Czech-Japanese Seminar in Applied MathematicsAbstracts |
| Home Registration Participants Abstracts Scientific programme Accommodation Transportation Food and dining |
Abstract: Material fracture due to deformation caused by contact that changes with time is a complex phenomenon. Here, we use numerical simulations with the phase-field approach to study crack propagation in several contact problems, such as cutting problems and arch collapse. FreeFEM with the IPOPT package is very useful for analyzing problems with such inequality constraints.
Abstract: Fracture problems in modern science and technology are prevalent and severe issues nowadays. For the numerical simulation of crack propagation, the fracture phase field model (PFM) is commonly used in fracture mechanics. We examine the numerical simulation for crack propagation due to the various scenarios, such as mode I crack propagation, mixed mode crack propagation, and compression in the domain with inclined and horizontal cracks, by using the finite element method (FEM) for PFM. Here, we applied PFM to investigate the two-dimensional and three-dimensional problems in the original model and the model with unilateral contact condition. Using the numerical technique FEM, in every instance, we observed the realistic fracture propagation feature. For the case of unrealistic, we mentioned the PFM with unilateral contact condition. Here, we also proposed the model with a unilateral contact condition to avoid unrealistic cases of crack propagation. Using both cases, we noticed that mode I and crack propagation due to compressing in the domain with horizontal crack yield realistic results. The model with unilateral contact condition gave us a realistic crack path for the mixed mode, whereas the original PFM gave us a branching crack path, which is not possible. For crack propagation due to compressing in the domain with inclined crack, the original is not realistic, but the crack profile is realistic in the model with unilateral contact condition. We also discussed and theoretically proved the energy dissipation identity, more commonly known as the waste energy identity for both the original and unilateral contact condition. This identity helps to formulate fracture criteria like Griffith’s criterion, which states that a crack grows when the energy release rate exceeds the toughness of the fracture. Thus, we may conclude that compared to the original model, crack propagation resulting from unilateral contact circumstances produces more realistic findings. In some circumstances, we also discuss the driving force profile and surface energy of the original and unilateral contact condition. Here, we used FreeFem++ software for numerical simulation and symmetric mesh generation with adaptation.
Abstract: We propose a numerical scheme for evolution inclusions involving an elasto-plastic model with a time-dependent threshold function. The proposed scheme projects the stress within constraints after solving the linear elliptic PDE at each step, avoiding the need to solve nonlinear problems. We derive the existence of an exact solution to the original problem using the stability of the scheme. The presentation is based on collaborative work with Yoshiho Akagawa (Kyoto University of Education).
Abstract:
The Hele-Shaw problem is a popular model of the flow of an incompressible
fluid between two closely spaced parallel plates, known as the
Hele-Shaw cell. In its classical form, the Hele-Shaw problem is homogeneous
as it does not involve a time-dependent coefficient. However, we
focus on a Hele-Shaw problem in an inhomogeneous medium, where the
coefficient in the free boundary velocity depends on both space and time.
The homogenization behavior in such a situation exhibits interesting effects,
and finding the average velocity of free boundary movement is an
interesting problem. We develop a new BBR-like scheme for the Hele-Shaw
problem with a time-dependent coefficient. The scheme is formally derived
as the large conductivity limit of the scheme introduced by [Berger-Brezis-
Rogers, 1979] for the Stefan problem. To discretize in time, the BBR-like
scheme is used, while quadtree and octree structures are applied for spatial
discretization. We use adaptive mesh refinement with quadtree structures
in two dimensions and octree structures in three dimensions to make computing
more efficient. Numerical experiments demonstrate that our method
yields a more robust and precise estimation of average free boundary velocity
in comparison with the results presented in [Palupi-Pozar, 2018]. We
also compare the performance of our new scheme to that of a state-of-theart
first-order level set method.
This is a joint work with Prof. Norbert Pozar from the Faculty of
Mathematics and Physics, Institute of Science and Engineering, Kanazawa
University, Kanazawa, 920-1192, Japan.
References
[1] Berger, A. E., Brézis, H., and Rogers, J. C. W. A numerical method
for solving the problem ut − Δf(u) = 0. RAIRO Anal. Numér., 13(4), 297–
312, 1979.
[2] Palupi, I. and Požár, N. An efficient numerical method for estimating the
average free boundary velocity in an inhomogeneous Hele-Shaw problem. Sci.
Rep. Kanazawa Univ., 62, 69–86, 2018.
Abstract: Free boundary problems like the Hele-Shaw problem require solving an elliptic PDE on a moving domain, and need an accurate gradient of the solution at the free boundary. We show an application of a discrete Aleksandrov--Bakelman maximum principle to obtain an improvement of an error estimate in the max norm for linear elliptic finite difference schemes on adaptive grids. This is joint work with Shahadat Ali and Alam Md Joni from Kanazawa University.
Abstract: A method of adjusting nuclear reactor kinetic parameters to experimental data is proposed. The fractions of neutrons delayed via different precursor groups are of interest. Their values originally calculated by Monte Carlo simulations are modified to bring the power output of the reactor predicted by the point kinetics equations closer to the measured values. The measurements were performed on the VR-1 zero-power training reactor in the Czech Republic. Three reactivity patterns were investigated to account for the different reactor transients. The resulting ODE-constrained optimization problem is solved numerically, using the adjoint equations to obtain the gradient of the loss functional and applying a specifically tailored gradient descent technique. The performance of our approach is compared to other variants of gradient-based optimization. As a side result, a gradient descent step size adaptivity algorithm is proposed. Finally, discussion on the physical relevance of the obtained results is provided.
Abstract: The problem of electrohydrodynamic stability in an external time-periodically varying field is formulated as simply as possible. The aim is to perform an analysis of the fastest forming instabilities having an appearance of a jetting onset. The analysis is focused on the relationship between average inter-jet distances and electric field intensity as well as its frequency. Theoretical results will be compared with an experiment based on AC electrospinning method. We consider an idealization of a realistic AC electrospinning set-up. This set-up consists of a disc spinning-electrode connected to a frequency and voltage tuneable AC high voltage supply. The set-up works without any collector, i.e., counter-electrode. The lover part of the rim of the slowly rotating disc is in touch with a polymeric solution that fills an opened container. This arrangement of the experimental enables: (i) localization of the polymeric solution on the narrow disc perimeter only, (ii) homogeneous field strength distribution on the liquid gas interface, and (iii) thorough observation of distances between jets using a high-speed camera. The electrohydrodynamic problem under such conditions has not been addressed in literature, to the best of the authors’ knowledge.
Abstract: TBA
Abstract:
This study examines the structure of shock waves in two-phase flow of a
dilute mixture of non-ideal gas and small solid particles employing the NavierStokes-Fourier model.
The gas is assumed to follow the reduced van der Waals' equation of state while the solid dust particles are treated as pseudo-fluids. The mass and volume fraction, radius, and specific heat of the dust particles and the pre-shock Mach number and the non-idealness of the gas have been taken as the parameters for the flow. Initially, gas and dust particles are assumed to be in equilibrium, moving at a uniform speed. Upon the initiation of the shock formation process, the gas and solid particles transit into a non-equilibrium state, exhibiting different profiles and relax finally to achieve the equilibrium at distinct relaxation rates. It is found that only when the size of dust particles approaches the order of mean free path of the gas, the mixture exhibit behavior akin to single-phase fluid. The non-equilibrium intensifies as the size of the dust particles increases. The profiles of the gas and dust particles are found to differ considerably with the presence, size, and thermodynamic properties of dust particles, the nonideal nature of the gas, and the Mach number of the flow.
Keywords: shock wave structure; dusty gas; Navier-Stokes-Fourier approach; continuum model; non-ideal gas; two-phase flow
Abstract: We consider type II blow-up solutions for a quasilinear parabolic equation that appears in curve shortening problems. In this talk, we investigate eventual monotonicity for solutions of the quasilinear parabolic equation and asymptotic behavior on the boundary of the blow-up set for type II blow-up solutions. This is a joint work with T. Ishiwata (Shibaura Institute of Technology) and T. Ushijima (Tokyo University of Science).
Abstract: For various types of partial differential equations, standard finite element discretizations are often unstable, which can be cured by adding suitable stabilization terms. Typically, these terms contain user-chosen parameters whose optimal choice is usually not known but which considerably influence the quality of the approximate solution. In this talk, we will consider stabilized methods for steady convection-diffusion equations. A typical example is the streamline upwind Petrov-Galerkin (SUPG) method. It is possible to compute the stabilization parameters a posteriori in an adaptive way by minimizing a target functional characterizing the quality of the approximate solution, however, this functional is often difficult to design. Moreover, the solution of this high-dimensional constrained nonlinear optimization problem is usually very time-consuming. Therefore, we proposed a method based on techniques from machine learning in order to select (nearly) optimal stabilization parameters in a cheap way. The idea is to compute these parameters locally based on properties of the SUPG solution obtained with standard (nonoptimal) parameters. The training phase will use parameters computed by the mentioned minimization approach employing accurate approximate solutions which can be obtained using nonlinear (and hence again costly) approaches. We will report our first experiences with this strategy. This is the joint work with Manoj Prakash (Charles University, Prague).
Abstract:
This paper investigates the conditions that guarantee unique solvability and unsolvability
for the generalized absolute value equations (GAVE) given by Ax − B|x| = b. Further, these
conditions are also valid to determine the unique solution of the absolute value matrix equations
(AVME) AX − B|X| = F. We give the possible revised version of the unique solvability
conditions for the two incorrect results that appeared in the published paper by Wu et al. (Appl
Math Lett 76:195-200, 2018). Finally, certain aspects related to the solvability and unsolvability
of the absolute value equations (AVE) have been deliberated upon.
Keywords. Generalized absolute value equation, Absolute value matrix equations, Unique
solution, Sufficient condition, Unsolvability.
2020 MSC. 15A18, 65H10, 90C05, 90C30.
Abstract: Spectral computed tomography (CT) extends conventional CT by acquiring energy-resolved data, enabling better material differentiation and quantitative imaging. These datasets are often affected by high noise and structural complexity. Using diffusion equations with tailored boundary conditions, we reduce noise and separate structures while preserving local intensity statistics, improving segmentation accuracy and maintaining the physical relevance of reconstructed data.
Abstract: In my contribution, I will summarize the exploration of using the lattice Boltzmann method for flow datasets measured on various phantoms using MRI over the last decade. I will highlight the challenges encountered and present the key findings of each stage. In particular, I will address the following questions: the reliability of flow measurements in the turbulent regime; the need to use a non-Newtonian flow model to simulate haemodynamics; or the use of 3D printing to develop and produce MR flow phantoms.
Abstract:
This research aims to generate uniform distributions of points on compact set by minimizing
Coulomb energy, defined as the sum of the inverses of Euclidean distances between pairs of points,
in combination with the horizontal winding number method. Achieving uniformity of points is crucial
in diverse fields such as robotics (e.g., explosion distance analysis), computer vision, healthcare,
and scientific computing, where applications include mesh reconstruction and geometric modeling.
The proposed approach addresses the need for evenly spaced points, which enhances the accuracy
and reliability of computational methods in these domains.
Keywords: Coulomb energy, Horizontal winding number, uniform distribution of points
Abstract: We study the amoeboid movement of a eukaryotic cell by modeling its boundary as a closed planar curve evolving under constrained curvature flow. The motion law incorporates protrusion, retraction, membrane tension, and frictional forces, all projected onto the normal direction of the evolving curve. To solve the governing equations, we employ a parametric method. For maintaining a stable numerical representation of the curve, we use either natural redistribution or uniform redistribution techniques to ensure an appropriate distribution of discretization points along the curve. Simulations demonstrate how these forces influence the cell’s shape and movement, offering a simplified yet insightful representation of real cell motility mechanisms. In addition, we solve a transport equation to study how a density of quantity evolves along the moving curve over time.
Abstract: Motivated by pattern formations, many evolution equations incorporating spatial convolution with suitable integral kernel have been proposed. Numerical simulations of these nonlocal evolution equation can reproduce various patterns depending on the kernel shape. In this talk, we consider the relationship between these nonlocal evolution equations and a reaction-diffusion system. By controlling parameters and taking a singular limit in the reaction-diffusion system, we show that nonlocal interactions that satisfy dimensional conditions can be approximated by the reaction-diffusion systems in general. This research is a collaboration with Hiroshi Ishii of Hokkaido University.
Abstract: A computational fluid dynamics-based framework for optimizing 3D geometry in an idealized total cavopulmonary connection (TCPC) is presented. The TCPC is a surgical procedure designed to treat congenital heart defects involving a single functional ventricle. The presented custom optimization framework integrates Python-based geometry generation, lattice Boltzmann method (LBM) simulations, and gradient-free optimization algorithms, including Nelder-Mead and the Mesh Adaptive Direct Search methods. The three optimization steps – generation of parameterized 3D geometry, simulation of incompressible Newtonian fluid flow with a rigid wall, and evaluation of objective functions – are executed automatically. The massively parallel implementation of LBM on GPUs allows the use of a spatial resolution suitable for optimizing the flow metrics sensitive to the actual resolution, such as the turbulent kinetic energy or near-wall shear rate. A simplified, parameterized model of the TCPC geometry was used to test the framework, demonstrating its feasibility and effectiveness. While this study focuses on idealized geometries with simplified assumptions, the results provide a foundation for extending the framework to patient-specific data and more complex physiological scenarios. This work represents a step in applying computational optimization to cardiovascular surgery, with the potential to improve clinical outcomes and patient-specific treatment planning.
Abstract: Template Numerical Library (TNL) is a collection of building blocks that facilitate the development of efficient numerical solvers and HPC algorithms. It is implemented as a free and open-source project in modern C++ and provides a flexible and user-friendly interface similar to the STL library, for example. TNL provides native support for modern hardware architectures such as multicore CPUs, GPU accelerators, and distributed systems, which can be managed via a unified interface. The TNL project also comprises a growing number of high-level modules that implement various numerical methods used in computational fluid dynamics, namely the mixed-hybrid finite element method (MHFEM), lattice Boltzmann method (LBM), and smoothed particle hydrodynamics (SPH). As is typical for our research group, the solvers were developed with a focus on high-fidelity modeling of selected phenomena and careful validation using experimental data. We are also interested in coupling these methods with others to cover more complex multi-physics problems and to explore new possibilities for computational enhancements based on improved implementation. For example, we can couple MHFEM with LBM to introduce heat and/or component transport into incompressible flow model solved by LBM, or incorporate the immersed boundary method with LBM to solve incompressible flow around objects that can move and be subject to potentially elastic deformations. This contribution will provide an overview of the Template Numerical Library, its components, and recently implemented features. The main focus will be on the TNL-LBM module, which is the basis for many applications involving fluid flow modeling and simulations in our research group.
Abstract: In this talk I would like to propose a A threshold-type algorithm for the fourth order geometric motions, in particular Willmore-type flows. The threshold-type algorithm was first proposed by Bence, Merriman and Osher to compute mean curvature flows. We apply their algorithm to the fourth order flows by using the fourth-order linear parabolic equations. This talk is based on my joint work with Professors Kohsaka, Miyake and Sakakibara.
Abstract: A mathematical model describing dynamics of particle-fluid two-phase fluids with low particle volume fractions flowing down the slope with low inclination angles and non-flat bottoms is constructed. The proposed model is based on the dilution approximation system derived by Murisic et al. (2013) corresponding to the experiment by Zhou et al. (2005) in which a suspension of glass beads and silicone oil was poured onto an acrylic slope with a fixed angle. In addition to the complete model, we have derived a simplified model as systems of conservation laws / balance laws taking non-flat bottoms and dilution approximation into account.In this talk, we provide the derivation details and sample numerical simulations of fluid morphology.
Abstract: Our contribution deals with the phenomenon in material science called multiple cross-slip of dislocations in slip planes. The numerical model is based on a mean curvature flow equation with additional forcing terms included. The curve motion in 3D space is treated using our tilting method, i.e., mapping of a 3D situation onto a single plane where the curve motion is computed. The physical forces acting on a dislocation curve are evaluated in the 3D setting.
Abstract: Generating coherent textures for 3D models by leveraging large-scale, pre-trained 2D text-to-image diffusion models presents significant technical challenges, particularly in maintaining multi-view consistency. StableGen introduces an algorithmic framework that adapts novel concepts to address these problems. The framework utilizes ControlNet to condition the diffusion process on the underlying 3D geometry and employs IPAdapter to enforce stylistic coherence through image-based prompting. These conditioning methods are applied across two primary generation methodologies. A parallel "Grid Mode" processes all viewpoints simultaneously as a single image to achieve consistency, while an iterative "Inpainting Mode" sequentially synthesizes the texture by leveraging visibility masks and rendered context from preceding views. To address regions untextured due to complex occlusions, a "UV Inpainting" algorithm, which operates directly in texture space as a targeted post-processing step, is also introduced. The technical implementation, performance trade-offs, and efficacy of these methods will be discussed and demonstrated through visual results.
Abstract: Quantum computers are promising hardware that is rapidly evolving. Quantum qubits can have several physical realizations. One such realization is using a superconducting transmon qubit. In this type of qubit, there are still open problems in terms of the accurate quantum state preparation. There are still open problems with this type of qubit, such as the precise preparation of the quantum state. In this paper we will discuss one possible way of preparing the qubit state and focus on the numerical aspects of numerical optimization.
Abstract: In this talk, we present a structure-preserving finite element method for addressing the multi-phase Mullins-Sekerka problem with triple junctions. This sharp interface formulation is designed to handle networks of evolving curves, driven by the reduction of total surface energy while preserving the areas of the enclosed phases. Our scheme guarantees unconditional stability and exact volume conservation. We demonstrate the efficacy of our method through several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow. This talk is based on joint works with Harald Garcke (University of Regensburg, Germany) and Robert Nürnberg (University of Trento, Italy).
Abstract: Langmuir films are molecular monolayers whose dynamics are governed by a coupling between surface incompressibility and subfluid Stokes flow. The ILLSS (Isothermal Langmuir Layer–Stokes Subfluid) model captures this interaction, yielding a nonlocal geometric evolution driven by curvature. We derive a boundary integral formulation of the model using the Dirichlet-to-Neumann map, resulting in a degenerate parabolic equation for the evolving interface. The main result is the local well-posedness of this equation in Sobolev spaces. To overcome the degeneracy, we apply a DeTurck-type reparametrization, which restores parabolicity. The analysis relies on maximal $L^2$-regularity theory and careful estimates of the nonlinear terms. This talk is based on joint work with Yoichiro Mori (University of Pennsylvania) and Shinya Okabe (Tohoku University).
Abstract:
Discrete dislocation dynamics (DDD) has become a standard tool for analyzing deformation microstructures. In our approach to
DDD, a single dislocation carrying the plastic flow in its respective slip plane is represented as an evolving planar curve. The
experimentally observed fact is that the motion of the dislocations is of a non-Newtonian type and can be schematically
described as
normal velocity = curvature + external force.
Here the curvature term approximates the self-stress of the dislocation generated by a line tension. The external force term is
responsible for other dislocation mechanisms. We treat this kind of evolution equation by the direct (Lagrangian) approach
resulting in the system of two degenerate parabolic equations for parametrization of the single dislocation curve. We solve this
problem numerically by means of the flowing finite-volume method, and we improve the stability of the numerical scheme by the
suitable choice of the external tangential velocity appropriately redistributing the discretization points along the
dislocation. We demonstrate our approach in several computational examples covering the fundamental dislocation mechanisms
including interaction with obstacles and treating the cross-slip effect by means of geodesic description of curves.
Abstract: It is well known that time delays sometimes cause instability and oscillation in the solutions of differential equations. In this talk, we mainly focus on delay differential equations with distributed delay and discuss the effects of time delay for such instabilities from the viewpoint of a finite time blow-up of the solutions. This is joint work with Y. Ichida and Y. Nakata.
Abstract: Motile eukaryotic cells establish polarity through the spatial segregation of their internal components. To describe this phenomenon, Mori et al. proposed the wave-pinning model, a bistable reaction-diffusion system. Our matched asymptotic analysis of the model reveals that the segregation front is governed by two-timescale dynamics: a slow evolution driven by mean curvature and a fast wave propagation whose speed depends on inactive chemical species. This is a joint work with Koya Sakakibara (Kanazawa University) and Shunsuke Kobayashi (University of Miyazaki).
Abstract: Geometric flows are often formulated as gradient flows of certain energy functionals on appropriate spaces. In this work, based on this variational structure, we propose a numerical scheme which computes the evolution of the curve via minimizing movement in Reproducing Kernel Hilbert Space. We demonstrate the effectiveness of the proposed scheme through numerical simulations such as the curve-shortening flow. This is a joint work with Prof. Koya Sakakibara (Kanazawa University) and Prof. Yuka Hashimoto (NTT Corporation).
Abstract: In my contribution, I will present the use of the Lattice Boltzmann Method for shape optimization in fluid dynamics simulations. I will outline the overall workflow, from geometry setup to flow simulation and iterative optimization. Particular attention will be given to the advantages of LBM for handling complex shapes. I will also highlight the main challenges and possible future directions of this approach.
Abstract: As quantum technologies continue to advance, tools from dynamical systems theory are becoming increasingly relevant for analyzing quantum processes. This talk explores selected concepts from complex dynamics—particularly Julia sets, Fatou sets, and the Mandelbrot set—and their application in studying the behavior of quantum purification protocols. These protocols involve iteratively processing two imperfect copies of quantum states through a series of operations, aiming to produce states of progressively higher quality. The presentation also touches on geometric properties of these sets via Hausdorff measure and dimension. Visualizations based on implemented algorithms will be shown to illustrate the underlying mathematical structures.
Abstract: In wildfires and forest fires, flying sparks are one cause of the spread of fire. Firebrands fly randomly from the canopy fire and ignite when they land. The sudden spread of fire to locations far from the forest where the fire is burning, such as the opposite bank, makes forest firefighting even more difficult. The first step in modeling is to construct a mathematical model of the process from the moment multiple firebrands land on a flat surface until ignition occurs. This research is a joint study with Kazunori Kuwana (Tokyo University of Science).
Abstract: We study Turing instabilities in reaction-diffusion systems defined on metric graphs composed of two finite-length edges and containing at least one vertex of degree three or higher. Such graphs can be classified into two distinct types, and for each type, we derive reduced systems on the center manifolds. Our analysis reveals that the bifurcation structures depend on the topology of the graphs. In particular, we show that the parity of the wavenumber determines the type of primary bifurcation. If time permits, we will also present wave patterns exhibiting time-periodic motion in addition to the Turing patterns.
Abstract: We study the existence of a periodic solution for differential equations with distributed delay. It is shown that, for a class of distributed delay differential equation, a symmetric period-2 solution is obtained via a Hamiltonian system of ordinary differential equations, where the period is twice the maximum delay. This work extends the result of Kaplan and Yorke (J. Math. Anal. Appl., 1974) for a discrete delay differential equation with an odd nonlinear function. We present distributed delay differential equations that have periodic solutions expressed in terms of Jacobi elliptic functions. We also discuss cases where the nonlinear function is not necessarily an odd function.
Abstract: In this talk, we numerically compute plane curves evolved in time by hyperbolic curvature flow, using the appropriate tangential velocity, such as the de Turck trick, which was devised by previous research, the uniform distribution, and the curvature-adjusted method. It is well known that the tangential velocity of the time-evolving curve does not affect its shape. Depending on how the tangential velocity is determined, the points are sparse or dense while the curve evolves in time, which will cause the numerical computation to break down. For this reason, we can utilize the appropriate tangential velocity to redistribute points along the curve and achieve stable numerical computation.
Abstract: This contribution studies the multi-speed entropic lattice Boltzmann method (ELBM). A generalization of a single-speed LBM wall boundary condition to multi-speed models is proposed and compared. A numerical study of drag and lift coefficients for a cylinder and NACA profile is perfomed using a single-speed LBM model, single-speed ELBM model, multi-speed ELBM model and the finite volume method.
Abstract: In this talk, I will present a numerical scheme to simulate injection CO2 into an oil reservoir. The viscosity of CO2 is altered by adding a small amount of a specially engineered molecule. Numerical simulations will be presented to show the effect of CO2 viscosification on the transport of CO2 in the reservoir. The viscosification may find useful applications also in hydrogen storage or geothermal energy production.
Abstract: The central theme of this presentation is the mathematical modeling of water pollution in water bodies, including rivers, oceans, reservoirs, lakes, and wetlands. This study aims to clarify the qualitative properties of the model’s solutions and to mathematically resolve water pollution problems by applying numerical simulations to real-world data. We first derive a theoretical model of water quality change based on ordinary differential equations. Specifically, we build a model that considers the relationship between water pollutants such as organic matter and nutrients, microorganisms that decompose and produce pollutants, and dissolved oxygen. The model’s validity is then examined using experimental data from previous studies. Subsequently, we discuss stability analysis and bifurcation structure of the model equations.
Abstract: In my work, I focus on deriving the weak formulation of the FitzHugh–Nagumo system of reaction-diffusion equations in a two-dimensional domain. The aim is to find a weak solution in Sobolev spaces and to approximate it numerically using the finite difference method. The model simulates the propagation of electric signals in cardiac tissue. The results show how the system’s dynamics depend on model parameters and the domain type.
Abstract: Ordinary differential equations that are non-linear yet autonomous can exhibit rich local behaviour near equilibrium points. My talk surveys two powerful tools for investigating their behaviour—stable manifolds and central manifolds. We begin with a concise review of equilibrium, Lyapunov and asymptotic stability for autonomous ODEs and outline fundamental properties of a system’s flow. Building on Lyapunov–Perron methods, the Stable Manifold Theorem is presented. Part of its proof is sketched - the fact that the local stable manifold is a C1 graph of a function tangent to the stable eigenspace of the linearized problem. Then the Central Manifold Theorem is discussed and non-uniqueness of the central manifold is highlighted. Lastly we touch on the fact that the reduction principle lets us infer the full system’s stability from the dynamics restricted to a sufficiently accurate approximation of a central manifold. The theoretical discussion is supported by two illustrative examples. First, an elementary three-dimensional system with spectrum {±i,−1} reveals a one-parameter family of distinct C1 central manifolds. Second, an extended four-dimensional Lorenz model demonstrates dimensionality reduction in practice: after transforming into central–stable coordinates, the dynamics collapse to a two-dimensional subsystem whose cubic Taylor’s approximation captures the onset of instability and a bifurcation at ρ=1.
Abstract: An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.
Abstract: This talk revisits a toy inverse geometry problem formulated via shape optimization. We present a simple numerical method and study a related Hele-Shaw-type moving boundary problem, establishing short-time existence and uniqueness of a classical solution. The analysis provides a conceptual framework that also supports the stability of numerical schemes for similar shape optimization problems. This is joint work with Professor Masato Kimura (Kanazawa University, Japan).
Abstract: Combustion experiments in a narrow channel have been conducted. When a paper which is combustion material is ignited in these experiments, according to the experimental settings, many types of combustion char patterns are generated. In this talk, we introduce three-component combustion model and report our recent results obtained through theoretical analysis.
Abstract: The presentation is intended as an overview of activities at the Technical University of Liberec, in numerical modelling of problems related to safety of the radioactive waste disposal. Examples contain e.g. reactive transport near the interface of the corroding carbon steel in clay, with mineral dissolution and precipitation, the thermal dimensioning problem, with heat transfer in multi-scale geometry, or the hydro-mechanical behaviour of fractures in granite, using detailed measured data of the fracture surface.
Abstract:
Plasticity with softening and ductile fracture lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions
are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete failure of a material.
Moreover, spatially continuous models with softening suffer from localization of strains to measure-zero submanifolds.
We formulate a problem of quasistatic evolution of elasto-plastic spring networks with a plastic flow rule which describes linear hardening, linear
softening and perfectly plastic springs in a uniform manner. The problem reduces to the mathematical framework of a sweeping process with state- and
time-dependent convex constraint. A numerical solution can be found for each time-step by iterating a metric projection on a convex set.
We will discuss the construction of the model, and the nonsmooth bifurcation which happens when the parameters change between the three cases above. The
model is also helpful to study the phenomena connected to instability, such as snapback.
Abstract: Hydrogen-atom abstraction (HAA) is central to life and its importance in synthetic chemistry continues to grow. This contribution summarizes progress on a project concerning the prediction of activation energy in HAA reactions. The goal of this project is to develop an easy-to-apply, quantitatively predictive theory that will guide chemists in designing new, economical syntheses based on C−H (and X−H in general) bond cleavage. We are investigating the applicability and limitations of Marcus-type models enhanced by recently formulated thermodynamic quantities (asynchronous behavior and frustration) across proton-coupled electron transfer and hydrogen-atom transfer reaction datasets.
Abstract: In this talk, we consider the stability of the non-zero equilibrium state for the viscous conservation laws with a delay effect. The linear stability is analyzed by using the characteristic equation of the corresponding eigenvalue problem. If our equation does not have a delay effect, the characteristic equation is given by a polynomial equation. On the other hand, if our equation has a delay effect, the characteristic equation becomes a transcendental equation, and it is not easy to analyze it. In this situation, we apply the useful known result concerned with the characteristic equation for the ordinary delay differential equations and try to get the sharp stability condition for our equation. In addition, the nonlinear stability will be discussed in the second half of the talk.
Abstract: The soap film can take on various shapes, even when the wire remains unaltered in shape. Can we estimate the number of all shapes in soap films from the wire’s shape? To rephrase mathematically, this can be described as determining the number of all solutions to the Plateau problem based on geometric quantities representing the boundary shape. Although there are numerous related studies in geometric analysis, it remains a challenging problem to determine the exact number of solutions, even in simple settings. In this research, by leveraging complex analysis of minimal surfaces, a numerical scheme for calculating multiple solutions to the Plateau problem has been developed, grounded in the method of fundamental solutions. The talk will highlight the application of complex analysis in the context of minimal surfaces. After quickly overviewing the entire research, we will focus on insights into the relationship between the number of solutions and the boundary shapes obtained through numerical experimentation. Joint work with Prof. Koya Sakakibara (Kanazawa University).