Student Workshop on Scientific Computing

June 15-18, 2012
Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

Implementation of the multigrid solver for the Navier-Stokes equations on GPU

Vladimír Klement


Abstract:Talk deals with the problem of implementing numerical model for 2D flow over an urban canopy on GPU. This problem is disctretised by means of the mixed finite element method with semi-implicit time stepping and solved by the multigrid method with the Vanka type smoother. Both the solver and system creation were parallelized and implemented on GPU. GPU implementation turns to be 5 times faster than OpenMP parallel version.


Petr Pauš


Abstract:The interpretation of the experimentally determined critical distance for the screw dislocation annihilation in persistent slip bands is still an open question. We attempt to analyze this problem using discrete dislocation dynamics simulations. Glide dislocations are represented by parametrically described curves. The model is based on the numerical solution of the dislocation motion law belonging to the class of curvature driven curve dynamics. We focus on the simulation of the cross-slip of two dislocation curves of the opposite signs where each evolves in a different primary slip plane in a channel of a persistent slip band. The dislocations move under their mutual interaction, the line tension and an applied stress forming a screw dislocation dipole. A cross-slip leads to annihilation of the dipolar parts. In the changed topology each dislocation evolves in two slip planes and the plane where cross-slip occurred. The goal of our work is to determine the conditions under which the cross-slip occurs. The simulation of the dislocation evolution and merging is performed by improved parametric approach and numerical stability is enhanced by the tangential redistribution of the discretization points. The critical annihilation distance determined by the simulations is close to the experimental value.

Mathematical modelling of the nucleation rates using the gradient theory

Barbora Planková


Abstract:The talk introduces a homogeneous nucleation as a process of formation of the microscopic nuclei of a new phase inside a homogeneous mother phase. This process is described by the nucleation rate which can be computed using the classical nucleation theory and the gradient theory. Both the theories are described and their results are compared. Two equations of state (EoS) are used for the computations: the classical Peng-Robinson EoS and the modern PC-SAFT EoS; their comparism is also given.

Application of the nonlinear Galerkin FEM method to the solution of the reaction diffusion equations

Jan Mach


Abstract:Complex long-term dynamics of reaction-diffusion problems requires finer approach leading to quantitatively reliable numerical schemes. One of approaches proposed by M. Marion and R. Temam is known as the nonlinear Galerkin method. The motivation for this method was to capture the effect of some of the terms that are neglected in the usual Galerkin method. The idea of the method is further extended to finite element method. In our talk we present an application of this method to the numerical solution of a particular reaction-diffusion system and give example numerical results.

Application of a Degenerate Diffusion Method in 3D Cardiac MRI Data Segmentation

Radek Máca


Abstract:The contribution presents a 3D (2D+time) segmentation of the real cardiac MRI data using an algorithm based on a numerical solution of a partial differential equation of the level set type. The algorithm is derived from the level set equation using a semi-implicit complementary volume numerical scheme approximation. In particular, the application is focused on the segmentation of the heart ventricles from the cine MRI data.

OpenMP Parallelization of a Multigrid Solver for the Incompressible Navier-Stokes Equations

Vítězslav Žabka


Abstract:We are concerned with the numerical simulation of the air flow over a simplified urban canopy. The flow is modeled by the two-dimensional incompressible Navier-Stokes equations, which are solved numerically by means of the finite element method and a geometric multigrid method based on Vanka-type smoothers. This contribution deals with the parallelization of the multigrid method. Two parallel OpenMP implementations of the method are presented. One of them uses the Jacobi iteration, while the other relies on the red-black Gauss-Seidel method.


David Fabian, Radek Mařík, and Tomáš Oberhuber


Abstract: oftware configuration often studies two issues: firstly, how to merge various software components together to create a program with a fixed structure that fits the requirements, and secondly, how to effectively set up the remaining (usually installation specific) configuration options when deploying the program. Nowadays, the user demands a simple and well arranged way to set up these options, possibly through a graphical user interface (GUI). There are various tools designed to assist the user with these tasks. In this contribution, a general multi-platform configuration tool Freeconf is introduced. Our technique to simplify GUI, which has been incorporated into Freeconf, is described. This technique is based on a set of properties that allow splitting the universe of configuration options into several categories with a clear semantics, and rules that control the dynamics of options distribution to these categories in response to the user’s actions. The rules are currently only implemented in the source code of Freeconf as a proof-of-concept without any formal proof of correctness or completeness. The results from the domain of Rule-Based Constraint Programming have been applied to develop a formal description of the rules.

Numerical Analysis of Bacterial Colony Model

Ondřej Pártl


Abstract:Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this contribution, we shall discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon. Numerical solutions of this system obtained by the method of lines and the semi-discrete Galerkin finite element method will be compared, and theoretical and experimental estimates of the orders of convergence of these methods will be presented.

Combined Method for Computation of Multicomponent Compressible Flow in Porous Media

Ondřej Polívka


Abstract:The mathematical modeling of transport of multicomponent mixtures in porous media is important for many applications including oil recovery or CO2 sequestration. The traditional approaches use either the fully implicit method or a sequential method. The fully implicit method is stable, allows for long time steps, but leads to extremely large systems of linear algebraic equations whose size is proportional to the number of mixture components. Alternatively, in sequential solution procedures like IMPEC (implicit pressure, explicit concentrations), the size of the system is reduced significantly to a size independent of the number of components. However, this approach is conditionally stable and the time step has to be chosen prohibitively small in many cases. We propose a new scheme for the numerical modeling of multicomponent compressible flow. Our method is based on a combination of the mixed-hybrid finite element method for the discretization of total flux, and the finite volume method for the discretization of transport equations. The scheme is almost fully implicit, allowing for long time steps, yet, it is possible to reduce dimensions of the final system of linear algebraic equations to a size independent of the number of components. We will show several simulations using our new approach.

Numerical Solution of Anisotropic Mean Curvature Flow of Graphs

Dieu Hung Hoang and Michal Beneš


Abstract:This contribution deals with the motion by anisotropic mean curvature of graphs within the context of epitaxial crystal growth. 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 present the numerical results for various types of anisotropy.

Numerical solution of deformation of elastic bodies by finite element method

Jan Valášek


Abstract:This presentation deals with the practical application of the finite element method to the problem of planar linear elasticity. The work includes physical fundamentals of plane elasticity, mathematical description of the problem being solved, its variational formulation and the method of finite element approximations including the main results of the convergence. Further the implementation of the finite element method is carried out. There are used linear triangular finite elements, the derivation of the stiffness matrix method is described in detail. The described method is implemented in a C program and its functionality is verifed on a number of test examples, on which the convergence speed of the method is numerically evaluated.

Model of soil freezing and thawing

Alexandr Žák


Abstract:In the contribution, we present our progress in developing the soil freezing and thawing model. The 2D model comprises of a heat equation with phase change and the Navier equations linked by a temperature-expansion term. The first part of the model is mathematically analyzed providing the information on the solution existence, and the entire model is solved numerically with emphasis on both thermally and mechanically heterogeneous properties of saturated soil material.

Two-Phase Flash Computation in Multicomponent Mixtures

Tereza Jindrová


Abstract:In the contribution we present investigation of two-phase equilibrium at constant volume, temperature and moles (so called VT-flash) in the multicomponent mixture which is described using unusual thermodynamic variables, i.e. volume, temperature and moles. The problem is formulated using the Helmholtz free energy and the Peng-Robinson equation of state. We introduce a numerical algorithm based on the Newton method in which modified Cholesky method is used to converting the Hessian matrix to a positive definite form. The results are compared with those obtained from an algorithm based on the fixed point iteration and Newton method, and the classical flash at constant pressure, temperature and moles (PT-flash).