Thursday, March 21, 2013

Stiff Systems - III

Several changes made to the entire codes. In order to solve every type of stiff system effectively, the only was seems to be to completely recalculate not only the A, B and E matrices but the KVL loops and their interactions.

At the highest level, I defined another layer called the branch. Currently, it is another list, but it may be an object depending on the later use. I have defined the branch as follows (click on "view raw" to see the code in a new window):

There was a problem where some branches were the reverse of the other and these were difficult to handle when loops were being manipulated. So I added the code to delete the redundant branches and also remove them from the system_loops matrix.

Another concept planned for the next stage was that each branch can have an "event" flag. That is if there is a change in the parameters like a diode starting or stopping conduction, there will be an event generated. When an event is generated on a branch, an event is generated to the entire system and only then will the system matrices A, B and E be recalculated. Otherwise, leave them as it is and just solve the equations for new inputs. Even in a power electronics circuit, this may result in significant savings as a change may happen once in 20-50 microseconds so that with a time step of 1 microsecond, the computation can be 20-50x faster if all the recalculation are left out.

The branches are updated as follows (click on "view raw" to see the code in a new window):

Also, each class has another method called "transfer_to_branch".

The really huge change has been in the file. The basic concept is described here. The file has comments as far as possible.

1. The way to find out if a loop is stiff is to check the L/R ratio. If the L/R ratio of the diagonal element is low, the offdiagonal elements are checked. If they are also stiff, a row manipulation is done with the lower loops so that the stiff elements are removed from the lower loops.

2. Once the stiff elements have been isolated to individual loops by row operations, the loops are now recalculated.

3. Once again the system matrices A, B and E are then calculated from the new loops and now the ODE is solved.

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