Dynamic Boundary Conditions Research Articles

Double horizon peridynamics

In this study, a new double horizon peridynamics formulation was introduced by utilising two horizons instead of one as in traditional peridynamics. The new approach allows utilisation of large horizon sizes by controlling the size of the inner horizon size. To demonstrate the capability of the current approach, four different numerical cases were examined by considering static and dynamic conditions, different boundary and initial conditions, and different outer and inner horizon size values. For both static and dynamic cases, it was observed that as the inner horizon decreases, peridynamic solutions converge to classical continuum mechanics solutions even by using a larger horizon size value. Therefore, the proposed approach can serve as an alternative approach to improve computational efficiency of peridynamic simulations by obtaining accurate results with larger horizon sizes.

Impulse controllability for the heat equation with inverse square potential and dynamic boundary conditions

Abstract In this paper, we investigate the null approximate impulse controllability of the heat equation with an inverse square potential subject to dynamic boundary conditions in the ball $B(0, R_{0})$ of radius $R_{0}=\left (\frac{4}{3}\right )^{\frac{3}{2}}$. To that purpose, we use the Carleman commutator approach to show a logarithmic convexity estimate traducing an observability inequality at one instant of time.

Dissipation-preserving discretization of the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with time stepping schemes for the Cahn–Hilliard equation with three different types of dynamic boundary conditions. The proposed schemes of first and second order are mass-conservative and energy-dissipative and – as they are based on a formulation as a coupled system of partial differential equations – allow different spatial discretizations in the bulk and on the boundary. The latter enables refinements on the boundary without an adaptation of the mesh in the interior of the domain. The resulting computational gain is illustrated in numerical experiments.

A Simplified Numerical Method for Simulating the Generation of Linear Waves by a Moving Bottom

In this study, simplified linear numerical method that can simulate wave generation and transformation by a moving bottom is introduced. Numerical analysis is conducted in wave number domain after continuity equation, linear dynamic and kinematic free surface boundary conditions and linear kinematic bottom boundary condition are Fourier transformed, and the results are expressed in space domain by an inverse Fourier transform. In the wavenumber domain, the dynamic free water surface boundary condition and the kinematic free water surface boundary condition are numerically calculated, and the velocity potential in the mean water level (<i>z</i> = 0) satisfies the continuity equation and the kinematic bottom boundary condition. Wave generation and transformation are investigated when the triangular and rectangular shape of bottoms move periodically. The results of the simplified numerical method are compared with the results of previous analytical solutions and agree well with them. Stability of numerical results according to the calculation time interval (△<i>t</i>) and the calculation wave number interval (△<i>k</i>) was also investigated. It was found that the numerical results were appropriate when △<i>t</i>≤<i>T</i>(period)/1000 and △<i>k</i>≤π/100.

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: gd(t,x), gfr(t,x), U0(x) and ζ0(x), we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space Wp1,2(Q)×Wp1,2(Σ). Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation.

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

Dynamic boundary conditions for divergence form operators with Hölder coefficients

Dynamic boundary conditions for divergence form operators with Hölder coefficients

Gas and star kinematics in cloud–cloud collisions

ABSTRACT We model the collision of molecular clouds to investigate the role of the initial properties on the remnants. Our clouds collide and evolve in a background medium that is approximately ten times less dense than the clouds, and we show that this relatively dense background is dynamically important for the evolution of the collision remnants. Given the motion of the clouds and the remnants through the background, we develop, implement, and introduce dynamic boundary conditions. We investigate the effects of the initial cloud mass, velocity, internal turbulence, and impact angle. The initial velocity and its components have the largest effect on the remnant. This affects the spatial extent of the remnant, which affects the number of resulting star clusters and the distribution of their masses. The less extended remnants tend to have fewer, but more massive, clusters. Unlike the clusters, the gas distributions are relatively insensitive to the initial conditions, both the distribution of the bulk gas properties and the gas clumps. In general, cloud collisions are relatively insensitive to their initial conditions when modelled hydrodynamically in a dynamically important background medium.

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.

Slot coating flows with a Boussinesq–Scriven viscous interface

We present a computational study of free surface flows with rheologically complex interfaces in the film formation region of a slot coater. The equations of motion for incompressible Newtonian liquids in the bulk flow are coupled with the Boussinesq–Scriven constitutive equation for viscous interfaces in the dynamic boundary condition at the liquid-air free surface and solved with a mixed finite element method. We show that the interfacial viscosity plays a major role in the flow dynamics and operating limits of slot coating. We find that the interfacial viscosity makes viscous interfaces generally stiffer than their simple counterparts, affecting both the normal and the tangential stress jumps across the free surface. As a result, the interfacial viscosity counteracts the meniscus retraction and slows down the film flow, increasing the development length over the substrate and changing the topology of the recirculation region in the coating bead. Remarkably, we also find that the interfacial viscosity can substantially broaden the operating boundaries of the coating window associated with the low-flow limit, suggesting that surface-active components can be suitably designed to allow for the stable production of thinner films at higher speeds by tuning interfacial material properties in slot coating applications.

Dynamic simulation on laser-metal interaction in laser ablation propulsion considering moving interface, finite thermal wave transfer, and phase explosion

Despite extensive investigations on the mechanism of laser-irradiated solid propellants, formulating an accurate model that characterizes more details of laser-target interactions remains a challenging task. In this paper, a thermal model for nanosecond pulsed laser ablation of aluminum was developed using the finite volume method which considers the instant recession on the propellant surface and the non-Fourier effect during heat conduction. The robust coupling iteration scheme adopted in the model tracks the dynamic boundary condition due to instant propellant removal. In particular, the pre-qualified threshold is used as discriminator for phase explosion and the thermal relaxation time is introduced to quantify the non-Fourier effect on heat conduction. In addition, the temperature-dependent physical and optical parameters of the propellant were also regarded in the proposed model. The numerical implementations for Gaussian laser ablation of aluminum were performed at different laser fluences and thermal relaxation times. The theoretical computation of the ablation depth coincides well with the experimental results to validate the feasibility of the model.

Lebeau–Robbiano Inequality for Heat Equation with Dynamic Boundary Conditions and Optimal Null Controllability

Lebeau–Robbiano Inequality for Heat Equation with Dynamic Boundary Conditions and Optimal Null Controllability

Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of p-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and mathbb {R}^N with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allows us to cover the nonlocal counterparts of a large scope of local diffusion problems like, for example, Stefan problems, Hele–Shaw problems, diffusion in porous media problems and obstacle problems. Nonlinear semigroup theory is the basis for this study.

Newton's Method to Determine Fourier Coefficients and Wave Properties for Deep Water Waves

Since Chappelear developed a Fourier approximation method, considerable research efforts have been made. On the other hand, Fourier approximations are unsuitable for deep water waves. The purpose of this study is to provide a Fourier approximation suitable even for deep water waves and a numerical method to determine the Fourier coefficients and the wave properties. In addition, the convergence of the solution was tested in terms of its order. This paper presents a velocity potential satisfying the Laplace equation and the bottom boundary condition (BBC) with a truncated Fourier series. Two wave profiles were derived by applying the potential to the kinematic free surface boundary condition (KFSBC) and the dynamic free surface boundary condition (DFSBC). A set of nonlinear equations was represented to determine the Fourier coefficients, which were derived so that the two profiles are identical at specified phases. The set of equations was solved using Newton’s method. This study proved that there is a limit to the series order, i.e., the maximum series order is N=12, and that there is a height limitation of this method which is slightly lower than the Michell theory. The reason why the other Fourier approximations are not suitable for deep water waves is discussed.

Sigma mapping for drainage problems with a time-dependent water table

Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.

A Richards’ equation-based model for wave-resolving simulation of variably-saturated beach groundwater flow dynamics

This study introduces a model based on Richards’ equation to describe variably-saturated beach groundwater flow. The surface wave propagation is computed by the phase-resolving non-hydrostatic SWASH code. The SWASH data are used to make a suitable dynamic boundary condition at the beach face to force Richards’ equation. The latter is solved by a weighted discontinuous Galerkin method together with adaptive mesh refinement. The model is validated by comparison with a laboratory experiment of a transient water table recharge problem. Then, the BARDEX II prototype-scale experiment is considered to assess the model abilities for beach groundwater dynamics. The barrier beach is studied for three cases with different lagoon levels. Steady-state results with no-wave conditions show excellent agreement. Transient waves simulations are evaluated in terms of pressure heads, saturations, water table position and groundwater velocities for time-averaged, swash-resolving and spectral analysis. Results bring interesting insights about beach groundwater modelling by comparison with the experimental data as well as a Darcy’s equation-based model. A first investigation is carried out to assess the groundwater effect on the bed sediment dynamics through the modification of sediment relative weight.

A two-layer viscous Boussinesq-type model for surface waves: Derivation, analysis, numerical implementation, and verification

A two-layer viscous Boussinesq-type model is developed to simulate the wave energy dissipation during wave propagation in deep water. The viscous terms are incorporated into both the dynamic and kinematic boundary conditions at the free surface, and the corresponding analytical solution of the second-order amplitude has been derived for the first time. The linear and nonlinear properties of the model are analyzed with different viscosity coefficients. When the viscosity coefficient is 1 × 10−4 m2/s, the linear phase velocity, decay rate, second-order amplitude, and velocity profiles of the viscous model are accurate for up to h/L0 (h is water depth, L0 is characteristic wavelength) ≈ 8.66, 5.86, 3.60, 3.60, and 7.51 within 1% error, respectively. The finite difference method is adopted for the numerical implementation of the model. To verify the linear and nonlinear properties of the model, computed results for linear waves and focused wave group in deep water are compared with linear analytical solutions and experimental data, respectively.

Second order stabilized semi-implicit scheme for the Cahn–Hilliard model with dynamic boundary conditions

We study the numerical algorithm and error analysis for the Cahn–Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized approach. The corresponding energy stability and convergence analysis of the scheme are derived theoretically. Some numerical experiments are performed to verify the effectiveness and accuracy of the second-order numerical scheme, including numerical simulations under various initial conditions and energy potential functions, and comparisons with the literature works.

Study on heat flow transfer characteristics and main influencing factors of waxy crude oil tank during storage heating process under dynamic thermal conditions

In response to the energy conservation and emission reduction targets proposed by China's Fourteenth Five-Year Plan, it is necessary for oilfield enterprises to achieve the purpose of reducing energy consumption and carbon emissions of crude oil storage under the premise of ensuring safe production of oil depots. As the preferred oil storage facility for crude oil depots, the internal heat flow transmission process of storage tanks plays a vital role in the overall energy consumption of oil depots. In this paper, a three-dimensional theoretical model of the heating process of large waxy crude oil tank storage is established, and the influence law of the circumferential effect of the coil on the heating process of waxy crude oil is put forward. The external environment influence area, the internal temperature rise heat flow influence area and the transition area are put forward. The multi-nonlinear regression method is innovatively used to establish the main influencing factor model for each influence area so as to realize the quantitative characterization of the influence mechanism between the heat flux at the tank boundary and the dynamic thermal environment and the variable physical properties parameters of crude oil. The results show that the convective heat transfer of oil in the tank bottom area is greatly influenced by the circumferential effect of the coil, and the circumferential effect of the coil is enhanced with the gradual proximity of the coils on both sides, and the low temperature area at the tank bottom is obviously improved. The heat loss of the storage tank is mainly concentrated at the boundary of the tank. According to the change of heat flux in each boundary area, the external environment influence area, the internal heating heat flow influence area and the transition area are divided. Among them, the top of the storage tank forms an external environmental impact area dominated by solar radiation heat transfer, supplemented by natural convection of temperature difference, and the bottom of the storage tank forms an external soil impact area dominated by heat conduction to the external soil. Due to the effect of the thermal insulation layer, the influence of the external dynamic thermal environment on the position of the tank wall is weakened, and only the influence area of the internal temperature rise heat flow dominated by the natural convection of crude oil is formed. At the same time, there is a transition area with different thickness between the internal and external influence areas, which is affected by the external dynamic thermal boundary conditions and the physical properties of crude oil. The internal heat transfer direction is different, which causes the influence weight of each factor to fluctuate, and with the reduction of the external dynamic thermal environment, the transition area becomes the external condition influence area.

Effect of boundary conditions on particle spurious movement in Smoothed Particle Hydrodynamics method

SPH method due to its Lagrangian particle nature suffers from non-physical particle motion resulting from its numerical properties. We study these effects associated with solid walls realization.We used stationary test case to investigate behavior of multiple formulations of boundary conditions for solid walls. Multi-layer boundaries with ghost particles are considered (Dynamic Boundary Conditions, Modified Dynamic Boundary Conditions) as well as one-layer boundary conditions based on discretization of boundary terms and usage of boundary integrals. Some aspects of each realization with respect to the stationary case and particle spurious motion are discussed. Moreover we examined the levels of spurious motion caused by different sources as the free surface, straight walls, sharp corners, … The results show that for the boundary conditions good enough the most significant portion of spurious motion arises in place where the water surface connects to the wall and the spurious motion caused by this interaction is several orders of magnitude higher, then artificial motion caused by other sources.