1,896 research outputs found
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
Convergence of discrete duality finite volume schemes for the cardiac bidomain model
We prove convergence of discrete duality finite volume (DDFV) schemes on
distorted meshes for a class of simplified macroscopic bidomain models of the
electrical activity in the heart. Both time-implicit and linearised
time-implicit schemes are treated. A short description is given of the 3D DDFV
meshes and of some of the associated discrete calculus tools. Several numerical
tests are presented
Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology
In standard models of cardiac electrophysiology, including the bidomain and
monodomain models, local perturbations can propagate at infinite speed. We
address this unrealistic property by developing a hyperbolic bidomain model
that is based on a generalization of Ohm's law with a Cattaneo-type model for
the fluxes. Further, we obtain a hyperbolic monodomain model in the case that
the intracellular and extracellular conductivity tensors have the same
anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is
equivalent to a cable model that includes axial inductances, and the relaxation
times of the Cattaneo fluxes are strictly related to these inductances. A
purely linear analysis shows that the inductances are negligible, but models of
cardiac electrophysiology are highly nonlinear, and linear predictions may not
capture the fully nonlinear dynamics. In fact, contrary to the linear analysis,
we show that for simple nonlinear ionic models, an increase in conduction
velocity is obtained for small and moderate values of the relaxation time. A
similar behavior is also demonstrated with biophysically detailed ionic models.
Using the Fenton-Karma model along with a low-order finite element spatial
discretization, we numerically analyze differences between the standard
monodomain model and the hyperbolic monodomain model. In a simple benchmark
test, we show that the propagation of the action potential is strongly
influenced by the alignment of the fibers with respect to the mesh in both the
parabolic and hyperbolic models when using relatively coarse spatial
discretizations. Accurate predictions of the conduction velocity require
computational mesh spacings on the order of a single cardiac cell. We also
compare the two formulations in the case of spiral break up and atrial
fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
We present a numerical method for approximating the solutions of degenerate
parabolic equations with a formal gradient flow structure. The numerical method
we propose preserves at the discrete level the formal gradient flow structure,
allowing the use of some nonlinear test functions in the analysis. The
existence of a solution to and the convergence of the scheme are proved under
very general assumptions on the continuous problem (nonlinearities, anisotropy,
heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the
efficiency and of the robustness of our approach
Geometric partial differential equations: Theory, numerics and applications
This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications
Studying Turbulence Using Numerical Simulation Databases. 5: Proceedings of the 1994 Summer Program
Direct numerical simulation databases were used to study turbulence physics and modeling issues at the fifth Summer Program of the Center for Turbulence Research. The largest group, comprising more than half of the participants, was the Turbulent Reacting Flows and Combustion group. The remaining participants were in three groups: Fundamentals, Modeling & LES, and Rotating Turbulence. For the first time in the CTR Summer Programs, participants included engineers from the U.S. aerospace industry. They were exposed to a variety of problems involving turbulence, and were able to incorporate the models developed at CTR in their company codes. They were exposed to new ideas on turbulence prediction, methods which already appear to have had an impact on their capabilities at their laboratories. Such interactions among the practitioners in the government, academia, and industry are the most meaningful way of transferring technology
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