1. 非線性分析的收斂性探討 Convergence of Nonlinear Analysis
Chapter 16
非線性分析的收斂性探討
Convergence of Nonlinear Analysis

2. Contents 16.1 何謂非線性分析？ What Are Nonlinear Analyses? 16.2 解題方法
16.1 何謂非線性分析？
What Are Nonlinear Analyses?
16.2 解題方法
Solution Methods
16.3 收斂性控制的一些重要觀念
Important Concepts for Convergence Controls
16.4 收斂性控制的基本參數
Basic Solution Options
16.5 收斂性控制的進階參數
Advanced Solution Options
16.6 練習題：橡膠密封墊分析
Exercise: Hyperelastic Planar Seal

3. 何謂非線性分析？ What Are Nonlinear Analyses?
第16.1節
何謂非線性分析？
What Are Nonlinear Analyses?

4. 16.1.1 What is a Nonlinear Structure
Displacements
Forces

5. 16.1.2 Causes of Nonlinearities
Geometric Nonlinearity
Material Nonlinearity
Status Nonlinearity

6. Geometric Nonlinearity
Force
Displacement
Moment arm
Forces
Displacements

7. Material Nonlinearity
Strain
Stress
Mild Steel
Rubber

8. Status Nonlinearity Force
The contact area depends on the applied force
Force

9. 16.1.3 Consequences of Nonlinearities
Principle of superposition no longer applicable
Solution may depend on loading history

10. 第16.2節
解題方法
Solution Methods

11. 16.2.1 Equations for a Nonlinear StructureD
K(D)
F(D)

12. 16.2.2 Incremental Method Error 3 F 2 Calculated Response 1 Actual

13. 16.2.3 Newton-Raphson Method
D = D4
F = F4
Actual
response 1 2 3 4 D1 D2 D3 F1 F2 F3

14. 16.2.4 Convergence Criteria in ANSYS
D = D4
F = F4
Actual
response 1 2 3 4 D1 D2 D3 F1 F2 F3

15. 收斂性控制的一些重要觀念 Important Concepts for Convergence Controls
第16.3節
收斂性控制的一些重要觀念
Important Concepts for Convergence Controls

16. 16.3.1 Radius of Convergence Radius of convergence F Actual responseDo
Radius of convergence

17. 16.3.2 Consistent Stiffness Matrix
Tangent stiffness
Additional stiffness due to deformation (geometric change)
Additional stiffness due to stress stiffening
Additional stiffness due to change of loading direction.

18. 16.3.3 Load Steps, Substeps, and Equilibrium Iterations
Time
Load
Load step 2
Load step 1
D = D4
F = F4
Actual
response 1 2 3 4 D1 D2 D3 F1 F2 F3

19. Concepts of Time
The ends of load steps or substeps can be identified by time.
For dynamic problem, time is used as a real-world clock.
For static problem, time is used as a counter.

20. 16.3.5 Automatic Time Stepping
For nonlinear problem, the user-input Dt is used as initial incremental time.
ANSYS adjusts Dt automatically according to the convergence behavior of the solution.

21. 16.3.6 Text Output Information2 3 4 5

22. 16.3.7 Graphical Output Information

23. 16.3.8 Automatic Nonlinear Solution Control
SOLCONTROL command can be used to activate automatic nonlinear solution control algorithm.
The default is ON.

24. 收斂性控制的基本參數 Basic Solution Options
第16.4節
收斂性控制的基本參數
Basic Solution Options

25. Basic Solution Options01 02 03 04 05 06 07 08 09 10
SOLCONTROL, Key
ANTYPE, Antype
NLGEOM, Key
TIME, TIME
NSUBST, NSBSTP, NSBMX, NSBMN, Carry
DELTIM, DTIME, DTMIN, DTMAX, Carry
AUTOTS, Key
KBC, KEY
OUTRES, Item, FREQ, Cname
MONITOR, VAR, Node, Lab

26. 16.4.1 Analysis Type (ANTYPE)
For nonlinear analysis, only two analysis types applicable: STATIC and TRANS.

27. 16.4.2 Nonlinear Geometry (NLGEOM)
NLGEOM turns ON/OFF the effects of geometric nonlinearity.

28. Load Step Time (TIME)
TIME specifies the time at the end of the following load step.

29. 16.4.4 Number of Substeps (NSUBST, DELTIM)
NSUBST, NSBSTP, NSBMX, NSBMN, Carry
DELTIM, DTIME, DTMIN, DTMAX, Carry
NSUBST specifies number of substeps for the following load step.
DELTIM specifies time for each substep for the following load step.
The two commands are equivalent, i.e.,

30. 16.4.5 Automatic Time Stepping (AUTOTS)
AUTOTS turns ON/OFF the functions of auto time stepping.

31. 16.4.6 Ramped/Stepped Loading (KBC)
Time
Load
Stepped load
Substeps

32. 16.4.7 Output Controls (OUTRES)
OUTRES, Item, FREQ, Cname
OUTRES controls the amount of data stored in Jobname.RST.

33. Monitor File (MONTOR)

34. 收斂性控制的進階參數 Advanced Solution Options
第16.5節
收斂性控制的進階參數
Advanced Solution Options

35. Advanced Solution Options01 02 03 04 05 06 07 08 09 10 11 12 13
EQSLV, Lab, TOLER, MULT
SOLCONTROL, Key1, Key2
NEQIT, NEQIT
CNVTOL, Lab, VALUE, TOLER, NORM, MINREF
NROPT, Option,, Adptky
LNSRCH, Key
PRED, Sskey, --, Lskey
SSTIF, Key
PSTRES, Key
CUTCONTROL, Lab, VALUE, Option
TIMINT, Key, Lab
TINTP, GAMMA, ... NCNV, KSTOP, DLIM, ITLIM, ETLIM, CPLIM

36. 16.5.1 Equation Solver (EQSLV)
EQSLV lets you choose the algorithm solving the equations.

37. 16.5.2 Time Step Prediction Based on Contact Status (SOLCONTROL)
SOLCONTROL, Key1, Key2
The second key of SOLCONTROL command is to ask ANSYS to adjust time steps according to contact status.

38. 16.5.3 Number of Equilibrium Iterations (NEQIT)
NEQIT specifies the number of iterations beyond which ANSYS would start another “attempt”.

39. 16.5.4 Convergence Criteria (CNVTOL)
CNVTOL overrides the default convergence criteria.

40. 16.5.5 Newton-Raphson Option (NROPT)
NROPT, Option,, Adptky
(a) Full Method
(b) Modified Method
(c) Initial Method

41. Line Search (LNSRCH)
LNSRCH turns ON/OFF the “line search” algorithm in ANSYS.

42. Predictor (PRED)

43. 16.5.8 Adaptive Descent (NROPT)
NROPT, Option,, Adptky

44. 16.5.9 Stress Stiffening (SSTIF, PSTRES)
SSTIF/PSTRES turns ON/OFF the computation of stress stiffness (Ks) and storing on file.

45. 16.5.10 Cutback Control (CUTCONTROL)
CUTCONTROL specifies the conditions in which ANSYS would decrease time step during auto time stepping.

46. 16.5.11 Time Integration Effects (TIMINT, TINTP)
TIMINT, Key, Lab
TINTP, GAMMA, ...
TIMINT turns ON/OFF transient effects.
The parameter GAMMA of TINTP command can be used to introduce a “numerical damping”.

47. NCNV, KSTOP, DLIM, ITLIM, ETLIM, CPLIM
Stop Control (NCNV)
NCNV, KSTOP, DLIM, ITLIM, ETLIM, CPLIM
NCNV specifies the stop (failure) conditions of the computation.

48. 16.5.13 Terminating an Analysis

49. 16.5.14 Restarting an Analysis
An analysis terminated normally can be restarted.

50. 16.6 Exercise: Hyperelastic Planar Seal