This document summarizes an application of inverse analysis to the material point method (MPM) for modeling geotechnical problems. The key points are: - Inverse analysis can be used to calibrate constitutive model parameters in MPM simulations by comparing results to field or laboratory observations. - Two examples are provided where inverse analysis is used to estimate Mohr-Coulomb parameters from CPT data and calibrate friction parameters to match a granular flow experiment. - The main case study involves calibrating a constitutive model to simulate liquefaction and slope failure observed in a laboratory experiment using Northwich Park Coal. Parameters are fitted to triaxial test data and the slope failure stages are modeled.

1. Application of Inverse analysis to
the Material Point Method
Pooyan Ghasemi and Mario Martinelli
Geo Klantendag 2018
Workshop Anura3D Modelleren met MPM

2. dykes, dams, landslides
installation, impact, SSI
flowslides, erosion, liquefaction, SWI
Applications

3. Applications
• How can we benefit? What can we gain?
• study beyond “initial” failure
> progressive failures and dynamic processes
• prediction of consequences
> catastrophic failures
• study the efficiency of remedial actions
> location, type and strength of reinforcement
• or defence structure
• improve risk assessment

4. dykes, dams, landslides
installation, impact, SSI
flowslides, erosion, liquefaction, SWI
Applications
small
deformation
onset of
failure

5. dykes, dams, landslides
installation, impact, SSI
flowslides, erosion, liquefaction, SWI
Applications
small
deformation
onset of
failure
Results depend on the value of the CONSTITUTIVE MODELS PARAMETERS
->> parameter calibration is crucial! <<-

6. Applications
Most of the time multiple sets of data are available:
• Lab tests (TX tests, …. )
• Field tests (SPT, CPT…)
• Field monitoring observations
CPT Real field data

7. Applications
Most of the time multiple sets of data are available:
• Lab tests (TX tests, …. )
• Field tests (SPT, CPT…)
• Field monitoring observations
CPT Real field data
?

8. Outline
• What is inverse analysis
• Inverse analysis in Geotechnical problem
• Example 1
• Example 2
• Main case study
• Remarks and conclusion

9. What is the Inverse Analysis
Much research in science and engineering is devoted to constructing numerical models of physical
systems. Given a complete description of the physical system, these models can be used to predict
how the system affects its surroundings. This is called the direct or forward problem. When some
aspects of physical system are unknown, however, it is sometimes possible to infer these
characteristics from a known system response; this is the inverse problem.
Inverse problem
Forward problem Regular Simulation, Physical or Numerical
Parameter estimation
Design optimization Infer the optimal design
Infer the system characteristics
from experimental Data
More popular in Geotechnical engineering
Definition:

10. What is the Inverse Analysis
We are already using inverse analysis in our mind
Base on the our observation
we are looking for the cause
This is a difficult task if many types of
dragon produce similar footprints, and
becomes impossible if the footprints are
even slightly smeared in mud.

11. The roles of prior Information
• Numerical geotechnical models are mathematically ill-
posed due to an information deficit. Sometimes,
measurements barely provide sufficient information
• Some models’ parameters do not have physical meaning
Curve Fitting + Prior Information +
(e.g. max or min value of
parameter, relation between
parameters)
Combinations of the observations =
(e.g. choose the most appropriate
observation for the corresponding
parameter)
Inverse Analysis
• In inverse Analysis method , we are trying to get closer to the solution step by step,
considering only the correct parameters and the corresponding tests observations
• Adding the new observations together with other remaining parameters and complete the
inverse analysis -> step by step procedure
• Care must be taken to ensure that the information available are used appropriately!
(i.e. reasonable parameter value?, correct test to calibrate that parameter?)

12. Elementary soil behavior
Triggering stage
Propagation Stage
• mobilized volume?
• mode of collapse?
• Yielding surface ?
• Final run-out
• final geometry
What we need to do in order to calibrate a simulation of large deformation phenomenon
Inverse analysis in Geotechnical problems
Different Stage of Simulation and Calibration
Eckersley, D. (1990). Instrumented laboratory flowslides. Geotechnique, 40(3), 489-502.

13. Inverse analysis in Geotechnical problem
Observations :
Result of Laboratory testing
Calculation model
testtest
Model
test
Model
Optimization
algorithm
Update
parameters
Input
Parameters
test
BEST
Fit
Is min error?
No
Yes
End and report
the optimal
parameters
Error
• Particle swarm method
• Genetic algorithm
• Neural Network
Fuzzy Methods:

14. Inverse analysis in Geotechnical problem
Soil Test toolbox element test
Large deformation tests suite

15. Landslide
basal surface
Observation
Inverse analysis in Geotechnical problem
Numerical model
SPH, MPM, FEM
Optimization
algorithm
Update
parameters
Input
Parameters
Is min error?
No
Yes
End and report
the optimal
parameters
Observations :
Physical experiments or
Site inspecting
Landslide
basal surface
Observation
Simulation
Landslide
basal surface
Observation
Simulation
Landslide
basal surface
Observation
Best Fit
Error
Calvello M, Cuomo S, Ghasemi P (2017). The role of observations in the inverse analysis of landslide propagation. Computers and Geotechnics, 92:11-21)
• Kalaman Filter
• Nonlinear Regression
• Modified Guass Newoton
Gradient base method:

16. Elements of Inverse Analysis
• There are many options for components of the procedure, for each specific problem , the best
option might be varied
Optimization algorithm Objective Function
Examples for optimization methods :
Optimization algorithm Problem Ref.
• Ucode algorithm Excavation Calvello (2002)
• Genetic algorithms Excavation Levasseur (2007)
• PSO Constitutive model of Clay Knabe (2013)
• Differential evolution (DE) Excavation Zhao (2014)
• Artificial bee colony algorithm Excavation Zhao (2016)
• Nelder-Mead Excavation Tian (2016)
• Genetic algorithms constitutive model of rock salt Khaledi (2016)
• Ensemble Kalman Filter Slope Stability Varden (2016)
• hybrid methods Excavations De Santus (2015)
Objective Function
Observation
Which Type of observation
How many observation?
Error Calculation
Weighted Least squared Error
(WSE)
Robust and interpolation-free
technique (Horn 2015)
Calculation the area between two figures
Gradient base method:
• Give the Sensitivity of each Parameters
• Fewer number of Iteration
• Weak in the case of large number of
parameters and High correlated parameters
Fuzzy Method :
• Robust in the Case of high correlated Parameters
• Compatible for large number of parameters
• Large number of Iteration is needed

17. Implementation in Anura 3D
Numerical
Calculation
Input Parameters
Output
Desired Results
(observation)
Input.txt
OutPut.txt
Regression
(Optimization Engine)
Updated Parameters
Model
optimized?
Anura3D
INV. Interface (1st Phase of the Project)
Initial Values
No
Yes
End
Start
2nd Phase of the Project
Matlab and Python, Ucode
Anura3D

18. Example 1: Inverse Analysis of CPT
example to check the software performances
• Creating the synthetic observation:
MPM model while using the
a set of assumed parameter value
Mohr Coulomb Constitutive model:
Parameters Value
Porosity 0.2
Density Solid 2650
Young Modules (Kpa) 6000
Poisson Ratio 0.2
Cohesion(KPa) 1
Frictional angle 30
Dilatancy Angle 0
• Forward Model : MPM-2D Axisymmetric
• Cone Diameter = 3.57 Cm
GIP SIMON Project , MPM 2D Code Provided By Mario Martineli , Vahid Galavi and Faraz Sadeghi Tehrani

19. Iteration No Young
Module
KPa
Frictional
angle
Weighted
Error
1 4289 27.11 211.15
2 3921 34.12 20.788
3 4352 34.92 2.0565
4 4585 33.83 1.9234
5 4918 32.45 1.6839
6 5306 31.52 0.88127
7 5512 31.04 0.86326
8 6035 29.79 0.15969
9 6040 29.9 0.10034
Example1: Estimation of Mohr-Coulomb Parameters by CPT
• Parameters of Interest :
Frictional angle and young modules are assumed as unknown parameters
• Optimization Algorithm : Modified Gauss Newton Method

20. Example1: Estimation of Mohr-Coulomb Parameters by CPT
3000
3500
4000
4500
5000
5500
6000
6500
1 3 5 7 9
YoungModule(KPa)
Iteration No
Parameters Value
Desired Value
20
22
24
26
28
30
32
34
36
1 3 5 7 9
FrictionalAngle
Iteration No
-50
0
50
100
150
200
250
1 3 5 7 9
weightedResidualError
Iteration No
Iteration friction angle Young Modulus
1 10.1798 7.60181
2 11.2 6.398
3 13.7643 7.99513
4 13.08 7.5636
5 14.3756 7.272
6 15.1558 5.94192
7 14.6878 6.25564
8 16.1729 6.73256
9 15.7852 6.35672
Composite Scaled Sensitivity :
the importance of different
parameters to the calculation
Convergence criteria:
• The maximum parameter change of a
given iteration is less than a user-defined
tolerance
• The objective function, S(b), changes
less than a user-defined tolerance for
three consecutive iterations

21. Example2 : The run out of a dry granular flow
Small scaled flow test conducted by (DENLINGER AND IVERSON 2001)
• The experiment used a small flume with a bed surface inclined
31.4° adjoined to a horizontal runout plane.
• After material release, the flow accelerates gradually
spreading and reaching the end of the inclined plane
Denlinger, R. P., & Iverson, R. M. (2001). Flow of variably fluidized granular masses across three‐dimensional terrain: 2. Numerical predictions and experimental
tests. Journal of Geophysical Research: Solid Earth, 106(B1), 553-566.)
Experiment outcomes:

22. Example2 : The run out of a dry granular flow
Ceccato and Simonini 2016:
• only run out distance was
considered as the benchmark
• Parametric analysis
• The best agreement with the
experimental results were obtained
when :
basal friction angle equal to 26.6°
Static friction angle equal to 40°
Ceccato, F., & Simonini, P. (2016). Study of landslide run-out and impact on protection structures with the Material Point Method. In INTERPRAEVENT 2016-
Conference Proceedings
• MPM Model with Anura 3D
12555 Element
4 particles per element
• Parameters of Interest :
Soil Frictional Angle
Basal frictional Coefficient
Frictional contact

23. Example2 : The run out of a dry granular flow
Quantitative calibration based on run out distance and also soil thickness :
Benchmark : Soil thickness after propagation
𝜑 = 43
𝑡𝑎𝑛𝜑 𝑏 = 0.47
Cross Section

24. Example2 : The run out of a dry granular flow

25. Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material
Scaled slope failure experiment conducted by Eckersley 1990
• Instability was induced by raising the water level
• Water entered the slope from constant head tank
• To avoid surface erosion , water is injected through a
wire cage filled with coarse gravel
• Instability starts when the water level at the rear
of the model reached at 0.4 m
• Glass Tank
• Basal surface
Plywood floor
• Water proof sand paper glued to the floor
in order to inhabit direct sliding along the coal /
floor Interface, Angle of shearing resistance was
reported in the range of 30 to 36
Eckersley, D. (1990). Instrumented laboratory flowslides. Geotechnique, 40(3), 489-502.

26. Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material:
Material Properties
Name = Northwich Park Coal
4 ICU triaxial tests with confining
stress equal to 50 KPa
Different initial void ratio :
e = 0.41 , Contractive behavior
e = 0.38 , contractive behavior
e = 0.34 , contractive behavior
e = 0.32 , Dilative behavior
Specific Gravity = 1.34
𝑑10 = 0.06 − 0.3 𝑚𝑚
Minimum Density = 0.6 gr/cm3
Maximum density = 1.1 gr/cm3
Hydraulic conductivity = 0.01 cm/s2 Typical particle size distribution
Effective path stress
Saturated CU triaxial tests State diagram
Effective critical frictional angle = 40 degree
Available laboratory test Data:

27. Toe sliding duo to water seepageStage 1
Stage 2 Shallow failure along the slope surface
Stage 3 Deep failure and mobilize fairly all of
the domain
Geometry after flow slide
Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material:
Over view of the experience : Failure modes
Eckersley, D. (1990). Instrumented laboratory flowslides. Geotechnique, 40(3), 489-502.

28. Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material:
Selection of of constitutive models
Evaluation of constitutive models capability to simulate the material behavior through simulating an undrained
consolidated Triaxial test by Anura 3D
 Mohr Coulomb Strain Softening
Knownorreported
Parameters
Density solid (gr/cm3)
Porosity
Intrinsic Permeability
Peak Frictional angle
Residual Frictional
angle
Peak Cohesion (Kpa)
Residual cohesion
Young Module
Dilation angle
Shape factor
Value
1340
0.26
10 e-09
40
40
0.00
0
Clearly reported in case study
Unknown Parameters ?????????
?????????
?????????
FittingParameters
Parametric Study

29. Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material:
Selection of of constitutive models
 Mohr Coulomb Strain Softening
Fitting Parameters
Parameters SET01 SET02 SET03 SET04 SET05 SET06
E KPa 1000 1000 1000 1000 1000 1000
Ψ -5 -5 -5 -15 -15 -15
𝛽 4 40 400 4 40 400
.0
20.0
40.0
60.0
80.0
100.0
120.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12
DeviatoricStress(KPa) Axial Strain
SET 01 SET 02 SET 03
SET 04 SET 05 SET 06
.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
0 20 40 60 80 100
DeviatoricStress(KPa)
Mean effective Stress (KPa)
.0
10.0
20.0
30.0
40.0
50.0
60.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Porewaterpressure(KPa)
Axial Strain
SET 01 SET 02 SET 03
SET 04 SET 05 SET 06
• The model can not take into the
account the effect of density
(a) (b)
Cuomo, Ghasemi, Martinelli, Calvello, (2017) Simulation of liquefaction and retrogressive slope failure in loose cohesionless material
• one set of parameters can not
predict mechanical features
of all three tests at the same
time
pore water pressure
Shear strength
Effective Stress Path

30. Stress Initialization (quasi-static convergence)Removing fixities, applying liquid traction at the rear of the slope
Main Case Study : Slope Model : Geometry, boundary and Initial Condition
• One Particle formulation
• Mixed Integration
• For the Frist time Layered Soil in one particle formulation
• Special Consideration for pressure smoothing in Mixed
Element
• Number of Element = 3,336
• Number of Particle per element = 4
Elastic material
Dry
Saturated Material
liquid horizontal fixity
Frictional Contact
Liquid pressure
Tracking Points and Zone:
Elastic Material Zone1 Zone2 Zone3
Frictional contact
Sat
dry
F1 F2 F3
F4
F5
Point X Y
F1 0.7769 0.09424
F2 1.458 0.125
F3 1.762 0.09192
F4 0.9624 0.6130
F5 0.6833 0.6541
Liquid Fixity
Linearliquidtraction

31. Slope Model : Simulation By Strain Softening
Movie Time!!!
• Slope Simulation MC_ Strain Softening
• Progressive Collapse

32. Slope Model : Simulation By Strain Softening
Parameters Values
Density solid
(gr/cm3)
1340
Porosity 0.26
Intrinsic
Permeability
10 e-09
Peak Frictional
angle
40
Residual
Frictional angle
40
Peak Cohesion
(Kpa)
0.00
Residual
cohesion
0
Young Module 1000
Dilation angle -15
Shape factor 40
• Progressive failure
• Similar to stage 1 and 2 reported in experiment
• 3rd stage is missing
• Excess pore water pressure was not built enough to make further collapse
Time =5=0.3
Time =0.6
Final _scheme Time =1.2
Deviatoric Strain
Zone 1
Zone 2
Zone 3
Time (sec)
PWP(Kpa)PWP(Kpa)PWP(Kpa)
(a) (b)
Zone 3
Zone 2
Zone 3

33. The model by von Wolffersdorff (1996) is now considered as a reference hypoplastic model for granular materials. It
requires 8 material parameters:
𝜑𝑐 is the critical state friction angle.
ℎ 𝑠 and n control the shape of limiting void ratio curves (normal
compression lines and critical state line).
Ec0 is critical void ratio at zero stress
Ei0 is maximum void ratio at zero stress
Ed0 is minimal void ratio at the state of maximum density
𝛼 controls the dependency of peak friction angle on relative
Density.
𝛽 controls the dependency of soil stiffness on relative density.
Reported By case Study
Obtainable from State Diagram
Fitting parameters
Intergranular Parameters :
The intergranular strain concept (Niemunis and Herle 1997) enables to model small-strain-stiffness effects in
hypoplasticity. It requires 5 more material parameters:
mR: parameter controlling the initial (very-small-strain) shear
modulus upon 180 strain path reversal and in the initial loading
mT : parameter controlling the initial shear modulus upon 90
strain path reversal
R: the size of elastic range (in the strain Space)
𝛽𝑟 and 𝑥 control the rate of degradation of the stiffness with strain.
Main Parameters :
Fitting parameters
Main Case Study : Liquefaction and Retrogressive failure in Cohesion less Material:
Selection of of constitutive models
• Hypoplasticity
𝜑𝑐 40
ℎ 𝑠 93KPa
n 0.07
Ec0 0.93

34. i) Deviatoric stress versus axial strain
ii) pore water pressure versus axial strain
𝑒𝑟𝑟𝑜𝑟 =
𝑦′
𝑖 − 𝑦 𝑖 2
[0.01 × 𝑤𝑒𝑖𝑔ℎ𝑡 ]2
𝑛
𝑖=1𝑔𝑟𝑎𝑝ℎ𝑠
𝑛, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
𝑦′
𝑖 , 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖
𝑦 𝑖 , 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖
Optimization algorithm
New method called SQPSO, it is a modification of
conventional Particle Swarm Optimization method
adopted for the functions in which the large number of
parameters are going to be estimated.
(Hosseinnezhad et al 2014)
Objective function:
Parameters of interest: 10 Parameters
𝜑𝑐 , Ei0 , Ed0 , 𝛼 , 𝛽 , mR, mT , R, 𝛽𝑟 and 𝑥
PWP
q
Axial deformationAxial deformation
Graph discretization
(Eckersley 1986)
Observation
Main Case Study : Inverse analysis in order to obtain Hypo plasticity Parameters
2000
2050
2100
2150
2200
2250
0 10 20 30 40 50 60 70 80
ModelErrorFuncrion
Number of Iteration

35. Test NP-02 (Initial void ratio = 0.410)
Test NP-13 (Initial void ratio = 0.34)
Test NP-5 (Initial void ratio = 0.38)
Parameters Estimated Values
𝜑𝑐 [○] 40
ℎ 𝑠 [Kpa] 93.27195
n 0.0768
Ec0 0.93
Ei0 1.3
Ed0 0.397758
𝛼 0.349126
𝛽 0.562619
𝑚 𝑅 1.33593
𝑚 𝑇 7.05643
R 0.000142
𝛽𝑟 0.0773067
𝑥 0.951017
Main Case Study : Hypo-plasticity: Results of inverse Analysis

36. Main Case Study : Hypo-plasticity: Check the obtained Parameters
Parameters Estimated Values
𝜑𝑐 [○] 40
ℎ 𝑠 [Kpa] 93.27195
n 0.0768
Ec0 0.93
Ei0 1.3
Ed0 0.397758
𝛼 0.349126
𝛽 0.562619
𝑚 𝑅 1.33593
𝑚 𝑇 1.05643
R 0.000142
𝛽𝑟 0.0773067
𝑥 0.951017
Adding Prior knowledge to the results
is recalibrated
𝛼 = 0.21
0
5
10
15
20
25
30
35
40
45
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
FrictionalAngel
Axial strain
Alpha = 0.05
Alpha = 0.34
Alpha = 0.2
Alpha = 0.54

37. Main Case Study : Hypo-plasticity: Check the obtained Parameters
Comparing the results without and with modification:
f1
f3
f4
f2 Point 3
Point 2
Point 4
Point 2
f1
f3
f4
f2
Point 1
f1 f3f4 f2
f1 f3
f4 f2
2nd Stage(time = 0.6 sec)
1st Stage (time = 0.3 sec)
3rd Stage(time = 1.3 sec)
Final Shape (Time =1.7)
(b)
2nd Stage (time = 0.6 sec)
Final Shape (Time =1.4)
1st Stage (time = 0.3)
3rd Stage (time =1.1 Sec)
(a)
Deviatoric Strain
Without modification With modification TEST results

38. Slope Model : Simulation By Hypo plasticity
Movie
• Slope simulation with Hypo plasticity
• Retrogressive failure:

39. Slope Model : Simulation By Hypo plasticity
Movie
• Slope simulation with Hypo plasticity
• Static Liquefaction :

40. -2
-1
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
PoreWaterPressure(KPa)
Time (sec)
-2
-1
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
PoreWaterPressure(KPa)
Time (sec)
-2
-1
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
PoreWaterPressure(KPa)
Time (sec)
Porewaterpressure(KPa)
Zone1
Mean effective stress
Zone2
Zone3
Stage 1
Stage 2
Stage 3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
MeanEffectiveStress(KPa)
Time (sec)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
MeanEffectiveStress(KPa)
Time (sec)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
MeanEffectiveStress(KPa)
Time (sec)
Stage 1 Stage 2 Stage 3
Zone3
Zone2
Zone1
Meaneffectivestress(KPa) Overall Time (sec)
Stage 1 Stage 2 Stage 3
Overall Time (sec)
Zone3
Zone2
Zone1
f1
f3
f4
f2 Point 3
Point 2
Point 4
Point 2
f1
f3
f4
f2
Point 1
f1 f3f4 f2
e initial = 0.60
Deviatoric Strain Solid
1st Stage
2nd Stage
3rd Stage
2nd Stage(time = 0.8 sec)
1st Stage (time = 0.4 sec)
3rd Stage(time = 1.1 sec)
Slope Model : Simulation By Hypoplasticity
• Geometric Validation
• Soil Response in Fixed Coordinate

41. Remarks
Most advantage of Anura 3D: Capable to simulate all of the stage of phenomenon . appropriate for both of
inverse modeling scale; to calibrate small and large deformation parameters
Advanced numerical method are being developed, the validation of their capability is extremely needed
• Large deformation phenomenon usually involves complex mechanism
• Simulation of complex system usually involves complex constitutive model
• Many advanced constitutive models are already implemented in Geotechnical Codes but the difficulties in
choosing their parameters undermines their capability and applications
• Inverse Analysis feature could simplify the usage of advanced model
Strain Softening Mohr-coulomb
Modified Cam clay
Hypoplasticity
Mohr-coulomb
• Inverse Analysis could easily estimates Numerical Parameters which doesn't have physical meaning :
Shape factor
Contact coefficient
And ………
Modified hypoplasticity
Norsand
• Using the inappropriate constitutive model could lead to wrong prediction of the phenomenon

42. Thank you
Pooyan Ghasemi and Mario Martinelli
Deltares

43. Appendix
Comparing the results without and with modification:
f1
f3
f4
f2 Point 3
Point 2
Point 4
Point 2
f1
f3
f4
f2
Point 1
f1 f3f4 f2
f1 f3
f4 f2
2nd Stage(time = 0.6 sec)
1st Stage (time = 0.3 sec)
3rd Stage(time = 1.3 sec)
Final Shape (Time =1.7)
(b)
2nd Stage (time = 0.6 sec)
Final Shape (Time =1.4)
1st Stage (time = 0.3)
3rd Stage (time =1.1 Sec)
(a)
Deviatoric Strain
Without modification With modification