# Adaptive numerical algorithms for cardiac modelling

Ross Nobes,
James Southern,
Bruno Carpentieri,
Giorgio Fotia,
Fabio Maggio

Technical Report , Deliverable 5.4 EU Project PREDICT (ICT-2007-2-224381) - september 2009

This report examines three adaptive methods that might lead to higher computational performance for simulation of the electrophysiology of the heart: i) refinement/coarsening of the spatial mesh, ii) adapting the time step, and iii) use of spectral elements. Based on previous work in the literature and our perceived needs for adaptive meshing methods for cardiac electrophysiology, we have decided to employ a non‐hierarchical anisotropic approach with a relatively simple error indicator and techniques that minimize the number of adaptive steps required. A survey of available libraries for adaptive meshing led to the choice of libadaptivity from the Applied Modelling and Computation Group at Imperial College London for use in conjunction with the heart simulator Chaste. An initial code combining libadaptivity and Chaste has been developed. Initial tests using a tissue slab and a realistic heart geometry were performed. For the latter, the adaptive code is 10.9 times faster than the unmodified Chaste with a fine fixed mesh that gives comparable accuracy. Further development will lead to even higher ratios. A simple time adaptivity strategy has also been implemented in Chaste, leading to a speed‐up of 2--3 over the non‐adaptive code. Spectral element methods have been compared with finite elements for solution of the bidomain equations via development of two prototype codes. Spectral elements are shown to provide an improvement in CPU time by a factor of 2--3 over finite elements, albeit with an increased memory requirement. A discussion of possible further work in the three areas is given. This will be pursued in combination with implementation on massively parallel computers as the next phase of the work in preDiCT.

## BibTex references

@TechReport{NSCFM09a,
author = {Nobes, R. and Southern, J. and Carpentieri, B. and Fotia, G. and Maggio, F.},
title = {Adaptive numerical algorithms for cardiac modelling},
institution = {Deliverable 5.4 EU Project PREDICT (ICT-2007-2-224381)},
month = {september},
year = {2009},
type = {techreport},
keywords = {Cardiac electrophysiology, mesh adaptivity, temporal adaptivity, finite elements, spectral elements, bioinformatics},
url = {http://publications.crs4.it/pubdocs/2009/NSCFM09a},
}

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