Statc/Dynamc Dormaton wth Fnt Elmnt Mthod Graphcs & Mda Lab Sol Natonal Unvrsty
Statc/Dynamc Dormaton Statc dormaton Dynamc dormaton ndormd shap ntrnal + = nrta = trnal dormd shap statc qlbrm dynamc qlbrm
How to Smlat Statc Dormaton?
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
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
For Now, Us Cachy s n. Stran nsor Cachy Lt s dn a colmn vctor... ε Cachy Cachy ε
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
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
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
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
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
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
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
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
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
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
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
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?
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
Wth Cachy Stran nsor For an lmnt U t U K
Assmblng Elmnt Stnss Matrcs For an lmnt U t U K ndormd dormd
Assmblng Elmnt Stnss Matrcs For a sngl lmnt... For th ntr body... K K n n K 9 7 4 9 7 4 K n n K j K j
Assmblng Elmnt Stnss Matrcs For a sngl lmnt... For th ntr body... K K n n K 9 7 4 9 7 4 K n n K j K j Chck Ponts What dos K j mans? How many non-zro blocks ar thr?
How to Smlat Dynamc Dormaton?
Dynamc Dormaton Govrnng qatons: Elr-Lagrang qatons kntc nrgy lastc potntal nrgy d dt U t trnal orc, t, t ndormd dormd
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
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
What s th nttv Manng? Govrnng qatons: Elr-Lagrang qatons t U dt d K M
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
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,, 9 7 4 9 7 4 K C M,,
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
Statc/Dynamc Dormaton wth Fnt Elmnt Mthod HE END