an cnaaian ~ Q a -ieninal eh. ~ Unfunael he cnaaian bai ec ~ Q ~ ` bai ec f he eaen face f a cell in (hn in igue 5.5) hich ae nal he eh uface ae unknn. Hee, an appxiain he caian bai ec, hich ae angenial he uface ~ ~ Q ~ ` (5.14) can be eail calculae f he eh gee. Thu e ih eiae he cnaaian bai ec f hei caian cunepa. B cnieing a paallelepipe fe a he ea face f a cell b he caian ec (igue 5.), e can appxiae he cnaaian ec f he ai f he face aea he lue f he paallelepipe. Thu ~ ~ Q ~ ` Q ` (5.15) The face aea ae eail calculae b aking he c puc f he ~ caian bai ec, hil he lue i calculae f he puc f he nal aea an, ~ Q B ~ ` Q ~ ` B ~ ` ~ B ~ Q (5.1) ~
Q ` Q ` N N N N N N N Q ` Q ` CHAPTE 5. NN-THGNAL MESHES 120 i equialen he cell face nal peiul ene a A 2. 1 A Ω igue 5. The paallelepipe fe a he ea face f a cell b he caian ec. Subiuing he appxiain in Equain (5.15) in (5.12) gie b h h Q b h h ` b h h hee ha been ubiue f h h hee i i een ha he eiaie uliplie b he face aea (5.1). ne final iplificain i aailable f Equain (5.8) ae equie. Subiuing in Equain (5.1) gie he expein f he geeic iffuin cefficien f he ea face ( face 1), hee h h e b Q ` fe Q ` Q fe ` fe (5.18) (5.19) Siila expein can be eie f he nh an p face (face 2 an 3) f he cell, an he eiaie nal hee face can be appxiae a QQ `` e b e b e b hee b i he Œ nal f he ˆŠ face, cefficien ae gien b fe Q e ` fe Q fe QQ e Q` fe (5.20) ` fe `Q e `` fe Œ i he aea f he ˆŠ face, he geeic iffuin Ž Ž (5.21)
Q Q Q ` ` ` T e CHAPTE 5. NN-THGNAL MESHES 121 an he flling appxiain ae ue f he eiaie in cpuainal pace fe fe fe fe fe fe e e e e p e ez T e T ez T ep T ez T e p T e e e e e up ez p eup eqz ez p ez e e{ e e{ e e e p e{ p e8 eq{ ez ez (5.22) (5.23) (5.24) The aea an cell geeic iffuin cefficien ae ppeie f he face f he cell. Thu hee ae aea ec lcae a bh he ea an e face f he cell, ih he aea in bh cae being efine pining u f he cell. The aea e f he ea face f he cell ae he negaie f he aea f he e face f he cell. The geeic iffuin cefficien i ala piie, an i he ae f he face egale f hehe he face i being cniee he ea face f cell f he e face f cell. he ipleenain f he nn-hgnal eh le he face geeic ppeie f he high ie face f each cell (he ea, nh an p face) ae calculae an e. Thee ppeie ae Q ` ene Q an ` Q. The alue f he l face ae hen aken f he high face f he neighbuing cell. Uing he appxiain f he eiaie nal he cell face, he iffuie e in he anp equain (Equain (2.1)) c (5.25) can be iceie an facie in a linea e e (5.2) hee he iniiual equain can be expane a e p e p e up e up z ez z p ez p e e z eqz z ez e z5eqz 1p e p { e{ { p eq{ p {ge{ e { e{ { e{ (5.2)
p z { e CHAPTE 5. NN-THGNAL MESHES 122 an hee he equain cefficien ae gien b up p z z { { z p z { p { Q T Q Q T Q QQ T Q Q QQ T Q Q `` T ` ` `` T ` ` QQ T Q Q T Q Q T Q Q T Q Q T Q` `Q Q` `Q T T Q` `Q Q` `Q T T ` ` T ` ` T ` ` T ` ` ` ` `Q `Q Q` Q` QQ ` ` Q` Q` Q` Q` `` a ieninal e he lighl e anageable iceiain i e p e p e up e up e e e p e p `` e (5.28) (5.29) ih he equain cefficien p up p QQ QQ T Q T Q T Q T Q T T T T Q Q Q Q QQ Q Q Q Q Q Q Q Q QQ (5.30) The cpuainal lecule f he Caeian equain gien in Secin 2.2 cnaine 5 pin f he -ieninal eh, an pin in hee-ienin. ih he aiin f he exa e f he nn-hgnal iffuin hee lecule hae expane 9 an 19 ebe in - an heeienin epeciel. T all he ue f he linea le eelpe f Caeian ehe, he e Ž L can be calculae uing a efee cecin chee, ih he exa iffuie e f he e being abbe in he uce e f he equain. Thu he e in Equain (5.2) can be eien a ih cnaining he hgnal cpnen, an e (5.31) he nn-hgnal cpnen. a hee
p z { p z { p p CHAPTE 5. NN-THGNAL MESHES 123 ieninal e he equain cefficien ae =p 8 1p z z { { z p z { p { QQ QQ `` `` T T T T T T T T T T T T T T T T T T Q Q Q Q ` ` Q Q Q Q Q` Q` Q` Q` ` ` ` ` hil f a ieninal e he e f he hi axi ae ppe an he equain ae up p an hgnal eh he puc QQ QQ T T T T T T T T U Œ Q Q Q Q ` ` Q Q Q Q `Q `Q `Q `Q ` ` ` ` QQ Q Q Q Q Q Q Q Q QQ ` ` `Q `Q Q` Q` QQ Q Q Q Q Q Q Q Q `` ` ` Q` Q` Q` Q` QQ i nl nn-ze hen ˆ `` (5.32) (5.33). Thu f uch a eh
p z { p Q ` Q ` 3 3 3 Q ` CHAPTE 5. NN-THGNAL MESHES 124 he geeic iffuin cefficien ae nl nn-ze f ˆ euce he ahe e anageable e f ih uch a e Ž QQ `` fe fe fe fe b fe b fe b ep e ez e QQ e `` e, an Equain (5.20) (5.24) (5.34) e (5.35) e (5.3) e (5.3), an he iceie equain can be ien a, e (5.38) hee he equain cefficien f a ieninal iceiain ae gien b hil f a hee ieninal iceiain he bece, QQ QQ `` `` an hgnal eh ha i Caeian, he geeic iffuin cefficien ae QQ QQ QQ `` QQ QQ QQ 3 3 3 QQ `` `` U (5.39) (5.40) f he abe e (5.41) an he equain euce he gien in Secin 2.2.
e N e ž H H CHAPTE 5. NN-THGNAL MESHES 125 5.3 The Diceiain f he Aecie an Suce Te n a Nn-hgnal Gi I n nl eain iceie he aecin an uce e cplee he iceiain f he anp equain n a nn-hgnal eh. The uce e ae eae in an ienical anne ha ue f a Caeian eh, ih he cell cene alue being aken a he aeage f he cell a a hle, an he inegal f he uce e he cell i appxiae a œ Ÿž N (5.42) The aecie fluxe in he anp equain T e (5.43) ae appxiae in he ae anne a a ue ih he Caeian iceiain gien in Secin 2.2, ih he face alue f e being ineplae n he egula eh in cpuainal pace. The nl iffeence beeen he Caeian an nn-hgnal gee iceiain H i he calculain f he face a flux. he Caeian eh he a flux i calculae a, hee i he aea an i he elci nal he face. he nn-hgnal eh he a flux i inea calculae f he puc f he elci an he face aea nal, an all elci cpnen u be ineplae he face. The a flux f ea, nh an p face ae hen gien b, a Caeian eh hee expein euce hich ee he expein gien in Equain (2.1). T QQ T T `` H H V V (5.44) (5.45) The full iceiain f he anp equain n a nn-hgnal eh can n be gien. B appxiaing he aecie fluxe ih he cenal iffeence chee, hich a ecibe f a Caeian eh in Secin 2.2.3, he ea anp equain can be iceie a a iceiain f he -ieninal anp equain he iniiual cefficien f he «ª (5.4)
z { p ž T p CHAPTE 5. NN-THGNAL MESHES 12 ª an aa ae, p Q Q QQ Q QQ Q `` Q `` Q N Q T Q Q T Q T Q Q T Q Q =p T Q Q 8 T Q Q 1p T Q Q T Q Q V T 5 (5.4) a iceiain f he hee ieninal anp equain he equain cefficien ae gien b
p z { p z { ž T p e e z { CHAPTE 5. NN-THGNAL MESHES 12 Q Q QQ Q QQ Q `` Q `` Q T N Q T Q ` ` Q T Q ` ` T Q Q `Q `Q T Q Q `Q `Q T ` ` Q` Q` T ` ` Q` Q` up T Q Q T Q Q p T Q Q T Q Q z T Q` `Q z Q` `Q T { T Q` `Q { Q` `Q T z p T ` ` z T ` ` { p T ` ` { T ` ` (5.48) Thee e a be cpae he equialen e gien in Equain (2.25) an (2.2) f a iceiain upn a Caeian eh. The e i le uing a efee cecin chee, ih he aa being uliplie b he peiu ieain alue f he e fiel an abbe in he igh han ie, ha a an gien ieain e being le b he linea le i hee e ± i he peiu ieain luin. an hgnal eh «ªŸ eq ± (5.49), an he an hee ieninal anp equain euce }ª (5.50)
p z { p z { ž T p T p ž e z { CHAPTE 5. NN-THGNAL MESHES 128 hee he cefficien f a -ieninal iceiain ae Q ² Q QQ Q QQ Q `` Q `` ² Q N T an f a hee ieninal iceiain, he equain cefficien ae gien b N Q Q QQ Q QQ ² Q `` Q `` ² Q T (5.51) (5.52) 5.4 Buna Cniin f a Nn-hgnal Meh The buna cniin f a iceiain n a nn-hgnal eh ae iila he f a Caeian eh, excep f Neuann bunaie hich hae exa c eiaie cpnen in he calculain f he gaien nal he buna. Cnieing he cae f an eaen buna, a illuae in igue 5., a Diichle buna cniin hee he alue f he cala i pecifie a he buna a eul in an equain f he buna ne f he f e p }³ e e (5.53) e (5.54) hich can be ubiue in he equain f pin gien in Equain (5.2) an (5.29). Neuann buna cniin he gaien nal he face i pecifie, fe b e (5.55) The nal gaien i gien in Equain (5.20) abe, an hi i ubiue in he e a he buna ne. The nn-hgnal e inling c-eiaie a he face ae nall le uing he efee cecin chee a a gien in Equain (5.31), an a an ieain he buna gaien a n be a pecifie b Equain (5.55), bu f a full cnege luin he buna cniin ill appl. icninuiie in he buna cniin, uch a a ep change in he pecifie buna alue f Diichle bunaie alng a face, a change f Neuann Diichle bunaie n a face, a
T T T ³ T T CHAPTE 5. NN-THGNAL MESHES 129 P b E igue 5. The buna f a nn-hgnal eh. cne f he eh, he cenal iffeence appxiain f he c eiaie in he eh (uch a he gien in he ecn an hi equain in Equain (5.10)) ill be in e, an inea f uing cenal iffeence, ne ie iffeence hul be ue a apppiae. 5.5 The Sluin f he Naie Ske Equain n a Nn-hgnal Meh ling he Naie Ske equain n a nn-hgnal eh hee ae hee e iue cnie. The fi cncen he iceiain f he peue e in he enu equain, he he e in he equain being eae in he ae anne a hei cunepa in he geneic anp equain. The ecn iue i he peue-elci cupling an he geneain f an equain f he peue cecin. An finall he ineplain chee ue calculae he elci a he cell face nee be fulae. 5.5.1 Ealuain f he Peue Gaien Te The peue fce n a cell can be appxiae b he inegal f he peue e he face f he cell, he peue fce n each face acing nal ha face. The peue a he face can be appxiae b an ineplain f he alue a he cene f he cell n eihe ie f he face. Thu he alue a he eaen face can be appxiae a, The aea f he eaen face i gien abe a appxiae b µ µ u µ µ p (5.5), an he peue fce n he eaen face i µ ³ p ¹ (5.5) he negaie ign being ue he peue fce acing ina n he cell, hil he face aea i efine pining ua. Inegaing e he ix face f he cell gie he peue fce n he cell a Q»º QQ `` µ µ µ µ p µ µ z µ QQ µ `` µ T µ µ µ { ¼ (5.58)
Å Á º ƒ ƒ ƒ ƒ ¼ ƒ a ƒ ƒ ƒ a ¼ ¼ ¼ CHAPTE 5. NN-THGNAL MESHES 130 Thi can be facie Q»º µ p µ µ Q µ º QQ QQ µ QQ QQ µ `` z µ `` `` { `` ¼G (5.59) The u f he aea n he ecn line f Equain (5.59) ill be ze f a cle lue, an he expein euce Q5º µ p µ µ QQ µ QQ µ z `` µ { `` ¼ (5.0) The iniiual Caeian cpnen f hi fce can hen be ae he uce e f each cpnen f he enu equain. 5.5.2 Ineplain f he ace Velciie The elci ineplain a he face i pefe in an ienical anne he eh ue f Caeian ehe, ecibe in Secin 4.2.2. The eh, aibue hie an Ch[139], ineplae he u f he elciie an peue fce, ahe han ineplaing he elciie alne. The equain f he cneain f enu a he pin can be ien in ec f a, a¾ µ À (5.1) hee i he elci ec a he pin, Á µ he cell cene a, an i he iagnal eleen f he e f linea equain Á i he u f peue fce n he face f. Such an equain exi f each cell in he ain, an ih hie Ch ineplain i i aue ha a iila equain can be appxiae a he face f a cell. Thu f he eaen face f a cell hee i an equain f he f, V ¹ a¾ µ ¹ ¹ ¹ (5.2) ih iila equain being able be ien f he nh an p face f he cell. The lef han ie f equain (5.2) a he face f he cell i aue be equal a linea ineplain f he lef han ie f he equain f he an cell n eihe ie f he face,  ¹ a¾ µ ¹ V a¾ µ ¹ (5.3) hee an eba T ignifie a linea ineplain f he alue a he an cell cene. The alue f Ã\Ä a he eaen face in he lef han ie f Equain (5.3) can be appxiae b a linea ineplain, an an expein f he elci a he eaen face i u ¹ am µ ¹ a¾ µ ¹ (5.4) The u f he peue fce a he eaen face i appxiae b µ Æ µ p µ º `` µ z º QQ µ `` Ç µ z p QQ Ç µ up `` µ { QQ µ QQ JÇ µ p `` Ç µ { p (5.5)
a a a ƒ ƒ ƒ ƒ ¹ º º º Å Á Å Á Å Á a µ µ T CHAPTE 5. NN-THGNAL MESHES 131 QQ Ç QQ hee i he face aea f he cell. The peue fce n he cell an ae gien b Equain (5.0), an all he e in Equain (5.4) ae efine an he face elci can be calculae. The elciie f he nh an p face can be ineplae uing iila expein, an he face a fluxe can be calculae uing Equain (5.44), QQ `` ¼ ¼ ¼ 5.5.3 The Nn-hgnal Peue Cecin Equain (5.) The cupling f he elci an peue equain n a nn-hgnal eh i pefe in an ienical anne ha ecibe in Secin 4.2.1 f Caeian e. The iceiain f he Naie Ske ha been ecibe abe, a ha he eh ue f ineplaing he elci he cell face. The nl eaining ak i he eiain f he peue an elci cecin. A a ecibe in Secin 4.2.1 he SIMPLE cupling chee inle he calculain f an iniial eiae f he elci fiel, GÈ, f hich he face elciie ae ineplae. The geneae fiel ill n geneall aif he cninui equain, an a peue cecin µ i hen calculae, hich hen ae he enu equain eul in a elci cecin hich hich aifie he cninui equain. The iniial calculain f he elci fiel i ih an equain f he f f È È am (5.) hee b ignifie a uain f he neighbuing cell, hil É i a uain e he cell face. The luin hi equain ill n geneall aif he cninui equain. e heefe ih a he elci cecin geneae f a peue cecin µ, uch ha he euling elci fiel È aifie he cninui equain. µ an ÊÈ in he enu Equain (5.) gie Subiuing µ T È T È Y Appxiaing he ff-iagnal e in he enu equain b Equain (5.) a T^ È an hen ubacing he enu Equain (5.) gie a. a. µ µ a È µ (5.8) (5.9) (5.0) (5.1) The cecin elciie f he ea, nh an p face f he cell ae hen fun b ineplain a ¹ ¹ Æ µ Æ µ Æ µ (5.2)
È p z { T T p µ ƒ ƒ ƒ ¹ z { CHAPTE 5. NN-THGNAL MESHES 132 lling he eiain f hie an Ch[139] he c eiaie e a he face ae igne, ince f nea-hgnal ehe he ae all, an he µ fiel anihe f a cnege luin. Theefe he expein f he face elci cecin can be ien a µ p µ The iceie cninui equain can be ien a ¹ ¹ QQ µ `` µ z ³ µ ³ µ ³ (5.3) (5.4) Subiuing in, hee È i he ineplae elci f he È elci fiel, an i he a flux cecin efine a, an eaanging gie µ p µ µ µ QQ QQ µ µ QQ QQ µ µ `` `` µ z µ `` `` µ µ { V Thi can be facie a hee µ p µ p µ µ µ QQ QQ QQ QQ `` `` `` `` z µ z { µ { (5.5) (5.) (5.) Thi e can be le f he peue cecin µ, hich i hen ue cec he elciie aif he cninui equain, he elci an peue fiel being upae b SËÍÌÏÎ ËÍÌÏÎ GÈ µ µ (5.8) Aie f he ue f elci ec a cell face, ahe han ingle Caeian cpnen f elci, an he ue f aea ec ahe han he agniue f he cell face aea, he e i ienical Equain (4.23) eie f a Caeian e in Secin 4.2.1.
Ñ Ò Ó Ò CHAPTE 5. NN-THGNAL MESHES 133 5. Exaple f Nn-hgnal Pble T e he ipleenain f he nn-hgnal iffeencing chee he ce a ue le hee e pble. The fi el -ieninal cnucin in a ke ain. he gee chen a Caeian finie iffeence luin a eail calculae ha cul be ue a a cpain f he luin calculae uing he nn-hgnal finie lue iceiain. The ecn pble elle he -ieninal ien cai fl ecibe in Secin 4.4.1. The fl a le f a quae cai, ih ne luin being geneae n a quae eh, an he he n a nn-hgnal eh ha fie he cai. A he eh i efine he luin calculae n he Caeian an nn-hgnal ehe hul bh cnege a he benchak luin pie b Ghia e al[50]. The final e cae a he ien cai fl pie a a benchak e cae b Deižić, Lilek an Peić[3]. Thi pble el a ien fl in a ke cai, iila in hape he ain ue in he cnucin e. 5..1 Cnucin in a Ske Dain T e he iceiain f he iffuin e f he anp equain, an he ipleenain f Diichle an Neuann buna cniin, he iniial pble elle a ne f ea ae cnucin in a ke ain. The gee f he pble i gien in igue 5.8 alng ih he e f buna cniin ue. Ô Õ Ö Ñ Ò ƒ b ؃ b Diichle Bunaie Neuann Bunaie igue 5.8 The ea ae cnucin in a ke ain pble, hing he pble gee an he buna cniin elle. The cai i f uni lengh an heigh, ih he ie all angle a Õ Ö.
e Ù CHAPTE 5. NN-THGNAL MESHES 134 ²ƒ The fi pble applie Diichle cniin n all bunaie, ih he li f he ain being e, hil he he hee bunaie ae e e. The ecn e ipe Diichle cniin n he p an b bunaie f he ain, ih alue f e ²ƒ an e epeciel, hil he ie bunaie ha a ze flux Neuann buna cniin ipe, ÙÚl, ih b being he iecin nal he uface. T pie a benchak luin, he pble a al elle uing a finie iffeence iceiain n a Caeian eh. The pe f ehe ha ee ue ae hn in igue 5.9, ih he Caeian eh ue b he finie iffeence le being n he lef, hil he buna fie nn-hgnal eh ue b he finie lue le i hn n he igh. igue 5.9 The ehe ue f he ke cnucin pble. n he lef he Caeian eh ue b he finie iffeence ce. n he igh he nn-hgnal eh ue b he finie lue ce. The luin calculae b he eh ae hn in igue 5.10 an 5.11, he Caeian fi- Ö ƒ Q eh, hil he nn-hgnal finie lue luin ae calculae n a eh. The cnu f he luin lk iila f bh he Neuann an Diichle pble. nie iffeence luin being calculae n a ƒüûúƒ Q igue 5.10 The calculae epeaue fiel f he ke cnucin ih Diichle bunaie. n he lef he luin calculae ih he finie iffeence ce, n he igh ha calculae ih he finie lue ce. igue 5.11 The calculae epeaue fiel f he ke cnucin ih Neuann bunaie. n he lef he luin calculae ih he finie iffeence ce, n he igh ha calculae ih he finie lue ce.
Þ CHAPTE 5. NN-THGNAL MESHES 135 T bee cpae he eh, he epeaue pfile f each pble, calculae n ƒýûxƒ Q ehe, ae gien in igue 5.12 an 5.13. igue 5.12 h he epeaue pfile ac he ain alng he hiznal ceneline (ie ia up he ain) calculae f he Diichle buna cniin pble. The finie lue luin i hn a a li line, ih ha calculae uing he finie iffeence chee being gien b he ce. A can be een he luin agee quie ell. Siilal igue 5.13 h he luin f he Neuann buna cniin alng he ceneline unning up he cene f he cai. Again he eh ae in excellen ageeen ih each he. 0.14 0.12 Nn-hgnal inie Vlue Caeian inie Diffeence 0.1 Scala 0.08 0.0 0.04 0.02 0 0.5 0. 0. 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 x igue 5.12 The calculae epeaue pfile alng he hiznal ceneline f he Diichle ke cnucin pble. inall, he cnegence f he luin calculae b he eh a he eh i efine ae hn in igue 5.14 an 5.15. he Diichle pble, he epeaue a he cene f he ain i ple a a funcin f he cell ize, hil f he Neuann pble he epeaue a he iheigh f he lef all i ple. bh pble he luin cnege a he eh i efine. N anking f he elaie accuac f he chee can be ae, ince f he Neuann pble he finie lue luin i he e accuae, hil f he Diichle pble he eee i ue. T uaie, f he pble ee he nn-hgnal finie lue chee gie a luin ha i cpaable in accuac ha geneae b a Caeian finie iffeence eh. A he eh i efine he luin gien b bh chee cnege a cn luin.
Þ ß CHAPTE 5. NN-THGNAL MESHES 13 1 Nn-hgnal inie Vlue Caeian inie Diffeence 0.8 Scala 0. 0.4 0.2 0 0 0.2 0.4 0. 0.8 1 igue 5.13 The calculae epeaue pfile alng he eical ceneline f he Neuann ke cnucin pble. 0.114 0.113 Nn-hgnal inie Vlue Caeian inie Diffeence 0.112 Scala Value a Cene 0.111 0.11 0.109 0.108 0.10 0 0.01 0.02 0.03 0.04 0.05 0.0 0.0 Gi Size igue 5.14 The cnegence f he luin calculae ih he finie iffeence an finie lue le a he eh i efine. The epeaue in he cene f he ain f he Diichle pble i ple a a funcin f he gi ize 3.
à CHAPTE 5. NN-THGNAL MESHES 13 0.252 0.25 Scala Value a Cene f Sie all 0.248 0.24 0.244 0.242 0.24 0.238 Nn-hgnal inie Vlue Caeian inie Diffeence 0.23 0 0.01 0.02 0.03 0.04 0.05 0.0 0.0 Gi Size igue 5.15 The cnegence f he luin calculae ih he finie iffeence an finie lue le a he eh i efine. The all epeaue a he i-heigh f he ain f he Neuann pble i ple a a funcin f he gi ize 3.
h h Ñ ã Ò â Ó á CHAPTE 5. NN-THGNAL MESHES 138 5..2 Dien Cai l ih a Die Meh The ecn pble ha a elle a he Dien Cai fl f a quae ain, peiul encunee in Secin 4.4.1. hi fl hee exi a nube f benchak luin, an a nube f luin f he fl ee calculae in he e ecibe in Chape 4. æ æ å" " " " â â igue 5.1 The ien cai pble. T e he nn-hgnal iceiain ue in he fl le, he pble a elle n a ie eh, a hn in igue 5.1. Sluin ee al calculae n a Caeian eh f cpain, an he benchak luin f Ghia, Ghia an Shin[50] a al ue a a e f he luin. l a elle n a ange f ehe, f a fl ih a enl nube f 1000. Sluin ee calculae uing he MSU an U iffeencing chee. igue 5.1 The ehe ue f he ien cai pble. n he lef he Caeian eh, n he igh he ie nn-hgnal eh. ƒ Q The luin f a fl calculae n an ç nn-hgnal eh uing he MSU iffeencing chee i hn in igue 5.18, ih cnu pl f he eafuncin è, he agniue f elci, an he an cpnen f elci being gien. Cnu leel ae ienical he ue in Secin 4.4.1 an he pape f Ghia e al.
CHAPTE 5. NN-THGNAL MESHES 139 he bulk f he fl he ageeen beeen he Caeian an nn-hgnal luin i exeel g, ih he cnu in igue 5.18 elaing he gien in igue 4.5. Hee he eciculain bubble in he le lef an igh cne f he cai ae incecl ele, ih he eaachen pin being incecl lcae. Thi pble ccu ih luin calculae uing he U, MSU an QUICK iffeencing chee, an he Auh i unceain if hi i ue he appxiain ue in he nn-hgnal iceiain, i a eh epenen e ha ul iappea ih eh efineen. The elciie n he eical an hiznal ceneline ae hn in igue 5.19 an 5.20, calculae uing he nn-hgnal ehe uing he MSU iffeencing chee. Sluin ae gien f a ange f eh eniie, ih he luin being cpae ih he benchak elciie f Ghia e al. A ih he Caeian cae he calculae elciie cnege he benchak luin, giing an excellen ageeen ih he fine eh luin. inall he elci fiel calculae uing he Caeian an nn-hgnal ehe ae cpae in igue 5.21. each eh ize ue, he Caeian elci fiel ee ineplae n he ie eh uing bicubic ineplain, ih he MS e beeen he Caeian an nn-hgnal luin being calculae. The e beeen he luin elci fiel i ple a a funcin f he ean cell ize 3 f luin calculae uing he U an MSU iffeencing chee. A he eh i efine he MS e eceae an he luin cnege. The e beeen he MSU luin ee lage han he beeen he U luin. Hee, a he eh i efine he MSU e euce a a fae ae han ha f he U luin.
è CHAPTE 5. NN-THGNAL MESHES 140 igue 5.18 The luin fiel f he ien cai calculae n a ç cell ie eh, f a enl nube f 1000 uing MSU iffeencing. h lef igh, p b, he fiel hn ae he ea h funcin, ƒ elci agniue ƒ ƒ é é, elci an elci. Seafuncin cnu ae a ineal f, ih fuhe cnu a ê ƒ ë ` ê ƒ ë ê ƒ Úëì ê ƒ ëí an ê ƒ Úëî. Velci cnu ae a Ö ineal beeen ê ƒ. The luin hul be cpae ih he gien in igue 4.4 an 4.5. ƒ Q
ï ï CHAPTE 5. NN-THGNAL MESHES 141 1 0.8 0. 0.4 0.2 0-0.4-0.2 0 0.2 0.4 0. 0.8 1 U elci igue 5.19 The pfile f he elci alng he eical ceneline calculae n he ie eh a a enl nube f 1000 uing MSU iffeencing. Calculae aa cpae ih he benchak f Ghia, Ghia an Shin[50]. 15x15 2x2 39x39 51x51 9x9 81x81 Ghia 0.4 0.3 0.2 0.1 15x15 2x2 39x39 51x51 9x9 81x81 Ghia 0-0.1-0.2-0.3-0.4-0.5-0. 0 0.2 0.4 0. 0.8 1 V elci igue 5.20 The pfile f he elci alng he hiznal ceneline calculae n he ie eh a a enl nube f 1000 uing MSU iffeencing. Calculae aa cpae ih he benchak f Ghia, Ghia an Shin[50].
CHAPTE 5. NN-THGNAL MESHES 142 0.025 0.02 U U Velci U V Velci MSU U Velci MSU V Velci 0.015 MS E 0.01 0.005 0 0 0.02 0.04 0.0 0.08 0.1 x igue 5.21 The MS e beeen 3 he luin calculae n a Caeian an ie eh, a a funcin f aeage eh ize. E gien f an elci fiel, f luin calculae ih U an MSU iffeencing chee.
,,,,,,,,, Ñ Ò æ æ ã â, â, â,, $, $, $, á $, $, $ $ $ $ $ $ñ CHAPTE 5. NN-THGNAL MESHES 143 5..3 Dien Cai l in a Ske Cai A a final e, he nn-hgnal ce a cpae he benchak luin gien b Deižić, Lilek an Peić[3] f a ien cai fl in a ke cai. The gee f he pble i gien in igue 5.22, an luin ee baine f a cai ih uni lengh all (Ò á ²ƒ ), a all angle f Õ Ö, a a enl nube f ƒ, uing he QUICK iffeencing chee. ò å" " " " $$ð igue 5.22 The ke ien cai pble. igue 5.23 The eh ue f he calculain f he ke ien cai fl fiel. The luin fiel calculae uing a eh f ƒýûúƒ B ƒýûxƒ pin i hn in igue 5.24, ih cnu pl f eafuncin, elci agniue, an he an cpnen f elci being gien. The cnu leel ue ae ienical he ue b Deižić e al in hei pape, an he ageeen beeen he publihe an calculae luin i excellen. The elciie calculae n he eical an hiznal ceneline ae ple in igue 5.2, ih he luin being gien f a nube f ehe. The ae cpae ih he benchak alue f Deižić e al, an again he ageeen i excellen n he fine ehe. In aiin he e in he cae eh luin h he ae en a he cae eh luin f Deižić e al.
ï CHAPTE 5. NN-THGNAL MESHES 144 igue 5.24 The luin fiel f he ke ien cai, calculae a a enl nube f 1000 uing QUICK iffeencing. h h lef igh, p b, he fiel hn ae he ea funcin, elci agniue, elci an Ö Û ó Ö elci. Seafuncin cnu ae a ineal Õ ó Ö beeen ê an ƒ 1 ³ 1, an a ineal beeen an ç Ö. Velci cnu ae a ineal beeen. 0.8 0. 0. 0.5 0.4 0.3 0.2 0.1 Deizic 0-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0. 0. 0.8 0.9 1 1.1 U elci 19x19 35x35 x 131x131 igue 5.25 The pfile f he elci alng he ceneline f he ke ien cai, calculae f a enl nube f 1000 uing QUICK iffeencing. Calculae aa cpae ih he benchak f Deižić e al[3].
ô CHAPTE 5. NN-THGNAL MESHES 145 0.08 0.0 0.04 19x19 35x35 x 131x131 Deizic V elci 0.02 0-0.02-0.04-0.0 0.3 0.4 0.5 0. 0. 0.8 0.9 1 1.1 1.2 1.3 1.4 x igue 5.2 The pfile f he elci alng he ceneline f he ke ien cai, calculae f a enl nube f 1000 uing QUICK iffeencing. Calculae aa cpae ih he benchak f Deižić e al[3].
ó CHAPTE 5. NN-THGNAL MESHES 14 5. Cncluin The finie lue iceiain gien in Chape 2, an he SIMPLE peue elci cupling chee ecibe in Chape 4 hae been exene f Caeian ehe ucue nn-hgnal ehe. The iceiain f he iffuin e in he anp equain ha been eie, hil he iceiain f he aecie e i hanle in an ienical anne he iceiain n Caeian ehe ecibe in Chape 2. The peue elci cupling chee a hen aape nn-hgnal ehe, ih he peue cecin equain being e-eie. The ich f a Caeian a nn-hgnal eh cplicae he calculain f he geeic ppeie f he eh, uch a face aea an cell lue, an he echnique ue f calculaing hee alue i ecibe in Secin 5.1. In aiin, he ich f an hgnal a nn-hgnal eh inceae he nube f e in he linea equain geneae b he finie lue iceiain, ƒ Ö ih he nube f e inceaing f f -ieninal iceiain, an f in iceiain n hee-ieninal ehe. A efee cecin chee ha all he ue f he linea le eelpe in Chape 3 i ecibe an ue. inall, e he ce i a ue le hee benchak pble. The fi, a -ieninal cnucin pble, ee he iceiain an ipleenain f he iffuin e in he anp equain, an he ipleenain f he Neuann an Diichle buna cniin. The ecn an hi pble elle -ieninal ien cai fl, ih he eul being cpae again publihe benchak. In all cae he ageeen beeen he nn-hgnal ce an he benchak i excellen, ih cnegence being enae ccu ih efineen f he eh.