Czech-Japanese Seminar in Applied MathematicsAbstracts |
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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.