A SCALAR AUXILIARY VARIABLE (SAV) FINITE ELEMENT NUMERICAL SCHEME FOR THE CAHN-HILLIARD-HELE-SHAW SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS

In this paper,we consider the Cahn-Hilliard-Hele-Shaw(CHHS)system with the dynamic boundary conditions,in which both the bulk and surface energy parts play important roles.The scalar auxiliary variable approach is introduced for the physical system;the mass conservation and energy dissipation is proved for the CHHS system.Subsequently,a fully discrete SAV finite element scheme is proposed,with the mass conservation and energy dissipation laws established at a theoretical level.In addition,the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

SAV Finite Element Method for the Peng-Robinson Equation of State with Dynamic Boundary Conditions

In this paper,the Peng-Robinson equation of state with dynamic boundary conditions is discussed,which considers the interactions with solid walls.At first,the model is introduced and the regularization method on the nonlinear term is adopted.Next,The scalar auxiliary variable(SAV)method in temporal and finite element method in spatial are used to handle the Peng-Robinson equation of state.Then,the energy dissipation law of the numerical method is obtained.Also,we acquire the convergence of the discrete SAV finite element method(FEM).Finally,a numerical example is provided to confirm the theoretical result.

Long Time Behavior of the Cahn-Hilliard Equation with Irregular Potentials and Dynamic Boundary Conditions

The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered.The existence of the global attractor is proved and the long time behavior of the trajectories,namely,the convergence to steady states,is studied.

A Decay Result to an Elliptic Equation with Dynamical Boundary Condition

The decay estimations of the solution to an elliptic equation with dynamical boundary condition is considered.We proved that,for suitable initial datum,the energy of the solution decays "in time" exponentially if p=0,whereas the decay is polynomial order if p>0.

Dynamic Control of Defective Gap Mode Through Defect Location

A one dimensional model is developed for defective gap mode（DGM）with two types of boundary conditions：conducting mesh and conducting sleeve.For a periodically modulated system without defect,the normalized width of spectral gaps equals to the modulation factor,which is consistent with previous studies.For a periodic system with local defects introduced by the boundary conditions,it shows that the conducting-mesh-induced DGM is always well confined by spectral gaps while the conducting-sleeve-induced DGM is not.The defect location can be a useful tool to dynamically control the frequency and spatial periodicity of DGM inside spectral gaps.This controllability can be potentially applied to the interaction between gap eigenmodes and energetic particles in fusion plasmas,and optical microcavities and waveguides in photonic crystals.

BLOW-UP OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH SEMILINEAR DYNAMICAL BOUNDARY CONDITIONS OF HYPERBOLIC TYPE

In this paper, we present some sufficient conditions for blow-up of solutions to elliptic equations under semilinear dynamical boundary conditions of hyperbolic type.

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.

A Novel Approach for the Numerical Simulation of Fluid-Structure Interaction Problems in the Presence of Debris

A novel algorithm is proposed for the simulation of fluid-structure interaction problems.In particular,much attention is paid to natural phenomena such as debris flow.The fluid part(debris flow fluid)is simulated in the framework of the smoothed particle hydrodynamics(SPH)approach,while the solid part(downstream obstacles)is treated using the finite element method(FEM).Fluid-structure coupling is implemented through dynamic boundary conditions.In particular,the software“TensorFlow”and an algorithm based on Python are combined to conduct the required calculations.The simulation results show that the dynamics of viscous and non-viscous debris flows can be extremely different when there are obstacles in the downstream direction.The implemented SPH-FEM coupling method can simulate the fluid-structure coupling problem with a reasonable approximation.

Analysis of Heat Transfer Phenomena inside Concrete Hollow Blocks

During both hot and cold seasons,masonry walls play an important role in the thermal performance between the interior and the exterior of occupied spaces.It is thus essential to analyze the thermal behavior at the hollow block’s level in order to better understand the temperature and heat flux distribution in its structure and potentially limit as much as possible the heat transfer through the block.In this scope,this paper offers an experimental and numerical in-depth analysis of heat transfer phenomena inside a hollow block using a dedicated experimental setup including a well-insulated reference box and several thermocouples and fluxmeters distributed at the boundaries and inside the hollow block.The block was then numerically 3D modelled and simulated using COMSOL Multiphysics under the same conditions,properties,and dimensions as the experimentally tested block.The comparison between the numerical and experimental results provides very satisfactory results with relative difference of less than 4%for the computed thermal resistance.

Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis:Global Dynamics and Diffusivity Limit

In this paper,we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate.Utilizing energy methods,we show that under time-dependent Dirichlet boundary conditions,long-time dynamics of solutions are driven by their boundary data,and there is no restriction on the magnitude of initial energy.Moreover,the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions,which has not been observed in previous studies on related models.

Representation of a Class of Linear Operators with Coupled Domain in a Product Space and Its Applications

This paper is concerned with the representation problem of a coupled operator in a product space.A necessary and sufficient condition is given for a class of operators with closed range to have a one-sided coupled operator matrix representation.The applications of this result in a delay equation and in a diffusion-transport system with dynamical boundary conditions are further presented.

Long Wavelength Approximation to Peristaltic Motion of Micropolar Fluid with Wall Effects

Peristaltic motion of an incompressible micropolar fluid in a two-dimensional channel with wall effects is studied.Assuming that the wave length of the peristaltic wave is large in comparison to the mean half width of the channel,a perturbation method of solution is obtained in terms of wall slope parameter,under dynamic boundary conditions.Closed form expressions are derived for the stream function and average velocity and the effects of pertinent parameters on these flow variables have been studied.It has been observed that the time average velocity increases numerically with micropolar parameter.Further,the time average velocity also increases with stiffness in the wall.