These examples use coupled electro-mechanical cellular models to generate the active fibre tension component for use in ZCTG5A, as part of the continuum mechanics solution process. The finite element strains are then used to compute the new extension ratios at each of the grid points ("cells").

The procedure used goes something like this...

  1. Define the model geometry, fibres, etc;
  2. Define the continuum/distributed mechanics equations, material parameters, solution method, etc;
  3. Define the cellular model, material parameters, solution method, etc;
  4. Time loop:
    1. Compute the cellular extension ratios from the continuum strains/geometry;
    2. Solve the cellular model for the current time step;
    3. Compute the active fibre tension component of the tension tensor at each gauss point from the grid points;
    4. Solve the continuum mechanics for the current time step.

The trouble so far...

So far, including the fading memory equations of the HMT model leads to severe instability in the solution. This has been avoided by simply using the isometric tension generated at each of the cells - i.e., treating each of the cellular time steps as a semi steady state problem.

Due to the distributed nature of finite elements, it is entirely possible that the wrong values will be obtained when computing the extension ratios at each of the grid points. This is due to the electrical wave taking a finite time to cross a given element, which should mean that the different parts of the element begin developing tension at different times causing the element to contract. However, since the extension ratios are interpolated from nodal strains, the value computed at a grid point depends on the element's basis - for example, a 2D bilinear element will give the same extension ratio at every grid point in the element resulting in cells generating tension before they are actually activated by the electrical wave. One way to fix this would be to use cubic Hermite elements on the same scale as the grid points, which due to the continuity of strain between elements, would give the correct extension ratio at each grid point. Another way is to use fairly small elements and a large continuum mechanics time step such that you can assume that the whole element is in the same state for each distributed mechanics solution. Or better yet, come up with some other interpolation scheme...

If the distributed mesh is held too rigid (e.g., fixing all of the boundary nodes in space) the distributed and cellular mechanics tend to be very unstable. This instability is caused by a feedback loop where a element contracts, causing another element with one or more fixed edges being unrealistically stretched which causes that element to contract against the applied load which then stretches another element...