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
3 - Code_Aster and Salome-Meca course material GNU FDL Licence
Definition of continuation methods
4 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
5 - Code_Aster and Salome-Meca course material GNU FDL Licence
Continuation methods
Isotropic fragile damage law (ENDO_ISOT_BETON)
Example 1 : tensile stress test of the
notched specimen
6 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
7 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
8 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
9 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
10 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
11 - Code_Aster and Salome-Meca course material GNU FDL Licence
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.
12 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
13 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving mechanical problems with continuation
14 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
17 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
18 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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
19 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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
20 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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
21 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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
23 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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
24 - Code_Aster and Salome-Meca course material GNU FDL Licence
Solving non-linear problems with continuation
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)
25 - Code_Aster and Salome-Meca course material GNU FDL Licence
Using continuation methods in Code_Aster
26 - Code_Aster and Salome-Meca course material GNU FDL Licence
Using continuation methods in Code_Aster
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
27 - Code_Aster and Salome-Meca course material GNU FDL Licence
Using continuation methods in Code_Aster
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
28 - Code_Aster and Salome-Meca course material GNU FDL Licence
Fixed charge
Drived charge
Resulting charge
application
Continuation
parameters
Pseudo-time
29 - Code_Aster and Salome-Meca course material GNU FDL Licence
Continuation methods : PRED_ELAS
Isotropic fragile damage law (ENDO_ISOT_BETON)
Example 1 : tensile stress test of the
notched specimen
30 - Code_Aster and Salome-Meca course material GNU FDL Licence
Continuation methods : PRED_ELAS
Anisotropic fragile damage law (ENDO_ORTH_BETON)
Example 2 : Impact on damaged concrete
31 - Code_Aster and Salome-Meca course material GNU FDL Licence
Continuation methods : PRED_ELAS
Cohesive Zone Model (CZM)
Example 3 : tensile stress test for the perforated plate
32 - Code_Aster and Salome-Meca course material GNU FDL Licence
Continuation methods : LONG_ARC
Buckling of a shell/column
F
Example 4
A
B
F
u
33 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
34 - Code_Aster and Salome-Meca course material GNU FDL Licence
Using continuation methods in Code_Aster
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)
35 - Code_Aster and Salome-Meca course material GNU FDL Licence
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.