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Periodic Boundary Conditions

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\nAuthor: Edward Vigmond edward.vigmond@u-bordeaux.fr\ \ \n<h2 id=\"periodic-bc-tutorial\">Periodic Boundary Conditions\n<div\ \ class=\"sectionauthor\">\n

Edward Vigmond <<a href=\"mailto:edward.vigmond@u-bordeaux.fr\"\

edward.vigmond@u-bordeaux.fr>

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Periodic boundary conditions\ \ connect the left edge of a sheet to the right, or the top to the bottom. This,\ \ for example models a cylinder very simply while allowing visualization of the\ \ whole domain. The intact atria or ventricles are better approximated by sheets\ \ with periodic boundary conditions and not the simpler no flux boundaries. Indeed,\ \ reentries last longer on periodic domains since they do not annilhilmate at boundary\ \ edges.

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The implementation of periodic b.c.'s is not straightforward since\ \ adding the equations equating nodes on the left boundary to those on the right\ \ boundary (or top and bottom) is mathematically trivial but destroys the symmetry\ \ of the system, and, thereby, invalidates many conjugate gradient preconditioners.\ \ A workaround is to define line connections linking the edges of the tissue, and\ \ assigning ridiculously large conductivity values to these links, effectively short\ \ circuiting the nodes. The mesher program will generate these links.

\n<h2 id=\"\ experimental-setup\">Experimental Setup\n<div id=\"fig-period-bcs\">\n
\n\ <img src=\"07b_periodic/perCnnx.png\" class=\"align-center\" style=\"width:30.0%\"\ \ alt=\"\" />
A sheet showing periodic boundary conditions in X implemented\ \ with linear elements (in pink) joining the left and right edges.
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\n<img src=\"07b_periodic/perbcs.gif\" class=\"align-center\"\ \ style=\"width:30.0%\" alt=\"\" />
A point stimulus delivered off-center\ \ on a piece of tissue with periodic boundary conditions in both directions.
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\n<h2 id=\"initiating-reentry\">Initiating Reentry\n

The standard\ \ cross-shock method of reentry initiation will fail with periodic b.c.'s. The initial\ \ line stimulus will produce two wavefronts which collide, and the quarter sheet\ \ S2 will quickly go everywhere and annihilate itself. There are two simple ways\ \ to induce reentry.

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  • We define a mesh without the periodic\ \ connections and run an intial portion of the silulation on that. Just after the\ \ S2, we save the state and restart on the mesh with the periodic b.c.'s enforced.
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  • We temporarily block propagation across an edge by clamping the transmembrane\ \ voltage and then turning off the clamp after the S2.
  • 07A_augmentation 07_extracellular