Student workshop on scientific computing 2020

December 4, 2020. Online only.

Departments of Software Engineering and Mathematics
FNSPE CTU in Prague, Czech Republic

List of abstracts

Mathematical Methods in Fractal Geometry
Michaela Diasová
FNSPE, CTU in Prague

Abstract:

This thesis deals with basic terms of fractal geometry, presents considerable geometrically complex sets and describes its properties. These sets are analyzed in terms of topology, measure theory and geometry. The large and small inductive dimensions are defined and their sum attributes are studied there. Afterwards, the Hausdorff measure and dimension are defined together with other used fractal dimensions. Subsequently, the iterated function systems and their rucurrent version are described there. In the final part can be found visualization of some of the geometrically complex sets using the chaos game algorithm, which exploits these systems, and two methods of coloring these sets are introduced.

Cumulant LBM simulations of turbulent boundary layer flow above rough surfaces
Pavel Eichler
FNSPE, CTU in Prague

Abstract:

Dynamics of wall-bounded flow is an open problem in the theory of turbulence although it has been studied for decades. To investigate the complicate fluid flow in the boundary layer region, solution of numerical methods can be used. The main aim of this contribution is to present the investigation of the cumulant lattice Boltzmann method (CuLBM) simulations of the turbulent fluid flow in the boundary layer region above rough surfaces as the direct numerical simulator (DNS), i. e., with the numerical method without any turbulent models, etc.

To investigate the CuLBM, two benchmark tests are selected. The first benchmark test deals with the turbulent fluid flow above a rough plate with regularly distributed protrusions. The second benchmark test is inspired by the street-canyon flow experiment. The CuLBM results are compared both with the standard finite difference method (FDM) and with experimental results. The CuLBM results were in a good agreement with both FDM and experimental results. Thanks to the good agreement, CuLBM can be considered as an appropriate for the turbulent fluid flow in the boundary layer above rough surfaces.

Application of Interval Arithmetic for Solving the Phase Stability Problem
Jan Fejtek
FNSPE, CTU in Prague

Abstract:

In this presentation, we will discuss a method of testing phase stability of multicomponent mixtures at constant temperature, volume and moles. The phase stability criterion is formulated by the means of the Tangent plane distance (TPD) function. To study the TPD function, we will use the methods of interval arithmetic. Finally, we will present current findings.

Blood-like fluids and in vitro flow experiments
FNSPE CTU in Prague, IKEM in Prague

Abstract:

In vitro experiments are important for computational fluid dynamics modelling of blood flow. If non-Newtonian flow is not insignificant in the studied phenomena, the need for non-Newtonian blood-like medium arises. Several fluids have been suggested. In this work, several blood-like fluids are measured for viscosity and used in an experimental setup with artificial aortic valve in order to decide the optimal setup for future experiments.

Mathemathical modeling of multicomponent flow in porous media
Petr Gális
FNSPE, CTU in Prague

Abstract:

Numerical scheme of second order approximation in space for the single-phase multicomponent fow in porous media is presented. Mathematical model consists of Darcy velocity, transport equations for components of a mixture, pressure equation and associated relations for physical quantities such as viscosity or density. Discrete problem is obtained via discontinuous Galerkin method for the discretization of transport equations and pressure equation with the combination of mixed-hybrid finite element method for the discretization of Darcy velocity with the use of Raviart-Thomas element of higher order. Subsequent problem is solved with the iterative IMPEC method. Numerical experiment of 2D flow is carried out to validate the code.

Particle Movement over Discrete Approximation of Fractal Sets
František Gašpar
FNSPE, CTU in Prague

Abstract:

The diffusion is studied in the spaces where dimension can also be of non-integer value. The solution of the diffusion equation is known for such spaces and serves as a basis of the presented time, and the diffusion coefficient invariant dimension estimate. Presented Monte Carlo simulations of diffusion in the discrete model of fractal sets serve as the source of data to study the derived dimension estimated. Properties of the dimension estimate are presented and the quality of the discrete model simulations is discussed.

Pre-interventional planning in patients with congenital heart disease by using biomechanical heart modeling
Maria Gusseva, Radomir Chabiniok, Dominique Chapelle, Tarique Hussain
Inria and LMS (Ecole Polytechnique), France, Inria, LMS (Ecole Polytechnique, France), UT Southwestern Medical Center (Dallas, TX)

Abstract: Coupling biomechanical model with clinical data has a potential to augment clinical markers derived by medical exams and contribute to optimise the therapy management. We will explore the role of patient-specific biomechanical models in the assessment of heart function in patients with some congenital pathologies prior to intervention.

Traveling wave solutions for degenerate nonlinear parabolic equations
Yu Ichida, Takashi Sakamoto
Meiji University

Abstract: We consider the traveling wave solutions of the degenerate nonlinear parabolic equation $u_{t}=u^{p}(u_{xx}+u)$ which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation $v_{\tau}=v^{p}(v_{xx}+v-v^{-p+1})$ related with it. We report on the results obtained for traveling wave solutions.

Phase stability testing of multi-component mixtures with application of global optimization methods
Martin Jex
FNSPE, CTU in Prague

Abstract:

In this contribution we examine the phase equilibrium of mixtures in VTN variables. More precisely, the initial molar concentration c and the temperature of mixture is known and our task is to determine whether the mixture stays in one phase or splitting to more phases occurs. This physically motivated problem can be rendered to a mathematical problem of optimization of a non-convex function (TPD) on a convex set (interior of simplex). The optimization problem is solved by the branch and bound algorithm, in which the convex lower estimate of TPD function is used. This estimate is derived by using a convex-concave split and is optimized to be closest to original TPD function in certain sense. The algorithm is implemented in Mathematica and is illustrated on 3 examples with various level of difficulty. Furthermore, the computational complexity is discussed. The implemented algorithm gives the consistent results with a recent paper, which are more precise than solutions obtained by previously used approaches.

TNL: Numerical library for modern parallel architectures and distributed meshes
FNSPE, CTU in Prague

Abstract:

We present the Template Numerical Library (TNL) which provides native support for modern parallel architectures, such as multi–core CPUs and GPUs. The library offers an abstract layer for accessing these architectures via unified interface which facilitates rapid development of high-performance algorithms and numerical solvers. The library is written in C++ and benefits from template meta–programming techniques. In this contribution, we present the most important data structures and algorithms in TNL together with scalability on multi–core CPUs and speed–up on GPUs supporting CUDA.

In addition to providing an abstraction for computational architectures, TNL provides implementations of common parallel algorithms and data structures for numerical computing. The most important data structures are those representing a numerical meshes which are necessary for finite volume or finite element methods. Currently, TNL supports regular orthogonal structured grids and unstructured conforming homogeneous meshes. Utilizing the design of the abstract layer for accessing computational architectures, all data structures provided by TNL support computations on multi-core CPUs as well as GPUs. Additionally, the most recent development has focused on adding support for distributed MPI computations with decomposed unstructured meshes.

Mathematical modeling of fluid flow and transport of contrast agent in vessels
Jan Kovář
FNSPE, CTU in Prague

Abstract:

This contribution deals with mathematical modeling of problems associated with myocardial perfusion examination using a contrast agent. Initially, the problem of fluid flow and transport of contrast agent is divided into three benchmark problems, two of which will be analyzed within this contribution. The audience will be briefly introduced to a mathematical model of Newtonian incompressible fluid flow in an isothermal rigid porous medium and in an isothermal free flow system. The results of the benchmark problem of single-phase flow in a porous medium representing the myocardium and in a two-dimensional domain representing a blood vessel from the vascular bed using the lattice Boltzmann method will be shown and further compared to the results acquired from the finite difference method or the mixed-hybrid finite element method.

Lattice Boltzmann simulations of phase-field equation
Michal Malík
FNSPE, CTU in Prague

Abstract:

This contribution presents the short introduction to the phase-field equation (PFE) describing the evolution of the interface between inmiscible phases. Then, the lattice Boltzmann method (LBM) for the simulation of PFE is presented. Based on the numerical simulations, two different approximations of the normal vector to the phase interface were suggested and tested. The presented results were obtained by the in house implementation using C++ with support of CUDA. Based on the numerical results, experimental order of convergence in space was estimated. LBM converges second in the case of diagonal disk translation. In the case of Zalesak's Disk, the speed of convergence is dependent on the approximation of the normal vector. This observation is an impetus for a further study of this method.

A projection method for the Navier-Stokes equations with a total pressure boundary condition
Kazunori Matsui
Kanazawa University

Abstract:

The projection method has been used for a numerical computation of the time-dependent Navier--Stokes equations. We propose a projection method for the Navier--Stokes equations with a Dirichlet-type boundary condition for the total pressure. Here, the total pressure is the sum of the pressure of the fluid and the fluid's kinetic energy per unit volume. We show the stability of our projection methods and establish error estimates in suitable norms between the solutions to the projection method and the original Navier--Stokes equations.

Mathematical Modelling in Electrocardiology
Niels van der Meer, Michal Beneš
FNSPE, CTU in Prague

Abstract:

Cardiovascular diseases account for more than thirty per cent of all deaths which makes them the most common cause of decease worldwide. It is therefore understandable that considerable effort has been exerted to treat and prevent these conditions. This talk(based on a thesis of the same name) probes for the potential contributions of mathematics and its tools developed from the theory of reaction-diffusion equations. The main area of interest is electrocardiology which studies heart rhythm disorders as well as their causes. Some of the mathematical models describing the propagation of a signal in an excitable medium are introduced. One such example is the FitzHugh–Nagumo model whose several variations were numerically analyzed and the results are presented in this talk.

Nondegenerate homotopy and geometric flows of space curves
Jiri Minarcik
FNSPE, CTU in Prague

Abstract:

This contribution deals with problems related to geometric flows of space curves that depend on the existence of the Frenet frame. Using results from homotopy theory we can predict some aspects of the long term behaviour of these motion laws. We introduce new quantity which is invariant to nondegenerate homotopies and use its value to classify connected components of the space of locally convex curves on which these motion laws operate.

The Method of Fundamental Solutions applied for some Hele-Shaw problems
Yusaku Shimoji, Shigetoshi Yazaki
Meiji University

Abstract:

The Method of Fundamental Solutions (MFS) is a mesh-free numerical method which is often applied for Laplacian problems. In MFS, we can find the approximate solution as a linear combination of fundamental solutions. Here, we will introduce a Hele-Shaw boundary value problem and show some numerical results. Then we apply MFS for a Hele-Shaw problem of magnetic fluid.

Solution of the VTN-phase stability problem using the Sherman-Morrison iterations
Tomáš Smejkal, Jiří Mikyška
FNSPE, CTU in Prague

Abstract: In this contribution, a simple procedure for solving systems of linear equations arising from the linearization of the VTN-phase stability problem will be derived. Using the structure of the Hessian matrix, the solution is obtained by sequential usage of the Sherman-Morrison formula. The great advantage of the method is that the system of equations can be solved without even assembling the system. The performance and the speed-up of the computation will be demonstrated on several examples of different complexity.

BDDC for MHFEM discretization of two-phase flow in porous media
Jakub Solovský
FNSPE, CTU in Prague

Abstract:

This work deals with the application of the Balancing Domain Decomposition based on Constrains (BDDC) method to two-phase flow problems in porous media. We briefly describe the spatial discretization of the problem which is based on the mixed-hybrid finite element method (MHFEM) and semi-implicit time discretization. Then in detail, we describe the BDDC method, discuss the differences between the 2D and 3D cases, and present necessary modifications of the algorithm to improve its efficiency for a more complicated 3D case. We describe the parallel implementation of the method and highlight the critical steps of the algorithm that affect the performance and scalability. The parallel implementation is then tested on benchmark problems in 2D and 3D and its efficiency is investigated on various meshes. The numerical results indicate that the method preserves high computational efficiency for increasing number of processes and, therefore, allows solving problems on very fine meshes.

Geometric Flow of Curves and Its Applications
Monika Suchomelová
FNSPE, CTU in Prague

Abstract:

The work studies dynamics of curves by means of differential geometry. Firstly, the thesis deals with the heat equation and its properties. Then, the essential terminology from mathematical analysis of curves is introduced. The main studied problem is the curvature flow of planar curves. To solve the problem, the numerical schemes are developed using parametric approach. In the last section, these schemes are tested on specific examples.

Application of maximal monotone operator method for solving Hamilton-Jacobi Bellman equation arising in optimal portfolio selection problem
Cyril Izuchukwu Udeani, Daniel Sevcovic
FMPI, Comenius University in Bratilava

Abstract:

In this paper, we investigate a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati method for transformation. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach's fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space. Some financial applications of the proposed result are presented in one-dimensional space. However the paper is still under preparation for publication.