Interactive Simulation of Elasto-Plastic Materials using the Finite Element Method

Transcription

1 Otline Interctie Simltion of Elsto-Plstic Mterils sing the Finite Element Method Moie Mtthis Müller Seminr Wintersemester FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion Mss-Spring s. FEM Pros nd Cons of FEM. Discretition of n object into mss points. Representtion of forces beteen mss points ith springs. Compttion of the dnmics. Discretition of n object into s tetrhedr. Discretition of continos energ eqtions into lgebric eqtions for forces cting t ertices. Compttion of the dnmics Pros: o indiidl spring constnts needed onl knon mteril prmeters Eν o inersion problems inerted tetrhedr prodce forces Stress nd strin tensors llo frctre nd plsticit simltions deformble mss-spring sstem deformble FEM sstem Cons: Pre-compte stiffness mtri Store stiffness mtri per edge Store originl nd ctl positions of ertices

2 Otline One-dimensionl Spring FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion f f k 5 hree-dimensionl Object 6 hree-dimensionl Object Finite Element Mesh 9 tetrhedr 9 ertices 9 79 dof.... n n n f f f f... fn f fel K Mti K Rnn fn fn fel F Fnction F : Rn Rn 7 8

3 Sttic Deformtion Dnmic Deformtion fet fel K K- fet M C K f et Copled sstem of n liner ODEs Eplicit integrtion: o soler needed Implicit integrtion: Liner soler per time step Sole liner sstem Conjgte Grdients fet fel F F-fet M C F f et f Copled sstem of n non-liner ODEs Eplicit integrtion: o soler needed Implicit integrtion: Linerie t eer time step: K dfd Sole non-liner sstem eton-rphson - generlied eton-method 9 Otline Continos Elsticit -d FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion fn l stress [m] orml-spnnng l fn E ll strin [] Dehnng Elsticit Yong s Modls [m] Metl: ~ m Soft mteril: ~6 m

4 Continos Elsticit -d Deformtion: Continos -d ector field : R R Defined ithin ndeformed object Liner -d Strin norml strin in - direction: d d d d d- d d sher strin: d d d d- d d 5 Liner -d Strin Liner strin tensor: Smmetric mtri 6 ector: 6 on-liner -d Strin Green-Sint-Vennt strin tensor: rnsformtion e se: : Use cobin of trnsformtion: I Green Interprettion:

5 -d Stress Constittie Reltion isotropic Stress is force per oriented re: df df d d df n d n n f n he stress tensor is smmetric mtri 6 ector: Hooke s l: ν c E ν c E E c G ν ν ν ν c G G G Onl to sclr prmeters: E: Yong s modls ν: Poisson rtio 7 8 Ptting it ll together Energ Formltion Gien e cn compte strin nd stress E t eer point ithin the object. Energ U is sclr U t point is gien b displcement force : U elstic E Find sch tht corresponding stresses re in blnce ith eternl forces f eerhere ithin object: f Strong formltion f Copled sstem of prtil differentil eqtions! f he totl Energ of the deformed bod: U bod bod E f dv Gien f E e cn compte U bod for n Find sch tht U bod is minimm δu 9 5

6 6 Otline FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion Finite Element Formltion So fr e looked for continos field o e look for.. n t fied loctions:.. n nd interpolte ith fied bsis fnctions: Σ i i φ i i i φ i Liner Displcement etrhedron nknons.. eqtions i i i re ribles i i i re gien nmbers Displcements he displcement fnction cn be epressed s mtri of bsis fnctions H times ector of displcements: H à

7 Strin Stress nd Energ Liner displcements ield constnt strin: Stress s fnction of the displcements: H à B à Mtri B R 6 is constnt independent of B depends on the originl geometr of the tetrhedron onl E EB à Energ s fnction of the displcements: U E dv à B EB dv à à B EB dv à à [ V B EB ] à à K à 5 6 Mtri ssembl of s U bod à K à Forces re the derities of the energ ith respect to the degrees of freedom: U bod à K à f Single : f f f f f f K e he mtri K R is the stiffness mtri of the! K B EB dv V B EB Entire bod: f.. f n.. n 7 8 7

8 Implementtion Otline f.. f n K.. n K is sprse block t ij describes ho j inflences f i eer erte stores djcenc list of -mtri erte-reference pirs FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion 9 Plstic Strin Plstic Updte Rles n is nder strin û de to displcements û: û [ ] B û Initilition plstic : Updte rle eer time step: plstic stores strin in stte rible: plstic he elstic strin tht cses internl forces is no: Compte û B û from ctl displcements Compte elstic û - plstic if elstic > ield then plstic plstic creep elstic if plstic > m then plstic plstic m plstic elstic û - plstic û o internl forces re present hen û plstic Might be for û! plstic elstic before fter 6 dimensionl 8

9 Implementtion Otline Since B û the displcements tht correpsond to the plstic strin re: û plstic B - plstic nd the correpsonding forces re: f plstic KB - plstic [B EB] B - plstic B E plstic dd plstic forces to eternl forces FEM s. Mss-Spring he Mtri StticDnmic Deformtion Continm Mechnics nd FEM Strin nd Stress ensors Continos PDE s FEM Discretition Plsticit Plstic Strin Updte Rles Frctre Principl Stresses Crck Compttion Frctre criterion Principl Stresses Brek if internl elstic force eeeds threshold stress is tensor the force.r.t. norml n is: he stress tensor is smmetric there is rottion mtri R sch tht df n d n p R p R p Find n m sch tht dfd is miml! n m is direction of miml tensile stress the digonl entries re the eigenles of the colmns of R re the corresponing eigenectors there is ls rotted coordinte sstem here is digonl! 5 6 9

10 Principl Stresses Crck compttion p for ll s: compte miml tensile stress p m p p the p re the principl etreml stresses find miml eigenle of corresponding eigenector is the direction of miml stress 7 8 Crck compttion Crck compttion for ll s: compte miml tensile stress p m if p m eceeds ield stress for ll s: compte miml tensile stress p m if p m eceeds ield stress select erte crck tip rndom p m p m 9

11 Crck compttion Crck compttion for ll s: compte miml tensile stress p m if p m eceeds ield stress select erte crck tip rndom set plne α norml p m to throgh p m α for ll s: compte miml tensile stress p m if p m eceeds ield stress select erte crck tip rndom set plne α norml p m to throgh α mrk tetrs.r.t. α p m - - Crck compttion he End for ll s: compte miml tensile stress p m if p m eceeds ield stress select erte crck tip rndom set plne α norml p m to throgh α mrk tetrs.r.t. α split erte p m hnk o for or ttention! - -