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W. Auzinger, C. Fabianek:
"Iterative Solution of Large Linear Systems Arising in the 3-Dimensional Modelling of an Electric Field in Human Thigh";
Report for ANUM Preprint No. 12/04; 2004.

Functional electrical stimulation (FES) of denervated skeletal muscles found in recent years an entry into rehabilitation of paraplegics. The aim of current research is the optimization of stimulation parameters as well as studying the ramifications of stimulating the muscles in the thigh.

To simulate the distribution of the electric field a 3-dimensional model of the thigh is created which is based on the theory of activation functions from Rattay 1990 and the muscle model from Reichel 1999. With this model it is possible to simulate the electrical activity of muscle fibers. In the current implementation conductivity values based on grey values obtained from CT data are used in the Poisson equation. Via discretization by the method of finite differences and solution of the arising systems this leads to the voltage respectively current distribution. This work aims to implement and integrate an efficient solver for the large linear systems arising and shorten the time of the solution process.

After a thorough analysis of the numerical properties of these systems, various solution strategies have been investigated. Beginning with direct solvers which take the symmetric and sparse nature into account, iterative solvers and finally Krylov subspace methods have been implemented and tested. In the sequel we focused on the preconditioned Conjugate Gradient method.

Because of the bad condition of the given systems the solvers mentioned above led to unsatisfactory results and special preconditioning methods became mandatory. Again various methods have been implemented and tested until multigrid methods finally led to outstanding results in terms of convergence speed. However, the quite high memory requirements could not be satisfied on the target computer and therefore tools for remote computing were used to leverage external machines. Now these multigrid methods are performed remotely on high-end computers and the solution time from previously 9 hours shortens to about an hour.

In course of the evaluation of the methods described above it became necessary to work with various model problems, and a new simple and especially small thigh model was developed. Using this model existing results could be verified and new insight was gained. In particular, this allows to evaluate new methods easier and quicker than before.

http://www.math.tuwien.ac.at/~winfried/papers/anumpp1204.pdf