594
V. Skorych et al.
3.3 Waveform Relaxation Method
To compute partitions with tear streams, an iterative procedure is applied in Dyssol. Dynamic modelling is usually performed for a large time interval, during which
process parameters can vary significantly. Moreover, the number of parameters that
participate in the convergence detection is not limited and depends on the complexity
of the scheme. Consequently, the calculation of such partitions may be computationally intensive. To overcome this problem, the simulation framework Dyssol applies a
calculation approach based on the waveform relaxation method (WRM). It was originally proposed to accelerate calculation of systems of differential algebraic equations
by decomposing it into several subsystems, which can be then calculated separately
[33].
To apply it to the solution of the convergence problem in dynamic flowsheet modelling, this method has been slightly modified [16, 53]. The entire simulation interval
[t START ; t END ] (Fig. 5) is sampled into smaller frames—so called time windows. The
length of these windows (t W INDOW ) is selected such that the changes in the material
parameters inside each of them are small, which ensures faster convergence. To calculate each partition, the first window [t START ; t START + t W INDOW ] is selected, all the
parameters of tear streams inside the partition are initialized, and iterative calculation of all units is performed until convergence. In the simplest case, the difference
between the parameters of tear streams at two consecutive iterations can serve as a
convergence criterion. When the convergence is reached, the system initializes the
next time window [t START + t W INDOW ; t START + 2 · t W INDOW ] and proceeds to its calculation according to the same procedure. Sequential calculation of all time windows
from t START to t END for all partitions will lead to a rapid convergence of the entire
solution.
The application of this approach imposes some restrictions on dynamic models:
their internal simulation time step cannot be larger than the window size. Thus,
t W INDOW is an important parameter that affects the overall calculation performance.
On the one hand, large time windows lead to an increase in deviations between
successive iterations, which, in turn, can lead to poor convergence rates. On the
other hand, too short time windows will lead to correspondingly small internal steps
in the models, which can slow down the calculations. Therefore, the size of the
time window in the Dyssol system is not a constant value, but may be changed
dynamically, according to changes in the behaviour of the simulated processes. The
faster convergence is achieved on a given time window, the larger the next one can
be, and the other way around.
3.4 Convergence Methods
Iterative calculations of partitions on each time window are performed in the
following order:
V. Skorych et al.
3.3 Waveform Relaxation Method
To compute partitions with tear streams, an iterative procedure is applied in Dyssol. Dynamic modelling is usually performed for a large time interval, during which
process parameters can vary significantly. Moreover, the number of parameters that
participate in the convergence detection is not limited and depends on the complexity
of the scheme. Consequently, the calculation of such partitions may be computationally intensive. To overcome this problem, the simulation framework Dyssol applies a
calculation approach based on the waveform relaxation method (WRM). It was originally proposed to accelerate calculation of systems of differential algebraic equations
by decomposing it into several subsystems, which can be then calculated separately
[33].
To apply it to the solution of the convergence problem in dynamic flowsheet modelling, this method has been slightly modified [16, 53]. The entire simulation interval
[t START ; t END ] (Fig. 5) is sampled into smaller frames—so called time windows. The
length of these windows (t W INDOW ) is selected such that the changes in the material
parameters inside each of them are small, which ensures faster convergence. To calculate each partition, the first window [t START ; t START + t W INDOW ] is selected, all the
parameters of tear streams inside the partition are initialized, and iterative calculation of all units is performed until convergence. In the simplest case, the difference
between the parameters of tear streams at two consecutive iterations can serve as a
convergence criterion. When the convergence is reached, the system initializes the
next time window [t START + t W INDOW ; t START + 2 · t W INDOW ] and proceeds to its calculation according to the same procedure. Sequential calculation of all time windows
from t START to t END for all partitions will lead to a rapid convergence of the entire
solution.
The application of this approach imposes some restrictions on dynamic models:
their internal simulation time step cannot be larger than the window size. Thus,
t W INDOW is an important parameter that affects the overall calculation performance.
On the one hand, large time windows lead to an increase in deviations between
successive iterations, which, in turn, can lead to poor convergence rates. On the
other hand, too short time windows will lead to correspondingly small internal steps
in the models, which can slow down the calculations. Therefore, the size of the
time window in the Dyssol system is not a constant value, but may be changed
dynamically, according to changes in the behaviour of the simulated processes. The
faster convergence is achieved on a given time window, the larger the next one can
be, and the other way around.
3.4 Convergence Methods
Iterative calculations of partitions on each time window are performed in the
following order: