arXiv subjects

Michael Eden

Publications and source records attributed to Michael Eden.

16 recordsLinked to original sources

arXiv2026

Reactive Flow around Spherically Evolving Particles in Critical and Supercritical Dilute Regimes

We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order $\varepsilon^\alpha$ for $\alpha\in(1,3]$. Their centers are separated on the order of $\varepsilon$ but are not assumed to form a periodic lattice. The inclusions grow or shrink through an interfacial adsorption-desorption mechanism, so that the evolution of the geometry is coupled to the Stokes-reaction-diffusion-advection system. We first establish well-posedness on arbitrary large time intervals for sufficiently small $\varepsilon$. We then derive quantitative homogenization limits in terms of the limiting empirical measure describing the spatial distribution and size of the inclusions. Its evolution is governed by a continuity equation in the radius variable, and the convergence of the empirical measures is controlled in the $2$-Wasserstein distance. In the critical case, $\alpha=3$, the Stokes system converges to a Brinkman system, while non-vanishing concentration boundary layers modify the microscopic exchange law for the reaction-diffusion-advection equation. For $\alpha\in(1,3)$, the limit is of Darcy type and the original exchange law is retained. The homogenization results are quantitative and provide explicit error estimates.

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arXiv2026

A Two-Component Poro-viscoelastic System for Fibre-Reinforced Hydrogels: Analysis and Homogenization

We study the multiscale behavior of a coupled visco-poroelastic system arising in the modelling of fibre-reinforced hydrogels (FIHs) used in tissue engineering scaffolds. The composite material consists of a periodic fibre scaffold, which is governed by quasi-static linear elasticity, and a hydrogel phase saturating the interstitial space, which is modelled as a Biot linear poroelastic medium enhanced with Kelvin--Voigt structural damping. The two phases are coupled through continuity of displacement and traction across their shared interface. Both the fibre scaffold and the hydrogel phase are connected, so that mechanical forces can be transmitted through the composite and interstitial fluid can flow directly through the hydrogel network. Starting from a microscopic ($\varepsilon$-scale) model, we derive uniform a-priori estimates and establish well-posedness via a Rothe time-discretisation argument for both the case of standard Biot fluid content $\eta=0$ and the case of viscous fluid content $\eta=\alpha\delta>0$. We then perform a rigorous two-scale homogenization in the limit $\varepsilon \to 0$ using periodic unfolding. In addition to the usual effective elasticity, storage, coupling, and permeability coefficients, the homogenized constitutive laws contain nonlocal-in-time memory terms generated by the microscopic viscoelastic relaxation. All effective coefficients and memory kernels are explicitly characterized in terms of the microscale geometry and material parameters.

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arXiv2026

A mathematical model for colloids deposition in porous media combined with a moving boundary at the microscale: Solvability and numerical simulation

We study a reaction-diffusion model posed on two distinct spatial scales that accounts for diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles within a porous material. In this model, the macroscopic transport of the particles is described by an effective equation whose transport coefficients are determined by cell problems posed on the underlying pore scale. The internal pore geometry can change over time due to deposition or detachment of colloidal particles. We represent the evolving microstructure as solid cores whose phase boundaries can grow or shrink over time. As deposition progresses, neighbouring growing cores may come into contact, leading to local clogging of the pore space. We investigate how such evolving microstructures influence the effective transport and storage properties of porous layers. We establish basic analytical results concerning the weak solvability of the resulting multiscale evolution problem, which takes the form of a strongly non-linear parabolic system, in the non-clogging regime. For the numerical approximation of weak solutions we propose a two-scale finite element discretization. Numerical experiments illustrate how local clogging affects the effective dispersion tensor and quantify the resulting trade-off between transport efficiency and storage capacity.

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arXiv2024

Analysis of a Hydrodynamic Thermoelastic Mixture Model

This study proposes and explores a hydrodynamic thermo-elasticity system within the context of mixture models, encompassing fluid and solid phases, with particular emphasis on biological tissues and tumor-related phenomena. While tumor growth dynamics are not explicitly modeled yet, this work examines the interaction between thermal effects and hydrodynamics on short-time scales where the tumor size typically remains stable. We establish the existence of a unique weak solution within the framework of implicit evolution equations where we exploit the intricate coupling mechanisms inherent to the PDE system. To further investigate the model, we then study the one-dimensional model and explore in detail the complex interplay between fluid flow, solid deformation, and heat transfer. This complex coupled system of equations is then reduced over a short time scale to obtain semi-analytical solutions.

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arXiv2024

Precomputing approach for a two-scale phase transition model

In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries. The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint inclusions. The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information. We use the Hanzawa transformation to transform the problem onto a fixed reference domain. Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically. We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters. Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem. We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations.

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arXiv2024

Analysis and Simulation of a Fluid-Heat System in a Thin, Rough Layer in Contact With a Solid Bulk Domain

We investigate the effective coupling between heat and fluid dynamics within a thin fluid layer in contact with a solid structure via a rough surface. Moreover, the opposing vertical surfaces of the thin layer are in relative motion. This setup is particularly relevant to grinding processes, where cooling lubricants interact with the rough surface of a rotating grinding wheel. The resulting model is non-linearly coupled through(i) temperature-dependent viscosity and (ii) convective heat transport. The underlying geometry is highly heterogeneous due to the thin, rough surface characterized by a small parameter representing both the height of the layer and the periodicity of the roughness. We analyze this non-linear system for existence, uniqueness, and energy estimates and study the limit behavior within the framework of two-scale convergence in thin domains. In this limit, we derive an effective interface model in 3D (a line in 2D) for the heat and fluid interactions inside the fluid. We implement the system numerically and validate the limit problem through direct comparison with the micromodel. Additionally, we investigate the influence of the temperature-dependent viscosity and various geometrical configurations via simulation experiments. The corresponding numerical code is freely available on GitHub.

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arXiv2024

Thermo-Elasticity Problems with Evolving Microstructures

We consider the mathematical analysis and homogenization of a moving boundary problem posed for a highly heterogeneous, periodically perforated domain. More specifically, we are looking at a one-phase thermo-elasticity system with phase transformations where small inclusions, initially periodically distributed, are growing or shrinking based on a kinetic under-cooling-type law and where surface stresses are created based on the curvature of the phase interface. This growth is assumed to be uniform in each individual cell of the the perforated domain. After transforming to the initial reference configuration (utilizing the Hanzawa transformation), we use the contraction mapping principle to show the existence of a unique solution for a possibly small but $\varespilon$-independent time interval ($\varespilon$ is here the scale of heterogeneity). In the homogenization limit, we discover a macroscopic thermo-elasticity problem which is strongly non-linearly coupled (via an internal parameter called height function) to local changes in geometry. As a direct byproduct of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective precomputing strategy that is very much needed as the limit problem is computationally expensive.

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arXiv2024

Numerical Exploration of Nonlinear Dispersion Effects via a Strongly Coupled Two-scale System

The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.

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arXiv2023

Homogenization and simulation of heat transfer through a thin grain layer

We investigated the effective influence of grain structures on the heat transfer between a fluid and solid domain using mathematical homogenization. The presented model consists of heat equations inside the different domains, coupled through either perfect or imperfect thermal contact. The size and the period of the grains are of order $\varepsilon$, therefore forming a thin layer. The equation parameters inside the grains also depend on $\varepsilon$. We considered two distinct scenarios: Case (a), where the grains are disconnected, and Case (b), where the grains form a connected geometry but in a way such that the fluid and solid are still in contact. In both cases, we determined the effective differential equations for the limit $\varepsilon \to 0$ via the concept of two-scale convergence for thin layers. We also presented and studied a numerical algorithm to solve the homogenized problem.

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arXiv2023

Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation

We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems. The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term. This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). We prove the existence and uniqueness of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor. Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.

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arXiv2022

Effective Heat Transfer Between a Porous Medium and a Fluid Layer: Homogenization and Simulation

We are investigating the effective heat transfer in complex systems involving porous media and surrounding fluid layers in the context of mathematical homogenization. We differentiate between two fundamentally different cases: Case (a), where the solid part of the porous media is assumed to consist of disconnected inclusions and Case (b), where the solid matrix is assumed to be connected. For both scenarios, we consider a heat equation with convection where a small scale parameter epsilon characterizes the heterogeneity of the porous medium and conduct a limit process via two-scale convergence for the solutions of the epsilon-problems. In Case (a), we arrive at a one-temperature problem exhibiting a memory term and, in Case (b), at a two-phase mixture model. We compare and discuss these two limit models with several simulation studies both with and without convection.

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arXiv2021

A multiscale quasilinear system for colloids deposition in porous media: Weak solvability and numerical simulation of a near-clogging scenario

We study the weak solvability of a quasilinear reaction-diffusion system nonlinearly coupled with an linear elliptic system posed in a domain with distributed microscopic balls in $2D$. The size of these balls are governed by an ODE with direct feedback on the overall problem. The system describes the diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of the problem relies on a suitable application of Schauder's fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

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arXiv2020

Homogenization of a Poroelasticity Model for Fibre-Reinforced Hydrogels

In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical response of fibre-reinforced hydrogels is considered. Here, the medium in question is considered to be a highly heterogeneous two-component media composed of a connected fibre-scaffold with periodically distributed inclusions of hydrogel. While the fibres are assumed to be elastic, the hydromechanical response of hydrogel is modeled via \emph{Biot's poroelasticity}. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior (in the sense of two-scale convergence) of the solutions with respect to a scale parameter, characterizing the heterogeneity of the medium. Letting this scale parameter tend to zero, we arrive at an effective model where the micro variations of the pore pressure give rise to a micro stress correction at the macro scale.

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arXiv2019

Homogenization of a moving boundary problem with prescribed normal velocity

The analysis and homogenization of a moving boundary problem for a highly heterogeneous, periodic two-phase medium is considered. In this context, the normal velocity governing the motion of the interface separating the two competing phases is assumed to be prescribed. Parametrizing the boundary motion via a height function, the so-called Direct Mapping Method is employed to construct a coordinate transform characterizing the changes with respect to the initial setup of the geometry. Utilizing this transform, well-posedness of the problem is established. After characterizing the limit behavior (with respect to the heterogeneity parameter $\varepsilon\to0$) of the functions related to the transformation, the homogenized problem of the heterogeneous two-scale problem is deduced.

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arXiv2017

Corrector Estimates for the Homogenization of a Two-Scale Thermoelasticity Problem With a Priori Known Phase Transformations

We investigate corrector estimates for the solutions of a thermoelasticity problem posed in a highly heterogeneous two-phase medium and its corresponding two-scale thermoelasticity model which was derived in an earlier paper by two-scale convergence arguments. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing a priori known phase transformations. While such estimates seem not to be obtainable in the fully coupled setting, we show that for some simplified scenarios optimal convergence rates can be proven rigorously. The main technique for the proofs are energy estimates using special reconstructions of two-scale functions and particular operator estimates for periodic functions with zero average. Here, additional regularity results for the involved functions are necessary.

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arXiv2017

Homogenization of a Fully Coupled Thermoelasticity Problem for a Highly Heterogeneous Medium With a Priori Known Phase Transformations

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two-phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an a priori known interface movement due to phase transformations. After transforming the moving geometry to an $\varepsilon$-periodic, fixed reference domain, we establish the well-posedness of the model and derive a number of $\varepsilon$-independent a priori estimates. Via a two-scale convergence argument, we then show that the $\varepsilon$-dependent solutions converge to solutions of a corresponding upscaled model with distributed time-dependent microstructures.

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