Special Session 83: New Aspects of Mathematical Modeling and Analysis in Materials Science

Analysis of nonlinear parabolic equations via the finite volume method

Toyohiko Aiki Japan Women’s University Japan Co-Author(s):    Akiko Morimura

Abstract: The finite volume method (FVM) is well-known as the numerical techniques for partial differential equations. Particularly, since the conservation law is guaranteed locally, it is widely employed for physic-based problems. In this talk, we apply FVM to the analysis for the moisture transport in the one-dimensional porous materials. The transport process is given as the quasi-linear partial differential equations of parabolic type with non-monotone boundary conditions, which poses significant mathematical challenges for establishing the existence of strong solutions, as the standard evolution equation theory is not directly applicable. To overcome these difficulties, we succeed in showing strong solvability by employing FVM. Moreover, we derive error estimates for the approximate solutions. The aims of this talk are to explain our approach and to highlight the theoretical and numerical advantages of the finite volume method for quasilinear parabolic equations accompanying the non-monotone boundary conditions.

Homogenization of multi-species Poisson-Nernst-Planck equations in a porous medium

Apratim Bhattacharya NISER India Co-Author(s):

Abstract: We derive a homogenized model for the multi-species Poisson-Nernst-Planck (PNP) equations defined on a periodic porous medium. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by a new approach involving cut-off functions.

Homogenization of a Stokes-Reaction System with Evolving Microstructure

Michael Eden University of Regensburg Germany Co-Author(s):    Richard H\{o}fer

Abstract: We study a coupled Stokes-reaction-diffusion system posed in a non-periodically perforated domain where the size of solid ball inclusions of size $\sim\varepsilon^\alpha$ evolves through a surface reaction mechanism. We consider the scaling regime $\alpha\in(1,3]$ for the size of the inclusions and show local-in-time well-posedness for the micro-problem via a contraction mapping argument. We then analyze the asymptotic behavior $\varepsilon\to0$ as the microstructure size tends to zero. Depending on the scaling parameter $\alpha$, we get different limiting regimes, including a critical case and subcritical regimes

Quasi-variational inequality approach to modeling superconductivity

Maria Gokieli Cardinal Stefan Wyszynski University in Warsaw Poland Co-Author(s):    Nobuyuki Kenmochi, Marek Niezg\`{o}dka

Abstract: The Maxwell equation $u_t +\nabla \times (\rho\; \text{curl}\, u) = f$ is well known as a fundamental equation, where $u$ is a magnetic field, $\rho$ is a nonnegative function of resistivity and $f$ is a given vector field. In this context, $ \text{curl}\, u$ stands for the current density, In some situations, it cannot exceed some critical valuue, which may depend on the temperature or the magnetic field $u$. THis leads to a quasi-variational inequalty form of the Maxwell equation, which should also allow us to treat the superconductivity case, when the resistivity $\rho$ drops near zero at some critical temperature.

A threshold-type algorithm for fourth order geometric motions

Katsuyuki Ishii Kobe University Japan Co-Author(s):

Abstract: In this talk, we are concerned with a numerical algorithm for the motion of hypersurfaces/curves by the fourth order geometric equations such as the Willmore flow equation. The algorithm we introduce here was first proposed by Bence, Merriman and Osher in 1991 to numerically compute mean curvature flows. Our algorithm is an application of theirs and we use the fourth order heat equation instead of the second order one. We present the asymptotic expansion of the solution for the fourth order heat equation and the consistency result. This talk is based on my joint work with Professors Y. Kohsaka (Kobe U.), N. Miyake (Kyushu U.) and K. Sakakibara (Kanazawa U. \& RIKEN).

Morphology formation in three space dimensions for a coupled system of parabolic equations with nonlinear and nonlocal drift

Nicklas J\\"{a}verg\\{aa}rd Karlstad University Sweden Co-Author(s):    Emilio N M Cirillo, Nicklas J\{a}verg\{aa}rd, Rainey Lyons, Adrian Muntean and Stela Andrea Muntean

Abstract: We are interested in the formation of morphologies as seen in the production of organic solar cells as it arise in ternary mixtures. These mixtures consists of two active components and a passive solvent, and we are interested in facilitating their separation in space. We construct a semi-discrete finite-volume scheme that approximates the weak solution of our coupled system of parabolic partial differential equations with nonlocal and nonlinear drift posed in three space dimensions. We explore how initial conditions and model parameters affect the behavior of the solution, both with and without solvent evaporation through boundary conditions. Through numerical experiments we approximate the convergence rate of our scheme. Theoretical stability results are recovered numerically.

Behavior of the free boundary to two-phase Stefan problems for the bread baking process

Hana Kakiuchi Japan Women`s University Japan Co-Author(s):    Toyohiko Aiki

Abstract: We consider a free boundary problem for a system of two parabolic equations on the one-dimensional space interval. The problem has been proposed as a mathematical model for a baking bread process in a hot oven. In the model we assume that the region consists of crumb, crust, and the evaporation front. For the model, we add a free boundary condition obtained from the energy conservation law. The unknown functions of our problem are the position of the evaporation front, the temperature field, and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary contains the water content and the boundary condition for the water content depends on the unknown temperature at the boundary. For these issues, we can establish existence of a strong solution locally in time and its uniqueness under high regularity assumptions on the initial data. Moreover, based on analysis for stationary solutions, we present the behavior result on the free boundary in this talk. The key of the proof is that a comparison principal for the solutions holds, even if the latent heat coefficient depends on time and lacks differentiability with respect to the time variable.

On solvability of a time-fractional semilinear heat equation

Mizuki Kojima Kanagawa University Japan Co-Author(s):    Kotaro Hisa

Abstract: We are concerned with a semilinear parabolic problem with the Caputo type fractional derivative with respect to time. The problem where the fractional derivative is replaced with the classical one can be considered as the usual parabolic problem. The main interest is to clarify the solvability when the order of derivative approaches integer.

Existence of solutions for the elastic curve model generated by the free energy depending on the curvature

Chiharu Kosugi Yamaguchi University Japan Co-Author(s):    Chiharu Kosugi, Toyohiko Aiki

Abstract: In this talk, we consider the initial and boundary value problem for the beam equation, which is known as one mathematical model for elastic materials. As the first step in modeling, we have introduced the stress function having the singular point. In our previous results, we have proved the existence and uniqueness of weak and strong solutions. Moreover, we obtained the lower bound of the strain. This estimate guarantees that the elastic curve does not shrink to the one point. This result indicates advantage of our model. We remark that in our model, an unknown function is vector-valued function and representing the position. For this reason, we need to treat the nonlinear strain function, and we adopt the singular stress function. For the construction of a model representing the characteristics of the elastic curve, we derive the initial and boundary value problem from the free energy including the curvature. For the mathematical difficulties, we approximate the equation by adding the sixth derivative term with respect to space variable. Based on the Galerkin method and the Aubin compact theorem, we can prove the existence of weak solutions. This is a joint work with Prof. Aiki from Japan Women's University, Japan.

Asymptotic behavior of inhomogeneous damped total variation flows with time-dependent coefficients

Daisuke D Kubota Graduate School of Science and Engineering, Chiba University Japan Co-Author(s):    Daiki Mizuno, Ken Shirakawa and Naotaka Ukai

Abstract: In this study, we consider an inhomogeneous PDE with time-dependent damping coefficients, derived as the pseudo-parabolic gradient flow of a total variation energy with perturbation. Such perturbed total variation energies arise in applications including grain boundary motion and image denoising, and their stationary points play a fundamental role in physical and engineering contexts. In our PDE, the Euler--Lagrange equation for the perturbed total variation energy appears as the associated steady-state problem. In contrast, this Euler--Lagrange equation is formulated as a singular diffusion type equation, which admits many discontinuous solutions. Hence, it involves significant analytical difficulties. The aim of this study is to establish a method for asymptotically approaching the singular Euler--Lagrange equation through a smooth framework provided by our PDE. Moreover, with time-dependent damping coefficients, this framework enables control of the rate of time evolution. Based on these, we focus on the relationship between the set of steady-state solutions and the $\omega$-limit set of solutions to our PDE as time tends to $\infty$. In particular, by analyzing the decay rate of the time-dependent damping coefficients, we show that the decay rate essentially determines the relationship between the steady-state solutions and the asymptotic behavior of solutions.

Construction of a strong solution to a one-dimensional free boundary problem via the finite volume method

Kota Kumazaki Kyoto University of Education Japan Co-Author(s):

Abstract: This talk presents the construction of a strong solution to a one-dimensional free boundary problem modeling the penetration of a diffusing agent into rubber. The model consists of a parabolic equation describing the diffusion of the substance and an ordinary differential equation governing the motion of the free boundary. A central difficulty arises from the non-monotone boundary conditions at the moving edge, which occur because it is not known whether the concentration of the diffusing substance vanishes at the boundary. Due to this non-monotonicity, abstract evolution equation theory cannot be applied, and only weak solutions have been obtained so far. In this talk, we demonstrate that a strong solution can be constructed by employing the finite volume method, in which the spatial domain is divided into control volumes and discretized by enforcing conservation laws on each volume.

A projection approach to elastoplasticity with kinematic hardening

Kazunori Matsui Tokyo University of Marine Science and Technology Japan Co-Author(s):    Yoshiho Akagawa

Abstract: Engineering materials such as metals exhibit elastic deformation under small loads, but may develop permanent (plastic) deformation once the load exceeds a threshold. In a variational framework, elastoplasticity can be formulated through a stress constraint of variational inequality type. Under repeated loading and unloading, the yield surface evolves due to strain hardening; one common modeling choice is kinematic hardening, where the admissible stress set translates in stress space as plastic deformation accumulates. We study an elastoplastic model with kinematic hardening in a variational inequality setting, coupled with an equation of motion including Kelvin--Voigt viscosity. A key analytical and computational difficulty is to construct solutions while respecting the stress constraint at all times. We propose a projection-based time discretization. At each time step we first compute an unconstrained trial stress and then project it onto the translated admissible set. This trial-projection strategy preserves the constraint while avoiding a nonlinear fully implicit problem, leading to a simple implementation. We prove stability of the resulting discrete solutions under appropriate norms. Moreover, these stability estimates yield existence of a solution to the original continuous coupled problem.

Mathematical analysis of a model for tuberculosis granuloma formation

Masaaki Mizukami Kyoto University of Education Japan Co-Author(s):    Mario Fuest, Johannes Lankeit, and Yuya Tanaka

Abstract: This talk is concerned with a model for tuberculosis granuloma formation proposed by Feng (2024). In that work, Feng not only introduced the model but also analyzed the corresponding ODE system, showing that the basic reproduction number plays a crucial role in determining the qualitative behavior of solutions. In contrast, for the PDE setting, no analytical results seem to be available so far. The purpose of this talk is to analyze the PDE system and to establish global existence of solutions as well as their grow-up properties. This talk is based on joint work with Mario Fuest (Leibniz University Hannover), Johannes Lankeit (Leibniz University Hannover), and with Yuya Tanaka (Kwansei Gakuin University).

A stochastic particle-continuum model for the chemical-induced corrosion of marble

Adrian Muntean Karlstad University Sweden Co-Author(s):    Nicklas Javergard, Daniela Morale, Giulia Rui, Stefania Ugolini

Abstract: We present a nonlocal evolution system of SDEs describing marble degradation by acid particles, coupled with ODEs for the random fields evaluating the current damage at a continuum scales. The coupling particle-continuum is done via non-local operators, while the chemical reaction is assumed to be a Poisson counting process. Strong solutions to this system were found earlier and they are unique. We illustrate numerically their behavior, and finally, we point out a couple of currently open mathematical questions mostly concerning the model averaging (periodic homogenization) and reduction.

Numerical simulations and analysis of mathematical modeling for moisture transport in porous media

Yusuke Murase Meijo University Japan Co-Author(s):    Yusuke Murase

Abstract: In this talk, we discuss numerical simulations and analysis of mathematical modeling for moisture transport in porous media. Our model is a system of partial differential equations with free boundary motion, coupled with a second order differential equation which presents moisture transport in macro region its outer force depends upon the position of free boundary in micro region, and usual heat equation with free boundary motion its depends on the volume of moisture at the point in the macro region, which presents moisture adsorption in porous. By these dependences, we can see our model is a multi-scale model. I will show you some numerical simulations of the model of adsorption phenomena and moisture transport full model, solvability and numerical stability of approximated equations, and some related topics.

Scale size-dependence and homogenization in elasticity

Grigor Nika Karlstad University Sweden Co-Author(s):

Abstract: Classical homogenization is insufficient for finite-sized structures as it does not account for crucial scale-size effects. We will present a thermodynamically consistent model and subsequent two-scale expansion homogenization framework for strain-gradient elasticity that derives effective models with scale-dependent homogenized coefficients. Our key result is that the homogenized mechanical properties depend not only on the micro-geometry and volume fraction but also on the absolute size of the underlying constituents. Hence, the homogenized coefficients are not constant (as in classical homogenization) but rather functions of the microstructural size. Numerical validation confirms that the homogenized coefficients converge to the classical ones as the scale-size effects become vanishingly small, providing a critical tool for designing micro-architected materials.

Derivation and Analysis of Macroscopic Models Describing Reaction and Diffusion in Porous Media

CHRISTOS NIKOLOPOULOS University of the Aegean Greece Co-Author(s):

Abstract: The derivation and analysis of mathematical models describing diffusion and reaction of chemical components in porous media is presented. More specifically there is emphasis on the variation of the overall diffusion resulting from the changing shape of a single pore due to reaction process. These models are derived by averaging, with the use of the multiple scales method. Aspects such as the volume expansion of the material, clogging, etc, due to the material transformation are investigated. The derived macroscopic models are solved numerically and a series of simulations are presented.

Wave propagation through a time-varying heterogeneous interface

Vishnu Raveendran University of Bonn Germany Co-Author(s):    B. Verfuerth

Abstract: We discuss the homogenization and dimension reduction of a wave-type equation with multiple scales in space and time. To perform the homogenization together with dimension reduction, we use the method of multi-scale convergence for thin layer, which is the generalization of the two-scale convergence for a thin heterogeneous layer [M. Neuss-Radu, W. Jager (2007)]. Special attention is given to deriving the transmission condition along the interface. The study should be seen as the preliminary work to design a time-varying metasurface to control wave propagation

Global existence of weak solutions to an $N$-dimensional moisture transport model for porous materials

Yutaka Tsuzuki Department of Economic Informatics, Hiroshima Shudo University Japan Co-Author(s):    Tomomi Yokota, Yutaro Chiyo

Abstract: This talk deals with the moisture transport model for porous materials, $$ \dfrac{\partial}{\partial t}\psi(u) = \nabla\cdot\big(\lambda(u)\nabla(u+g)\big), \quad t \in (0,T),\ x \in \Omega $$ under the Neumann boundary condition $\frac{\partial}{\partial n}(u+g)=0$ for $t \in (0,T)$ and $x \in \partial \Omega$ and the initial condition, where $T>0$, $\Omega\subset\mathbb R^N$ ($N\in\mathbb N$) is a bounded domain with smooth boundary, $\psi, \lambda, g$ are given functions satisfying some conditions. In the previous work, global weak solvability and uniqueness were shown under the restriction that $\frac{\lambda}{\psi`}$ is constant in the one-dimensional setting. The purpose of this talk is to relax these restrictions and to give an elementary proof of global weak solvability of the above $N$-dimensional initial-boundary value problem by using the theory for linear operators.

Parabolic gradient flows of energy functionals with state dependent coefficients

Naotaka Ukai Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University Japan Co-Author(s):    Harbir Antil, Daiki Mizuno, Ken Shirakawa

Abstract: In this talk, we study a parabolic gradient system that integrates two models: the free-energy functional for anisotropic-orientation-adaptive image processing; and the phase-field model of grain-boundary motion. Through the analysis of this gradient system, we aim to construct a unified mathematical framework bridging the two research areas of image processing and materials science. Recently, several attempts have been made to construct a unified framework for the pseudo-parabolic gradient system obtained by incorporating the energy-dissipation operator $-\Delta \partial_t$ into our system. However, the mathematical models for image processing and grain boundary motion are originally formulated as parabolic gradient flows. Therefore, establishing a unified analytical framework for these parabolic models still remains an open problem. In this talk, we address this open problem. Our main objective is to further advance the unification of this mathematical framework through the analysis of our system. Building on the time-discretization method for the pseudo-parabolic system, we carry out the mathematical analysis of the corresponding parabolic system. As a main result, we clarify conditions that ensure the well-posedness of the parabolic system with energy-dissipation. Additionally, in the proof of the main result, we construct an energy-dissipating time-discrete scheme that reduces the cost associated with higher-order derivatives.

On the modelling of the dissolution of a solid particle

Michael Vynnycky University of Limerick Ireland Co-Author(s):

Abstract: The dissolution of a solid spherical particle in a surrounding quiescent solvent is a canonical problem that is found in many industrial and consumer applications, ranging from pharmaceutical and food products, to chemicals, detergents, and paints. Mathematically, this constitutes a moving-boundary problem, akin to a classical Stefan problem. However, analysis of a time-dependent, spherically symmetric diffusion-dominated problem indicates a variety of possible pitfalls with the modelling of this problem, even when adopting the seemingly uncontroversial common assumption that the dissolution (reaction) kinetics at the interface are fast compared to mass transfer from the interface, i.e. the limit of infinite Damkohler number. Much of the discussion centres on the appropriate boundary condition that should be used at the dissolution surface in the situation when the density of the solvent is different to that of the particle, as is almost always the case in practice. In particular, we find that the radial solvent flow which dissolution sets into motion results in velocity and pressure singularities when dissolution begins, although the first of these is removed if the Damkohler number is finite.

Qualitative properties of entropy solutions to one-dimensional scalar parabolic-hyperbolic conservation laws

Hiroshi Watanabe Oita University Japan Co-Author(s):    Hiroshi Watanabe

Abstract: We consider one-dimensional problems (P) for scalar parabolic-hyperbolic conservation laws. These equations may be regarded as a combination of scalar hyperbolic conservation laws and porous medium type equations. Hence, they have both properties of hyperbolic equations and those of parabolic equations. In this talk, we study qualitative properties of entropy solutions to (P). In particular, we focus on the free boundary separating the parabolic and hyperbolic regions. We first review our previous results on the construction of traveling waves, one-sided Lipschitz estimates and decay estimates for entropy solutions. We then investigate the behavior of the free boundary, drawing on propagation estimates for the support of solutions to the porous medium equation and estimates for the interface of the mushy region in the enthalpy formulation of the Stefan problem.