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 {\bf E} and stresslet {\bf S} 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:

E_{xx}, E_{xy}, E_{xz}, E_{yy}, E_{yz}, \\
S_{xx}, S_{xy}, S_{xz}, S_{yy}, S_{yz}.

With any of the 5 independent components (or linear combinations thereof), the resistance couplings can be given as

\tilde{\bf R}_{\rm SU} = {\bf P}^{-1}_{S} \cdot {\bf R}_{\rm SU}, \\
\tilde{\bf R}_{\rm FE} = {\bf R}_{\rm FE} \cdot {\bf P}_{E}, \\
\tilde{\bf R}_{\rm SE} = {\bf P}^{-1}_{S} \cdot {\bf R}_{\rm SE} \cdot {\bf P}_{E},

where {\bf P}^{-1}_{S} is a permutation tensor that builds the full 9-component stresslet from the 5 independent components and {\bf P}_{E} 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.

\tilde{\bf R}_{SE} = \tilde{\bf R}_{SE}^{T}, \\
\tilde{\bf R}_{SU} = \tilde{\bf R}_{FE}^{T}.

It is straightforward to show that the permutation tensors that satisfy these conditions give

\begin{bmatrix} S_{xx}\\ S_{xy}\\ S_{xz}\\ S_{yy}\\ S_{yz} \end{bmatrix} = {\bf P}_{S} \cdot \begin{bmatrix}  S_{xx} & S_{xy} & S_{xz} \\ S_{yy} & S_{yy} & S_{yz} \\ S_{zx} & S_{zy} & S_{zz} \end{bmatrix}, \\
\begin{bmatrix} E_{xx} - E_{zz} \\ 2 E_{xy}\\ 2 E_{xz} \\ 2 E_{yz} \\ E_{yy} - E_{zz} \end{bmatrix} = {\bf P}_{E} \cdot \begin{bmatrix}  E_{xx} & E_{xy} & E_{xz} \\ E_{yy} & E_{yy} & E_{yz} \\ E_{zx} & E_{zy} & E_{zz} \end{bmatrix}.

Therefore, these 5 components of the stresslet and rate of strain are stored in the simulation.