Student workshop on scientific computing 2015

June 11-14, 2015
Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

CMR Data Segmentation Using Level Set and Allen-Cahn Approach

Radek Máca


Abstract:The contribution deals with the task of left heart ventricle segmentation from cardiac MR (CMR) images. In more detail, the CMR investigation consist of dozens of 2D images including both temporal and spatial information. The outcome of the segmentation is to find a contour representing the inner ventricle wall. The problem is solved using partial differential equation of level set and Allen-Cahn type. Numerical solutions of the equations is found using a semi-implicit
complementary volume discretization. Both 2D and 3D approaches are presented and detailed comparison study is made.

Modeling of infiltration experiment done on soil sample by TOUGH2

Tomáš Princ


Abstract:Ponded infiltration was performed on the composed heterogeneous sand sample while monitoring using neutron imaging was conducted to obtain quantitative spatiotemporal information about the water content distribution in the sample. The same experiment was numerically simulated by commercially available two-phase flow code TOUGH2. The aim was to reproduce the residual air redistribution between coarser and finer materials observed in the experiment. The talk presents the comparison of the modeling and experimental results.

Modelling of Gas Flow and NAPL Vapor Transport in Porous Medium and above its Surface

Ondřej Pártl


Abstract:In this contribution, numerical simulation of flow of a mixture of two gases in porous medium and above its surface will be discussed. In both media, the mixture is described by balance equations for mass, momentum and energy supplemented by the ideal gas equation of state, and at the interface, suitable coupling conditions are used. The systems of governing equations for the free flow and for flow in porous medium are solved by an explicit and semi-implicit time-stepping scheme, respectively. The spatial discretization is carried out by means of the control volume finite element method in both media.

Modelling of Interacting Dislocations in Persistent Slip Band

Miroslav Kolář


Abstract:This contribution deals with the material imperfections in nanoscale having the line character, i.e., the crystallographic arrangement of atoms is disturbed along the dislocation line. Dislocations are described as smooth planar curves evolving in time and space, and driven by the (mean) curvature and external forces.

We consider two unlike dislocations gliding in parallel slip planes in the channel of the persistent slip band. The interaction between the dislocation and the channel wall is represented by elastic field of rigid dipoles, which act as potential wells. The interaction stress field between two gliding dislocations is also incorporated. As the dislocations are pushed by the applied stress between two walls in the opposite directions, they bow out and attract each other forming a dipole. With the increasing stress the dislocations become more and more curved, until they separate. The objective is to determine the passing stress in the channel needed for the dislocations to escape one another and compare the generated stress field in the stress and strain controlled regimes. In the stress controlled regime the stress in the channel induced by boundary conditions is assumed to be uniform. In the strain controlled regime the sum of elastic and plastic strain is taken to be uniform. The stress control provides an upper estimate of the passing stress, whereas the strain control yields a lower estimate.

Mathematical model of this problem is formulated by means of the parametric approach resulting in the system of degenerate parabolic PDEs for the position of the dislocation. For numerical experiments we investigate semi-implicit flowing finite volume method enhanced by tangential redistribution.

Novel Approach of Fractal Dimension Estimation Based on Optimal Segmentation of Time Series

Martin Dlask


Abstract:Fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) are stochastic processes whose properties can be used for estimating Hurst exponent of investigated time series. Traditional methods generally provide only a point estimate of fractal dimension and therefore they do not hold information about estimation error. However, it is possible to collect these point estimates from several classical methods and consider them as multicriteria decision problem in order to determine time series with most predictable development. A novel approach of fractal dimension estimation is based on signal segmentation – using Bayesian analysis it is possible to determine mean and confidence interval for Hurst exponent in each segment and obtain its aggregate estimate by utilizi ng statistical features calculated at the interval level. The innovative methodology was proven to be useful for analysis of stock market indices and the efficiency of provided results together with statistical characteristics are compared with standard techniques for fractal dimension estimation.

Simulation of the 3D incompressible Navier-Stokes equations on the GPU

Vladimír Klement


Abstract:The presentation deals with the simulation of the 3D flow governed by the Navier-Stokes incompressible equations. This problem is discretized on regular mesh by the means of finite difference method. The solution is divided into three parts: advection, diffusion and pressure correction. Nonlinear advection is solved explicitly, diffusion implicitly and the pressure correction by the means of the projective SIMPLE algorithm. Computational part of the program was also implemented on the GPU using the NVidia CUDA technology which provided a considerable speed-up of the simulation.

Pore-Scale Modeling of Thermo-Mechanical Interaction within Freezing Saturated Soil

Alexandr Žák


Abstract:We present a pore-scale model describing a dynamics of the pore water phase transition and of the associated mechanical effects within a water-saturated soil subjected to freezing ambient conditions. In an effort to investigate the effects of the pore water density change during the propagation of the phase transition within cooled soil material, we have designed the 2D continuum pore-scale model which incorporates the system of the heat equation and the Navier equations for solids and the system of the modified heat equation involving the phase transition of the pore water and the Navier-Stokes equation for fluid.

On nonlocal transport based closure relations for plasma hydrodynamics

Milan Holec


Abstract:Nonlocal theory of the electron transport in plasmas of arbitrary ratio of the characteristic spatial scale length to the electron mean free path is applied to define closure relations of hydrodynamic system. The corresponding transport phenomena cannot be described accurately with the usual fluid approach dealing only with local values and derivatives. Thus the first order terms like viscous force and heat flux are calculated directly from the electron distribution function obtained from a simplified kinetic equation allowing one to take into account the effect of long-range particle transport. The key feature of the proposed method is use of the BGK collision operator delivering a calculation efficiency and an inherent coupling to the fluid plasma parameters.

A Massively Parallel Implementation of Mixed Hybrid Finite Element Method for Single-Phase Flow in Porous Media

Jakub Klinkovský


Abstract:The work deals with the mathematical modelling of flow of a single compressible phase in porous media. We devise a numerical scheme that is based on the mixed-hybrid finite element method with the semi-implicit approach for the time discretization in order to obtain a system of linear equations for which either a direct or iterative solver is used. We propose a parallel implementation of the numerical scheme using the TNL library and the CUDA architecture. We present a numerical analysis of the scheme and compare the efficiency and accuracy of its implementation on CPU and GPU.

Comparison of Semi Implicit and Fully Implicit Mixed-Hybrid Finite Element Method for Two-Phase Flow in Porous Media

Ondřej Pelech


Abstract:The contribution deals with two numerical schemes based on the mixed-hybrid finite element (MHFE) method for the simulation of flow of two immiscible, compressible or incompressible phases in unsaturated porous media. The first scheme uses a semi-implicit time discretization approach that leads to a system of linear equations with positive definite matrix. The second one is based on the fully implicit time discretization and a modified Newton's method with a linesearch strategy is used. The applicability of both schemes is demonstrated on a set of benchmark problems and the comparison of the methods in the terms of accuracy and computational complexity is presented.

Mathematical modeling of planar and spherical phase interfaces of liquid mixtures

David Celný


Abstract:Phase interface is common phenomenon that can be found in each glass of mineral water. It is beneficial to understand its behavior and use it for more pressing matters such as carbon capture and storage technologies (CCS). Our work deals with gas-liquid phase interfaces of two component systems. These systems can form two basic types of the phase interfaces on which we focus, namely the planar interface (water level) and the spherical interface (bubble). Both types of the interface can be described via a density function.
One of main task is obtaining molar density function profiles through the minimization of grand potential functional. Our approach is based on theoretical results of the Cahn-Hilliard gradient theory from 1957 that enables transformation of the problem into a system of second order differential equations. Due to the performed simplification, this system can be split into a non-linear algebraic system and one differential equation. The simplified formulation can be solved and provides resulting molar density function illustrating change of molar density over the change of coordinate (radius from center of the bubble). Then, from the obtained molar density function and further results it is possible to compute reference data that can be compared with experimental data such as nucleation speed.
We were able to obtain density functions for various binary mixtures (i.e. CO2 & C9H20). Comparison of the result characteristics with experimental data and data obtained from the molecular simulation will be performed.

Investigation of Field Scale Vapor Intrusion Problems in Porous Media using Mixed Hybrid Finite Element Method

Jakub Solovský


Abstract:This work deals with two phase compositional flow. We present equations describing two phase flow, component transport and interphase mass transfer. We propose a numerical method based on mixed hybrid finite element method, for solving this type of problems. The method is verified on problems for which exact solution is known or solution can be found in a literature. Numerical experiments show, that the method is convergent and order of convergence is slightly less than one. In the next part we focus on a compositional flow. Behavior of vapor plumes and vapor intrusion is studied on proposed scenario. We focus on a kinetic mass transfer, propose three models for mass transfer coefficient and perform sensitivity analysis. We examine effects of rain infiltration and watter table fluctuation on vapor intrusion. Solution strongly dependeds on mass transfer coefficient and it is affected by rain infiltration and water table fluctuation. In comprarison the effect of rain infiltration is more significant.

Numerical solution of a curvature driven flow of plane and space curves and its applications

Jiří Minarčík


Abstract:Tracking curves in motion can be useful for numerous applications, such as image segmentation, noise reduction, material dislocation etc. This contribution deals with the numerical aspects of curvature and external force dependent curve evolution. Different approaches to this problem will be discussed and several qualitative numerical experiments will be presented and compared with analytic solutions.