continuation methods for non-linear analysis - code-aster.org · pdf fileanalysis fr :...

Code_Aster, Salome-Meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)

Continuation methods for non-linear

analysisFR : Méthodes de pilotage du chargement

2 - Code_Aster and Salome-Meca course material GNU FDL Licence

Outline

Definition of continuation methods

Theoretical elements for continuation methods

Solving non-linear problems with continuation methods

Using continuation methods in Code_Aster


Definition of continuation methods


Continuation method – What ?

Linear mechanical problem definition:Unknowns: displacement, Lagrange multiplier for boundary conditions

Loadings : displacement (Dirichlet), forces (Neumann)

Unique solution (elliptical differential equation) :

Non-linear problem definition:Unknowns: displacement, Lagrange multiplier for boundary conditions, temperature,

pressure, stress and internal variables

Loadings : displacement (Dirichlet), forces (Neumann), contact/friction

Parameterization: t is not real time (quasi-static problem)

Sequence of linearized solutions

General non-linear continuation methodSome external loading and prescribed displacements are partially unknowns by user:

directions are known, intensity are unknown -> continuation method

si det 0Ku F K


Continuation methods

Isotropic fragile damage law (ENDO_ISOT_BETON)

Example 1 : tensile stress test of the

notched specimen


Continuation method – Where ?

Where to use :1. External loading and prescribed displacements are

partially unknowns by user: direction, application are

known, intensity are unknown

2. Solution of an unstable problem =>

impossibility to follow system evolution continuously

=> Newton method fails

u

F

sound state

completely damaged

state

21 Experimental setup controlled by extensometer Damage of the notched bar

Applied force at point A is controlled

by displacement at point B

η.F

A

B


Continuation method – Loading control

Nooru-Mohamed concrete fracture test :1) Loading is applied via rigid mobile platform by piston displacement

2) Efforts are controlled on the stable effort control cell

3) Unstable concrete fracture

Hydraulic Piston

Rigid mobile platform

Tested Sample

Effort control cell

2m


Continuation method – Instability = snap-back

Bar II

Bar I

e

Simplified damage law :

ec

Bi-material with close characteristics

l1

In 1d tensile test “weak-chain” is damaged first

2

2

1

1

2 21 1( ) ; f

l

l

l

lE Ee ee e e

2 1 2( ) /( )f ll lE l e

ef

si

( ) si

c

f c

E

E

e e e

e e e e

III

l2

1 12 2/ ( / )fl ll ll E E e

Solution post peak : equilibrium => stress equality

Global post peak force-displacement response :

l

Snap-back if l1> l2

l1<< l2


Continuation method – Instability = snap-back

softening

Elastic domain

e

Simplified damage law :

ec

Damage of the bar

l1

ef

si

( ) si

c

f c

E

E

e e e

e e e e

Global force-displacement response :

l

Snap-back

Impossibility to follow the solution

after this charge level


Continuation methods : General FU curve

F

u

Example : Snap-through for shell buckling

F

u

BC Prescription: forces

BC Prescription: displacements

Multiplicity: one force ->

several displacements

Multiplicity: one displacement

-> several forces

Horizontal tangent matrices:

singular (slope vanishing)

General form of the force-displacement curve


Continuation method – Why ?

Continuation method in non-linear problems:

Choosing a solution for incomplete model

Yield-point analysis : critical loading

Follow physical solution for “ill defined” problem :

Multiple solutions coming from computationFrom constitutive laws. Non-elliptic condition as softening, damage, geo-mechanic laws : ENDO_SCALAIRE,ENDO_ISOT_BETON,ENDO_ORTH_BETON,CZM_EXP,ROUSSELIER,

VENDOCHAB,…

From equilibrium equations: buckling, structural instabilities

From Coulomb’s friction

1 si det 0n n n nK u R K

.ext ext ext

impo piloF F F

. ext

piloF

1.

2.

3.


Continuation methods

Choosing single solution for partially defined loadingPartially unknown loadings: direction, application are known, intensity are unknown

• Goal: u

• Parameter: intensity of force

• Direction of force: known !

uη.F

η.p

• Goal: all domain is plastified

• Parameter: intensity of pressure

• Yield-point analysis

Unknown

force or

displacement

(intensity)

Yield-point

analysis


Solving mechanical problems with continuation


Solving linear problem with continuation

Boundary conditions: known and unknown parts of prescribed forces

Linear Elasticity :

Formal solution :

Cable length control new equation for

.impo pilo critu u u Δτ

• Goal: u

• Parameter: intensity of force

• Direction of force: known !

uη.F

where .ext ext ext ext

impo piloKu F F F F

Electric pylon stability control

1 1. .ext ext

impo pilo impo pilou K F K F u u


Solving non-linear problems with continuation

Continuation equation:Build on displacements, strain or stress

Should be easy to solve (linear, quadratic)

Using only one scalar parameter

Continuation methods list by goal function:Degree of freedom: DDL_IMPO

Norm of displacement: LONG_ARC

Displacement jump: SAUT_IMPO (XFEM)

Norm of displacement jump: SAUT_LONG_ARC (XFEM)

Work of exterior forces (yield-point analysis): ANA_LIM

Strain increment: DEFORMATION

Elastic prediction: PRED_ELAS

How to find unknown load parameter ?

C

tuP

.impo pilo critu u u Δτ



Continuation by degree of freedom - dof (DDL_IMPO)

Equation

Using rules:Control displacement increment of one dof

The controlled node must be important for movement

C is a constant given by user in STAT_NON_LINE

η.F

A

B

Good goal: what is for a given

vertical displacement of A node ?

Bad goal: what is for a given

displacement of B node ?

B doesn’t move !

C

tuuP dof



Continuation by norm of displacement (LONG_ARC) – Extended

RIKS method (1972)

Equation

Using rules:Control norm displacement increment of several dof and several nodes

The controlled nodes must be important for movement

C is a constant given by user in STAT_NON_LINE

Resulted equation is quadratic: two solutions -> need selection criterion RESIDU,

ANGL_INCR_DEPL, NORM_INCR_DEPL (see documentation)

C

tuuP



Continuation by norm of displacement (LONG_ARC) – Extended

RIKS method

A

B

F

u

Arc-length: construct successive circles to follow loading path

Very useful for complex path (snap-through for instance)

2

2

.impo pilo

tP u u u u

C

Quadratic

equation for



Continuation by strain increment (DEFORMATION)

Equation

Using rulesAt least, one point where strain is increasing

No indication on plasticity state

Need a reference state with deformation ( ): first computation without continuation

method to establish this state

Impossibility to follow the snap-back solutions : impossible loading-unloading transition.

Strains at the previous load step

Increment of strains at the current step

01 giε

1

1

:g g

i

ggaussi

ε Δε ΔtP u = Max =

Cε

giε 1

gΔε



Continuation method by elastic prediction (PRED_ELAS)

Available for Yield function constitutive laws : plasticity, damage

Equations :

t control either magnitude of Yield function overflow, or damage increment

Strains at the previous load step

Increment of strains at the current step

for elasto-plasticity laws

Damage at the previous step

giε 1

gΔε

gid 1

1 1,g g g

gauss i igauss

ΔtP u = Max d ε Δε =

C

1 1, 0g g g

gauss i igauss

ΔtP u = Max d ε Δε =

C

for damage laws



Continuation method by elastic prediction (PRED_ELAS)

Using rules

At least, one point which passes through initial yield surface

Depend on behavior law: ENDO_SCALAIRE, ENDO_FRAGILE, ENDO_ISOT_BETON,

ENDO_ORTH_BETON, VMIS_ISOT_*, CZM_* and BETON_DOUBLE_DP

Criterion C : increasing ratio of damage or strain

Resultant equation should have two solutions -> need selection criterion RESIDU, ANGL_INCR_DEPL, NORM_INCR_DEPL (see documentation)



As continuation methods is using parameter for determination of

loading path, you must avoid direct or indirect using of time in

your model:No dynamic (only STAT_NON_LINE where t is pseudo-time)

No time for loadings: no FONC_MULT, no AFFE_CHAR_MECA_F with parameter INST

No « command variables » as temperature in AFFE_MATERIAU/AFFE_VARC

Contact/friction is not possible except for specific XFEM

methods (with CZM, see documentation) or discrete element (DIS_CHOC)

Line search is possible only for some continuation methods



Definition of loads in AFFE_CHAR_MECA

Definition of continuation load in STAT_NON_LINE/EXCIT

EXCIT/TYPE_CHARGE='FIXE_PILO'

Definition of the parameters for continuation method in STAT_NON_LINE/PILOTAGE

Post-processing : parameter could been found in result (ETA_PILOTAGE)

Continuation method : command file


Fixed charge

Drived charge

Resulting charge

application

Continuation

parameters

Pseudo-time


Continuation methods : PRED_ELAS

Isotropic fragile damage law (ENDO_ISOT_BETON)

Example 1 : tensile stress test of the

notched specimen



Anisotropic fragile damage law (ENDO_ORTH_BETON)

Example 2 : Impact on damaged concrete



Cohesive Zone Model (CZM)

Example 3 : tensile stress test for the perforated plate


Continuation methods : LONG_ARC

Buckling of a shell/column

F

Example 4

A

B

F

u


Continuation methods : DDL_IMPO

Cohesive Zone Model (JOINT_MECA_RUPT)

Example 5 : stability of the gravity dam

upstream opening,m

Ap

plie

d p

ressu

re, P

a



Documentation:General documentation about non-linear solver [R5.03.01]

General documentation about continuation methods [R5.03.80]

Using continuation method, syntax in [U4.51.03]

Examples:

See [V6.03.114], forma03d test-case for general example

See [V6.01.101], ssna119b test-case for fragile damage (Elastic prediction)

See [V6.04.124], ssnv124 test-case for yield-point analysis

See [V6.05.101], ssns101 test-case for shell buckling (Riks method)


End of presentation

Is something missing or unclear in this document?

Or feeling happy to have read such a clear tutorial?

Please, we welcome any feedbacks about Code_Aster training materials.

Do not hesitate to share with us your comments on the Code_Aster forum

dedicated thread.

http://www.code-aster.org/forum2/viewtopic.php?id=17343

continuation methods for non-linear analysis - code-aster.org · pdf fileanalysis fr :...

Documents