Computational Tricks¶
This page contains a list of computational tricks and/or simplifications that have been used in writing the software. In some cases, they are based on physical intuition, and in others computational convenience. This may not be a complete list, but I have done my best to make sure it is.
Stresslet and Rate of Strain Storage¶
- The rate of strain
and stresslet
are rank-2 tensors with 9 components each. However, both tensors are symmetric and traceless, so only 5 of the components are independent, and can be represented as a rank-1 tensor. The components chosen for this plugin are:
- With any of the 5 independent components (or linear combinations thereof), the resistance couplings can be given as
- where
is a permutation tensor that builds the full 9-component stresslet from the 5 independent components and
projects the 5 independent components out of the full rate of strain tensor. These permutation matrices are chosen to maintain the symmetry relations of the resistance tensors, i.e.
- It is straightforward to show that the permutation tensors that satisfy these conditions give
Therefore, these 5 components of the stresslet and rate of strain are stored in the simulation.