The Development of XFEM Fracture and Mesh-free Adaptivity

Transcription

1 5. Anwndrorum, Ulm 6 Nu Mthodn Th Dlopmnt o XFEM Fractur and Msh-r Adaptiity C. T. Wu, Yong Guo and Hong Shng Lu LVERMORE SOFTWARE TECHNOLOGYCORPORATON (LSTC), Lirmor, USA 6 Copyright by DYNAmor GmbH G - -

2 Nu Mthodn 5. Anwndrorum, Ulm 6 LVERMORE SOFTWARE TECHNOLOGY CORPORATON Th Dlopmnt o XFEM Fractur and Msh-r Adaptiity C. T. Wu, Yong Guo and Hong Shng Lu LVERMORE SOFTWARE TECHNOLOGYCORPORATON (LSTC) ctwu@lstc.com, yguo@lstc.com, hslu@lstc.com Forum 6 Octobr, 6 Ulm Problm Looking or Solution Multi-Physics : shar band + history dpndnt larg dormation + ailur Out o Lagrangian dscription Numrical : multi-rsolution + aoid msh tangl + ailur mchanics Spctral lmnt mthod Th ariationl multiscal mthod Partition o unity mthod ALE Eulrian Msh-r Adaptiity Damag mchanics Cohsi modl Discrt lmnt mthod ntrac Elmnt, XFEM G Copyright by DYNAmor GmbH

3 5. Anwndrorum, Ulm 6 Nu Mthodn Adapti Msh-r Mthod Motiation mpro th larg dormation analysis that is byond th Lagrangian dscription. Currnt Practic Soling th orging and xtrusion problms. Formulation Adapti Msh-r Mthod EFG Fast Transormation Mthod + Adapti Lagrnagian Particls with Eulrian Krnl + Consistnt Msh-r ntrpolation or Stat Variabl Transr 3. EFG Fast Transormation Mthod Momntum quation x + b ρ m ( ) x xs sv s Continuity quation ρ ρ x r ( x,t) δ ρ d δ : ρ ρ s s h u ( x ) x [n] x ( x ) d ( x ) ˆ s + ( x δ bd ) x x + δ d ( x ) x Dirct ariabls computation Particls Strss points Acclration Vlocity Position Dnsity Strain rat Strsss ntrnal nrgy Particl Strss point 4 6 Copyright by DYNAmor GmbH G - - 3

4 Nu Mthodn 5. Anwndrorum, Ulm 6. Adapti Lagrangian Particls with Eulrian Krnl Concti locity C du to Eulrian krnl + ( X, t t (, tn t + C t χ n+ ) + ) χ X + ( C ) ( matrial tim ram) ( rrnc tim ram) Momntum quation i i ij + & j + bi t j j Continuity quation ρ t χ ρ j + C j ρ j j ( t ) n (, t ) x n + Eulrian Krnl ( t ) X, n+ Lagrangian Krnl Particl Strss point 5 Oprator Split A B -. Lagrnagian phas : ( x,tn ). Transport phas : χ + ( C ) t n + ( x, t ) + θδt( ) + - ~ - θ + ΔtC x x x x ; n ( x ) ( x ) ( x x ) + x + + ~ ( x ) : TB strss rcory schm consrati consistnt is ordr accurat; Currntly m Rmappd alu ~ ( x n ) t ~, ; Nodal alu : Ponthot and Blytschko G Copyright by DYNAmor GmbH

5 5. Anwndrorum, Ulm 6 Nu Mthodn Currnt EFG Mthodology or Larg Dormation Analysis Rubbr matrials : Lagrangian krnl ( ls97, ls97) Foam matrials : Smi-Lagrangian and Eulrian krnl (ls97) Mtal matrials : Adaptiity + Eulrian krnl ( ls97) Fluid matrials and E.Q.S. matrials : Eulrian krnl (trial rsion) ( ls97) EFG EFG + Smi-Lagrangian Krnl 7 3. Msh-r ntrpolation or Data Transr Currnt ariabl updat : n+ n+ ~ s Aα s α A (, t ) x n + ( x ) A n+ αs n αβ A β + (, t ) x n + S S + (, t ) x n+ Particl Old Particl Strss point Old Strss point Strss rcory schm is consrati, consistnt and monotonic! Nw Particl Nw Strss point 8 6 Copyright by DYNAmor GmbH G - - 5

6 Nu Mthodn 5. Anwndrorum, Ulm 6 ntrpolation o th Stat Variabl in -D d du ( ) + u x (,) dx dx u( ) u() 5sinh 4 x + sinh 4 x u x sinh 4 sinh 4 x Log (Error norm) Slop.68 EFG solution ntrpolation solution -4 Analytical Solution EFG Solution ntrpolation Solution Log (Nodal spac) Conrgnc in driati o solution 9 Whl Forging Simulation V EFG EFG Adaptiity 587 nods 366 nods Volum chang (.7%) Forc G Copyright by DYNAmor GmbH

7 5. Anwndrorum, Ulm 6 Nu Mthodn Whl Forging Simulation (Explicit) Moi Extrusion Simulation 4 nods 769 nods EFG Adapti EFG Volum chang (.6%) ntrnal nrgy 6 Copyright by DYNAmor GmbH G - - 7

8 Nu Mthodn 5. Anwndrorum, Ulm 6 Extrusion Simulation Moi 3 Th Upstting Procss with Thrmal nitial Tmpratur C Low carbon stl Ck5 No hat transr to th nironmnt ntrnal Enrgy Forc 4 G Copyright by DYNAmor GmbH

9 5. Anwndrorum, Ulm 6 Nu Mthodn Th Upstting Procss (mplicit with Thrmal) Plastic Strain 5 XFEM Fractur Motiation Modling arbitrary dynamic cracks. Currnt Practic Soling D plain-strain problms. Formulation XFEM Fractur Extndd FEM (Partition o Unity) with Ll St Thory + Phantom Nods Approach + nitially-rigid Cohsi Law - Td Blytschko t al. (Northwstrn Unirsity), Pablo D. Zaattiri (GM R&D) 6 6 Copyright by DYNAmor GmbH G - - 9

10 Nu Mthodn 5. Anwndrorum, Ulm 6 Modling Matrial Failur and Dlamination/Dbonding. Wak Discontinuitis : discontinuous dormation gradints Continuum damag constituti quation + Nonlocal strain smoothing + Matrial rosion mplicit Cracks: Crack is an assumd width Polynomial basis is inadquat to rprsnt th in scal. Tim stp tnts to b ry small in xplicit analysis with in msh. Strong Discontinuitis : discontinuous displacmnt Cohsi modl + (ntrac lmnt, or lmntal nrichmnt EFEM, or nodal nrichmnt XFEM, or EFG) Explicit Cracks : rmo th inlunc o msh siz and orintation No dirct corrlation btwn th strain sotning and critical nrgy rlas rat. Tim stp: k Δt ; ωmax ω ρh max 3. Wak + Strong Discontinuitis loss o uniqunss as a critrion or changing rom continuum damag mchanics to cohsi law 7 Litraturs in Strong Discontinuitis Ti-brak intrac (orc/strss-basd ailur + spring lmnt, rigid rods, or othr constraints) Cohsi ntrac Elmnt (Cohsi Zon modl + ntrac lmnt, or contact orcs) Trgaard, V. and Hutchinson,. W. (993) Ndlmam, A. (997) Ortoz, M. and Pandoli, A. (999) Borg, R., Nilsson, L. and Simonsson, K. () Espinos, H. D. and Zaattiri, P. D. (3) XFEM (Cohsi Zon modl + ll sts + xtndd init lmnt) Blystchko, T., Mos, S., Usui, S. and Parimi, C. () Sukumar, Huang, Z., Prost,. H. and Suo, Z. (4) Httich, T. and Ramm, E. (6) EFG (Cohsi Zon modl + Moing last-squar + EFG isibility) Rabczul, T. and Blytschko, T. (4) Simonsn, B. C. and Li S. (4) Brighnti, R (5) Othrs (Virtual Crack Closur tchniqu ) 8 G Copyright by DYNAmor GmbH

11 5. Anwndrorum, Ulm 6 Nu Mthodn. Extndd Finit Elmnt with Ll St Signd distanc unction α ( x) min x Γ α α ( x) α N ( x) Discontinuity x x sign( n ( x x) ) X Γ α i ( X ) and g( X, t) > α Approximation o crack in lmnt [ u ( t ) q ( t ) H ( ( )] h u ( X, t ) N ( X ) + α X ) n on-dimnsion u u H ( N + un + qnh + qn ) 9. Phantom Nods Approach u ( u + q ) N H + u N ( H ) + ( u u u lmnt u u q ; lmnt q ) N u u + q u u ( H ) + u N H u u N ( H ( X a)) + u N ( H ( X a)) + u N H ( X a) + u N H ( X a) Standard XFEM Phantom Nods - Td Blytschko t al. (6) 6 Copyright by DYNAmor GmbH G - -

12 Nu Mthodn 5. Anwndrorum, Ulm 6 n Two-dimnsion kin kin int xt coh Ω Ω Ω int ( ) T B σh (( ) Γ, t xt + coh T ρ N NH (( ) T ρ N bh (( ) T c N τ n dγ ( X )) dω u ( X )) dω ( X )) dω, t + Γ, t T N th (( ) ( X )) dγ, t 3. nitially-rigid Cohsi Zon Modl T max T T() δ nitially-lastic nitially-rigid δ G max C T ( δ ) dδ δ max δ : displacmnt jump Th potntial crack propagation plan is idalizd as a cohsi zon and is assumd to support a traction ild T. Th mchanical rspons o th cohsi intrac is dscribd through a constituti law rlating th traction ild T with a sparation paramtr. φ T ( δ n δ, q), t G Copyright by DYNAmor GmbH

13 5. Anwndrorum, Ulm 6 Nu Mthodn Mod- Failur nitially-lastic Cohsi lmnt nitially-rigid XFEM Siz: 5. X. Displacmnt-control (. in.3 sc) Trgaard-Law cohsi law Tmax 5. α. δ n. δ t. λcr./ 3. ρ. 6; E 67.; ν.3 3 Failur in Particl Rinorcd Rubbr ntrac C mprction Matrial B Matrial ailur Matrial A Dlamination/Dbonding. 4 6 Copyright by DYNAmor GmbH G - - 3

14 Nu Mthodn 5. Anwndrorum, Ulm 6 Ongoing Works nclud local rinmnt and rror stimator/indicator or solid adaptiity. Study th stability or XFEM Considr crack closur (contact) and mass scaling in XFEM. Extnd XFEM into thr dimnsional cas. mplmnt MPP rsion. 5 G Copyright by DYNAmor GmbH