a constrained large time increment method for modelling...

A constrained LArge Time INcrement method

for modelling quasi-brittle failure

Bram VANDORENa

K. De Profta A. Simoneb L. J. Sluysb

a Hasselt University, Belgium

b Delft University of Technology, The Netherlands

CFRAC 2013, 6th June, 2013, Prague, Czech Republic

Outline

introduction to LArge Time INcrement (LATIN) method

tracing snap-back behaviour within the LATIN method

numerical examples

A constrained LArge Time INcrement method for modelling quasi-brittle failure 1 / 20

LATIN method

non-incremental solution method for non-linear mechanics

[Boisse et al., IJNME 1990; Ladevèze, 1999]

an alternative to step-by-step solution methods

e.g. incremental-iterative Newton-Raphson method

limited range of convergence

requires continuation techniques in case of limit andsnap-back points, i.e. the addition of a constraint equation

to the FE equations [Riks, IJSS 1979; Crisfield, C&S 1981]

continuation techniques: which constraint function?

convergence problems in case of bifurcations


LATIN method

whole time domain is calculated in one iteration

converged solution is the exact solution

snap-back behaviour cannot be traced

F

u

F

u

t

t+1


Newton-Raphson LATIN

LATIN method

typically applied to hardening (visco-)plastic and brittle

materials [Abdali et al., JMPT 1996; Dolbow et al., CMAME 2001]

goals of this contribution

application to quasi-brittle materials

tracing snap-back points within LATIN method


LATIN method: Ingredients

two main solution stages, executed alternately

local solution stage (n + 1/2): local and non-linear

global solution stage (n + 1): global and linear

‘separating the difficulties’ [Ladevèze, 1999]


(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)

(σexact, εexact) admissible solutions

(global and linear)

solutions verifying constitutive behaviour

(local and non-linear)

E+E−

iterationiterationiterationn + 1n + 2n + 3

LATIN method: Local solution stage

verify on the whole space and time domains:

constitutive behaviour: σn+1/2 = Cεn+1/2

search equation:(

σn+1/2 − σn

)

+ E+(

εn+1/2 − εn

)

= 0


(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


LATIN method: Local solution stage


constitutive behaviour : σn+1/2 = Cεn+1/2

search equation :(

σn+1/2 − σn

)

+E+(

εn+1/2 − εn

)

= 0


(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


LATIN method: Global solution stage


admissibility conditions:

search equation:(

σn+1 − σn+1/2

)

− E−

(

εn+1 − εn+1/2

)

= 0


∫

Ωδε(σn+1 − σn) dΩ =∫

Γtδu

(

tn+1 − tn

)

dΓt

with σ S. A. and u K. A.

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−





search equation:(

σn+1 − σn+1/2

)

− E−

(

εn+1 − εn+1/2

)

= 0


Kn∆un+1 = ∆fext −∆fn

∆fn =∫

ΩBT((

σn+1/2 − σn

)

− E−

(

εn+1/2 − εn

))

dΩ

∆fext = 0

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−





search equation :(

σn+1 − σn+1/2

)

−E−

(

εn+1 − εn+1/2

)

= 0



∆fn =∫

ΩBT((

σn+1/2 − σn

)

− E−

(

εn+1/2 − εn

))

dΩ

∆fext = 0

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


LATIN method: Tracing snap-back behaviour

one option: switch to classical Newton-Raphson algorithmnear the snap-back point [Kerfriden et al., CM 2009]

complicates the algorithm

which constraint function?

an alternative: add a constraint function to the globalsolution stage of the LATIN algorithm

‘stand-alone’ LATIN algorithm

which constraint function?



classical constrained Newton-Raphson method

constraint function defined in terms of incremental

displacement field

constrained LATIN method

no previous converged load step

constraint function in terms of thetotal displacement field, e.g.

un+1TAun+1 − τ2 = 0

A selects a subset of DOF’s[Geers, IJNME 1999]


τ

u2u1

λ

u


classical constrained Newton-Raphson method

constraint function defined in terms of incremental

displacement field


no previous converged load step

constraint function in terms of thetotal displacement field, e.g.

unTAun+1 − τ2 = 0

A selects a subset of DOF’s[Geers, IJNME 1999]


τ

u2u1

λ

u




search equation:(

σn+1 − σn+1/2

)

− E−

(

εn+1 − εn+1/2

)

= 0



∆fn =∫

ΩBT((

σn+1/2 − σn

)

− E−

(

εn+1/2 − εn

))

dΩ

∆fext = 0∆λn+1 fext

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


Constrained LATIN method: Global solution stage



search equation:(

σn+1 − σn+1/2

)

− E−

(

εn+1 − εn+1/2

)

= 0

constraint equation: unTAun+1 − τ2 = 0



∆fn =∫

ΩBT((

σn+1/2 − σn

)

− E−

(

εn+1/2 − εn

))

dΩ

∆fext = ∆λn+1 fext

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


Constrained LATIN method: Global solution stage


admissibility conditions :

search equation :(

σn+1 − σn+1/2

)

−E−

(

εn+1 − εn+1/2

)

= 0

constraint equation: unTAun+1 − τ2 = 0



∆fn =∫

ΩBT((

σn+1/2 − σn

)

− E−

(

εn+1/2 − εn

))

dΩ

∆fext = ∆λn+1 fext

(

σn+1/2, εn+1/2

)

(σn, εn)

(σn+1, εn+1)


(global and linear)



E+E−


Numerical examples

masonry wall under shear loading

fracture of a perforated cantilever beam


Numerical examples: Masonry wall

linear elastic bricks and exponential damage law in mortarjoints (modelled as interface-like elements [Simone, CNME 2004])

tint = (1 − ω)dint[[u]]int

degenerated capped Drucker-Prager material model

[Vandoren et al., CMAME 2013]


0.30 N/mm2

990

70

70

1106

F bricks: 52 mm × 210 mm× 100 mm

hjoints = 10 mm

Ebrick = 16700 N/mm2

νbrick = 0.15

dn,int = 82 N/mm3

dt,int = 36 N/mm3

ft,int = 0.25 N/mm2

fc,int = 10.5 N/mm2

GfI,int = 0.018 N/mm

[Raijmakers and Vermeltfoort, 1992]


algorithmic variables

E+ = ∞

E− = (1 − ω)dint

A initially selects the loading DOF, after convergence the

most critical DOF is selected if∣

∣Ft − Ft−1

∣

∣ > 2∣

∣Ft−1 − Ft−2

∣

∣


0.30 N/mm2

990

70

70

1106

F bricks: 52 mm × 210 mm× 100 mm

hjoints = 10 mm

Ebrick = 16700 N/mm2

νbrick = 0.15

dn,int = 82 N/mm3

dt,int = 36 N/mm3

ft,int = 0.25 N/mm2

fc,int = 10.5 N/mm2

GfI,int = 0.018 N/mm


0 1 2 3 4 50

10

20

30

40

50iteration 1

displacement [mm]

forc

e [k

N]



0 1 2 3 4 50

10

20

30

40

50

displacement [mm]

forc

e [k

N]


Numerical examples: Perforated beam

non-linear interface element along horizontal axis of

symmetry: linear cohesive damage law

tint = (1 − ω)dint[[u]]int

damage driven by [[un]]int


0.2 0.375

1.5

1.0

F

F

Ebeam = 100 N/mm2

νbeam = 0.30

dn,int = 10000 N/mm3

dt,int = 5000 N/mm3

ft,int = 1.0 N/mm2

GfI,int = 2.5 · 10−3 N/mm

[Verhoosel et al., IJNME 2009]


algorithmic variables

E+ = ∞

E− = (1 − ω)dint

A initially selects the loading DOF, after convergence the

most critical DOF is selected if∣

∣Ft − Ft−1

∣

∣ > 2∣

∣Ft−1 − Ft−2

∣

∣


0.2 0.375

1.5

1.0

F

F

Ebeam = 100 N/mm2

νbeam = 0.30

dn,int = 10000 N/mm3

dt,int = 5000 N/mm3

ft,int = 1.0 N/mm2

GfI,int = 2.5 · 10−3 N/mm


0 0.01 0.02 0.030

0.02

0.04

0.06

0.08

0.1

0.12iteration 1

displacement [mm]

forc

e [N

]



0 0.01 0.02 0.03

0.02

0.04

0.06

0.08

0.1

displacement [mm]

forc

e [N

]


Conclusions


now possible to trace snap-backs within LATIN method

most critical DOF’s in the constraint function are

automatically detected

linear convergence rate (with the current search directions)

a robust alternative to conventional step-by-step methods

future work

optimise search directions


a constrained large time increment method for modelling...

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