Pavel Strachota

Towards clinical applicability of the diffusion-based DT-MRI visualization algorithm

For the purpose of DT-MRI data visualization, an algorithm based on a numerical model of texture diffusion is proposed. As a prerequisite of entering clinical use, its parameters need to be adjusted properly so that the procedure gives satisfactory results with limited computational resources and time available. This contribution introduces the principles of the method and reports on the results of extensive computational studies aimed at finding optimal settings of the numerical scheme and model parameters with respect to visualization purposes. Total variation is used as a measure of visual quality of the produced images. Further, we provide evidence that using the algorithm is fully feasible on state of the art hardware. Finally, high resolution visualizations based on real data are demonstrated.

A Multipoint Flux Approximation Finite Volume Scheme for Solving Anisotropic Reaction–Diffusion Systems in 3D

In [15], our DT–MRI visualization algorithm based on anisotropic texture diffusion is introduced. The diffusion is modeled mathematically by the problem for the Allen–Cahn equation with a space–dependent anisotropic diffusion operator. To preserve its anisotropic properties in the discretized version of the problem, an appropriate numerical treatment is necessary, reducing the isotropic numerical diffusion. The first part of this contribution is concerned with the design and investigation of the finite volume scheme with multipoint flux approximation. Its desirable properties are investigated by means of our technique based on total variation measurement. The second part presents the recent achievements in applying the same scheme to the phase field model of dendritic crystal growth.

Nonlinear Galerkin finite element method applied to the system of reactiondiffusion equations in one space dimension

We study the finite-element nonlinear Galerkin method in one spatial dimension and its application to the numerical solution of nontrivial dynamics in selected reactiondiffusion systems. This method was suggested as well adapted for the long-term integration of evolution equations and is studied as an alternative to the commonly used numerical approaches. The proof of the convergence of the method applied to a particular class of reactiondiffusion systems is presented. Computational properties are illustrated by results of numerical simulations. We performed the measurement of the experimental order of convergence and the computational efficiency in comparison to the usual finite-difference method.

Design and Verification of the MPFA Scheme for Three-Dimensional Phase Field Model of Dendritic Crystal Growth

As an alternative to the sharp interface formulation, the phase field approach is a widely used technique for modeling phase transitions. The governing system of reaction-diffusion equations captures the instability of the underlying physical problem and is capable of modeling the evolution of complicated crystal shapes during solidification of an undercooled melt. For its numerical solution, we propose our novel anti-diffusive multipoint flux approximation (MPFA) finite volume scheme on a Cartesian mesh. The scheme is verified against the analytical solution of the modified sharp interface model. Experimental order of convergence (EOC) is measured for the temperature field in the usual norms. In addition, EOC is also obtained for the phase interface through approximating the volume of the symmetric difference of the solid phase subdomains. In the anisotropic cases including unusual higher order symmetries, computational studies with various settings also confirm convergence of our MPFA scheme which is faster than in the case of the reference finite volume scheme with 2nd order flux approximation.

An Efficient and Robust Numerical Solution of the Full-Order Multiscale Model of Lithium-Ion Battery

We propose a novel and efficient numerical approach for solving the pseudo two-dimensional multiscale model of the Li-ion cell dynamics based on first principles, describing the ion diffusion through the electrolyte and the porous electrodes, electric potential distribution, and Butler-Volmer kinetics. The numerical solution is obtained by the finite difference discretization of the diffusion equations combined with an original iterative scheme for solving the integral formulation of the laws of electrochemical interactions. We demonstrate that our implementation is fast and stable over the expected lifetime of the cell. In contrast to some simplified models, it provides physically consistent results for a wide range of applied currents including high loads. The algorithm forms a solid basis for simulations of cells and battery packs in hybrid electric vehicles, with possible straightforward extensions by aging and heat effects.

A Quasi-1D Model of Biomass Co-Firing in a Circulating Fluidized Bed Boiler

We introduce an outline of the mathematical model of combustion in circulating fluidized bed boilers. The model is concerned with multiphase flow of flue gas, bed material, and two types of fuels (coal and biomass) that can be co-fired in the furnace. It further considers phase interaction resulting in particle attrition, devolatilization and burnout of fuel particles, and energy balance between heat production and consumption (radiative and convective transfer to walls). Numerical solution by means of the finite volume method together with a Runge-Kutta class time integration scheme is mentioned only briefly as the used methods are generic and well documented elsewhere. Some representative results are also presented.

Analysis of the Parallel Finite Volume Solver for the Anisotropic Allen–Cahn Equation in 3D

In this contribution, a parallel implementation of the finite volume solver is introduced, designated to numerically solve the initial boundary value problem for the Allen–Cahn equation with anisotropy on large 3D grids. The choice of a suitable numerical scheme is discussed and its convergence properties are investigated by means of evaluation of the experimental order of convergence. Afterwards, the consequent limitations for the theoretical error estimate are pointed out. Furthermore, the results of parallel algorithm efficiency measurements are shown, based on extensive tests performed on high performance computing systems. The final part gives a brief overview of a magnetic resonance tractography (neural tract tracking and visualization) method consisting in the solution of the above problem.