Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases

Evaluation of PV technology implementation in the building sector

This paper presents a simulation case that shows the impact on energy consumption of a building applying photovoltaic shading systems. In order to make photovoltaic application more economical, the effect of a photovoltaic facade as a passive cooling system can result in a considerable energy cost reduction, with positive influence on the payback time of the photovoltaic installation. Photovoltaic shading systems can be applied to both refurbishment of old buildings and to new-build, offering attractive and environmentally integrated architectural solutions

Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies

Weak formulation for singular diffusion equation with dynamic boundary condition

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.Comment: 23 page

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field systems, first-order necessary optimality condition

A parallel solver for reaction-diffusion systems in computational electrocardiology

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain %(AB) cardiac model, consisting of a system of two degenerate parabolic reaction-diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. %In case of equal anisotropy ratio, this system reduces to The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction-diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh-Nagumo (FHN) model to the more complex phase-I Luo-Rudy model (LR1). The solver employs structured isoparametric $Q_1$ finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to $O(10^7)$ unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains

A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

A system with equation and dynamic boundary condition of Cahn–Hilliard type is considered. This system comes from a derivation performed in Liu–Wu (Arch. Ration. Mech. Anal., 233:167–247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter

A philosophical context for methods to estimate origin-destination trip matrices using link counts.

This paper creates a philosophical structure for classifying methods which estimate origin-destination matrices using link counts. It is claimed that the motivation for doing so is to help real-life transport planners use matrix estimation methods effectively, especially in terms of trading-off observational data with prior subjective input (typically referred to as 'professional judgement'). The paper lists a number of applications that require such methods, differentiating between relatively simple and highly complex applications. It is argued that a sound philosophical perspective is particularly important for estimating trip matrices in the latter type of application. As a result of this argument, a classification structure is built up through using concepts of realism, subjectivity, empiricism and rationalism. Emphasis is put on the fact that, in typical transport planning applications, none of these concepts is useful in its extreme form. The structure is then used to make a review of methods for estimating trip matrices using link counts, covering material published over the past 30 years. The paper concludes by making recommendations, both philosophical and methodological, concerning both practical applications and further research

Population Genetic Structure and Species Delimitation of a Widespread, Neotropical Dwarf Gecko

Amazonia harbors the greatest biological diversity on Earth. One trend that spans Amazonian taxa is that most taxonomic groups either exhibit broad geographic ranges or small restricted ranges. This is likely because many traits that determine a species range size, such as dispersal ability or body size, are autocorrelated. As such, it is rare to find groups that exhibit both large and small ranges. Once identified, however, these groups provide a powerful system for isolating specific traits that influence species distributions. One group of terrestrial vertebrates, gecko lizards, tends to exhibit small geographic ranges. Despite one exception, this applies to the Neotropical dwarf geckos of the genus Gonatodes. This exception, Gonatodes humeralis, has a geographic distribution almost 1,000,000 km2 larger than the combined ranges of its 30 congeners. As the smallest member of its genus and a gecko lizard more generally, G. humeralis is an unlikely candidate to be a wide-ranged Amazonian taxon. To test whether or not G. humeralis is one or more species, we generated molecular genetic data using restriction-site associated sequencing (RADseq) and traditional Sanger methods for samples from across its range and conducted a phylogeographic study. We conclude that G. humeralis is, in fact, a single species across its contiguous range in South America. Thus, Gonatodes is a unique clade among Neotropical taxa, containing both wide-ranged and range-restricted taxa, which provides empiricists with a powerful model system to correlate complex species traits and distributions. Additionally, we provide evidence to support species-level divergence of the allopatric population from Trinidad and we resurrect the name Gonatodes ferrugineus from synonymy for this population