On a gradient flow for the anisotropic ratio and other nonlocal geometric flows

Daniel Ševčovič

Comenius University, Bratislava

Abstract:We analyze a system of governing PDEs for geometric quantities and propose a numerical method for computing the mean curvature flow with a nonlocal term. The numerical computation is stabilized by using the curvature adjusted tangential velocities for which the tangential velocity speed depends on the function of curvature and its curve average.

Model of Radiative Heat Transfer in Pusher Furnace - Comparative Analysis

Robert Straka

AGH University of Science and Technology, Krakow

Abstract:The radiative heat flux in a pusher furnace is described and solution is obtained by numerical methods. The modified zonal method is used. The load and walls are divided into isothermal zones however the flue gas is taken as the only one isothermal zone with mean temperature. Knowing the geometric configuration of the furnace and temperature together with emissivity of each zone, one can assemble the system of linear equations in terms of the radiosity of each zone. The matrix of the system is dense and contains configuration factors for each zone. By solving the system we obtain vector of radiosities for each zone. Flue gases radiation is described using various models (diathermal, black, gray, non-gray gas and wide band model by Edwards-Balakrishnan, with both black and gray bands approximation for H2O and CO2). Numerical method used for the solution of the resulting system is LU decomposition method with pivoting. Finally the radiative heat flux is computed from the radiosity vector and emissive properties of the load. This model will be incorporated to the other model used to compute the load heating.

Convex optimization methods in optimal experimental design

Mária Trnovská

Comenius University, Bratislava

Abstract:In the presentation, we solve the problem of computing the maximin efficient design with respect to the class of all orthogonally invariant criteria. It turns out that on a finite experimental domain, the maximin efficient design can be computed by the methods of semidefinite programming. Using this approach, we can deal with the non-differentiability inherent in the problem, due to which the standard iterative procedures cannot be applied. We illustrate the results on the models of polynomial regression on a line segment and quadratic regression on a cube. Keywords: optimal design; Ek-optimality; maximin efficiency; semidefinite programming, quadratic regression, polynomial regression

A plane curve evolution in a forest fire modeling

Karol Mikula

Slovak University of Technology, Bratislava

Abstract:We present a mathematical model for numerical modeling of a wind-driven wild forest fire front propagation. The model is based on evolution of plane curve (representing the fire front) in the outer normal direction by a speed given by the properties of a fuel bed scaled (exponentially) by a wind speed projected onto the normal to the front. The influence of the front shape on the speed of propagation is modeled by adding the curvature regularization which represents useful generalization of the current fire front propagation models. For numerical modeling we use Lagrangean approach. We use a stabilization of numerical solution by suitable tangential redistribution of grid points which prevents the moving front from forming spurious crossovers and swallow tails. Thanks to that, we also solve in a fast and simple way a detection of topological changes in moving front by considering a distance function from a given point along the curve. These two approaches makes our approach highly efficient and represent significant computational improvement of the existing numerical models used for the wild fire propagation. In our approach, we solve an intrinsic advection-diffusion partial differential equation with a driving force by a semi-implicit scheme in the curvature part accompanied by the inflow-implicit/outflow-explicit scheme in advective part which guarantee solvability of linear systems arising in discrete time moments without restriction on the computational time step.

Data filtering on the Earth's surface using linear and nonlinear diffusion equations

Róbert Čunderlík

Slovak University of Technology, Bratislava

Abstract:We deal with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method (SFVM) to approximate numerically parabolic partial differential equations on closed surfaces, namely on the Earth's surface. The Earth is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green's theorem to the Laplace-Beltrami operator. Then SFVM is applied to discretize the weak formulation, where we consider a piece-wise linear approximation of a solution in space and the backward in time discretization. Later on, we extend a linear diffusion on the surface to the regularized surface Perona-Malik model, which represents a nonlinear diffusion equation. In our numerical experiments we focuse on reducing the stripping noise from the satellite geopotential model due to the truncation error of spherical harmonics. We discuss advantages of the nonlinear diffusion that conserves main structures of the gravity field, while the stripping noise is effectively reduced.

4D Numerical Schemes for Cell Image Segmentation and Tracking

Mariana Remešíková

Slovak University of Technology, Bratislava

Abstract:We introduce new techniques for 4D (space-time) segmentation and tracking in time sequences of 3D images of zebrafish embryogenesis. Instead of treating each 3D image individually, we consider the whole time sequence as a single 4D image and perform the extraction of objects and cell tracking in four dimensions. The segmentation of the spatiotemporal objects corresponding to the time evolution of the individual cells is realized by using the generalized subjective surface model that is discretized by a 4D finite volume scheme. Afterwards, we use the distance functions to the borders of the segmented spatiotemporal objects and to the initial cell center positions in order to backtrack the cell trajectories. The distance functions are obtained by numerical solution of the time relaxed eikonal equation.

Finite volume method for solving geodetic boundary value problems

Zuzana Minarechova

Slovak University of Technology, Bratislava

Abstract:Since precise gravity field modelling is indispensable for a unification of a local vertical datums, quality of terrestrial gravimetric measurements and precision of their positions have significant impact on accuracy of geoid and quasigeoid models. In this study we present how inconsistencies of vertical positions of input terrestrial gravity data can influence numerical solutions by finite volume method. We solve the geodetic BVP with the mixed boundary conditions (BCs) in 3D domain above the Earth's surface. This domain is bounded by the Earth's surface at the bottom, one spherical artificial boundary outside the Earth at altitude of a satellite mission and four side artificial boundaries. All numerical solutions are fixed to the satellite only geopotential model on all artificial boundaries, where the Dirichlet BCs are supposed. On the Earth's surface the oblique derivative BC in the form of surface gravity disturbances is prescribed. In our numerical experiments we compare numerical solutions with and without considering the corrections from the shifts and tilts of local vertical datums in the input surface gravity disturbances.

Some applications of level set methods

Peter Frolkovič

Slovak University of Technology, Bratislava

Abstract:In this talk we will discuss some new developments of level set methods like extrapolation along characteristics and some recent applications like numerical modelling of groundwater flow with free top surface.

Numerical schemes based on finite volume method for regularized mean curvature flow level set equation in image processing

Angela Handlovičová

Slovak University of Technology, Bratislava

Abstract:New numerical methods for regularized mean curvature flow level set equation are derived. Numerical analysis such as stability and convergence results for approximated solution to the weak solution of a problem is presented. Some numerical experiments for comparing new methods with previous numerical schemes are included.

Discontinuous Galerkin and Mixed-Hybrid Finite Element Method for Two-Phase Flow in Heterogeneous Porous Media with Different Capillary Pressures

Radek Fučík and Jiří Mikyška

FJFI ČVUT v Praze

Abstract:A modern numerical scheme for simulation of flow of two immiscible and incompressible phases in inhomogeneous porous media is proposed. The method is based on a combination of the mixed-hybrid finite element (MHFE) and discontinuous Galerkin (DG) methods. The combined approach allows for accurate approximation of the flux at the boundary between neighboring finite elements, especially in heterogeneous media. In order to simulate the non-wetting phase pooling at material interfaces (i.e., the barrier effect), we extend the approach proposed in Hoteit and Firoozabadi (2008) by considering the extended capillary pressure condition. The applicability of the MHFE-DG method is demonstrated on benchmark solutions and simulations of laboratory experiments of two-phase flow in highly heterogeneous porous media.

Applications of a Phase-field Crystal Model

Simon Praetorius

TU Dresden

Abstract:The Phase-Field Crystal model - a density-field approach, that describes the dynamic of interacting Brownian particles - can be coupled to other physical models, e.g. flowing solvents, physical boundaries/obstacles, hydrodynamic interactions, two-phase flows, i.e. particles restricted to one phase. Some ideas and first results will be presented.

Mathematical modelling of dispersion of chemical agents in the atmospheric boundary layer

Petr Bauer

AV ČR

Abstract:Our goal is to model the dispersion and persistence of chemical agents in the environment. The model is based on parametrized advection-diffusion equation with turbulent diffusivity. The velocity profile corresponds to the neutral stratification of the boundary layer. The chemical agent evaporates from the surface, which is represented by the boundary condition for the flux of concentration. The model is solved by the finite volume method. We present numerical studies of the given problem.