Continuation methods for non-linear analysis

Transcription

1 Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (

2 Outline Definition of continuation methods Theoretical elements for continuation methods Solving non-linear problems with continuation methods Using continuation methods in Code_Aster 2 - Code_Aster and Salome-Meca course material GNU FDL Licence

3 Definition of continuation methods 3 - Code_Aster and Salome-Meca course material GNU FDL Licence

4 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 method Ku F si det K 0 Some external loading and prescribed displacements are partially unknowns by user: directions are known, intensity are unknown -> continuation method 4 - Code_Aster and Salome-Meca course material GNU FDL Licence

5 Continuation methods Example 1 : tensile stress test of the notched specimen Isotropic fragile damage law (ENDO_ISOT_BETON) 5 - Code_Aster and Salome-Meca course material GNU FDL Licence

6 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 1 Experimental setup controlled by extensometer Damage of the notched bar B η.f F 2 sound state Applied force at point A is controlled by displacement at point B A completely damaged state u 6 - Code_Aster and Salome-Meca course material GNU FDL Licence

7 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 Rigid mobile platform Hydraulic Piston Tested Sample Effort control cell 2m 7 - Code_Aster and Salome-Meca course material GNU FDL Licence

8 Continuation method Instability = snap-back I II Bi-material with close characteristics In 1d tensile test weak-chain is damaged first l 1 l 2 Simplified damage law : Bar I Bar II e Solution post peak : equilibrium => stress equality l 1 Ee E( e e ) ; e e 1 f E( l l e ) /( l l ) 2 f 1 2 l 1 l l l l l / E l ( e / E) Global post peak force-displacement response : f l 2 2 e c e f Snap-back if l 1 > l 2 Ee si e ec E( e f e) si e ec l 1 << l 2 l 8 - Code_Aster and Salome-Meca course material GNU FDL Licence

9 Continuation method Instability = snap-back Damage of the bar l 1 Global force-displacement response : Simplified damage law : Elastic domain softening e Snap-back l e c e f Ee si e ec E( e f e) si e ec Impossibility to follow the solution after this charge level 9 - Code_Aster and Salome-Meca course material GNU FDL Licence

10 Continuation methods : General FU curve F General form of the force-displacement curve Example : Snap-through for shell buckling F u u BC Prescription: forces Multiplicity: one force -> several displacements BC Prescription: displacements Multiplicity: one displacement -> several forces Horizontal tangent matrices: singular (slope vanishing) 10 - Code_Aster and Salome-Meca course material GNU FDL Licence

11 Continuation method Why? Continuation method in non-linear problems: Choosing a solution for incomplete model ext. F pilo F F. F ext ext ext impo pilo Yield-point analysis : critical loading Follow physical solution for ill defined problem : 1 K n u n R n si det K n 0 Multiple solutions coming from computation From 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 Code_Aster and Salome-Meca course material GNU FDL Licence

12 Continuation methods Choosing single solution for partially defined loading Partially unknown loadings: direction, application are known, intensity are unknown Unknown force or displacement (intensity) u η.f Goal: u Parameter: intensity of force Direction of force: known! Yield-point analysis η.p Goal: all domain is plastified Parameter: intensity of pressure Yield-point analysis 12 - Code_Aster and Salome-Meca course material GNU FDL Licence

13 Solving mechanical problems with continuation 13 - Code_Aster and Salome-Meca course material GNU FDL Licence

14 Solving linear problem with continuation Electric pylon stability control Boundary conditions: known and unknown parts of prescribed forces u η.f Goal: u Parameter: intensity of force Direction of force: known! Linear Elasticity : Formal solution : Ku F where F F. F ext ext ext ext impo pilo u K F. K F u. u 1 ext 1 ext impo pilo impo pilo Cable length control new equation for u u. u Δτ impo pilo crit 14 - Code_Aster and Salome-Meca course material GNU FDL Licence

15 Solving non-linear problems with continuation How to find unknown load parameter? 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 u u. u Δτ P impo pilo crit u t C 17 - Code_Aster and Salome-Meca course material GNU FDL Licence

16 Solving non-linear problems with continuation Continuation by degree of freedom - dof (DDL_IMPO) Equation Using rules: P u u 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 dof t C B η.f A 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! 18 - Code_Aster and Salome-Meca course material GNU FDL Licence

17 Solving non-linear problems with continuation Continuation by norm of displacement (LONG_ARC) Extended RIKS method (1972) Equation P u u t C 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) 19 - Code_Aster and Salome-Meca course material GNU FDL Licence

18 Solving non-linear problems with continuation Continuation by norm of displacement (LONG_ARC) Extended RIKS method F A u Arc-length: construct successive circles to follow loading path Very useful for complex path (snap-through for instance) B 2 t P u u uimpo. upilo C 2 Quadratic equation for 20 - Code_Aster and Salome-Meca course material GNU FDL Licence

19 Solving non-linear problems with continuation Continuation by strain increment (DEFORMATION) Equation Using rules At least, one point where strain is increasing No indication on plasticity state g g εi 1 : Δε Δt Pu = Max = gauss g ε i1 C g εi 1 Δε g Need a reference state with deformation ( method to establish this state Strains at the previous load step Increment of strains at the current step g 0 ε 1 i ): first computation without continuation Impossibility to follow the snap-back solutions : impossible loading-unloading transition Code_Aster and Salome-Meca course material GNU FDL Licence

20 Solving non-linear problems with continuation Continuation method by elastic prediction (PRED_ELAS) Available for Yield function constitutive laws : plasticity, damage Equations : Δt P u = Max g 1, g g gauss di ε i1 Δε = gauss C g Δt g g Pu = Max gauss i 1, i 1 0 gauss d ε Δε = C g ε i 1 g Δε g di 1 Strains at the previous load step Increment of strains at the current step Damage at the previous step for elasto-plasticity laws for damage laws t control either magnitude of Yield function overflow, or damage increment 23 - Code_Aster and Salome-Meca course material GNU FDL Licence

21 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) 24 - Code_Aster and Salome-Meca course material GNU FDL Licence

22 Using continuation methods in Code_Aster 25 - Code_Aster and Salome-Meca course material GNU FDL Licence

23 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 26 - Code_Aster and Salome-Meca course material GNU FDL Licence

24 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) 27 - Code_Aster and Salome-Meca course material GNU FDL Licence

25 Continuation method : command file Fixed charge Drived charge Pseudo-time Resulting charge application Continuation parameters 28 - Code_Aster and Salome-Meca course material GNU FDL Licence

26 Continuation methods : PRED_ELAS Example 1 : tensile stress test of the notched specimen Isotropic fragile damage law (ENDO_ISOT_BETON) 29 - Code_Aster and Salome-Meca course material GNU FDL Licence

27 Continuation methods : PRED_ELAS Example 2 : Impact on damaged concrete Anisotropic fragile damage law (ENDO_ORTH_BETON) 30 - Code_Aster and Salome-Meca course material GNU FDL Licence

28 Continuation methods : PRED_ELAS Example 3 : tensile stress test for the perforated plate Cohesive Zone Model (CZM) 31 - Code_Aster and Salome-Meca course material GNU FDL Licence

29 Continuation methods : LONG_ARC Example 4 F F A B Buckling of a shell/column u 32 - Code_Aster and Salome-Meca course material GNU FDL Licence

30 Applied pressure, Pa Continuation methods : DDL_IMPO Example 5 : stability of the gravity dam upstream opening,m Cohesive Zone Model (JOINT_MECA_RUPT) 33 - Code_Aster and Salome-Meca course material GNU FDL Licence

31 Using continuation methods in Code_Aster Documentation: General documentation about non-linear solver [R ] General documentation about continuation methods [R ] Using continuation method, syntax in [U ] Examples: See [V ], forma03d test-case for general example See [V ], ssna119b test-case for fragile damage (Elastic prediction) See [V ], ssnv124 test-case for yield-point analysis See [V ], ssns101 test-case for shell buckling (Riks method) 34 - Code_Aster and Salome-Meca course material GNU FDL Licence

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