Motivated by questions related to morphology formation in 3D involving interacting ternary mixtures, we propose a finite volume scheme to approximate numerically the unique weak solution to a coupled system of parab olic equations with nonlinear and nonlocal drift. The special feature of our system is that the coupling takes place precisely via the structure of the drift terms. We prove the convergence of the scheme towards the unique solution of the target evolution system and explore as well the stability of the solution with respect to selected parameters. We illustrate numerically in 3D the appearance of the wanted morphologies and compute as well the empirical order of convergence of the numerical approximations towards their limit.

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 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 ε independent time interval (ε is here the scale of heterogeneity). In the homogenization limit, we recover 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 by-product of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective pre-computing strategy that is very much needed as the limit problem is computationally expensive.

We employ analytical and numerical techniques to examine a phase tran-sition model with moving boundaries. The model displays two relevant spatial scales: amacroscopic scale which governs the overall heat conduction inside the dominant phaseand a microscopic scale consisting of small inclusions of a second phase which may shrinkor grow. We use the Hanzawa transformation to transform the problem onto a fixed reference domain and utilize a Schauder fixed-point argument to demonstrate the well-posedness of this system for a finite time interval. Due to the model’s nonlinearities andthe macroscopic parameters, which are given by differential equations that depend onthe size of the inclusions, the problem is computationally expensive to solve numerically.We introduce a precomputing approach that solves multiple cell problems in an offlinephase and uses an interpolation scheme afterwards to determine the needed parameters.Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearityof the problem. We investigate the errors of both the precomputing and time-steppingprocedures and verify the theoretical results via numerical simulations.

Film formation from solvent evaporation in polymer ternary solutions is relevant for several technological applications, such as the fabrication of organic solar cells. The performance of the final device will strongly depend on the internal morphology of the obtained film, which, in turn, is affected by the processing conditions. We are interested in modeling morphology formation in 3D for ternary mixtures using both a lattice model and its continuous counterpart in the absence of evaporation. In our previous works, we found that, in 2D, both models predict the existence of two distinct regimes: (i) a low-solvent regime, characterized by two interpenetrated domains of the two polymers, and (ii) a high-solvent regime, where isolated polymer domains are dispersed in the solvent background. In the significantly more intriguing 3D case, we observe a comparable scenario both for the discrete and the continuous model. The lattice model reveals its ability to describe morphology formation even in the high solvent content 3D case, in which the three-dimensional nature of space could have prevented cluster formation. To realize the simulations we have written specific codes using the languages C and julia. The codes closely follows the algorithmic dynamics governing the lattice and the continuum model.

We consider a classical model of non-equilibrium statistical mechanics accounting for non-Markovian effects, which is referred to as the Generalized Langevin Equation in the literature. We derive reduced Markovian descriptions obtained through the neglection of inertial terms and/or heat bath variables. The adopted reduction scheme relies on the framework of the Invariant Manifold method, which allows to retain the slow degrees of freedom from a multiscale dynamical system. Our approach is also rooted on the Fluctuation–Dissipation Theorem, which helps preserve the proper dissipative structure of the reduced dynamics. We highlight the appropriate time scalings introduced within our procedure, and also prove the commutativity of selected reduction paths.

We consider a two-scale parabolic problem describing the water-induced swelling for a class of porous materials with elongated internal structures. The system of evolution equations we are considering here consists of a parabolic equation describing the evolution of the moisture content into a macroscopic domain coupled in a two-scale fashion to a free boundary problem capturing a microscopic swelling process. The macroscopic domain is a three-dimensional object (the target porous material), while the microscopic domains are a stack of elongated pores modeled as one-dimensional halflines connected at an edge to the macroscopic domain. By imposing a flux boundary condition at the edge of each pore, we allow the moisture content to intrude into the respective microscopic domain. In this work, we prove the existence and uniqueness of a solution to our two-scale problem. One key ingredient in our proof is the guarantee that the microscopic solution is measurable with respect to variable pointing out to the macroscopic domain. By using the Banach’s fixed-point theorem, we establish the local-in-time well-posedness of our two-scale problem.

This contribution is concerned with the well-posedness and homogenization of an ordinary differential equation (ODE) of Arrhenius-type coupled with a doubly nonlinear parabolic partial differential equation (PDE) with rapidly oscillating coefficients and taking into account disparate diffusion-reaction time scales, including regularly as well as singularly perturbed problems. The ODE-PDE system is spatially dependent and is subjected to Robin-type boundary conditions. Such problems are used to model a variety of processes and phenomena such as combustion and exothermal chemical reactions. We will have a special look at the questions of the existence, uniqueness, boundedness, and the asymptotic limit of the microscale problem by applying the two-scale convergence and unfolding method. A numerical example illustrates both the expected behavior of the approximated solutions as well as the capability of the proposed upscaled models.