Charles Pierre

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.

The Generalized Graetz Problem in Finite Domains

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of

Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems

L^2

Preconditioning the bidomain model with almost linear complexity

. In the case of finite domains a similar basis can be exhibited, and the modes amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection ima...

Benchmark 3D: CeVe-DDFV, a Discrete Duality Scheme with Cell/Vertex Unknowns

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards

Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

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MATHEMATICAL ANALYSIS OF PARALLEL CONVECTIVE EXCHANGERS WITH GENERAL LATERAL BOUNDARY CONDITIONS USING GENERALIZED GRAETZ MODES

and the other towards

In vivo contact EP data and ex vivo MR-based computer models: registration and model-dependent errors

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CONVERGENCE OF THE GENERALIZED VOLUME AVERAGING METHOD ON A CONVECTION-DIFFUSION PROBLEM: A SPECTRAL PERSPECTIVE ∗

, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonse...

Exponential Adams Bashforth integrators for stiff ODEs, application to cardiac electrophysiology

The bidomain model is widely used in electro-cardiology to simulate spreading of excitation in the myocardium and electrocardiograms. It consists of a system of two parabolic reaction diffusion equations coupled with an ODE system. Its discretisation displays an ill-conditioned system matrix to be inverted at each time step: simulations based on the bidomain model therefore are associated with high computational costs. In this paper we propose a preconditioning for the bidomain model either for an isolated heart or in an extended framework including a coupling with the surrounding tissues (the torso). The preconditioning is based on a formulation of the discrete problem that is shown to be symmetric positive semi-definite. A block LU decomposition of the system together with a heuristic approximation (referred to as the monodomain approximation) are the key ingredients for the preconditioning definition. Numerical results are provided for two test cases: a 2D test case on a realistic slice of the thorax based on a segmented heart medical image geometry, a 3D test case involving a small cubic slab of tissue with orthotropic anisotropy. The analysis of the resulting computational cost (both in terms of CPU time and of iteration number) shows an almost linear complexity with the problem size, i.e. of type nlog^@a(n) (for some constant @a) which is optimal complexity for such problems.