596
V. Skorych et al.
applying this method, for each three iterations, two steps of the direct substitution
method and one step of the Aitkens acceleration formula are used:
x k+3 = x k −
(x k+1 − x k )
2
x k+2 − 2x k+1 + x k
.
(5)
The choice of the convergence methods for a specific simulation depends entirely
on the dynamics of the process under consideration.
3.5 Extrapolation Methods
After convergence has been achieved on the current time window, the simulation
system proceeds to calculate the next one. To do so, the parameters of tear streams
must be initialized again. The number of iterations and, hence, the convergence rate
in this case will significantly depend on how close these initial values are to the final
solution. The Dyssol system uses data extrapolation algorithms to calculate the initial
assumption for each subsequent time window based on the results of the calculation
of the current interval. One of the following methods can be applied for simulations
in Dyssol [53]:
• Nearest-neighbour extrapolation that initializes the whole time window TW i+1
with values obtained at the last time point of TW i .
• Linear extrapolation, which uses the first and the last time points of time window
TW i to extrapolate its values to TW i+1 .
• Cubic spline extrapolation [8] uses three points from TW i to derive cubic polynomials describing its behaviour, and then extrapolates the resulting function to
the next time interval TW i+1 .
The effectiveness of each method depends on the behaviour of a particular process:
the nearest-neighbour is well suitable for nearly constant behaviour, the linear gives
a good response if the data changes monotonously, and the cubic spline is the most
advanced method that can well predict non-monotonic changes.
3.6 Validation of the Simulation Algorithm and Performance
Analysis
To verify the developed calculation algorithms, several specially designed modelling
experiments were performed. First, to test the basic calculation algorithm, flowsheets consisting of purely mathematical functions were simulated, and the results
were compared to the analytical solutions from Simulink (MATLAB). For instance,
Eq. (6), which was represented by a scheme including three sine operations (marked
V. Skorych et al.
applying this method, for each three iterations, two steps of the direct substitution
method and one step of the Aitkens acceleration formula are used:
x k+3 = x k −
(x k+1 − x k )
2
x k+2 − 2x k+1 + x k
.
(5)
The choice of the convergence methods for a specific simulation depends entirely
on the dynamics of the process under consideration.
3.5 Extrapolation Methods
After convergence has been achieved on the current time window, the simulation
system proceeds to calculate the next one. To do so, the parameters of tear streams
must be initialized again. The number of iterations and, hence, the convergence rate
in this case will significantly depend on how close these initial values are to the final
solution. The Dyssol system uses data extrapolation algorithms to calculate the initial
assumption for each subsequent time window based on the results of the calculation
of the current interval. One of the following methods can be applied for simulations
in Dyssol [53]:
• Nearest-neighbour extrapolation that initializes the whole time window TW i+1
with values obtained at the last time point of TW i .
• Linear extrapolation, which uses the first and the last time points of time window
TW i to extrapolate its values to TW i+1 .
• Cubic spline extrapolation [8] uses three points from TW i to derive cubic polynomials describing its behaviour, and then extrapolates the resulting function to
the next time interval TW i+1 .
The effectiveness of each method depends on the behaviour of a particular process:
the nearest-neighbour is well suitable for nearly constant behaviour, the linear gives
a good response if the data changes monotonously, and the cubic spline is the most
advanced method that can well predict non-monotonic changes.
3.6 Validation of the Simulation Algorithm and Performance
Analysis
To verify the developed calculation algorithms, several specially designed modelling
experiments were performed. First, to test the basic calculation algorithm, flowsheets consisting of purely mathematical functions were simulated, and the results
were compared to the analytical solutions from Simulink (MATLAB). For instance,
Eq. (6), which was represented by a scheme including three sine operations (marked