Discretization for a 2D-viscoelastic wave equation with dynamic boundary conditions

In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations.

In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schrödinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in $L^2$ and $H^1$ are proved for the associated semidiscrete and fully discrete Crank–Nicolson–Galerkin approximations. The problem involving the Schrödinger equation is motivated by considering the standard “parabolic” (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long-range sound propagation in the sea, in the specific case of a domain with a rigid bottom of variable topography. This model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations. In the (real) parabolic case, numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type.

We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system.We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both. This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well-posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed. The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of varioustypes of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.

Core Ideas Impact of dynamic boundary conditions on solute transport in heterogeneous materials Identification of transport mechanisms by conductivity function and boundary flux Predictions using one‐dimensional effective transport models with constant flux We investigate the effect of dynamic boundary conditions on solute transport in unsaturated, heterogeneous, bimodal porous media. Solute transport is studied with two‐dimensional numerical flow and transport models for scenarios where either (i) solely infiltration or (ii) more realistic dynamic (infiltration–evaporation) boundary conditions are imposed at the soil surface. Travel times of solute are affected by duration and intensity of infiltration and evaporation events even when cycle‐averaged inflow rates of the scenarios are identical. Three main transport mechanisms could be identified based on a criterion for the infiltration rate that is related to the hydraulic conductivity curves of the media. If, based on this criterion, infiltration rates are low, the transport paths for upward and downward transport do not differ significantly, and the breakthrough curves of solute are similar to the one obtained under stationary infiltration. If infiltration rates are moderate, travel paths deviate between upward and downward flow, leading to a trapping of solute and strong tailing of the breakthrough curves. If infiltration and evaporation rates are very high, lateral advective–diffusive transport can lead to very efficient and fast downward transport. Thus, solute breakthrough depends strongly on lateral flow paths enforced by the boundary conditions at the soil surface. If heterogeneity of the materials is not strong and the structure is tortuous, dynamic boundary conditions mainly lead to increased macrodispersion. We test simplified upscaled transport models based on stationary flow rates to estimate breakthrough curves and demonstrate how the transport mechanisms are captured in the model parameters.

A novel imbibition model of a coating fluid in a paper was developed. The microstructure of the paper was represented by a network of interconnected channels or throats formed between the paper's fibers. Geometrical characteristics of the channels, such as effective radius and length, were selected from representative statistical distributions. The interconnectivity of the channels was characterized by an average coordination number treated as a parameter of the model, and its effect on the imbibition process was studied. The study dealt with a dynamic (time‐dependent) boundary condition, in which a convection‐driven pressure was applied to the coating fluid on the external surface of the paper or driven only by capillary forces. The flow parameters used in the model represent a high‐speed coating process. Extensive computer simulations were carried out to study the effect on the imbibition process of three distinct sets of parameters: microstructural parameters of the paper; the fluid's properties; and the dynamic boundary condition. The mean coordination number and average throat size of the paper's pore space, the coating fluid's viscosity, and the dynamic boundary condition all strongly influenced imbibition of the coating fluid.

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.

When modeling the cardiovascular system, the use of boundary conditions that closely represent the interaction between the region of interest and the surrounding vessels and organs will result in more accurate predictions. An often overlooked feature of outlet boundary conditions is the dynamics associated with regulation of the distribution of pressure and flow. This study implements a dynamic impedance outlet boundary condition in a one-dimensional fluid dynamics model using the pulmonary vasculature and respiration (feedback mechanism) as an example of a dynamic system. The dynamic boundary condition was successfully implemented and the pressure and flow were predicted for an entire respiration cycle. The cardiac cycles at maximal expiration and inspiration were predicted with a root mean square error of 0.61 and 0.59 mm Hg, respectively.

The vadose zone is subject to dynamic boundary conditions in the form of infiltration and evaporation. A better understanding of implications for flow and solute transport, arising from these dynamic boundary conditions in combination with heterogeneous structure, will help to improve the prediction of the fate of solutes. We present laboratory experiments and numerical simulations of heterogeneous porous media under unsaturated conditions where controlled, temporally varying precipitation and evaporation are applied to study the effect of dynamic boundary conditions on solute transport in the presence of material interfaces. Dye tracers Eosine Y and Brilliant Blue FCF are utilized to visualize solute transport and analyze redistribution processes in a flow cell. Water and solute fluxes in and out of the flow cell are quantified. While in dynamic experiments application of small infiltration rates (significantly below the saturated hydraulic conductivities of the materials) led to a reversal of transport paths between infiltration and succeeding evaporation, larger infiltration rates altered downward transport such that flow and transport paths differed from those observed during evaporation. Differences in transport paths ultimately led to a redistribution and trapping of solute in one material which manifested as pronounced tailing in breakthrough curves. Trapping was induced not by the formation of a stagnant zone as result of large parameter contrast but by an interplay of dynamic boundary conditions and material heterogeneity. This study thereby highlights the importance to consider dynamic boundary conditions in predictions of solute leaching.

A parabolic inverse source problem with a dynamical boundary condition

The kinematic and dynamic boundary conditions on the free surface of a fluid should be posed for water wave problems. In the framework of potential theory for an inviscid and incompressible fluid with an irrotational motion, the combined boundary condition, which involves the velocity potential only, is often used by eliminating the elevation terms mathematically. Such a combination is correct for the solutions in the frequency domain, and is not feasible for an initial-boundary-value problem in the time domain since it leads to a totally different physical formulation. The correct initial conditions for pure gravity waves and hydroelastic waves are presented.

The shallow water Canada Basin Acoustic Propagation Experiment (SW CANAPE) was conducted to study the effects of oceanographic variability on broadband acoustic fields in the Arctic. The physics of the acoustic waveguide on the northeastern edge of the Chukchi Shelf are influenced by dynamic boundary conditions and spatio-temporal fluctuations in the water column temperature and salinity profiles. Several oceanographic and acoustic receiving arrays were deployed across the Chukchi Shelf out to the shelf break region. Linear frequency modulated (LFM) signals were transmitted by two sources on the shelf for a long period of time. The influence of small scale, short-term water column variability, and dynamic upper boundary conditions including open water, marginal, and solid ice zones on shallow water propagation is shown for a 10 km source-receiver separation with well-defined water column properties measured at the source, receiver, and a mid-point along the cross-shelf acoustic path. [Work supported by ONR 321OA.]

We consider incompressible Navier–Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.

We consider a real Klein-Gordon field in the Poincar\'e patch of $(d+1)$-dimensional anti-de Sitter spacetime, PAdS$_{d+1}$, and impose dynamical boundary condition on the asymptotic boundary of PAdS$_{d+1}$ that depend explicitly on the second time derivative of the field at the boundary. These boundary conditions are of generalized Wentzell type. We construct the Wightman two-point function for the ground state of the Klein-Gordon theory whenever the parameters of the theory (the field mass, curvature coupling and boundary condition parameters) render such ground state admissible. In the cases in which the mass of the Klein-Gordon field and the curvature coupling term yield an effectively massless theory, we can define a boundary field whose dynamics are ruled by the dynamical boundary condition and construct, in addition to the Wightman function for the Klein-Gordon field, boundary-to-boundary, boundary-to-bulk and bulk-to-boundary propagators.

We show maximal regularity results concerning parabolic systems with dynamic boundary conditions and a diffusion theorem on the boundary in the framework of $$L^p$$ spaces, $$1<p<\infty $$. Analyticity results can be derived for the semigroups generated by suitable classes of uniformly elliptic operators with general Wentzell boundary conditions having diffusion terms on the boundary.

Рассмотрена задача с динамическим краевым условием, учитывающим наличие демпфера при закреплении, для гиперболического уравнения на плоскости и доказана ее однозначная разрешимость. Динамическое условие, содержащее производные первого порядка как по пространственной, так и по переменной времени, приводит к несамосопряженной задаче, что затрудняет применение методов спектрального анализа. Однако эти трудности преодолены и существование единственного решения поставленной задачи доказано. Основным инструментом доказательства являются априорные оценки в пространствах Соболева, выведенные в процессе работы над статьей. Предложены способы получения приближенного решения, в качестве частного случая рассмотрен пример одномерного волнового уравнения и получено точное решение задачи с динамическим условием.