Transcript شبيه سازی - Ferdowsi University of Mashhad

‫شبيه سازی‬
Simulation
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!‫مقايسه‬
• Experimental
– Provide useful quantitative information
– Are common as they use real system
– Considerable Time and cost usage (repetition!)
• Numerical simulation (Computer Simulation)
– Virtual systems
– To predict the behaviour of a real system
– More flexible in application
– Micro and Macro scale results at any time
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Experimental Simulation
•In-lab experiment that is as much like some
real situation as possible.
•Small scale equipment
•Example:
 ground-based flight (Pesticide application), Dam,
rainfall and Silo simulators
 behaves as closely as possible to a real one
 still under researcher control
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Experimental Simulation
 Still fairly precise.
More realistic than in-lab experiment.
Not a natural setting – interaction may not be normal.
Extrapolation of Results may lead to uncertainty and
ERRORS
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Computer (Numerical) Simulation
• Creating a complete & closed system that
models the operation of the real system
without users.
• Example:
– Plant growth simulations (Agronomy
researchers)
– Engineering Models
• Continuum Models
• Discrete Element Models
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Continuum approach
• The behaviour of a mechanical system
can be expressed by differential equations
• Mechanical system is divided into Finite
elements
• Derived constitutive equations for
elements are linked together to solve the
problem
• Application for Stress and heat analysis
• Finite Element Method (FEM)
• Boundary Element Method (BEM)
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What is FEM?
• Full name: Finite Element Method
General Concepts
 FEM cuts a structure into several elements of the structure
 The nodes at the each end of an element are reconnected as if nodes were
pins that hold elements together
 This results in a set of simultaneous algebraic equations
Applications of FEM
Desire to understand how various elements behave with arbitrary shape,
loads, and support conditions
 Can be contained within a single computer program for users to input data
such as geometry, boundary conditions, and element selection
 Handle complex restraints, which allow indeterminate structures to be
solved

Disadvantages of FEM


FEM obtains only approximate solutions
Many input data are required
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Discrete Element Method
• Continuum models based on Continuity
• Increases in computer speed
• Calculation of the position of individual
particles
• DEM Useful for particulate materials
– Grains, Soil , Powder, Fruits
– Solid systems also can be modelled
• Need good Programming skills
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Main steps of DEM
• Particle and environment generation
• Search for contact
• Contact detection between pairs of
discrete particles
• Calculation of contact force
• Update particle motion due to unbalanced
force
• Circulation
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DEM Development
• Appeared in 1979 by Cundall and Strack
• Shape representation
– Circle (2D)
– Sphere (3D)
– Ellipse (2D)
– Ellipsoid (3D)
– Polygon (2 & 3D)
– Combined Primitives
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DEM Application
•
•
•
•
•
•
Geomechanics (Soil & rock)
Granular storage & flow (Silo)
Powder Technology
Fruits and Vegetable (handling)
Processing Operation ( ball mills)
Continuous System BUT composed of
individual ( say particles) in Microscopic
level ( Asphalt, Biomaterials, solid
structures)
• Combined DEM & FEM
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DEM Limitations
• Matching Real and Model particle shape
• Not for very large spatial domain where
millions of particles involved
• Need for physical properties
• Running time concern
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Theoretical aspects
• Contact models between contacting
bodies
• Contact area
• Contact point
• Contact vector
• Contact Displacement and deformation
– Normal
– Tangetial
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Particle centroid
A
A
Contact parameters: a) for
smooth non-spherical, convex; b)
circular particles
fA
lA
fB
A
n
fB
fA
BlB
B
B
n,
l
A
B
(b)
(a)
st
n
sr
(a)
Normal (n) and tangential (st and sr) displacements
for two particles in contact due to: (a) relative
translation and (b) relative rotation of the particles.
(b)
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Principle of DEM
The DEM procedure for contact force
calculations and updating the
dynamic situation of particles; a)
Recognising the formation of contact
points due to relative velocities and
position of particles. b) Application of
force-displacement law for each
contact point to calculate the contact
force. c) The moment of contact
forces about the particle centroid is
calculated and the resultant force
and moment on particle centroid is
determined.
d) Application
of
Newton’s second law of motion to
calculate the particle acceleration
and velocity.
(a)
F1
F3
(b)
F2
F4

M
F

a
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(c)
(d)
v
Fn  K 
3
2
Contact Models
1
4
K  R2E
3
1
1
1


R R1 R2
kn
1 
1 1


E
E1
E2
2
1
2
2
kt 

2 3G (1   ) Re
2 
Kn
Remove when
sliding occur
kt
2Gs
kn 
2R


(1   ) 2 ln( e )  1
A


2
s
cn
cn

1
3
m
1
3
n
ct
kt
m
ct
(a)
(b)
F
DEM contact models for cohesionless materials: a) the maximum
frictional force based on the sum of spring and dashpot; b) the
maximum frictional force calculated only from spring (elastic) force
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Damping
• Contact Damping
Fdn= Cn . ń
Fds = Ct . ś

ln e
(ln e) 2   2
cn  k n
ct  k t
Cn  2 k n m*
• Global Damping (act on Absolute Velocity & rotation)
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Time step
• In DEM the time step is the time during which force
is transmitted from one contact point to another
along the particle boundary.
• The time step should be as large as possible to
increase the efficiency of simulation and still be
smaller than the critical time step to justify the
assumption of constant acceleration within each
time step and to ensure stability of the calculations
• The idea is based on the assumption that the
selected time step is small enough so that no new
contacts take place in the current time step except
those that have already been recognised at the
beginning of the time step.
t cr 

k
ln 2 e
(1  2
)
2
m
  ln e
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Contact Detection
• the most important step prior to any
mechanical calculation is determination of
which surfaces are in contact and the type of
contact.
• It is estimated that more than 80% of the
computational time can be spent on this task.
• In a very simple approach each particle is
checked against every other particle to
determine any probable contact.
• The computational time for this simple
procedure with n particles will be
proportional to , which is too long if there are
hundreds of particles in the simulation
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Contact Detection (cont.)
• For densely pack a “link list“ algorithm
• the simulation space is divided into relatively large
cells
• A separate list of particles for each cell is provided,
including the particles in the home cell and
surrounding cells.
• The particles within a cell and its neighbouring
cells are considered as potential contacting bodies.
• Therefore, contact detection for such list would be
an efficient process regarding time consumption
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Contact Detection (cont.)
• For loosely pack a grid search
– Small cells, so that in each cell one particle
can be occupied
m
R1
R2
L
L
L  R1  R2
(b)
(a)
Contact detection between particles; a)
Circular shape b) Polygonal shape.
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Calculation cycle
2
3
(a)
F2
6
1
4 5
(b)
1
F3
F4
F5
(c)
p
F6
1
ap
(d)
• Simulation steps in DEM; a) Particle and
environment generation, b) Contact search and
detection, c) Calculation of contact force, d)
update the particle accelerations.
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START
Interpret the user commands and
material parameters from input file
Random particle generation
or restoring parameters and
arrays.
Reboxing
and update
contacts
RESTART
Calculation cycle and
update of the position
of particles and contact
forces
Out put of numerical results
STOP
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visualisation
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