Static/Dynamic Deformation with Finite Element Method. Graphics & Media Lab Seoul National University

Transcription

1 Statc/Dynamc Dormaton wth Fnt Elmnt Mthod Graphcs & Mda Lab Sol Natonal Unvrsty

2 Statc/Dynamc Dormaton Statc dormaton Dynamc dormaton ndormd shap ntrnal + = nrta = trnal dormd shap statc qlbrm dynamc qlbrm

3 How to Smlat Statc Dormaton?

4 Govrnng Eqaton W s th dsplacmnt ld Contnos D vctor ncton a : R R a At vry matral pont, nd sch that lastc potntal nrgy U t trnal orc Lagrang Eqaton

5 Ovrvw o Statc Dormaton U t. Drv th lastc potntal nrgy n trms o stran and strss. Solv th partal drntal qaton wth nt lmnt dscrtzaton

6 For Now, Us Cachy s n. Stran nsor Cachy Lt s dn a colmn vctor... ε Cachy Cachy ε

7 Hook s Law n a Matr Form z yz y zz yy z yz y zz yy z z yz yz y y G G G yy zz zz zz yy yy zz yy E E E

8 Hook s Law n a Matr Form ε E σ z yz y zz yy z yz y zz yy G G G E E E E E E E E E Not that th stran nrgy dnsty U can now b dnd as ε σ U yy zz zz zz yy yy zz yy E E E z z yz yz y y G G G

9 Now, th Elastc Potntal Enrgy U V U dv dv ε σ ε E ε V V dv yy zz y yz z yy zz y yz z Now, th govrnng qaton: U t ε V Eε dv t

10 Ovrvw o Statc Dormaton U t. Drv th lastc potntal nrgy n trms o stran and strss. Solv th partal drntal qatons wth nt lmnt dscrtzaton

11 Ovrvw o Statc Dormaton U t. Drv th lastc potntal nrgy n trms o stran and strss. Solv th partal drntal qatons wth nt lmnt dscrtzaton

12 Fnt Elmnt Dscrtzaton h abov govrnng qaton s or th contnm. t holds or ach matral pont. For smlaton, w nd to prorm nt lmnt dscrtzaton. W s lnar tntrahdral lmnts ndormd dormd t V Eε dv ε n n n n

13 Eprssng n trms o Nodal Ponts Eprss an arbtrary pont n trms o,,, = b + b + b + b, wth b + b + b + b = hy ar calld th barycntrc coordnats b b b b b X sx X sx A HW: Drv ths X s ndormd dormd

14 Eprssng n trms o b b b b vctor o nodal dsplacmnts b b b b H matr H th lnar shap ncton H b b b b b b b b b b b b ndormd dormd

15 HW: Calclat th matr B dpnds on th orgnal ttrahdron only! Now, th Stran nsor 6 B R H B H ε z yz y zz yy

16 Strss and Potntal Enrgy Strss as a ncton o th dsplacmnts σ Eε EB ε B Enrgy as a ncton o th dsplacmnts U V ε σ dv V B EB dv V B EB V B EB dv

17 Stnss Matr and Elastc Forc Potntal nrgy U V B EB K stnss matr Forc d to lastc potntal nrgy U K K HW: Drv ths Wth ttrahdral lmnts and Cachy stran mtrc h stnss matr dpnds only on th rst stat msh h lastc orc s lnar wth th nodal dsplacmnt Do th abov hold whn Grn s sd or th stran mtrc? HW: stdy ths and rport n th nt class

18 Statc Eqlbrm ntrnal orc rsltng rom lastc potntal nrgy o th lmnt shold balanc wth smmaton o all th trnal orcs actng on that lmnt K trnal orcs Gravtatonal orc actng on an lmnt Dormd or ndormd, an lmnt s ndr gravty. So RHS o th qlbrm qaton shold ncld th trms contrbtd rom gravty. How mch dos a dormd lmnts gt th gravtatonal potntal?

19 Gravtatonal Forc Frst, th gravtatonal potntal nrgy o a dormd lmnt Now, th gravtatonal orc dv G V g g H dv G V g H 4 V V dv HW: Drv ths HW: Drv ths g H g dv dv V V

20 Wth Cachy Stran nsor For an lmnt U t U K

21 Assmblng Elmnt Stnss Matrcs For an lmnt U t U K ndormd dormd

22 Assmblng Elmnt Stnss Matrcs For a sngl lmnt... For th ntr body... K K n n K K n n K j K j

23 Assmblng Elmnt Stnss Matrcs For a sngl lmnt... For th ntr body... K K n n K K n n K j K j Chck Ponts What dos K j mans? How many non-zro blocks ar thr?

24 How to Smlat Dynamc Dormaton?

25 Dynamc Dormaton Govrnng qatons: Elr-Lagrang qatons kntc nrgy lastc potntal nrgy d dt U t trnal orc, t, t ndormd dormd

26 Kntc Enrgy h Kntc Enrgy V dv t dt d t dt d,,, t, t ndormd dormd M V dv H H th mass matr HW: Drv ths

27 Forc d to Kntc Enrgy h kntc nrgy Forc d to kntc nrgy dt d dt d M M V dv t dt d t dt d,, M V dv H H th mass matr HW: Drv ths M V HW: Drv ths

28 What s th nttv Manng? Govrnng qatons: Elr-Lagrang qatons t U dt d K M

29 Dampng Matr Govrnng qatons wth dampng trms d dt U C t M C K dampng matr Adopt proportonal Raylgh dampng: Or, ncorporat vsco-lastc ormlaton C M K

30 Assmblng Eqs or All th Elmnts Sngl lmnt Entr body K C M K C M n n C K M,, n n K C M,, K C M,,

31 Solvng th Lnar Systm M, C, K matrcs ar tm ndpndnt! Convrt t to a lnar systm Eplct ntgraton mplct ntgraton K C M h h K Cv M v v v K C M v v h dt d h h h h h Kv K Cv K C M v v v HW: Drv ths

32 Statc/Dynamc Dormaton wth Fnt Elmnt Mthod HE END