Dirichlet-Neumann Boundary Conditions Research Articles

On the structural stiffness maximisation of anisotropic continua under inhomogeneous Neumann–Dirichlet boundary conditions

This work discusses three aspects of topology optimisation (TO) problems dealing with structural stiffness maximisation of anisotropic continua under mixed inhomogeneous Neumann–Dirichlet boundary conditions (BCs). Firstly, the total potential energy (TPE) is introduced as intuitive measure of the structural stiffness, instead of the work of applied forces and displacements (WAFD). Secondly, it is proven that the WAFD under mixed BCs is not a self-adjoint functional, while the one related to the TPE is always a self-adjoint functional, regardless of the BCs nature. Thirdly, the influence of the anisotropy, of the applied BCs and of the design requirement on the volume fraction on the optimised topology is investigated: depending on these features, the optimal solutions of the two problem formulations, i.e., minimisation of the functional involving the TPE or minimisation of the WAFD subject to a constraint on the volume fraction, can coincide. The problem is formulated in the context of a special density-based TO approach wherein a Non-Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor, i.e., the pseudo-density field. The properties of NURBS entities are exploited to derive the gradient of the physical responses involved in the problem formulation and to easily satisfy the minimum length scale requirement (related to manufacturing needs). The differences between TPE-based and WAFD-based formulations and the effectiveness of the proposed method are shown on 2D and 3D problems.

Solving Mixed Boundary Value Problems using Fourier Transform and Dual Integral Equations Method

The research considered the solution of the system of dual integral equations involving Fourier transform occurring in mixed boundary value problems for Laplace’s equation with mixed Dirichlet-Neumann boundary conditions. The dual integral equations involving trigonometric function kernel have been solved using Fourier sine transforms. Graphical solutions with the help of Mathematica were used to demonstrate the effectiveness of this method.
 Keywords: Boundary value problem; Laplace’s equation; Mixed boundary value problems; Fourier transform; Dual integral equations.

On the initial-boundary value problem for the three-dimensional compressible viscoelastic fluids with the electrostatic effect

We investigate the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids with the electrostatic effect, in which the Dirichlet-Neumann mixed boundary condition for the electrostatic potential is imposed. We prove that there exists a unique global-in-time small strong solution in H2 Sobolev space. Moreover, we show that such a solution converges to the constant equilibrium state with an exponential decay rate as time tends to infinity.

Monte Carlo simulations of two-component Coulomb gases applied in surface electrodes

In this work, we study the gapped surface electrode (SE), a planar system composed of two-conductor flat regions at different potentials with a gap between both sheets. The computation of the electric field and the surface charge density requires solving Laplace’s equation subjected to Dirichlet conditions (on the electrodes) and Neumann boundary conditions over the gap. In this document, the gapless surface electrode is modeled as a two-dimensional classical Coulomb gas having punctual charges +q and −q on the inner and outer electrodes, respectively, interacting with an inverse power law 1/r-potential. The coupling parameter Γ between particles inversely depends on temperature and is proportional to q 2. Precisely, the density charge arises from the equilibrium states via Monte Carlo (MC) simulations. We focus on the coupling and the gap geometry effect. Mainly on the distribution of particles in the circular and the harmonically-deformed gapped SE. MC simulations differ from electrostatics in the strong coupling regime. The electrostatic approximation and the MC simulations agree in the weak coupling regime where the system behaves as two interacting ionic fluids. That means that temperature is crucial in finite-size versions of the gapped SE where the density charge cannot be assumed fully continuous as the coupling among particles increases. Numerical comparisons are addressed against analytical descriptions based on an electric vector potential approach, finding good agreement.

Two-dimensional semi-analytical solution of land-derived solute transport under tidal fluctuations considering variable boundary conditions on the seaward boundary

Transport of contaminants in coastal aquifers subject to tidal fluctuations is an important topic and a growing problem due to increasing populations and development in coastal areas. Many studies have used variable boundary conditions to simulate transport behavior at the interface between seawater and groundwater. However, to the best of our knowledge, no analytical solution accounting for variable-concentration boundary conditions exists for plume migration of land-derived contaminants passing through the inland zone of aquifers under tidally induced periodic flow. Here, a two-dimensional (2D) semi-analytical solution for land-derived solute transport was developed with variable boundary conditions at the seaward boundary under tidal fluctuations. The proposed 2D semi-analytical solution was verified through comparison with an existing one-dimensional semi-analytical solution and a 2D numerical solution developed using the finite-element method. Sensitivity analysis was performed to explore factors including hydraulic properties, tidal amplitude, and the location and distribution of initial contaminants, which affect the characteristics of land-derived pollutant transport in coastal aquifers. Through comparison of the calculated masses remaining over time in the aquifer using the proposed semi-analytical solution under two types of variable boundary conditions (Dirichlet-Neumann and Cauchy-Neumann variable boundary conditions), we found that the Cauchy-Neumann variable boundary conditions reflected reality better than did the Dirichlet-Neumann variable boundary conditions. The proposed analytical solution will be useful for obtaining insights into the mechanisms that drive the plume behavior of contaminants such as radionuclides or toxic chemicals released into coastal aquifers of tidally affected systems.

Conformal boundary condition and massive gravitons in AdS/BCFT

According to Witten [1], the conformal boundary condition of gravity, which specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and leads to well-defined perturbation theory of gravity about any classical solution. The conformal boundary condition was previously considered in [2, 3] in the context of AdS/BCFT, wherein the equation of motion of the end-of-the-world was derived and emphasized. In this paper, we investigate further other consequences of the conformal boundary condition in AdS/BCFT. We derive the boundary central charges of the holographic Weyl anomaly and show that they are exactly the same for conformal boundary condition and Dirichlet boundary condition. We analysis the metric perturbation with conformal boundary condition (CBC), Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) imposed on the end-of-the-world brane and show that they admit an interpretation as the fluctuation of the extrinsic curvature (case of CBC and DBC) and the induced metric (case of NBC) of Q respectively. In all cases, the fluctuation modes are massive, which are closely relevant to the massive island formation in the literature. Our results reveal that there are non-trivial gravitational dynamics from extrinsic curvatures on the conformal and Dirichlet branes, which may have interesting applications to the island. We also discuss, in passing, the localization of gravitons in brane world theory. We find that, contrary to NBC, the graviton for CBC/DBC is located on the brane with non-positive tension instead of non-negative tension.

Transmission Problems for Parabolic Operators on Polygonal Domains and Applications to the Finite Element Method

We study linear parabolic equations \(\partial _t u+Lu=f\), where \(L = -{\text {div}}(A \nabla )\) is a second-order strongly elliptic operator, on non-smooth two-dimensional bounded domains. The domain is polygonal and not assumed to be convex. The coefficient matrix A is piecewise smooth and exhibits jump discontinuities across a finite number of piecewise smooth curves, collectively denoted the interface. We assume mixed Dirichlet–Neumann boundary conditions and standard transmission conditions at the interface. Under some additional assumptions, we establish well-posedness of the initial-value problem using suitable weighted Sobolev spaces. The solution admits a decomposition \(u = u_\mathrm{{reg}} + w_\mathrm{{s}}\), into a function \(u_\mathrm{{reg}}\) that belongs to the weighted Sobolev space and a function \(w_\mathrm{{s}}\) that is locally constant near the vertices, thus proving well-posedness in an augmented space. We use the theoretical analysis to devise graded meshes that give quasi-optimal rates of convergence for a fully discrete scheme that utilizes finite elements on a space grid and finite differences in time.

Brownian dynamics simulations for the narrow escape problem in the unit sphere.

The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.

Well-posedness of singular-degenerate porous medium type equations and application to biofilm models

We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an L1-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis.

Two-scale topology optimisation of cellular materials under mixed boundary conditions

This work proposes a theoretical/numerical framework for the topology optimisation of anisotropic architected cellular materials at different scales. In particular, the topological variable (i.e., the pseudo-density field) is defined at both the scale of the representative volume element (i.e., the unit cell) of the material and at the macroscopic scale of the structure. The two-scale topology optimisation problem is formulated in the most general sense, i.e., by considering non-zero Neumann-Dirichlet boundary conditions. The proposed method is based on: (a) non-uniform rational basis spline hyper-surfaces to represent the topological variable at each scale, (b) the solid isotropic material with penalisation approach, (c) a general numerical homogenisation scheme based on the strain energy to establish the link between scales. The proposed formulation exploits the properties of non-uniform rational basis spline entities to determine the relationships occurring among the topological variables defined at different scales to correctly state the optimisation problem and to satisfy the hypotheses at the basis of the homogenisation method. In particular, scale separation (a necessary condition to be met in order to apply the homogenisation method) and manufacturing requirements are implicitly ensured by introducing minimum length scale constraints on the topological variables defined at both macroscopic scale and unit cell scale, respectively. Furthermore, the sensitivity of the optimised topology (at each scale) to the applied boundary conditions and to the elastic symmetry group of the representative volume element is investigated by founding new and original results. The effectiveness of the approach is tested on 2D and 3D benchmark problems taken from the literature.

Stabilization of the critical semilinear wave equation with Dirichlet-Neumann boundary condition on bounded domain

We study the stability of the critical defocusing semilinear wave equation with a distributed locally damping and Dirichlet-Neumann boundary condition on a bounded domain. The main novelty is to establish a framework to study the stability of the damped critical semilinear wave equation on bounded domain. The unique continuation properties and the observability inequalities are proved by the Morawetz estimates in Euclidean spaces, and then the compactness-uniqueness arguments are applied to prove the main stabilization result.

Exponential stability in a Timoshenko system of type III

In this paper, a Timoshenko system with a delay term and a thermo-viscoelastic damping of type III subjected to Dirichlet–Dirichlet–Neumann boundary conditions will be considered. Exploiting energy method to produce a suitable Lyapunov functional, we establish the global existence and exponential decay of energy for Timoshenko system of type III.

Effect of phase lags of moisture–heat flow on the hygrothermoelastic field of hollow cylinders with convective surfaces

In this article, by considering the Dufour effect and Soret effect, a hygrothermal coupling model based on phase delay of heat and moisture fluxes is established for convective surfaces. The convective surfaces are described by Robin boundary conditions. The Dirichlet boundary conditions (prescribed temperature and moisture on the surfaces) and Neumann boundary conditions (prescribed heat and moisture fluxes) are two extreme cases of the Robin conditions. Using Laplace transform, the effects of different relaxation times and moist-heat coupling on the temperature, moisture, and stress of the hollow cylinder are analyzed from two aspects of time and space. For convective boundary, the dependence of temperature, moisture, and stress on the phase lag of heat and moisture fluxes is examined. The results show that the temperature and moisture distributions along with the hygrothermal stress fields exhibit different behaviors, depending on the coefficient of Robin condition.

Multiscale Numerical Modeling of Solute Transport with Two-Phase Flow in a Porous Cavity

This paper introduces dimensional and numerical investigation of the problem of solute transport within the two-phase flow in a porous cavity. The model consists of momentum equations (Darcy’s law), mass (saturation) equation, and solute transport equation. The cavity boundaries are constituted by mixed Dirichlet-Neumann boundary conditions. The governing equations have been converted into a dimensionless form such that a group of dimensionless physical numbers appear including Lewis, Reynolds, Bond, capillary, and Darcy numbers. A time-splitting multiscale scheme has been developed to treat the time derivative discretization. Also, we use the Courant-Friedrichs-Lewy (CFL) stability condition to adapt the time step size. The pressure is calculated implicitly by coupling Darcy’s law and the continuity equation, then, the concentration equation is solved implicitly. Numerical experiments have been conducted and the effects of the dimensionless numbers have been on the saturation, concentration, pressure, velocity, and Sherwood number have been investigated.

Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation

Many problems of continuum physics, applied physics, electrostatics, heat engineering, and structural mechanics are described by boundary value problems for the Poisson equation. The article is devoted to the construction of a numerical algorithm for solving plane boundary value problems for the Poisson equation. The algorithm is based on the transition to a polyharmonic equation, which is solved by the method of linear boundary elements. We consider two boundary value problems with different boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. To construct a numerical solution using the linear boundary element method, the boundary of the area is replaced by an inscribed polygon. The boundary conditions are satisfied at the middle (control) points of the elements. A polyharmonic equation is reduced to a system of linear integral equations, which can be reduced to a system of linear algebraic equations. Test numerical examples confirming the effectiveness of the proposed algorithm are given.

Validation of a finite element modelling approach for mitigated and unmitigated pile driving noise prognosis.

Piles are the state-of-the-art foundation type for offshore structures like offshore wind turbines. The pile driving process induces high sound pressure levels into the water, which are potentially harmful for the marine environment. To protect the marine life, regulations for these levels apply in many regions of the world. Therefore, detailed pile driving noise models are necessary to allow for both a prognosis of the underwater noise levels and the dimensioning and optimization of possible noise mitigation systems. In this paper, an established model based on a finite element approach is validated by means of three measurement campaigns. These have been conducted at different sites in the North Sea and include piling with and without noise mitigation measures. The noise mitigation systems are modelled as fully absorbing by applying a mixed Dirichlet-Neumann boundary condition at its position. Therefore, the computational results with noise mitigation measures are generally below the measured data and present the highest achievable noise reduction. The measurement campaigns have been conducted with a big bubble curtain and a noise mitigation screen. The occurring differences between the modelled and measured results with and without noise mitigation are shown.

New general and explicit stability result for a thermoelastic Timoshenko system

The goal of this article is to study a thermoelastic Timoshenko system with viscoelastic law acting on the shear force and thermoelastic dissipation acting on the bending moment. We prove a general decay as well as an explicit decay estimate for solution energy, from which the exponential and polynomial decay results are only special cases. The result is obtained without imposing equal-wave-speed condition, this is achieved by considering the Neumann–Dirichlet–Neumann boundary conditions . This result is therefore of great interest compared to majority of results in the literature that often time impose the equal-wave-speed condition.

Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.

Numerical investigation of the two– dimensional time–dependent diffusion equation using Radial basis functions

In this paper, we investigate the numerical solution of the two–dimensional time–dependent diffusion equation with non-local and mixed Neumann–Dirichlet boundary conditions. In the discretization process, the backward Euler as well as Crank–Nicolson schemes and radial basis function (RBF) collocation method are respectively used to discretize time derivative and spatial derivative terms. The accuracy and applicability of the presented methods are illustrated and compared by solving two examples.

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann–Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $$H^1$$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $$H^1$$ -type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.