4-几何非线性问题的有限元法2012-1

4.0Ⳃ󱔩

¾󰦬󰟚󰥳⧚¾󱳔󱇣󰢓㛑󰥳⧚

¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⡍⚍¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾

󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘUL⊩

34.1󰦬󰟚󰥳⧚

¾󰞴ԩ䴲㒓󱗻䯂乬󱥻󰠊󱮍⿟˖15h2Ͼ󱮍⿟•

󱑇㸵󱮍⿟˖

wª«§3«¨wui·¸¬¨©Gij󰀄wXSº0wWijh2Ͼ󱮍⿟wXjk»󰀄󰀄pi 0kj¸¹»¼pi 0wxj•

󰞴ԩ󱮍⿟˖

6h2Ͼ󱮍⿟Eij 1§2¨¨wui©wX󰀄wuj󰀄wukwuk·¸e1§¨wuiwujwukwuk·¸jwXiwXiwXj¸¹ij 2¨©wx󰀄󰀚jwxiwxiwxj¸¹•

󱴀󱵘󱮍⿟˖Wij OekkGij󰀄2Peij

Sij OEkkGij󰀄2PEij

6h2Ͼ󱮍⿟

•

䖍⬠󱴵ӊ˖S§¨wu0jk¨©Gik󰀄i·wX¸Nk¸¹j qiWijnj qi5

4.1󰦬󰟚󰥳⧚

¾㒧󱵘䴲㒓󱗻󰟚󱵤ⱘ⧚䆎㾷ҙ䰤Ѣㅔ󰤩󱴚ӊǃ󱵓ǃ󰻇ԧ㒧󱵘¾

󰻻䞣󱎹⿟㒧󱵘󰽖ḕ󱶊ǃṕǃ󱣅ǃḚ󱶊ǃ㓚㋶ǃ󱶨󱗻󰻇ԧǃ󱉽󰳳ㄝⱘ䴲㒓󱗻䯂乬󰦾㛑󰗳󰡽Ѣ󱳝䰤󰜗䕃ӊ䖯㸠󱭄󰘐󰟚󱵤

󰦬󰟚󰥳⧚

⧚䆎󱮍⿟

󱳝䰤󰜗󱮍⿟

7㄀󰲯ゴ󰞴ԩ䴲㒓󱗻󰟚󱵤ⱘ󱳝䰤󰜗⊩

󰀂󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󱮍⿟ⱘ󱓎ゟ䖛⿟˛󰀂󰞴ԩ䴲㒓󱗻󰟚󱵤ⱘTL⊩ǃUL⊩󰤎󰟿˛

2

4.1󰦬󰟚󰥳⧚

¾㒓󱔍󱗻䯂乬󱥻󰠊󱮍⿟˖15Ͼ󱮍⿟•

󱑇㸵󱮍⿟˖23Ͼ󱮍⿟Vij,i󰀄xj Uwujwt2•

󰞴ԩ󱮍⿟˖6Ͼ󱮍⿟

H1ij 2󰀇ui,j󰀄uj,i󰀈•

󱴀󱵘󱮍⿟˖6Ͼ󱮍⿟Vij OHkkGij󰀄2PHij

•

䖍⬠󱴵ӊ˖

xj Vijni4

4.1󰦬󰟚󰥳⧚

¾󱴤󱭭䴲㒓󱗻䯂乬󱥻󰠊󱮍⿟˖15Ͼ󱮍⿟•

󱑇㸵󱮍⿟˖23Ͼ󱮍⿟Vij,i󰀄xj Uwujwt2•

󰞴ԩ󱮍⿟˖6Ͼ󱮍⿟

H1ij 2󰀇ui,j󰀄uj,i󰀈•

󱴀󱵘󱮍⿟˖dV CepHep

6Ͼ󱮍⿟ijijkldkldHij DijkldVkl

•

䖍⬠󱴵ӊ˖

xj Vijni6

4.1󰦬󰟚󰥳⧚

¾

󱖂󰟚˖∖󰟑󱭄f(x)󰦪󱵕󰘐󱯊ⱘx

fc(x) 0

df(x) 0'f(x)x xi

dx'x 0fmax(x) f(xi)

¾

󰦬󰟚˖∖⊯󰟑F(f(x))󰦪󱵕󰘐󱯊ⱘf(x)

¾

∖“󰲴󰦬󰟑󱭄”󰦪󱵕󰘐󱯊㞾󰦬󰟑󱭄ⱘ󱔶󱓣ˈ䅽㞾󰦬󰟑󱭄󱳝Ͼ󰦬󰣪ˈ✊󰧢䅽“󰲴󰦬󰟑󱭄”ⱘ󰦬󰣪ㄝѢ䳊ˈ󱈅󱕫󰠄њ󱖂󰟚󱮍⿟ˈ✊󰧢∖㾷󰥇󰧃

¾

󰲴ѻ⫳󱵰ˈ󰜜䅽“󰲴”󱳝Ͼ󰦬󰣪ˈҢ㗠“󱵰”г󱳝󰦬󰣪ˈ䅽“󱵰”ⱘ󰦬󰣪ㄝѢ䳊ˈ󱈅󰧃ҹ∖󰟎“󰲴”ⱘԡ㕂DŽ󱇍Ѣ󱖂󰟚ˈ䖭Ͼԡ㕂󱰃⚍ˈ󱇍Ѣ󰦬󰟚ˈ䖭Ͼԡ㕂󱰃㞾󰦬󰟑󱭄

8

4.1󰦬󰟚󰥳⧚

¾

󱇍䖲㓁ҟ䋼䯂乬ⱘ㾷uˈ󰦪ϔϾ󱖂󱇣ⱘ󰦬󰣪GuˈՓ⊯󰟑3(u)󰦪偏󰘐ˈ󰥇⊯󰟑“󰦬󰟚”ㄝѢ䳊ˈ䖭⾡∖󱕫䖲㓁ҟ䋼䯂乬ⱘ󱮍⊩⿄Ў󰦬󰟚󰥳⧚󱟪󰦬󰟚⊩¾

󱇍䖲㓁ҟ䋼䯂乬­®A(u) 0in:¯B(u) 0on*¾

󰦪⊯󰟑3(u)ˈ󱇍ӏ󱛣Guˈ󰦪󰦬󰟚x2:A(u) = 0*

G3 ³GuA(u)d:󰀄³GuB(u)d* 0B(u) = 0

:*ox1x39

4.1󰦬󰟚󰥳⧚

¾⦄ҷⱘ󰦬󰟚󰥳⧚˖󱳝䰤󰜗⊩

󰀄ḍ󱥂㰮󰡳󰥳⧚˄㰮ԡ⿏󰥳⧚ǃ㰮󱑨󰡯󰥳⧚˅󱓎ゟ⊯󰟑󰀄󱇍⊯󰟑䖯㸠󱳝䰤󰜗⾏󱬷ˈ󰝡䖯㸠󰦬󰟚ˈ󱕫󰠄󱳝䰤󰜗󱮍⿟㒘󰀄

ҷ󰜹䖍⬠󱴵ӊˈ󱕫󰠄䖥Ԑ㾷

114.0Ⳃ󱔩

¾󰦬󰟚󰥳⧚¾󱳔󱇣󰢓㛑󰥳⧚¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⡍⚍¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾

󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘUL⊩

13

4.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚

•

󱓎ゟ⊯󰟑˖󰳼󱠔󱳝󰧃㛑ⱘuiǃEijЁˈՓ󱕫⊯󰟑3p󰦪󱵕󱇣󰘐㗙ˈ󱖙Ў󱳝䰤󰦬󱔶䯂乬ⱘ㾷

3p ³³³󰀇A(Eij)󰀚fiui󰀈d:󰀚³³piuid*:*p•

⊯󰟑󰦬󰟚

G3§wA·p ³³³¨GEij󰀚fiGui¸d:󰀚:¨©wE¹³³piGuid* 0ij¸*p󱑇㸵󱮍⿟

䖍⬠󱴵ӊ

15

4.1󰦬󰟚󰥳⧚

¾㒣󰝌ⱘ󰦬󰟚󰥳⧚˖

󰀄ḍ󱥂㰮󰡳󰥳⧚˄㰮ԡ⿏󰥳⧚ǃ㰮󱑨󰡯󰥳⧚˅󱓎ゟ⊯󰟑󰀄

䖯㸠⊯󰟑󰦬󰟚ˈ󱕫󰠄󱳔󱇣󰢓㛑󰥳⧚ǃ󱳔󱇣ԭ㛑󰥳⧚

󰳼󱠔󱳝󰧃㛑ⱘԡ⿏˄󱑨󰡯˅Ёˈⳳ󱅲ԡ⿏˄󱑨󰡯˅Փ㋏㒳ⱘ󱘏󰢓㛑˄ԭ㛑˅󰦪󱳔󱇣󰘐

󰀄ҷ󰜹䖍⬠󱴵ӊˈ󱕫󰠄⧚䆎㾷󰀄

󰽖Ritz⊩

10

4.1󰦬󰟚󰥳⧚

¾

ⳳ󱅲ԡ⿏˖󰳼󰡯ⱘ԰⫼ϟⱘⳳ󱅲䖤󰡼Ё㒣䖛ϔ󱅮󱯊䯈䯈䱨󰝙󰦥⫳ⱘԡ⿏ˈ󱔧't=0󱯊ˈdui=0DŽⳳ󱅲ԡ⿏󰧃ҹ󱰃󱖂󱇣ⱘˈг󰧃ҹ󱰃󱳝䰤ⱘˈ󰝋󱳝⹂󱅮ⱘ󱮍󰧥󱗻

¾

󰧃㛑ԡ⿏˖㒭󱅮󱯊䯈䯈䱨󰝙㑺󱴳󰜕䆌ⱘԡ⿏ˈ󱔧't=0󱯊ˈdui=0DŽ󰧃㛑ԡ⿏󱰃ϔ⾡⢊󱗕ˈ󱳝󰧃㛑󰦥⫳󱳝󰧃㛑󱇮󱳾󰦥⫳ˈ󱟪㗙󰦾󱰃԰ЎϔϾ󱢑䈵ⱘ󱦣䗄

¾

㰮ԡ⿏˖󱰃㑺󱴳䆌󰧃ϟ󱶤ⶀ󱯊󰧃㛑󰦥⫳ⱘ󱖂󱇣ԡ⿏ˈ󰦾󱰃ϔϾ󱢑䈵ⱘ󰞴ԩὖ󱗉ˈϢ㋏㒳󱟪䋼⚍ⱘ󱅲䰙䖤󰡼ǃ󰡯ⱘ԰⫼ǃ󱯊䯈󰥚⿟ǃ󰟱󰾟󱴵ӊㄝ󱮴󰝇

12

4.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳝䰤󰦬󱔶⧚䆎˄Lagrangian󱦣䗄˅•

󰞴ԩ󱮍⿟E 1ij2(ui,j󰀄uj,i󰀄uk,iuk,j)•󱑇㸵󱮍⿟((Gik󰀄ui,k)Skj),j󰀄fi 0󰳼:󰝙•

䖍⬠󱴵ӊ

󰀋󰀋GLN󰀄XL󰀏N󰀌6NM󰀌QM SL󰳼*SϞui ui󰳼*uϞ•

󱴀󱵘󱮍⿟

wA(Eij) SwB(Sij)wEijijwS EijijA˖󱑨󰦬㛑󱆚󱑺B˖ԭ㛑󱆚󱑺14

4.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚

•

⊯󰟑󱳝䰤󰜗⾏󱬷˖󱡞:⾏󱬷ЎNϾ󰤩󰜗ˈ󰦬󰟚3* ¦N­°p®¯³³³󰀇An(En󰀈d:󰀚n½°ij)󰀚fiuin 1°³³piuid*¾:n*np°¿G3* ¦N­°p1®°³³³§¨¨wAnnn·½°nGEij󰀚fiGui¸n¯:n©wEij¸¹d:󰀚³³piGuid*¾ 0n *np°¿󱑇㸵󱮍⿟䖍⬠󱴵ӊ

16

4.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚•󱳝䰤󰜗󱮍⿟˖•

󱑇㸵󱮍⿟

ª«wAnn«¬wEnGki󰀄uk,iij󰀇󰀈»ºn»¼󰀄fi 0󰳼:󰝙,jn•

䖍⬠󱴵ӊ

wAwEnGnki󰀄uk,innj󰀚pk 0󰳼*nϞij󰀇󰀈17

4.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚˖󱳝䰤󰜗󱮍⿟•

ҷ󰜹⊯󰟑˖3* ¦N­°®³³³1^d`nT>B@nT>C@>B@n^d`npd:󰀚^d`nT>N@nT^f`d:󰀚³³^d`nT>N@nT^p`d*½°n 1°¯:n2³³³¾:n*np°¿󰟮󱑺ⶽ䰉

󱑓Н㡖⚍󰡯ⶶ䞣{R}

³³³>B@nT

>C@>B@n

d: >K@n

󰀄>K@n

L

NL

:n

194.0Ⳃ󱔩

¾󰦬󰟚󰥳⧚¾󱳔󱇣󰢓㛑󰥳⧚

¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⡍⚍¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾

󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘUL⊩

214.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾󱳝䰤󰜗󰟚󱵤ϔ㠀ℹ偸˄1˅˖󰀄

㒧󱵘󰧜⧚ㅔ󰣪ǃ󱓎ゟ῵󰵟󱵘ӊǃ󰞴ԩ󱈎󱇌ǃ󱴤󱭭䖍⬠󱴵ӊˈ㑺󱴳ㄝ

234.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚˖󱳝䰤󰜗󱮍⿟•

󱓩󰜹ⶽ䰉㸼⼎˖

^u`n >u1n

un

T

2u3

n@{^u`n >N@n^d`n

{u[d} (N}󰤩󰜗)ԡ⿏ⶽ䰉]㡖⚍ԡ⿏ⶽ䰉󱦦󰘐󰟑󱭄ⶽ䰉

•

󰞴ԩ䴲㒓󱗻ⱘ󱑨󰦬>@—ԡ⿏ⶽ䰉󰟚㾷Ў㒓󱗻+䴲㒓󱗻ϸ乍

B >BL@󰀄>BNL@^E`n >B@n^d`n

•

󱇍󰻻ԡ⿏ǃ󱇣󱑨󰦬󱚙󰞉ˈ㒓󱗻󱴀󱵘󰝇㋏^S`n >C@^E`n >C@>B@n^d`n•

ҷ󰜹⊯󰟑

3*p ¦N{³³³󰀇An(Enij)󰀚fiuin 1󰀈d:󰀚:n³³pniuid*}*np184.2󱳔󱇣󰢓㛑󰥳⧚

¾󱳔󱇣󰢓㛑󰥳⧚˖󱳝䰤󰜗󱮍⿟•

㒘䲚󱟤󱭈ԧ󱳝䰤󰜗󱮍⿟˖3*p 12^d`T󰀇>KL@󰀄>KNL@󰀈^d`󰀚^d`T^R`•

󰦬󰟚

G3*p 0

󰀇>KL@󰀄>KNL@󰀈^d` ^R`󰻻ԡ⿏󱔍󱗻󱳝䰤󰜗󱑇㸵󱮍⿟

ϔ㠀⫼󰹲䞣󱔶󱓣

20

4.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾󱳝䰤󰜗󰟚󱵤ϔ㠀ℹ偸˖󰀄㒧󱵘ㅔ󰣪ǃ󱓎ゟ῵󰵟󰀄㒧󱵘⾏󱬷󰣪󰀄󰤩󰜗󰟚󱵤󰀄󰤩󰜗㒘䲚

󰀄

󱓩󰜹䖍⬠󱴵ӊˈ∖㾷

22

4.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾󱳝䰤󰜗󰟚󱵤ϔ㠀ℹ偸˄2˅˖

󰀄

㒧󱵘⾏󱬷󰣪˖󱇚㒧󱵘⾏󱬷Ў󱳝䰤󱭄䞣Ͼ󰤩󰜗ˈ󰤩󰜗䋼䞣󰟚󱏗󰳼催󱮃

⚍Ϟˈ󰤩󰜗䖲󱥹⬅㡖⚍󱡓󱢙ˈ㡖⚍П䯈󰦬󱔶󰤣䇗⬅󱔶󰟑󱭄󱥻󰠊

󱮴䰤䖲㓁ҟ䋼⾏󱬷󰤩󰜗Ͼ󰤩󱮴䰤Ͼ䋼⚍󱳝䰤Ͼ催󱮃⚍󰜗󱮴䰤Ͼ催󱮃⚍244.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾󱳝䰤󰜗󰟚󱵤ϔ㠀ℹ偸˄3˅˖

󰀄

󰤩󰜗󰟚󱵤˖󰟚󱵤󰤩󰜗П󰝙ⱘ󰝇㋏ˈ󱓎ゟ󰤩󰜗㡖⚍󰡯󰩠ԡ⿏ⱘ󰝇㋏

>K@e^d` ^f`e

>K@e ³T⾏󱬷󰤩󰜗V>B@>C@>B@dV󰤩󰜗⿃󰟚󱳝䰤Ͼ催󱮃⚍ ³111T󰀚1³󰀚1³󰀚1>B@>C@>B@Jdrdsdtㄝ󰦖󰜗⿃󰟚[K]ˉ󰟮󱑺ⶽ䰉 ¦NW{f}ˉ㡖⚍󰡯ⶶ䞣 1iFi(r,s,t)󰡴󱴗∖󰩠ҷ󱳓⿃󰟚i{d}ˉ㡖⚍ԡ⿏ⶶ䞣{u}ˉ󰤩󰜗ԡ⿏ⶶ䞣^H` >B@^u`^V` >C@^H`^u` >N@^d`[B]ˉ󰞴ԩԡ⿏ⶽ䰉[C]ˉ󰟮󱑺ⶽ䰉[N]ˉ󱔶󰟑󱭄ⶽ䰉

25

4.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘ⡍⚍˖󰀄

󰟮󱑺ⶽ䰉[K]󱰃󰦬󰣪ⱘ

>K(d)@^d` ^f`>K(u)@ ³T

V>B(u)@>C(u)@>B(u)@dV

^H` >B(u)@^u`[K]ˉ󰟮󱑺ⶽ䰉

^V` >C(H)@^H`{f}ˉ㡖⚍󰡯ⶶ䞣

{d}ˉ㡖⚍ԡ⿏ⶶ䞣

^u` >N@^d`{u}ˉ󰤩󰜗ԡ⿏ⶶ䞣

[B]ˉ󰞴ԩԡ⿏ⶽ䰉

[C]ˉ󱔍󱗻ⶽ䰉ˈ󱔍󰸥󱗻ⶽ䰉[N]ˉ󱔶󰟑󱭄ⶽ䰉

27

4.0Ⳃ󱔩

¾󰦬󰟚󰥳⧚¾󱳔󱇣󰢓㛑󰥳⧚

¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⡍⚍¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾

󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘUL⊩29

4.4󰞴ԩ䴲㒓󱗻󰹲䞣󱔶󱓣ⱘ󱑇㸵󱮍⿟

¾

UpdateLagrangian⊩˖UL󰝀󱓣ˈ󱅮Н󰳼󰥇󱯊ԡ󱔶tϞ

t󰀄'tijGEt󰀄'tijdVt Rt󰀄't

St󰀄't~t󰀄'tV³S

t

ijtSij

WijG'eijdVt

V³'WG'e

L

kl

dVt󰀄t

V³WtG'eNLttL

ijijkldV Rt󰀄't󰀚t

V³t

󱗑⬹䴲㒓󱗻󰹲䞣催䰊乍'WNL

ijG'ekl

󰟱󱑨󰡯󰟮󱑺ⶽ䰉

ϢWt

¾

󱳝䰤󰜗󱑇㸵󱮍⿟

󰀇>KL@󰀄>KNL@󰀄>KW@󰀈^'d` ^R`ij󱳝󰝇

󰀚^F`'W Ctijijkl'eklCt Uwx'S0mnpqij Cijkl'EklUmwxnwxpwxqC0ijkl0wXiwXjwXkwXl31

4.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾󱳝䰤󰜗󰟚󱵤ϔ㠀ℹ偸˄4˅˖󰀄

󰤩󰜗㒘䲚˖󰤩󰜗

㒧󱵘󰤩󰜗󰟮󱑺ⶽ䰉[K]e

㒧󱵘󱘏ԧ󰟮󱑺ⶽ䰉[K]

󰀄

󱓩󰜹䖍⬠󱴵ӊˈḍ󱥂󱏆ⶹ㡖⚍ⱘԡ⿏󰦞㡖⚍󰡯ˈ∖㾷󱮍⿟>K@^d` ^f`⾏󱬷󰤩󰜗󱳝䰤Ͼ催󱮃⚍催󱮃⿃󰟚䰊⃵n˖1˄㒓󰤩󰜗˅˗2X2˄4㡖⚍Ѡ㓈󰤩󰜗ǃ8㡖⚍ⶽ󱔶󰤩󰜗˅ˈ3X3˄8㡖⚍󱳆䖍󰲯䖍󱔶󰤩󰜗˅˗2X2X2˄ϝ㓈8㡖⚍󰤩󰜗˅DŽ󰧘󰤩󰜗催󱮃⚍󱭄˖n1˗n2˗n3

26

4.3󰞴ԩ䴲㒓󱗻󱳝䰤󰜗⊩

¾䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘ⡍⚍˖

•

󰤩󰜗󰟚󱵤˖󰟮󱑺䰉㗗㰥󱑨󰦬—ԡ⿏ⶽ䰉ⱘ催䰊乍ˈ󰧠󱯊⿃󰟚䖛⿟㽕㗗㰥ԧ⿃ⱘ󰦬󰣪

³³³>B(u)@T

>C@>B(u)@dV >KL@󰀄>KNL@V

•㒘䲚ǃ󱭈ԧ󰟚󱵤˖󰝭󱟤󰹲䞣ⱘ󱔶󱓣ˈiℹϢi+1ℹԡ󱔶ϡ󰧠•

䞛⫼󰹲䞣⊩ˈ↣ϔ󰹲䞣ℹ䖯㸠󰳼ϡ󰧠ԡ󱔶Ϟ

󱑇㸵䗁ҷˈ∖㾷䴲㒓󱗻󱮍⿟㒘

󱓎ゟ󱳝䰤󰜗󱮍⿟

28

4.4󰞴ԩ䴲㒓󱗻󰹲䞣󱔶󱓣ⱘ󱑇㸵󱮍⿟

¾

TotalLagrangian⊩˖TL󰝀󱓣ˈ󱅮Н󰳼󰟱󰾟ԡ󱔶t=0Ϟt󰀄'tt󰀄'tdV0 Rt󰀄't󰻪󰡯㰮󰡳ijGEijV³S0St󰀄't~t󰀄'0tijSijSt󰀄'ttEt󰀄'ttij Sij󰀄'Sijij Eij󰀄'EijRt󰀄't Rt󰀄'R'S0ij Cijkl'EklELij Eij󰀄ENLGEt󰀄'tijij G'Eij'SG'LdV0󰀄EtLkldV0 Rt󰀄'󰀚V³0³StNL³StijEklijG'ijG'EijdV0V0󱗑⬹䴲㒓󱗻󰹲䞣催䰊乍'SNL󰟱󱑨󰡯󰟮󱑺ⶽ䰉V0¾

󱳝䰤󰜗󱑇㸵󱮍⿟ijG'Ekl󰀇>ϢS0ij󱳝󰝇KL@󰀄>KNL@󰀄>KS@󰀈^'d` ^R`󰀚^F`Rt󰀄't V³ft󰀄't0iGuidV󰀄0A³pt󰀄't0iGuidA3004.4󰞴ԩ䴲㒓󱗻󰹲䞣󱔶󱓣ⱘ󱑇㸵󱮍⿟

¾󰹲䞣䗁ҷ󱔶󱓣ⱘ󱳝䰤󰜗󱑇㸵󱮍⿟¾

TL󰝀󱓣󰀇t

t0

>K

L

@󰀄0>KNL@󰀄t0>KS@󰀈^'d`i t󰀄't^R`󰀚t󰀄't0^F`i󰀚1

¾

UL󰝀󱓣

󰀇t>K@󰀄t>K'tW@󰀈^'d`i t󰀄't^R`󰀚t󰀄i󰀚1tLt>KNL@󰀄ttt󰀄't^F`㡖⚍󰤩󰜗󱑨󰡯ⱘ󰻪ㄝ󱬜㡖⚍󰡯

󰡯

t

0>KL@ >BTtL@0>C@0>BL@dV

t0

0>KtNL@ ³V0

0>BNL@T0>C@t0>BNL@dV

tttTt

>³tV0KtTL@ ³Vt

0>BL@t>C@tt>BL@dV

t

>KNL@ ³Vt

0>BNL@t>C@tt>BNL@dV

32

4.5󱇣㒧

¾󰞴ԩ䴲㒓󱗻󱳝䰤󰜗󱮍⿟ⱘ󱓎ゟ䖛⿟˛¾

󰞴ԩ䴲㒓󱗻󰟚󱵤ⱘTL⊩ǃUL⊩󰤎󰟿˛

33

㸹ϔǃ󰹲䞣󰟚󱵤⊩󱗱䏃

Ѣ󱰃󱳝˖

^F`|t>K@^U`t󰠄t+󰂍t󱯊󰠏󰝙ˈ󰤩󰜗㡖⚍ԡ⿏

˄4˅

󰟚䞣󰹲䞣󰟫䰉

¾

󱇚(3)ǃ(4)󱏺󰜹(2)ˈ󱕫˖

^F`|t>K@^U` t󰀄't^R`󰀚t^F`˄5˅

¾

⬅ℸ㾷󱕫{U}ˈҢ㗠󱕫󰠄t+󰂍t󱯊󰠏ⱘ˖t󰀄't

^U` t^U`󰀄^U`䖯㗠󱕫󰠄˖

t󰀄't

^E`,

t󰀄't

^S`,

t󰀄't

^F`⬅Ѣ(4)Ў䖥Ԑ㸼䖒󱓣ˈ󱬙(5)ⱘ㾷{U}гЎ䖥Ԑ󰘐DŽ󰝊󱏺󱴹ⱘ䇃󱏂󰽖ԩ󰞣󱇥󱟪⍜䰸˛󰧺󰟭ˈ󰧘󰹲䞣ℹ䇃󱏂ⱘ㌃⿃󰧃㛑Փ㒧󱵰󱅠󰜼󰘣⾏䯂乬ⱘ㾷DŽ

35

㄀Ѩゴ󱴤󱭭䴲㒓󱗻󰟚󱵤ⱘ󱳝䰤󰜗⊩

󰀂󱴤󱭭䴲㒓󱗻󱳝䰤󰜗󱮍⿟ⱘ󱓎ゟ䖛⿟˛󰀂󱴤󱭭䴲㒓󱗻󰟚󱵤ⱘTL⊩ǃUL⊩󰤎󰟿˛

375.1󱳔󱇣󰢓㛑󰥳⧚

¾󱴤󱭭䴲㒓󱗻䯂乬󱥻󰠊󱮍⿟˖15Ͼ󱮍⿟•

󱑇㸵󱮍⿟˖23Ͼ󱮍⿟Vij,i󰀄xj Uwujwt2•

󰞴ԩ󱮍⿟˖6Ͼ󱮍⿟

H1ij 2󰀇ui,j󰀄uj,i󰀈•

󱴀󱵘󱮍⿟˖

6Ͼ󱮍⿟dVepHep

ij CijkldHkldij DijkldVkl

•

䖍⬠󱴵ӊ˖

Vijni pj󰳼*pϞui ui󰳼*uϞ39

㸹ϔǃ󰹲䞣󰟚󱵤⊩󱗱䏃

¾

󰘛䆒˖԰⫼Ѣ󱳝䰤󰜗⾏󱬷㋏㒳ⱘ䞣㹿󱦣䗄Ў“󱯊䯈”ⱘ󰟑󱭄ˈ󰟭䆹㋏㒳t^󱯊󰠏ⱘ󱑇㸵󱮍⿟Ў˖tR`󰀚t^F` ^0`˄1˅

󰤩󰜗󱑨󰡯󱓩䍋ⱘ㡖⚍󰡯󰟚䞣󰟫䰉󰻪㥋䕑㡖⚍󰡯󰟚䞣󰟫䰉¾

󰹲䞣䗤ℹ∖㾷⊩ⱘ󰷎󱴀󱗱󱛇—t󱯊󰠏ⱘ󱑇㸵󱮍⿟ⱘ㾷Ў󱏆ⶹ

󰥇

t

^R`,t^F`,t^K`,t^U`,t^E`,t^S`󱏆ⶹ

¾

󱇍Ѣt+󰂍t󱯊󰠏ⱘ󱑇㸵󱮍⿟

t󰀄't

^R`󰀚t󰀄't^F` ^0`˄2˅

󱑨󰡯󱓩䍋ⱘ㡖⚍󰡯󰟚䞣󰟫䰉󰧃󰝭Ў˖

t󰀄't

^F` t^F`󰀄^F`t󰠄t+󰂍t󱯊󰠏󰝙ˈ󰤩󰜗󱑨󰡯󰹲䞣󱓩䍋ⱘ㡖⚍󰡯󰟚䞣󰹲䞣󰟫䰉˄3˅

34

㸹Ѡǃ󰹲䞣䗁ҷ⊩—Newton—Raphson⊩󱗱䏃

¾󰳼t+󰂍t󱯊󰠏󰹲䞣ℹЁˈ⬅˖^F`|t>K@^U` t󰀄't^R`󰀚t^F`󰀄't¾

󱳝˖

t

>K@'^U`󰀇i󰀈 t^R`󰀚t󰀄't^F`󰀇i󰀚1󰀈˄6˅

t󰀄't

^U`󰀇i󰀈 t󰀄't^U`󰀇i󰀚1󰀈󰀄'^U`󰀇i󰀈i=1ˈ2ˈ……Ў䗁ҷℹDŽ䗁ҷⱘ󰟱󰘐Ўt󱯊󰠏ⱘ㾷˄󱏆ⶹ˅

󰥇

t󰀄't

^U`󰀇0󰀈 t^U`,

t󰀄't

^F`󰀇0󰀈 t^F`⬅(6)㾷󱕫'^U`󰀇i󰀈䖯㗠󱕫󰠄

t󰀄't

^U`󰀇i󰀈t󰀄't

^E`󰀇i󰀈,t󰀄't

i󰀈,

t󰀄't

^F`󰀇i󰀈󱇚

t󰀄't

^^S`󰀇F`󰀇i󰀈󱏺󰲲(6)ⱘ󰧇䖍ˈ󰽖󱵰t󰀄't䎇󰻳

󱇣ˈ󰟭䗁ҷ䖛⿟㒧󱴳ˈ󰧺󰟭ˈ⬅(6)󱓣㒻㓁∖

^R`󰀚t󰀄't^F`󰀇i󰀚1󰀈t󰀄't

^U`󰀇i󰀄1󰀈36

5.0Ⳃ󱔩

¾󱳔󱇣󰢓㛑󰥳⧚¾

󱴤󱭭䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘMNO⊩¾󱴤󱭭䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾

󱴤󱭭䴲㒓󱗻󱳝䰤󰜗󰟚󱵤ⱘULJ⊩

38

5.1󱳔󱇣󰢓㛑󰥳⧚

¾󰹲䞣󱔶󱓣ⱘ󱳔󱇣󰢓㛑󰥳⧚

•

󰳼󱠔󱳝⒵䎇󱑨󰦬ԡ⿏󱮍⿟󰩠ԡ⿏䖍⬠󱴵ӊⱘduiЁˈՓ⊯󰟑

3dp󰦪󱵕󱇣󰘐㗙ˈ󱖙Ўԡ⿏󰹲䞣ⱘℷ⹂㾷

3dp ³³³A(dHij)d:󰀚³³dpiduid*:*p¾󰹲䞣󱔶󱓣ⱘ󱳔󱇣ԭ㛑󰥳⧚

•

󰳼󱠔󱳝⒵䎇󱑇㸵󱮍⿟󰩠󱑨󰡯䖍⬠󱴵ӊⱘdVijЁˈՓ⊯󰟑3dc󰦪󱵕󱇣󰘐㗙ˈ󱖙Ў󱑨󰡯󰹲䞣ⱘℷ⹂㾷

3du ³³³B(dVij)d:󰀚³³dVijnjduid*:*u40

5.2󱴤󱭭䴲㒓󱗻󱳝䰤󰜗⊩

¾ҙ󱴤󱭭䴲㒓󱗻ˈMNO󰝀󱓣•

󱇣ԡ⿏ǃ󱇣󱑨󰦬ˈ󰸥󱗻

³epV

C

ijkl

HijG'HkldV󰀄³'VijG'H't

V

kldV Rt󰀄¾TL󰝀󱓣

•

󰻻ԡ⿏ǃ󱇣󱑨󰦬ˈ󰸥󱗻

³

V

0

SijG'EkldV󰀄³0'St

V

ijG'EkldV Rt󰀄'41㄀󰝁ゴ䴲㒓󱗻󰡼󰡯䯂乬ⱘ󱳝䰤󰜗⊩

󰀂䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󱮍⿟ⱘ󱓎ゟ䖛⿟˛󰀂䴲㒓󱗻󰡼󰡯󰟚󱵤ⱘTL⊩ǃUL⊩󰤎󰟿˛

436.1󰡼󰡯󱄺䯂乬󱥻󰠊󱮍⿟

¾󰡼󰡯󱄺䯂乬󱥻󰠊󱮍⿟˖15Ͼ󱮍⿟•

󱑇㸵󱮍⿟˖2j3Ͼ󱮍⿟Vij,i󰀄xj Uwujwt2󰀄Kwuwt•

󰞴ԩ󱮍⿟˖6Ͼ󱮍⿟

H1ij 2󰀇ui,j󰀄uj,i󰀈•

󱴀󱵘󱮍⿟˖

dVepdHep

6Ͼ󱮍⿟ij CijklkldHij DijkldVkl

•䖍⬠󱴵ӊ˖Vijni pj󰳼*pϞui ui󰳼*uϞ•

󰟱󰾟󱴵ӊ˖

ui󰀇x1,x2,x3,0󰀈 ui󰀇x1,x2,x3󰀈ui,t󰀇x1,x2,x3,0󰀈 ui,t󰀇x1,x2,x3󰀈456.3䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘUL⊩

¾

䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘUL⊩ˈ󰹲䞣󱑇㸵󱮍⿟

󰀇tt

>K

L

@󰀄tt>K'tNL@󰀈^U` t󰀄't^R`󰀚tt^F`󰀚>M@t󰀄^U󰀁󰀁`󰀚>C@t󰀄't^U

󰀁`U 'd,U

󰀁󰀁 'd󰀁󰀁,U󰀁 'd󰀁¾

䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘUL⊩ˈ󰹲䞣䗁ҷ󱑇㸵󱮍⿟

󰀇t

󰀈t>K

L

@󰀄t>KNL@󰀈'^U`󰀇it

t󰀄't

^R`󰀚t󰀄't^F`󰀇i󰀚1󰀈t󰀄't

󰀚>M@t󰀄't

^U󰀁󰀁`󰀇i󰀈󰀚>C@t󰀄't^U

󰀁`󰀇i󰀈t󰀄't

^U`󰀇i󰀈

t󰀄't

^U`󰀇i󰀚1󰀈󰀄'^U`󰀇i󰀈47

5.2󱴤󱭭䴲㒓󱗻󱳝䰤󰜗⊩

¾ULJ󰝀󱓣

•

󰻻ԡ⿏ǃ󰻻󱑨󰦬ˈ󰸥󱗻

³

V

0

SijG'EkldV󰀄³0'St󰀄't

V

ijG'EkldV R42

6.0Ⳃ󱔩

¾󰡼󰡯󱄺䯂乬󱥻󰠊󱮍⿟¾

䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘTL⊩¾䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘUL⊩¾

⾏󱬷󰣪䯂乬

44

6.2䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘTL⊩

¾

䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘTL⊩ˈ󰹲䞣󱑇㸵󱮍⿟

󰀇t0

>KL

@󰀄t0>KNL@󰀈^U` t󰀄't^R`󰀚t0^F`󰀚>M@t󰀄't^U󰀁󰀁`󰀚>C@t󰀄't^U

󰀁`U 'd,U

󰀁󰀁 'd󰀁󰀁,U󰀁 'd󰀁•

䴲㒓󱗻󰡼󰡯䯂乬󱳝䰤󰜗󰟚󱵤ⱘTL⊩ˈ󰹲䞣䗁ҷ󱑇㸵󱮍⿟

󰀇t0

>K

L

@󰀄t>Ki󰀈tNL@󰀈'^U`󰀇 t󰀄't^R`󰀚t󰀄'ti󰀚10^F`󰀇󰀈󰀚>M@t󰀄'^U󰀁󰀁`󰀇i󰀈󰀚>C@t󰀄't^U

󰀁`󰀇i󰀈0t󰀄't

^U`󰀇i󰀈 t󰀄't^U`󰀇i󰀚1󰀈󰀄'^U`󰀇i󰀈,t󰀄't0^F`󰀇0󰀈 t0^F`,t󰀄't0^U`󰀇0󰀈 t0^U`46

6.4⾏󱬷󰣪䯂乬

¾

󱯊䯈⾏󱬷󰣪

0,t1,t2󰀃,t,t󰀄't

•

ⳳℷⱘ󱯊䯈ℹ䭓¾㒧󱵘⾏󱬷󰣪

•

㒧󱵘⾏󱬷Ў󱳝䰤Ͼ󰤩󰜗ˈ㥋䕑⾏󱬷Ў

0,t1,t2󰀃,t,t󰀄't

•䕑㥋ℹˈ㰮󱢳ⱘ󱯊䯈¾䗁ҷ

•

󰳼↣ϔϾ󱯊䯈ℹ󰝙䞛⫼䗁ҷ󱮍󱓣∖㾷ˈ󰝊㒧󱵰԰ЎϟϔϾ󱯊䯈ℹ䗁ҷⱘ󰟱󰘐ˈ󰦠䞡䗁ҷDŽ

48

㄀ϗゴ󱥹㾺䯂乬䴲㒓󱗻󱳝䰤󰜗󰟚󱵤

497.1󱓩㿔

¾󱐌㾕󱥹㾺䯂乬

•

啓䕂Ӵ󰡼ǃ󰥟󰡯󱆍󰰼ǃ䕈󱡓ǃ䞥󱈲󰞆󰥟󱟤󰵟ǃḹṕ󱬃󱑻ǃ䔺䕂Ϣ䏃䴶ⱘ󱥹㾺ǃ⺄󱩲ㄝ¾󱥹㾺䯂乬ⱘ䲒⚍

•

∖㾷П󰠡󱥹㾺󰤎󰶳󱳾ⶹˈ󰦬󱔶䖛⿟Ё󱥹㾺󰤎󰶳䱣䕑㥋ǃ󱴤󱭭ǃ䖍⬠󱴵ӊ󰦬󰣪

•

󰟚󱵤󱥹㾺䯂乬㽕㗗㰥󱨽󱪺ˈ󱨽󱪺Ў䴲㒓󱗻ⱘˈ󱇐㟈䯂乬䎳󱆊󰻡󱴖󰣪

51

7.2󱔍󱗻󱥹㾺䯂乬

ϡ㗗㰥󱨽󱪺ˈ󰝊䖛⿟󰧃䗚ˈ󱳝䰤󰜗ԡ⿏⊩∖㾷

•ϸϾ󱔍󱗻󱥹㾺ԧ󱳝䰤󰜗⾏󱬷󰧢ⱘ󱑇㸵󱮍⿟

>.󰀔

@^G󰀔

` ^)󰀔

`䕑㥋󰧥䞣

󱇍󱑨Ѣ󱶤󱥹㾺⢊󱗕ˈ󰦡󱯴󱥹㾺󱴵ӊⱘ󰟮󱑺ⶽ䰉ˈ䱣󱥹㾺⢊󱗕ⱘ󰦬

󰣪㗠󰦬󰣪˗󰟱󰾟ⱘ󱥹㾺⢊󱗕󱰃ḍ󱥂㒣偠󰩠󱅲䰙󱚙󰞉󰘛󱅮ⱘDŽ

•∖㾷䖛⿟

∖󰟎^G󰀔`ˈ✊󰧢∖󱕫󱥹㾺⚍ⱘ󱥹㾺󰝙󰡯^5󰀔`ˈ󰝡󱇚^G󰀔`ǃ^5󰀔`󱏺󰜹󱥹㾺󱴵ӊ↨䕗ˈ㢹ϡ⒵䎇ˈ󰥇󰗂󱬍󱥹㾺⢊󱗕Ў˖

>.󰀕

@^G󰀕

` ^)󰀕

`󱕾⦃ℸ䖛⿟ˈ󱔧^GQ`ǃ^Rn`󱏺󰜹󱥹㾺󱴵ӊ󱕫󰠄⒵䎇󰧢ˈ󰥇Ў󱥹㾺⢊

󱗕ⱘ㾷DŽ⬅ѢϞ䗄Ў󱨽󱪺󱥹㾺䯂乬󱰃󰧃䗚䖛⿟ˈϔℹ󰡴䕑󰥇󰧃∖󱕫

󱳔㒜㾷DŽ

53

7.3󱳝󱨽󱪺󱥹㾺䯂乬

󰀗ǃ󱥹㾺󰤎󰶳ⱘ󱥹㾺⚍󱇍󱰃䖲㓁ⱘˈ⬅󱥹㾺䴶ⱘⳌ󱆍󱮍⿟

󰀇GL󰀈$ 󰀇GL󰀈%󰀄󰀇GL󰀈󰀓

i󱥹㾺⚍󱇍ⱘ󰟱󰾟䯈䱭󰧃󱕫󰠄˖

¦P>&$@3P%QLMM󰀄¦>&LMM 󰀚M 󰀔

M 󰀔

@3¦&$LNIN󰀄N 󰀇GL󰀈󰀓

󰀔

>@5ǃ󱇍Ѣ㞾⬅⢊󱗕ⱘ⚍ˈPj=0˗󱇍󱥹㾺䴶Ϟ󰻘Ѣ⒥󰡼⢊󱗕ⱘ⚍ˈ󰘛䆒

⒵䎇Columb󱨽󱪺󱅮󱕟˖3WP3QM Mℸ󰻘3ª3WºM «M¬3Q»󰟛󰧥󰟚䞣«M¼»

⊩󰧥󰟚䞣󰀙ǃ⬅Ѣ>&@ϡ󰦬ˈϞ󱓣󰳼↣⃵䗁ҷЁ󰦾䳔㽕ḍ󱥂󱥹㾺⚍ⱘ󱥹㾺⢊󱗕ҷ󰜹Pj=0󱟪Ptj=­Pnjˈ∖󰟎Pjˈ󰝡ҷ󰜹󱶨󱑺󱮍⿟∖󰟎(¥i)Aǃ(¥i)B󰲴󱥹㾺⚍󱭄Ⳃ䖰󱇣Ѣ󱥹㾺ԧ㡖⚍󱭄ˈ䅵ㅫ󱬜⥛󱕫ҹ󱦤催DŽ󰀚ǃ󱇍Ѣ󰞴ԩ䴲㒓󱗻ˈ>&@󰳼↣Ͼ󰡴䕑ℹ󰝙ϡ󰦬DŽ

¾󰝊󱅗󱥹㾺󰤩󰜗󰻘⧚⊩˖䖲󱥹󰤩󰜗⊩ǃ䯈䱭󰤩󰜗⊩

55

7.0Ⳃ󱔩

¾󱓩㿔¾

󱔍󱗻󱥹㾺䯂乬¾󱳝󱨽󱪺󱥹㾺䯂乬¾󱥹㾺⢊󱗕ⱘ󰟸󰟿

¾

󱔍󰸥󱗻󱥹㾺䯂乬ⱘ󱳝䰤󰜗󰟚󱵤

50

7.1󱓩㿔

¾󱥹㾺󰟚㉏

•

󰟮ԧ—󱶨ԧ󱥹㾺˖󱥹㾺䴶ЁⱘϔϾ󱟪󰻮Ͼ㹿󱔧󰘮󰟮ԧˈϔ㠀ԧ⦄󰳼䕃󱴤󱭭Ϣ⹀󱴤󱭭󱥹㾺Ёˈ󰽖䞥󱈲󰞆󰥟󱟤󰵟DŽ•󱶨ԧ—󱶨ԧ󱥹㾺˖ϸϾ󰦬󱔶ԧ󰟮󱑺Ⳍ󱏂ϡ󰻻DŽ¾󱥹㾺󱮍󱓣•⚍—⚍󱥹㾺˗•⚍—⚍󱥹㾺˗•⚍—⚍󱥹㾺DŽ¾

∖㾷󱮍⊩

•

󱭄󱄺󱔦㒇⊩ǃ䖍⬠󰜗⊩ǃ󱳝䰤󰜗⊩

52

7.3󱳝󱨽󱪺󱥹㾺䯂乬

¾

㢹󱥹㾺䴶󱄬󰳼⒥󰡼󱨽󱪺ˈ󱥹㾺䯂乬󱰃ϡ󰧃䗚ⱘˈ䳔㽕⫼󰹲䞣󰡴䕑ⱘ󱮍⊩󱴹䅵ㅫDŽ

•

䞛⫼Ϟ䗄󱮍⊩󱯊ˈ󰟮󱑺ⶽ䰉󰳼↣⃵䗁ҷ󱯊㽕䞡󱮄䅵ㅫˈ䖭ḋ󰢓󱖙󱏺󱴹󱵕󰻻ⱘ䞡󰻡䅵ㅫ󱎹԰䞣ˈ󰲴ℸˈ󰧃ҹ󰦾󱬍󰦬󰝊ЁϢ󱥹㾺󰤎󰶳󱳝󰝇ⱘ䚼󰟚DŽ

¾

FrancavillaϢZienkiewicz󱦤󰟎ϔ⾡ㅔ󰖓󱮍⊩˖󱶨󱑺⊩

󰀔ǃ󱇍Ѣ󱥹㾺ԧ$ǃ%ˈ󰻪󰡯԰⫼󰳼$Ϟ󰀕ǃ$ǃ%󱥹㾺󰡯󰟚󰟿Ў

3M󰩠3Mcˈ󰟭3M 󰀐3Mc

󰀖ǃ%󱥹㾺⚍ⱘ󱶨󱑺󱮍⿟󰟚󰟿Ў˖

󰀇G󰀈P>&%L% ¦LM@3Mc

J=1~mˈЎ󱥹㾺⚍󱭄

M 󰀔Ⳃˈ(¥󰀇i)Aǃ(¥i)GPAǃ%󰳼󱥹㾺⚍BЎ⠽

QL󰀈$ ¦$

M 󰀔

>&LM@3¦N 󰀔>&$LⱘԡM󰀄LN@IԧN⿏ˈ>&@Ў󱶨󱑺ⶽ䰉DŽ7.4󱥹㾺⢊󱗕ⱘ󰟸󰟿

󱥹㾺䯂乬Ёˈ󱳔󱘏㽕ⱘ󱰃󱥹㾺󰤎󰶳ⱘ󱥹㾺󱴵ӊˈ󱣝✻󱥹㾺󰤎󰶳ԡ⿏ǃ󰡯П䯈ⱘ󰝇㋏ˈ󱥹㾺⢊󱗕󰣙󱣀ϝ㉏˖䖲㓁䖍⬠ǃ⒥󰡼䖍⬠ǃ㞾⬅䖍⬠DŽ

󰳒Ё8MLǃ5ML󰟚󰟿㸼⼎㄀MϾ󱥹㾺ԧ˄M $󰀏%˅󰳼㄀L󰧥ⱘԡ⿏ǃ󱥹㾺󰡯󰧥䞣DŽ⬅ℸ˖

5$L 󰀚5%L󰀋L [󰀏\\󰀏]󰀌

•䖲㓁䖍⬠󱴵ӊ˖

8$= 8%=󰀄G󰀓=󰀋⊩󰧥ԡ⿏Ⳍ󱆍󱮍⿟󰀌

8$L 8%L󰀄G󰀓=󰀋L [󰀏\\󰀏󰟛󰧥ԡ⿏Ⳍ󱆍󱮍⿟󰀌

㽕∖󰟛󰧥󱑇䴶ϡ⒥󰡼

5󰀓󰀏Ϩ5󰀕󰀕%=d%;󰀄5%7.4󱥹㾺⢊󱗕ⱘ󰟸󰟿

•

㞾⬅䖍⬠󱴵ӊ˖

5$L 5%L 󰀓󰀋L [󰀏\\󰀏]󰀌

8$=!8%=󰀄G󰀓=•

⒥󰡼䖍⬠󱴵ӊ˖

8$= 8%=󰀄G󰀓=󰀋⊩󰧥ԡ⿏Ⳍ󱆍󱮍⿟󰀌5$L 󰀚5%L󰀋L [󰀏\\󰀏]󰀌

󰝊Ё˖5%; P5%=FRVT󰀏5%< P5%=VLQTFRVT 5%;VLQ󰀏FRVT 5%<5󰀕󰀕󰀕%;󰀄5%<5%;󰀄5󰀕%<57

7.5󱔍󰸥󱗻󱥹㾺䯂乬ⱘ󱳝䰤󰜗󰟚󱵤

噝佪󰜜Ϣ󱔍󰸥󱗻󱳝䰤󰜗󰟚󱵤ϔḋ󰴛󱈲Ѣ䴲㒓󱗻󰟚󱵤噝󰟮󱑺ⶽ䰉Ўԡ⿏ⱘ󰟑󱭄ˈϡ󰝡󰖱󱣕Ў󱐌󰘐˗噝

󰳼󱔍󰸥󱗻󰹲䞣䗁ҷЁˈ↣ϔϾ󰡴䕑ℹ󰝙䖯㸠󱥹㾺䗁ҷDŽ

59

7.4󱥹㾺⢊󱗕ⱘ󰟸󰟿

¾

󱥹㾺䯂乬∖㾷ⱘ󰷎󱴀⌕⿟

噝󰀔ǃ㒭󱅮󱥹㾺䖍⬠ǃ󰻪㥋䕑˗噝

󰀕ǃ⬅󱥹㾺󱴵ӊ󰟸󱮁󱥹㾺⢊󱗕˗

噝

󰀖ǃ󰲴󱠔⫼ⱘԡ⿏ǃ󱥹㾺󰡯󱰃󱳾ⶹⱘˈ󱬙󰳼󱥹㾺∖㾷ⱘ㄀ϔℹ乏乘󰜜󰘛󱅮󱥹㾺⢊󱗕ˈ∖󰟎󱥹㾺䴶Ϟⱘԡ⿏ǃ󰦡󰡯㒧󱵰˗

噝

󰀗ǃҹ㾷󱕫ⱘ㒧󱵰󰟸󱮁󰩠󰗂ℷ󰥳󱴹ⱘ󱥹㾺⢊󱗕ˈϡ󱮁䖯㸠󱕾⦃ˈⳈ󰠄󱳔󰧢ⱘ󱥹㾺⢊󱗕〇󱅮ЎℶDŽ󰥇ℸ󱈔䚼󰞴ԩ䴲㒓󱗻ⱘ䗁ҷ䖛⿟󱕫ҹ󱅠󱟤DŽ

58

㒧󱵘䴲㒓󱗻󰟚󱵤󱘏㒧

󱴀䇒⿟󰣙󰧿󰞴ԩ䴲㒓󱗻ǃ󱴤󱭭䴲㒓󱗻ǃ䴲㒓󱗻󰡼󰡯䯂乬ǃ󱥹㾺䯂乬ㄝ󰝙󱆍ˈ󱑨Ң󰧘㉏䯂乬⡍󱕕ǃ󱦣䗄󱮍⊩ǃ䯂乬ⱘ∖㾷󱮍䴶󰡴ҹњ㾷DŽ

󰀔ǃ䴲㒓󱗻䯂乬ⱘ㉏󰵟ǃ⡍⚍ǃ󱦣䗄󱮍⊩ǃ󰡯󱄺䞣˗󰀕ǃ㒧󱵘䴲㒓󱗻󰟚󱵤ⱘ⡍⚍˗

󰀖ǃ󱴤󱭭󰸥󱗻󰦬󱔶ⱘ㉏󰵟ǃ󰸥󱗻⧚䆎ԧ㋏˗󰀗ǃ󰞴ԩ䴲㒓󱗻󱑨󰦬ǃ󱑨󰡯ⱘ󱅮Н˗

󰀘ǃ㒧󱵘䴲㒓󱗻󰟚󱵤ⱘ䖛⿟˄󱎹󰝋ǃ󱓎῵ǃ㒧󱵰󰟚󱵤ǃ󱇍

㒧󱵘䆒䅵ⱘ󰦖㗗Ӌ󰘐DŽ

60