Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies

SAJJAD KHUDHUR ABBAS
Ceo , Founder & Head of SHacademy
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
Episode 50 : Simulation
Problem Solution Approaches
Convergence Techniques
Simulation Strategies

2
Convergence Techniques
Atypical algorithm for Newton's method
i.Choose initial vector x(0)
ii. Set iteration counter i = 0
iii.Check convergence  f (xi)
iv.
v.
vi.
vii.
  , if so stop
Solve linear system J(xi) i = -f (xi ), for i
Update step x i+l = xi + i
i = i + 1
return to (iii)

3
3.2.3.3. Quasi-Newton (QN) Methods
These methods represent a very important class of techniques because of their
extensive use in practical alqorithms. They attempt to use an approximation to the
Jacobian and then update this at each step thus reducing the overall computational
work.
The QN method uses an approximation Hk
to the true Jacobian i and computes the
step via a Newton-like iteration. That is,
Hk
sk
= -fk
; x k+l
= xk
+ k
sk
(3.11)
or for the inverse
Sk
= (Hk
) –1
We have, sk
= -f k
(3.12)
We wish to update Sk
or Hk
in each iteration so that they approach the true Jacobian
(or inverse). The type of updating formula determines the various QN methods. As
previously defined we have,
(3.13)
and
sk
= p k+l
= xk+1
-xk
q k+1
= f k+l
- f k
For the QN process, p k+l
and
The Taylor series would give,
q k+l
= J k
p k+l
+ ………
qk+1
can only be calculated after the step calculation.
(3.14)
However, the matrix Hk
generally does not quantities correctly. That is,
q k+1
 H k
pk+1
or
sk
p k+l
 pk+1
(3.15)
Thus we want to update H k
to H k+1
so that, q k+1
= H k
p k+1
. This is
Newton (QN) condition.
the Quasi-

Main steps of a QN-method
A typical Broyden algorithm
1. Estimate H(O)
by differencing and invert to S(O)
. Choose initial starting point x(o)
.
2. Evaluate E(O)
= f(x(o)
)T
f(x(o)
). If E(O)
< , exit with solution x(o)
.
3. Estimate p (1)
= - S(o)
f(o)
. Set i = 0
4. Let x (i+l)
= x (i)
+ p (i+l)
. Evaluate f (i+l)
. Check Error E (i+l)
.
5. Evaluate denominator in update formula [(p k+l
) T
S k
q k+l
]. If equal to 10 -6
go to step 7.
6. Update inverse: Si+1
= Si
+ (p i +l
- S i
q i+1
)(p i+l
) T
/ [(p i+l
) T
S i
q i+l
]
7. Estimate new step. Set i = i + 1, then, set p i+l
= -s i
f i
.
8. Return to step 4.

Sparse Matrix Methods
Definition: A sparse matrix is a matrix in which zero components dominate.
Aim: To eliminate operations on the zeros and so increase computation
speed and reduce storage requirements.
Structured Matrices: Tridiagonal systems where, A x = b

Example of an Unstructured matrix
1 




1 

f5
1 1 1
f3 1 1 1
f 4 1 1
1
A  
f
1
2
In order to solve the problem Ax = b,
first we need to convert A into the structured form
 x1 x 2 x 3 x 4 x 5 

f1
1 1 


x = Q y (3.28)
Solving Sparse Linear Systems: A x = b
The general approach is to reduce A to block lower triangular form, although
the matrix A could be treated directly using sparse elimination techniques. For
the block lower triangular form we have permutation matrices P and Q such
that we write our original system of equations as:
PA Q y = b` = P b (3.26)
and PA Q has the form:






ANN

AN1



 A11
 A21 A22

. .
.
.
.
.
Matrices Aii ; i = 1(1)N are square diagonal block matrices. The above system of
equations given by (3.26) can then be solved for y via a series of block forward
substitutions,
A11 y 1 = b` 1
(3.27)
Aii y i = b i -  Aik y k ; i = 2(1)N
The solution of the original system is found from the vector y by a simple
permutation.

x4 x5 
1 

1 1 1 1 
1 1 1


1 1 

1 1 1 

f5
f4
A  
2
f3
f
x1 x2 x3
f1 1


We can regard this as permuting rows of A such that
B = R A R = 1, 3, 5, 4, 2
That is, we have,
0 0 1
1 1 1 
1 0 1 0A
  
 1 0 0 1 0
 1 0 1 0 1
1 
x1 x2 x3 x4
f1 1 1
1 1 1 1 
B  
2
f3 1 1 1
1 1
1
1
0
  1

0
1
x5 


 4
f5
f
f


1
Note the corrections for the lecture notes !

We can now associate the permutation matrix D with the output variable order, ie.,
D = 1 4 2 5 3. We can apply these permutations to B such that M = D B D T
to get,
0
0
1 0 0 0 0
 
0 0 1 0 0
DT
 0 0 0 0 1

1 0 0 1 0
 
1 1 0 1 0
B  1 0 1 0 1
1 0 0 0 0
 
0 0 0 1 0
D  0 1 0 0 0
 
2
We solve M y = D R b for y
and then, x = D y (3.34)
Note the corrections for the lecture notes !

0 0
0 0 1



1 0 0 1 0
 
0 1
0 0
0 0 1 0 0 0 1 1 1 1 0 0 0 1
 x1 x2 x3 x4 x5 

f1 1 1 0 0 0 

f 1 1 0 0 0 
f3 1 1 1 0 0


f4

0 1 1 1 1 

f5 1 0 0 1 1 

Simulation Strategy: What to select?
E-1
E-2
E-3
R-1
SP-1
D-1
C-1
variables Solve
x
M-1
x
x
Equation Oriented
x
equations
x
x represents variables of
the connecting streamsx
x
x
M-1 E-1 E-2 R-1 E-3 D-1 SP-1 C-1

What is Simulation Strategy ?
* Determine appropriate solution method by
analyzing the needs of the problem being solved
- Which approach to use ?
- Only one approach ?
- Choice of numerical method
- Initialization

What is Simulation Strategy ?
* Determine appropriate solution method by
analyzing the needs of the problem being solved
- Which approach to use ? Depends on the problem
- Only one approach ? Depends on the problem
- Choice of numerical method Depends on choices
made above
- Initialization

Selection of Simulation Approach
*Recycle-loops: How many are present ?
* Process Model: Linear or non-linear ?
*Type of simulation problem: Identify type
* Process information: What is known ?
*Robustness: Must always give a result?
* Computational efficiency: Must be very fast?

Modular approachversus equation orientedapproach
Strategy: Start with SM and switch to EO or
Two-Tier approach (simultaneous modular)
Sequential ModularApproach EquationOrientedApproach
Simulateone unit model at a time Solveall unit models together
Decompose flowsheet Order equations
Iteratein tear streams Update all unknown variables
simultaneously
Less flexible but morerobust Moreflexible but less robust
Initializationis important Initializationis veryimportant
Storage requirement not high Storagerequirement can be veryhigh

Convergence Techniques (Modular)
Table3.1:Theformof Jthat canbeusedinequations3.35–3.36for different convergencetechniques.
h (y) = y - w = 0
y i+l = y i – J h (y i)
Equation for tear-
stream convergence
Update method
Choice of the method defines J
Flowsheeting problem, use SS and then
WM, for specification problem, use Broydon
Method J
Successivesubstitution I
Wegstein D=diag{d}; djj =(yi
–yi-1
)/(hi
–hi-1
)j j j j
Dominant Eigen-value 1/(1-)I; =(wi
–wi -1
)/(yi
–yi-1
)
Broydon’srule Full matrixQN–update(see3.2.3.3)
Newton [F(yi
)/ y]–1

Table3.1:Theformof Jthat canbeusedinequations3.35–3.36fordifferentconvergencetechniques.
Convergence Techniques (Equation Oriented)
F  A y - b = 0
y i+l = y i – J h (y i)
Mathematical model
of process flowsheet
Update method
NM or QN-methods
solve EO &
optimization problems
Choice of the method defines J
Method J
Successivesubstitution I
Wegstein D=diag{d};djj =(yi
–yi-1
)/(hi
–hi-1
)j j j j
Dominant Eigen-value 1/(1-)I;=(wi
–wi-1
)/(yi
–yi-1
)
Broydon’srule FullmatrixQN–update(see3.2.3.3)
Newton [F(yi
)/y]–1

Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies

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Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies

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Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies