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University of LiègeAerospace & Mechanical Engineering
Non-linear mechanical solvers for GMSH
UCL – March 2012
Authors: J. Babu, G. Becker, V.-D. Nguyen, L. Noels, L. Wu
Aerospace & Mechanical engineering
Introduction
• Different projects going on– Fracture of composites
• ERA-NET• CENAERO, e-Xstream, IMDEA, Tudor
– Mean-field homogenization with damage• ERA-NET• CENAERO, e-Xstream, IMDEA, Tudor
– Fracture or MEMS• UCL
– Fracture of thin structures• MS3, GDTech
– Computational multiscale• ARC
• Different methods– Need of common computational tools– Need of maximum flexibility
2012 Non-linear mechanical solvers for GMSH 2
Flowchart
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
QuadratureRule
Geotools
Tools for linear finite element analysis
2012 Non-linear mechanical solvers for GMSH 3
Flowchart
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
IPVariable
Geotools
NLTermsPartDomain
Structure for non-linear finite element analyzes•Pure virtual classes
•Definition of classical material laws
•Time integration
•Parallel implementation
•…
2012 Non-linear mechanical solvers for GMSH 4
Flowchart
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
2012 Non-linear mechanical solvers for GMSH 5
Flowchart
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
dgshell2012 Non-linear mechanical solvers for GMSH 6
Flowchart
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
NonLocalDG HODG FE2dgshell2012 Non-linear mechanical solvers for GMSH 7
Material Law
NonLinearMechSolver
MaterialLawcreateIPState()=0
IPVariableget(int )
mlawJ2Linearconstitutive()
mlawNonLocalDamageconstitutive()
mlawVUMatconstitutive()
ipJ2LinearSTensor3 Fp, εp
ipNonLocalDamagedouble *
ipVUMatdouble *
mlawTwoLawsbulkLaw *cohesiveLaw *
ipTwoLawsipBulk *ipCohesive *
Structure for non-linear material laws•Defines constitutive model
•Interface with Abaqus, MFH (e-Xstream)
•Allows defining full coupled problems
•Allows considering fracture
2012 Non-linear mechanical solvers for GMSH 8
Interface for dG3D
NonLinearMechSolver
MaterialLawcreateIPState()=0
IPVariableget(int )
Applications
mlawJ2Linearconstitutive()
mlawNonLocalDamageconstitutive()
mlawVUMatconstitutive()
dG3DMaterialLawstress(IP0, IP1)=0
ipJ2LinearSTensor3 Fp, εp
ipNonLocalDamagedouble *
ipVUMatdouble *
dG3DJ2LinearMaterialLawmlawJ2Linear *mlaw
mlawTwoLawsbulkLaw *cohesiveLaw *
ipTwoLawsipBulk *ipCohesive *
dG3DIPVariableSTensor3 F0, F1, PSVector3 jump, N-
dG3DJ2LinearIPVariableipJ2Linear _ipJ2
2012 Non-linear mechanical solvers for GMSH 9
Interface for dG3D
• Discontinuous Galerkin formulation– Finite-element discretization– Same discontinuous polynomial approximations for the
• Test functions ϕh and • Trial functions δϕ
– Definition of operators on the interface trace:• Jump operator:
• Mean operator:
– Continuity is weakly enforced, such that the method• Is consistent• Is stable• Has the optimal convergence rate
(a-1)+ (a)+
x
(a+1)-
Fiel
d
(a+1)+(a)-(a-1)-
2012 Non-linear mechanical solvers for GMSH 10
• Discontinuous Galerkin formulation – // & fracture– Formulation in terms of the first Piola stress tensor P
&
– Weak formulation obtained by integration by parts on each element Ω e
Interface for dG3D
New interface terms
2012 Non-linear mechanical solvers for GMSH 11
• Interface term rewritten as the sum of 3 terms– Introduction of the numerical flux h
• Has to be consistent:
• One possible choice:
– Weak enforcement of the compatibility
– Stabilization controlled by parameter β, for all mesh sizes hs
– Those terms can also be explicitly derived from a variational formulation (Hu-Washizu-de Veubeke functional) [Noels & Radovitzky, IJNME 2006 & JAM 2006]
Interface for dG3D
2012 Non-linear mechanical solvers for GMSH 12
Interface for dG3D
• Taylor impact test– Copper bar impacting a rigid wall
2012 Non-linear mechanical solvers for GMSH 13
• Cohesive Zone Method for fracture– Based on the use of cohesive elements
• Inserted between bulk elements
– Intrinsic Law• Cohesive elements inserted from the beginning• Drawbacks:
– Efficient if a priori knowledge of the crack path – Mesh dependency [Xu & Needelman, 1994]
– Initial slope modifies the effective elastic modulus– This slope should tend to infinity [Klein et al. 2001]:
» Alteration of a wave propagation» Critical time step is reduced
– Extrinsic Law• Cohesive elements inserted on the fly when
failure criterion is verified [Ortiz & Pandolfi 1999]
• Drawback– Complex implementation in 3D (parallelization)
Interface for dG3D
2012 Non-linear mechanical solvers for GMSH 14
Interface for dG3D
• MATERA project: SIMUCOMP– CENAERO,
e-Xstream, IMDEA Materials, Tudor, ULg
– Application to
composites
– Representative
nature?
– First results
εxx
σxx (Pa)
2012 Non-linear mechanical solvers for GMSH 15
Interface for shells
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
dgshell2012 Non-linear mechanical solvers for GMSH 16
Interface for shells
NonLinearMechSolver
MaterialLawcreateIPState()=0
IPVariableget(int )
Applications
mlawJ2Linearconstitutive()
mlawNonLocalDamageconstitutive()
mlawVUMatconstitutive()
dgshellMaterialLawstress(IP0, IP1)=0
ipJ2LinearSTensor3 Fp, εp
ipNonLocalDamagedouble *
ipVUMatdouble *
dgshellJ2LinearMaterialLawmlawJ2Linear *mlaw
mlawTwoLawsbulkLaw *cohesiveLaw *
ipTwoLawsipBulk *ipCohesive *
dgshelIPVariablestd::vector<STensor3> Fstd::vector<STensor3> τ
dG3DJ2LinearIPVariablestd::vector <ipJ2Linear> _ipJ2
2012 Non-linear mechanical solvers for GMSH 17
Interface for shells
• Thin bodies– FRIA (MS3, GDTech)– C1 continuity required– Test functions
• New DG interface terms– Consistency– Compatibility – Stability
(a-1) (a)
x
(a+1)
Fiel
d
(a-1)+ (a)+
x
(a+1)-
Fiel
d
(a+1)+(a)-(a-1)-
Extension
for fracture
C0/DG formulation[Noels & Radovitzky, CMAME 2008]
DG formulation[Becker & Noels, IJNME 2011, CMAME2011]
2012 Non-linear mechanical solvers for GMSH 18
Interface for shells
• New cohesive law for thin bodies– Should take into account a through the
thickness fracture• Problem : no element on the thickness
• Very difficult to separate fractured and
not fractured parts
– Solution:• Application of cohesive law on
– The resultant stress
– The resultant bending stress
• In terms of a resultant opening
∆x
∆r
∆c=
N, M
N
M
N0
M0
∆*
2Gc σmax
heq for non-linear materials2012 Non-linear mechanical solvers for GMSH 19
Interface for shells
• Application– Notched elasto-plastic cylinder submitted to a blast
2012 Non-linear mechanical solvers for GMSH 20
Interface for shells
• Application– Fragmentation of a brittle ceramic ring submitted to centrifugal forces
• Weibull strength distribution
– Future application: Rupture of MEMS• UCL
2012 Non-linear mechanical solvers for GMSH 21
Interface for Non Local Damage
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
NonLocalDG2012 Non-linear mechanical solvers for GMSH 22
Interface for Non Local Damage
NonLinearMechSolver
MaterialLawcreateIPState()=0
IPVariableget(int )
Applications
mlawJ2Linearconstitutive()
mlawNonLocalDamageconstitutive()
mlawVUMatconstitutive()
dG3DMaterialLawstress(IP0, IP1)=0
ipJ2LinearSTensor3 Fp, εp
ipNonLocalDamagedouble *
ipVUMatdouble *
NonLocalDG3DMaterialLawmlawNonLocalDamage *mlaw
mlawTwoLawsbulkLaw *cohesiveLaw *
ipTwoLawsipBulk *ipCohesive *
dG3DIPVariableSTensor3 F0, F1, PSVector3 jump, N-
NonLocalDG3DIPVariabledouble dεp/dp ….ipNonLocalDamage _ip
2012 Non-linear mechanical solvers for GMSH 23
Interface for Non Local Damage
• MATERA project: SIMUCOMP– CENAERO, e-Xstream, IMDEA Materials, Tudor, ULg
• Mean Field Homogenization– 2-phase composite
– Mori-Tanaka assumption
• Extension to damage?
Macro Macro
Macro
MicroMFH
nε ε∆C σ
1010 ωωεεε vv +=
1010 ωωσσσ vv +=
11:1 ωωεCσ =
00:0 ωωεCσ =
???
2012 Non-linear mechanical solvers for GMSH 24
Interface for Non Local Damage
• Damage
• Implicit non-local approach– New equation on an internal variable
The numerical results change with the size of mesh and direction of meshHomogenous unique solution
Lose of uniqueness
Strain localized
The numerical results change without convergence
aaca =∇− 2∫=cV
c
awdVV
a 1
Green function as weight functions w[Peerlings et al., 1996]
2012 Non-linear mechanical solvers for GMSH 25
Interface for Non Local Damage
• Non-local damage– Lemaitre-Chaboche
• S0 and n are the material parameters• Y is the strain energy release rate• p is the accumulated plastic strain
– New equation in the system
ppcp =∇− 2
pSY
pcpcpSYD
n
n
&
&&&&
)(
...)()(
0
42
21
0
=
+∇+∇+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−
=⎥⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ppppup
puuu
dd
FFFF
pu
KKKK intext
2012 Non-linear mechanical solvers for GMSH 26
Interface for Non Local Damage
• MFH with Non-local damage– Based on Linear Composite Comparison [Wu, Noels, Adam & Dogrhi, CMAME2012]
&
– Finite elements with 4 dofs/node for homogenized material
related to matrix only
0011 σσσ δυδυδ +=
DD δδδ 00alg00 ˆ:)1( σεCσ −−= )1/(ˆ 00 D−= σσ
ppDDa δυδδ
∂∂
−= 00lg ˆ: σεCσ
Mori‐Tanaka
DDa δυδυδυδ 000alg00I
lgI1 ˆ:)1( σεCεCσ −−+=
⎩⎨⎧
=∇−=+∇
22 pplp0fσ
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−
=⎥⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ppppup
puuu
dd
FFFF
pu
KKKK intext
2012 Non-linear mechanical solvers for GMSH 27
Interface for Non Local Damage
• MFH with Non-local damage– Epoxy-CF (30%)
2012 Non-linear mechanical solvers for GMSH 28
Interface for Non Local Damage
• Application– Epoxy - CF (50%)– Laminate 45/-45/-45/45 with a hole– Finite element mesh in each layer, with appropriate MFH laws
External ply Inner ply
2012 Non-linear mechanical solvers for GMSH 29
Interface for FE2
GMSH(elasticity)
Solver
Linear System
DofManager
(Bi)linearterms
Function Space
NonLinearMechSolver
NLDofManagerQS & Explicit
QuadratureRule
NLSystemQS & Explicit
MaterialLaw
dG3D
FunctionSpace
MaterialLaw
IPVariable
InterfaceElement
TermsPartDomain
Interface
Solver –
projects
IPVariable
Geotools
NLTerms
Applications
PartDomain
HODG FE2
2012 Non-linear mechanical solvers for GMSH 30
Interface for FE2
NonLinearMechSolver
MaterialLawcreateIPState()=0
IPVariableget(int )
Applications
mlawJ2Linearconstitutive()
mlawNonLocalDamageconstitutive()
mlawVUMatconstitutive()
dG3DMaterialLawstress(IP0, IP1)=0
ipJ2LinearSTensor3 Fp, εp
ipNonLocalDamagedouble *
ipVUMatdouble *
hoMultiscaleMaterialLaw
mlawTwoLawsbulkLaw *cohesiveLaw *
ipTwoLawsipBulk *ipCohesive *
dG3DIPVariableSTensor3 F0, F1, P
hoMultiscaleIPVariablestd::string _microMeshnonLinearSolver *solverstd::vector<data *>
2012 Non-linear mechanical solvers for GMSH 31
Interface for FE2
• Computational Multiscale– Macro-scale
• High-order Strain-Gradient formulation • C1 weakly enforced by DG• Partitioned mesh
2012 Non-linear mechanical solvers for GMSH 32
Interface for FE2
• Computational Multiscale– Macro-scale
• High-order Strain-Gradient formulation • C1 weakly enforced by DG• Partitioned mesh
– Micro-scale• Usual 3D finite elements• Periodic boundary conditions [Nguyen, Béchet,
Geuzaine & Noels COMMAT2011]
– Non-conforming mesh– Use of interpolant functions
• Stability
2012 Non-linear mechanical solvers for GMSH 33
Interface for FE2
• Computational Multiscale– Macro-scale
• High-order Strain-Gradient formulation • C1 weakly enforced by DG• Partitioned mesh
– Transition• Gauss-Points on different processors• Each Gauss point is associated to
– One mesh– One solver– Data: IPStates, fields of microproblem
– Micro-scale• Usual 3D finite elements• Periodic boundary conditions [Nguyen, Béchet,
Geuzaine & Noels COMMAT2011]
– Non-conforming mesh– Use of interpolant functions
• Stability
FM PM
FM QM
∂QM/∂FM
∂QM/∂FM∂PM/∂FM
∂PM/ ∂FM
2012 Non-linear mechanical solvers for GMSH 34
Interface for FE2
• Application
2012 Non-linear mechanical solvers for GMSH 35
Conclusions
• NonLinearMechSolver– Generic tool to solve mechanical problems– // implementation based on DG
• Applications – Different projects, which include the solver
• Projects are independent– First results– More work coming …
• Efficient tool for collaborations– Can be downloaded– Allows defining a new project easily
2012 Non-linear mechanical solvers for GMSH 36