Student workshop on scientific computing 2018

June 14-17, 2018

Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

List of abstracts

Numerical analysis of immersed boundary - lattice Boltzmann method for interaction of fluid flow and elastic body
Pavel Eichler, Radek Fučík
FNSPE, CTU in Prague

Abstract: In this contribution, we deal with the numerical immersed boundary - lattice Boltzmann method for simulation of the fluid-structure interaction in 2D. We consider interaction of incompressible, newtonian fluid in isothermal system with the elastic fiber, which represent an immersed body boundary. First, a short introduction to the lattice Boltzmann and immersed boundary method will be presented and the combination of these two methods will be briefly discussed. Next, the choice of the smooth approximation of the Dirac delta function and the discretization of the immersed body will be discussed. The effect of the immersed body boundary discretization will be analyzed on the deformation of elastic fiber tightened in the domain with cavity flow. The results demonstrate that the restrictions placed on the discretization in literature are not necessary. In the last part, the deformation of elastic fiber with one fixed end behind a circular obstacle will be studied.

The simulation of Bloch equations for quantitative evaluation of relaxation times in MR
Kateřina Fricková
FNSPE, CTU in Prague

Abstract: This contribution deals with the reconstruction of T1 maps using simulation of the Bloch equations. The Bloch equations describe an evolution of magnetization vector in time where relaxation times T1 and T2, that are characteristic for each tissue, are present. The MOLLI (Modified Look-Locker Inversion) imaging sequence has been implemented, which is used for the measurement of moving objects -- hearth. The result of the measurement with this sequence is 11 pictures aquisited after the application of 3 inverse RF pulses and 3 or 5 series of RF pulses with smaller flip angle. These pictures are generated at the same phase of the cardiac cycle. Using the Bloch equations simulation, a database of curves with preconfigured parameters T1, T2 and a flip angle is obtained. The curves are then compared to the measured pictures. In this manner the correction of the measured T1 map which is erroneous due to the sequence by which they are taken can be obtained.

Current state of MPI support in TNL Library
Vít Hanousek
FNSPE, CTU in Prague


The Template Numerical Library (TNL) is developed at the Department of Mathematics at the Faculty of Nuclear Sciences and Physical Engineering. The TNL library is primarily focused on multi-core CPUs and Cuda compatible GPUs, nevertheless large computations exceeded capabilities of a single computer or a single GPU. Hence, our aim is to add support of multi-computer and multi-GPUs computations into this library. In this talk, we present parts of the library which now support distributed computations. First, we focus on a concept of TNL communicators and the MPI. Secondly, we describe implementation of a distributed rectangular grid and a simple explicit PDE solver. Finally, we present some performance measurements on the Heat equation problem.

Diffusive Learning of Kohonen SOM Network
Radek Hřebík
FNSPE, CTU in Prague

Abstract: Traditional self organised map (SOM) is learned by Kohonen learning. Novel model of self organisation is based on diffusion modelling in continuous space which is a good approximation of endorphins propagation in real brain. Therefore the structure of the system is described by neuron coordinates instead of neighbourhood relationship in traditional SOM. The contribution discusses neuron activation using diffusion process and novel diffusive learning algorithm is based on this activation mentioned above. Novel structure and algorithm are demonstrated on simple examples.

Mathematical model of excitable media
Jakub Kantner
FNSPE, CTU in Prague

Abstract: In this contribution, a model of signal propagation in excitable media based on a system of reaction-diffusion equations is studied. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. First the origin and the propagation of the cardiac action potential in the heart is discussed. Then, the model and its discretization by the means of finite-difference method is introduced. Finally, the results of a numerical study in both homogeneous and heterogeneous medium are presented with focus on the interactions of propagating signals with obstacles in the medium.

Statically configurable data structure for conforming unstructured homogeneous meshes on CPUs and GPUs
Jakub Klinkovský
FNSPE, CTU in Prague

Abstract: In the contribution we present a general data structure for an efficient representation of conforming unstructured homogeneous meshes for scientific computations on CPU and GPU-based systems. The underlying abstract representation of the mesh supports almost any cell shape. In the TNL library, where the data structure is implemented, common 2D quadrilateral, 3D hexahedron and arbitrarily dimensional simplex shapes are implemented. The data structure is statically configurable by means of C++ templates, which allows the data structure to be optimized for a given application. The internal memory layout is based on state-of-the-art sparse matrix storage formats that can be optimized for each hardware architecture in order to facilitate high performance computations. We investigate the efficiency of the implemented data structure on CPU and GPU hardware architectures using several benchmark problems. Finally, in order to demonstrate that the data structure is applicable to advanced numerical methods, we show results of MHFEM simulations of two-phase flow in porous media. We show speed-ups that rise above 32 in 2D and 59 in 3D when compared to sequential CPU computations, and above 5 in 2D and 11 in 3D when compared to a ten-threaded CPU computations.

Computational investigation of double cross-slip
Miroslav Kolář
FNSPE, CTU in Prague


We summarize our present results of dislocation dynamics modeling by means of the parametric method. 

In the talk we focus on development of the model of dislocation double cross-slip in FCC crystals. 

The cross-slip  is modeled as a deterministic and  stress-controlled process and the cross-slip critrion based on 

evaluation of exerted stresses is employed. The dynamics of dislocation curves is governed by the geodesic curvature driven flow on

surfaces and for the numerical solution, the flowing finite volume method is used. Our approach is tested on a scenario, in which 

a dislocation curve overcomes a  particle exerting spherically symmetric  repulsive  stress  field. Results of our simulations

are validated by analytical calculations and compared with results obtained by different approach.

Reaction-Diffusion Equations in Electrocardiology
Jana Lepšová
FNSPE, CTU in Prague


This contribution is an introduction to mathematical models in electrocardiology. It provides basic information on heart function and its mathematical description (in the form of the FitzHugh-Nagumo model) based on biological properties. The FitzHugh-Nagumo model is studied using mathematical methods for investigating dynamics of ordinary differential equations. A method for construction of invariant regions for a system of reaction equations is presented and the maximum principle for a linear diffusion equation is derived. Two different forms of the FitzHugh-Nagumo model are analyzed in detail emphasizing analysis of qualitative behaviour of the solution with respect to parameters.

Curvature driven flow of space curves in normal and binormal direction
Jiří Minarčík
FNSPE, CTU in Prague

Abstract: This contribution deals with problems associated with generalization of the curvature flow of curves into higher dimensional space. The motion in normal and binormal direction of closed curves embedded in $R^3$ is analyzed and compared to the standard two-dimensional case. We present several theoretical results including the Shrinking ball theorem which states that curves under the curvature flow cannot cross certain family of shrinking spheres.

The Kaldor-Kalecki model: A new approach
Zuzana Olajcová
FNSPE, CTU in Prague

Abstract: The Kaldor-Kalecki model is an extension of Kaldor model, which is a system of differential equations. By introducing the delays that express a gestation period between an investment decision and its delivery, it becomes a system of delay differential equations (DDE). As a dynamic system, it has been widely studied and special attention has been paid to its ability to generate chaotic dynamics. Unlike previous studies, in this paper a new specification of the model based on realistic economic explanation is proposed. After that estimation of its parameters is performed using U.S. historical data from 1950 to 2014. Since the objective function for parameter estimation is a multimodal optimization problem with weak software support, a more appropriate solution tool is also proposed and implemented. Consequently, the stability of the model with estimated parameters is studied. Conditions for the local stability and the existence of Hopf bifurcation at the equilibrium of the system are derived. Finally, to solve the estimated Kaldor-Kalecki model as a DDE system, several Runge-Kutta methods for constant delays are implemented, whose effectiveness is accordingly evaluated and verified the obtained results are.

Numerical Modeling of Non-isothermal Compositional Compressible Gas Flow in Zeolite Bed and Coupled Atmospheric Boundary Layer
Ondřej Pártl
FNSPE, CTU in Prague


I will present a mathematical and numerical model of non-isothermal compressible flow of a mixture of two ideal gases in a heterogeneous porous medium and in the coupled atmospheric boundary layer above the porous medium surface, where the balance equations for the flow in porous medium contain stiff source terms.

In our model, the domain in which the flow occurs is divided into the porous medium subdomain and the free flow subdomain. In each subdomain, the flow is described by corresponding balance equations for mass, momentum and energy. On the interface between the subdomains, coupling conditions are prescribed.

In both subdomains, the spacial discretization of the governing equations is carried out via the finite volume method, and the problems arising from the presence of stiff source terms are solved using an operator splitting combined with a complicated time-stepping procedure.

As an application of our model, I will consider humid air flow over a zeolite 13X bed.

Unified presentation of the phase stability problems
Tomáš Smejkal
FNSPE, CTU in Prague

Abstract: In this contribution we will present a general formulation of the phase-stability problems for multicomponent mixtures and verify that this formulation generalizes the problems of phase stability at constant volume, temperature, and mole numbers (VTN-stability), at constant internal energy, volume, and mole numbers (UVN-stability), and at constant pressure, temperature, and mole numbers (PTN-stability). Furthermore, we introduce a numerical method for solving the general formulation of the phase stability problems. This algorithm is based on the direct minimization of the objective function with respect to the constraints. The algorithm uses a modified Newton-Raphson method, along with a modified Cholesky decomposition of the Hessian matrix to generate a sequence of states with decreasing values of the objective function. The algorithm was implemented in C++ and using generic programing we have a single, portable solver for all three stability formulations. Properties of the algorithm will be shown on phase stability problems of multicomponent mixtures in different specifications.

Optical flow-based non-rigid registration of cardiac MRI images
Kateřina Solovská
FNSPE, CTU in Prague


This contribution deals with non-rigid registration of MRI data. The goal of this work is to propose and test a method for registration of segmented images of heart. Proposed method is based computing optical flow between the distance functions of segmented objects. Level set method and method of minimization of Mumford-Shah functional are used for the segmentation of myocardium and vertices. The proposed method was tested on artificial and real data. The results were compared with the results of method based on maximization of mutual information, which is often used for similar types of images. For registration of images with significantly different intensities, better results were obtained by optical flow method.

Numerical study of mass transfer and transport of CO2 in shallow subsurface
Jakub Solovský
FNSPE CTU in Prague


In this work, we investigate CO2 exsolution, transport, trapping and dissolution in shallow subsurface under various conditions.

First, we introduce mathematical model describing the system. For the mass transfer of CO2 the rate limited model is used.

Numerical results obtained using the model are compared to the experimental data obtained from two sets of experiments: 1D column experiments and intermediate scale 2D experiments. In all the experiments water with dissolved CO2 was injected into the tank and the fate of dissolved and gaseous CO2 was observed.

The experiments were conducted under various conditions including different heterogeneity configurations, flow rates and dissolved CO2 concentration.

We investigated the effects of the different conditions in the experiment on the studied processes of exsolution, transport, trapping, and dissolution and addressed these dependencies in the mathematical model.

Adaptive mesh refinement for FEM using a posteriori error estimates
Vojtěch Straka
FNSPE, CTU in Prague

Abstract: A posteriori error estimation provides fully computable upper bound on the error of numerical solution for many choices of partial differential equations and numerical methods. Therefore, these estimates can play a significant role in scientific computing, when computation of numerical solution with prescribed maximum error is needed. Also, they can be used for devising effective algorithms for solving PDEs. In this talk, a posteriori error estimates for Poisson equation will be presented. First, the general concept will be described and interesting properties will be shown, such as asymptotic exactness and local effectivity. Then, the performance of these estimates will be discussed. Finally, an adaptive algorithm based on FEM with local mesh refinement using a posteriori error estimates will be demonstrated on model problems in 2D. These results will be compared with FEM using global mesh refinement.

Mathematical modeling and numerical simulation of microstructure evolution during phase transitions
Aleš Wodecki
FNSPE, CTU in Prague


The phase field model is one of the possible ways of simulating anisotropic crystal growth. The mathematical model is briefly introduced and subsequently a novel algorithm that allows to simulate multiple grain growths with arbitrary orientations is presented. The pros and cons of this approach will be discussed along with possible future extensions. Lastly, simulations featuring realistic materials will also be shown.

Pore-Scale Thermomechanical Model of Solidification in Liquid-Saturated Porous Media
Alexandr Žák
FNSPE, CTU in Prague


This contribution analyses processes accompanying solidification of liquid-saturated porous media by means of thermomechanical balance and admitting difference in density of liquid and solid phases. Complexity of these processes is given by the inherent heterogeneity of the volume occupied both by the frame structure and pores of the medium. The developed mathematical model provides spatio-temporal dependencies of key variables and allows to evaluate the stresses caused by the difference in the specific volumes of the present phases.