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