Student Workshop on Scientific Computing

June 10-13, 2011
Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

Impact of Grid Geometry and Numerical Scheme on Crystalline Morphology in Three-Dimensional Phase Field Modeling of Pure Substance Solidification

Pavel Strachota


Abstract:This contribution is concerned with the phase field model of pure substance solidification with anisotropic surface energy. The governing system of reaction-diffusion equations captures the inherent instability of the underlying physical problem and is capable of modeling the evolution of complicated crystal shapes during solidification of an undercooled melt. Both theoretical and computational studies exist classifying the crystal growth type as "dendritic", "doublon", or "seaweed" by the settings of two relevant parameters: undercooling and anisotropy strength. We summarize the results of extensive computational studies aimed at convergence investigation of our novel anti-diffusive multipoint flux approximation (MPFA) finite volume scheme on a cartesian mesh, used for numerical solution of the problem. As opposed to most results, higher order symmetries are considered in addition to the usual four-fold symmetry. The original purpose was to confirm superiority of the MPFA scheme over the traditional 2nd order flux approximation in terms of convergence rate. Although this goal has been achieved, it turned out that trustworthiness of the numerical simulations with a given scheme and grid resolution is largely affected by the model settings. Moreover, modifying the grid geometry may represent an intervention significant enough to change the type of crystal growth, let alone the crystal shape and size. For most situations, sufficient grid refinement unified the behavior of the solution for all combinations of grid geometry and numerical scheme. However, examples have been found where convergence can only be anticipated and grid refinement beyond the capabilities of the used computing resources would be needed to prove it. Existing morphology classification efforts based on numerical simulations therefore need to be interpreted very carefully.

Numerical Solution for Anisotropic Surface Diffusion of Graphs

Dieu Hung Hoang


Abstract:This contribution is concerned with the motion of graphs driven by anisotropic surface diffusion. We study this motion within the context of epitaxial crystal growth. Accurate knowledge of morphological changes in epitaxial thin films is important for governing the properties of semiconductors. Due to the effects of stress, epitaxial films may undergo a morphological instability, known as the Asaro-Tiller-Grinfeld instability. Investigating such surfacial phenomena requires suitable computational tools. The numerical scheme is based on the method of lines where the spatial derivatives are approximated by finite differences. We then solve the semi-discrete scheme by means of the adaptive Runge-Kutta-Merson method. Finally, we show computational results with various anisotropies.

Fully Coupled Numerical Computation of Multicomponent Flow in Porous Medium

Ondřej Polívka


Abstract: We deal with the numerical modeling of the compressible single-phase flow of a mixture composed of several components in a porous medium which is suitable for description of multicomponent subsurface transport. We propose a new approach based on a combination of the mixed-hybrid finite element method with either the discontinuous Galerkin finite element method or the finite volume method. As the derived scheme is almost fully implicit, long time steps can be used to obtain the approximate solution. Similarly to the fully implicit schemes, our method leads to large systems of linear algebraic equations, but a non-trivial solution procedure makes it possible to reduce the size of the final system of equations to a size independent of the number of mixture components. Therefore, the solution cost is comparable with the traditional sequential approaches. Unlike in other sequential approaches, no pressure equation has to be formed as pressure is evaluated directly from the equation of state.

Computation of Equilibrium Phase Splitting at Constant Temperature, Moles and Volume

Kateřina Marková


Abstract:Computation of phase equilibrium in liquid-gas systems is a part of solution of many problems, e.g. carbon dioxide sequestration or oil extraction. When injecting carbon dioxide into oil reservoirs in the form of supercritical fluid, carbon dioxide tends to partially dissolve into the liquid and phase splitting of the mixture may occur. Models of carbon dioxide sequestration require robust and accurate thermodynamic description of the phase splitting. The traditional approach to phase-splitting modelling is based on the Gibbs energy minimization at given temperature, pressure and chemical composition of the mixture. A problem arises when trying to apply the traditional procedure in order to determine the phase compositions and phase amounts of a pure substance at the saturation pressure corresponding to given temperature. In this particular case it is not possible to distinguish between different states of the substance because pressure remains the same during the phase transition, while the phase compositions and phase amounts change. Therefore a different approach to phase splitting, seeking the equilibrium state of the system at given volume, temperature and mole numbers, has been developed. In this approach, the Helmholtz free energy (which is a function of volume, temperature and moles) is chosen to replace the role of Gibbs energy in the traditional approach. In the presentation, the phase equilibrium conditions at constant temperature, volume and moles will be formulated. Main features of the new formulation and its advantages in comparison with the traditional procedure will be discussed. An algorithm for computation of the phase compositions and phase amounts based on the Newton-Raphson method will be introduced and compared with another algorithm based on successive substitution iteration (SSI) method.

Several Numerical Results of Bacterial Colony Model

Ondřej Pártl


Abstract:It is well known that colonies of bacterial species Bacillus subtilis placed on a dish filled with agar medium and nutrient form various patterns while grow and migrate. This contribution deals with one of mathematical models based on a system of reaction-diffusion equations which is used for describing this phenomenon. We are going to compare the results of its numerical solution obtained by the finite difference method with the results obtained by the use of the method of lines and examine the dependency of the results on the fineness of the orthogonal mesh used. Further, the results of the discussed model considered in three-dimensional space are going to be presented. Nevertheless, without any discussion about suitability of this model for the description of the growth of the bacterial colony in three-dimensional space (e.g. in an aquarium).

Application of a Degenerate Diffusion Method in Medical Image Processing

Radek Máca


Abstract:Over the years many medical image analysis software products have been developed, but they still lack the fully-automatic segmentation algorithm. Accordingly, we try to propose the automatic algorithm for segmentation of the cardiac MRI data using a partial differential equation of the level-set type. Applying an appropriate modification of the level-set equation together with a convenient image thresholding, the objects in the image can be detected. The semi-implicit complementary-volume numerical scheme is used to solve this equation. We describe the algorithm parameters and their setting used segmentation of the left heart ventricle during the cardiac cycle. These parameters are set according to the image thresholds which are found using a special image thresholding algorithm. Finally, the results of the segmentation are presented.

Numerical Simulation of Air Flow over Urban Canopy

Vítězslav Žabka


Abstract:This contribution is concerned with the numerical simulation of the two-dimensional air flow over a simplified urban canopy. The mathematical model used is based on the Navier-Stokes equations for viscous incompressible flow, which are solved numerically by means of the finite element method. Because no explicit turbulence model is employed, the equations have to be resolved on fine enough computational grids to obtain reliable results. Such approach is very expensive in terms of computational cost. In order to be able to compute the solution in a reasonable time, a multigrid method based on smoothers of Vanka type is utilized to deal with the systems of linear equations arising from the finite element discretization. The method has been successfully implemented and tested. In addition, an OpenMP-based parallel implementation has been developed and compared with the serial implementation.

Graph Cuts for Segmentation of Images from MRI

Jakub Loucký


Abstract:This contribution is concerned with applications of graph cuts in image processing. The construction of a weighted directed graph based on an input image is explained and the connection of the minimal graph cut with image segmentation is shown. Implementations of algorithms for finding minimal cut in a network can utilize special characteristics of these networks. Two basic algorithms have been implemented for this purpose and compared in terms of computational costs and parallelization suitability.

Application of the nonlinear Galerkin FEM for the solution of the reaction diffusion equations

Jan Mach


Abstract:Experience with numerical studies shows that it is not always easy to approximate the dynamics of the reaction-diffusion equations. The usual error estimate contains a constant growing exponentially in time. One way of overcoming this difficulty is the approach known as the nonlinear Galerkin method. For discretization in space FEM is used. Modified Runge-Kutta method with time step adaptivity is used for integration in time. The method is compared with the FDM (in 1D) for solution of the initial-boundary value problem for selected reaction-diffusion equations.

Mathematical Modelling of Reaction - Diffusion Processes in Combustion

Petr Habásko


Abstract:The goal of this work is to solve two dimensional model which describes the combustion of premixed gases. The model is based on the laws of chemical kinetics and involves diffusion of mass and heat conduction. The formulation of these equations leads to the system of two partial differential equations - which are called reaction-diffusion equations exhibiting interesting behavior. The Merson numerical scheme with automatic choice of time step was used for integration of these equation. The experimental analysis of numerical approximation error was performed. In the contribution, the model equations, numerical results and comparison with real experiment will be presented.

Numerical Simulation of Brusselator Model

Miroslav Kolář


Abstract:This contribution deals with the numerical simulation of Brusselator model, which is a theoretical model of nonlinear chemical reaction described by a system of reaction-diffusion equations. It is well known that this model exhibits various types of solutions, e.g. stable fixed point or chaotic attractor. This behaviour depends on a single parameter of the system - the characteristic length of reactor. We study the possibility of using Galerkin method and nonlinear Galerkin method. Derived numerical schemes lead to the system of ordinary defferential equations and in addition to the system of linear algebraic equations in case of nonlinear Galerkin method. Then we solve described systems by the adaptive Runge-Kutta method and Gaussian elimination. Finally, the computational results are going to be presented.

Modeling of Radiation Transport in Laser Generated Plasma

Jiří Hanuš


Abstract:The radiative transfer plays an important role in direct-drive Inertial Confinement Fusion (ICF). This contribution covers the modeling of radiative transfer in plasma generated by laser irradiated target. Presented model is based on Euler equations in Lagrangian coordinates in 1D coupled with the radiation transfer equation. Finite difference methods are used to solve the system, explicit schemes for hydrodynamics, implicit for parabolic heat conductivity and for quasi-stationary elliptic equation describing the radiation transport. According to numerical results, even for low Z elements, the hydrodynamic motion of plasma is influenced by radiative transfer generating the radiation heat wave (RHW).

Implementation of the sparse matrix solvers on the GPU

Vladimír Klement


Abstract:This contribution is concerned with the use of modern graphics cards for solving system of linear equations with sparse matrix. Two sparse matrix solvers have been implemented on the GPU for this purpose and compared with their respective serial version. The first one of them was the SOR method, which we used to solve the problem of the image segmentation via the Levelset method, and the second one was the multigrid method, which we used to solve the numerical simulation of the two-dimensional air flow.

Mathematical Modeling of Dislocation Dynamics in Materials

Ladislav Zvoník


Abstract:Dislocations, which are defined as line defect of the crystalline lattice, strongly influence plastic and fatigued properties of crystalline solids. The purpose of this work is computation of eigenstrain caused by dislocations. The contribution deals with modeling of dislocation dynamics in micropattern called PSB (persistent slip bands) channel. Dislocations are treated as parametrically described curves moving in glide planes and this approach leads to system of two partial differential equations. For discretization of equations we use method of lines and we derive fourth-order numerical scheme in time and space. Some interesting properties of our scheme will be mentioned, namely accuracy, convergence and stability. At the end, numerical results will be presented.