Student Workshop on Scientific Computing

June 14-17, 2013
Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

Numerical solution of deformation of elastic bodies by finite element method

Jan Valášek

Czech Technical University in Prague

Abstract:This presentation deals with the practical application of the finite element method to the problem of planar elasticity. This work includes physical fundaments of planar elasticity and mathematical description by partial differential equations. The weak formulation of the problem in space was made by FEM and discretization in time by the Newmark method. In addition the analysis of time signal by Fourier transformation was carried out and characteristics frequencies were acquired. These frequencies were compared with results from the modal analysis using FEM discretization. The described methods were implemented partly in C language and Matlab program and their functionality was verified in a number of test examples.

Numerical Solution of Planar Curve Dynamics

Miroslav Kolář

Czech Technical University in Prague

Abstract:The contribution deals with the numerical solution of curve evolution in plane. The smooth curves are described by means of the parametric approach and their evolution is given by the mean curvature motion law, which reads as v=-k+F, where v is the normal velocity, k is the curvature and F is the spatially dependent force term bringing nonlinearities to the motion. The motion law is numerically solved by means of finite difference approach (semi-implicit and Crank-Nicholson schemes) and flowing finite volumes method. We improve the behaviour of the numerical algorithm by adding the tangential redistribution of curve discretization nodes. The numerical results contain the computational study of both open and closed curves, parametric study of the redistribution algorithm and numerical tests of area or length conservation.

Numerical Simulation of Soil-Air Pressure

Ondřej Pártl

Czech Technical University in Prague

Abstract:In our research, we try to simulate the soil-air pressure in a sand tank and we compare our results with the measurements conducted in a wind tunnel. For the description of this phenomenon, we employ a mathematical model based on the Navier-Stokes equations and we solve it numerically by means of the method of lines using the Galerkin finite element method and Runge-Kutta method. In this contribution, results of our numerical experiments will be presented and, moreover, they will be compared with real experimental data.

Numerical Modeling of Compositional Compressible Two-Phase Flow in Porous Media

Ondřej Polívka

Czech Technical University in Prague

Abstract:The reliable simulation of two-phase flow of a mixture in porous media is important for many applications including oil recovery or CO2 sequestration. We deal with the numerical modeling of two-phase compressible multicomponent flow in porous media. The mathematical model is formulated by means of partial differential equations representing the conservation laws, extended Darcy's laws, and by means of the conditions of local thermodynamic equilibrium in a form of algebraic relations describing the pressure and the distribution of components between the phases. Appropriate initial and boundary conditions are prescribed. The problem is solved numerically using the Mixed-Hybrid Finite Element Method (MHFEM) for Darcy's law discretization, and the Finite Volume Method (FVM) for the component transport equations discretization. To evaluate the component flux between the elements, we propose a special upwind technique for the discretization of the component fluxes across the boundaries of the elements. The time discretization is carried out by the backward Euler method. The resulting system of nonlinear algebraic equations is solved by the Newton-Raphson iterative method. The derived numerical scheme ensures the local mass balance and correct treatment of the phase fluxes between elements. The numerical model is used for a simulation of injection of methane or CO2 into a homogeneous 2D reservoir filled with different substances in two phases.

Numerical Simulation of Anisotropic Mean Curvature of Graphs in Relative Geometry

Hung Hoang Dieu

Czech Technical University in Prague

Abstract:This contribution is concerned with the graph formulation of anisotropic mean curvature flow in relative geometry [1]. The results of [2] are extended to our problem. The numerical scheme is based on the method of lines where the spatial derivatives are approximated by finite differences. We then solve the resulting ODE system by means of the adaptive Runge-Kutta-Merson method. Finally, computational results with various anisotropy settings are presented. [1] G. Bellettini, M. Paolini: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Mathematical Journal, {\it 25} (1996), 537--566. [2] K. Deckelnick, G. Dziuk: Discrete anisotropic curvature flow of graphs. ESAIM: Mathematical Modelling and Numerical Analysis, {\it 33} (1999), 1203--1222.

Mechanisms controlling the cyclic saturation stress and the critical cross-slip annihilation distance in copper single crystals

Petr Pauš

Czech Technical University in Prague

Abstract:The proposed model is inspired by Brown's suggestion that the saturation stress in cycling is controlled by the stress required to separate two screw dislocations of opposite signs, which are just on the point of mutual annihilation by cross-slip. Cross-slip is treated as the deterministic, athermal process governed by the line tension, the applied stress and the interaction force between dislocations. The saturation stress and the critical cross-slip annihilation distance predicted simultaneously by the model agree with the available experiments.

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

Jan Mach

Czech Technical University in Prague

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 the talk author present his progress in the study of this method and introduce numerical results for particular reaction-diffusion system in two spatial dimensions.

Configuration Dynamics Verification Using UPPAAL

David Fabian

Czech Technical University in Prague

Abstract:Modern software applications can have a very complicated internal dynamics. Most of the software tools are written in an imperative programming language which can quickly become impractical for describing a complex dynamics. Also, it is very hard to verify that the code actually covers fully all aspects of the tool’s dynamics. Propagation rules are suitable as a means for specification and verification of such dynamic systems. We have selected software tool from the domain of configuration for our study. Configuration wizards and tools are examples of software applications where even a small change made by the user can lead to a very complex outcome. In this paper, a configuration hierarchical model and a syntax of propagation rules are introduced. These constructs can be used to describe declaratively the dynamics that is typical for software configuration tools. The hierarchical model is then used for describing the internal dynamics of the configuration tool Freeconf. This specific model instance is then implemented in UPPAAL and verified by the UPPAAL model-checker.

Algebraic multigrid for Navier-Stokes Problems

Vladimír Klement

Czech Technical University in Prague

Abstract:Algebraic multigrid is an advanced method for solving systems of equations arising from various numerical problems. In this presentation we will describe this method and its implementation for the simulation of Navier-Stokes flow . We will also point out main differences compared to geometric multigrid and compare the speed of both methods.

Level Set Formulation of Geodesic Active Contours Model in Medical Image Segmentation

Radek Máca

Czech Technical University in Prague

Abstract:The contribution presents a 3D (2D+time) segmentation of the real cardiac MRI data using the level set formulation of the geodesic active contour model and its semi-implicit complementary volume discretization. In particular, the application is focused on the segmentation of the heart ventricles from the cine MRI data. Validation of achieved results is performed by comparing our algorithm with the Allen-Cahn approach and the software SEGMENT.

GPU implementation of a multigrid solver for the incompressible Navier-Stokes equations

Vítězslav Žabka

Czech Technical University in Prague

Abstract:We present a GPU implementation of a geometric multigrid method for the incompressible Navier-Stokes equations in 2D. The equations are discretized in space by means of the mixed finite element method on unstructured triangular meshes. For the time discretization a semi-implicit scheme is employed. The resulting systems of linear equations are solved using the multigrid V-cycle with a Vanka-type smoother. In the presented implementation, the solver runs completely on the GPU including the assembling of the linear systems. Its parallelization is based on the red-black ordering of the mesh triangles. We apply the solver to the numerical simulation of air flow in a simplified urban canopy where speedup up to five, compared to the corresponding multi-core CPU implementation, is achieved.

Modeling of Heaving in Freezing Saturated Uncemented Porous Media

Alexandr Žák

Czech Technical University in Prague

Abstract:Motivated by freezing soils in the cold regions of the Earth which often experience an occurrence of substantial potential for a heaving, we develop a model of porous medium with coupled elasticity and pore content freezing. The model is based on a heat equation with an additional term for heat of the phase transition and on the Navier equations with a coupling term. In this contribution, the model settings and computational studies of the model are presented, and also some driving principles of the heaving are reviewed.

Numerical methods for diffusive problems in laser plasma physics

Milan Holec

Czech Technical University in Prague

Abstract:Two numerical methods to solve 2D diffusion equation on a Lagrangien mesh will be presented: Mimetic Support Operators Method and Mixed Finite Element Method. In the field of laser plasma physics, electron heat conduction and radiation transport can be modeled by nonlinear diffusion. Particularly, Robin boundary condition is of great interest in the radiation transport phenomenon. The ability of both methods on an extensive set of test cases, including highly distorted meshes and nonlinear heat wave, will be discussed. A sample of laser-target interaction simulations calculated with our 2D PALE hydrodynamic code using the MSOM diffusion solver will be shown.

Solving stochastic control problems: HJB equation vs. stochastic maximum principle approach

Petr Veverka

Czech Technical University in Prague

Abstract:In the talk, a general form of a stochastic control problem will be given and two classical approaches to the solution will be presented. First, the Hamilton-Jacobi-Bellman PDE describing the infi nitesimal optimality principle and second, the stochastic maximum principle which follows ideas from analytic mechanics in terms of maximizing (or minimizing) the Hamiltonian of the problem. The two approaches will be compared in short and some deeper connections between them will be shown using Backward Stochastic Differential Equations. If there is time at the end of the talk, an application of maximum principle to stochastic controlled logistic model will be given. This application comes from joint work with prof. Bohdan Maslowski. Keywords: Stochastic control, HJB equation, Stochastic maximum principle, discounted control problem, stochastic logistic equation.