Extra Boundary Conditions Research Articles

State-Spatial Viewpoint of Stress in Thermally Postbuckled and Nonlinearly Bent FG Saturated Poroelastic Circular Plates

A dimensionless discrete state-space mathematical model is proposed for stress analysis of an axially-symmetric saturated poroelastic circular plate being loaded by temperature rise of one of the flat faces. Computation of stress by means of this model is free from a need for differentiation. The curves of thermal postbuckling, thermal nonlinear bending, radial stress, circumferential stress and resultant radial force have been achieved by means of six state-space variables and differential quadrature procedure. The effects of loading temperature, boundary conditions, porosity, pore distribution and fluid pore pressure on the thermal post buckling and nonlinear bending phenomena have been investigated. The critical postbuckling thermal load has been determined by means of a numerical algorithm that does not converge to trivial solution because it starts from a relatively great numerical thermal load. Contrary to the governing equation in usual form, the governing equations in state space together with the state equations equalize the number of the governing assembled difference equations and that of the unknowns, and eventually do not require to apply the so-called auxiliary extra boundary conditions at the nodes adjacent to the boundary nodes. As another advantage, in state-spatial viewpoint, there is not any necessity for computing the coefficients of derivatives with order higher than unity.

Novel low-order continuum models for the dynamic behaviour of microstructured plates based on a beam-grid lattice

In this paper, several non-classical continuum models are developed by applying different continualization techniques to a 2D beam-grid lattice. The purpose of these new models is to capture the size-effects that arise in certain dynamic problems, and which are not accounted by the Classic Kirchhoff plate model. All these models are evaluated via dispersion and natural frequency analyses, considering the discrete model as a reference. Interestingly, the non-standard models derived do not present any physical inconsistency, and capture the size-effect through space–time cross derivatives with low spatial order, thus not needing extra boundary conditions, an important feature ignored in other works.

De Sitter space, extremal surfaces, and time entanglement

We refine previous investigations on de Sitter space and extremal surfaces anchored at the future boundary $I^+$. Since such surfaces do not return, they require extra data or boundary conditions in the past (interior). In entirely Lorentzian de Sitter spacetime, this leads to future-past timelike surfaces stretching between $I^\pm$. Apart from an overall $-i$ factor (relative to spacelike surfaces in $AdS$) their areas are real and positive. With a no-boundary type boundary condition, the top half of these timelike surfaces joins with a spacelike part on the hemisphere giving a complex-valued area. Motivated by these, we describe two aspects of "time-entanglement" in simple toy models in quantum mechanics. One is based on a future-past thermofield double type state entangling timelike separated states, which leads to entirely positive structures. Another is based on the time evolution operator and reduced transition amplitudes, which leads to complex-valued entropy.

Robin boundary conditions in acoustic BTZ black holes

We introduce an analog model for the conformally coupled scalar field on the BTZ black hole. The model is based on the propagation of acoustic waves in a Laval nozzle. Since the BTZ black hole is not a globally hyperbolic spacetime, the dynamics of the scalar field is not well defined until extra boundary conditions are prescribed at its spatial infinity. We show that quasinormal modes (QNMs) satisfying Dirichlet, Neumann, and Robin boundary conditions in the BTZ black hole can be interpreted in terms of ordinary QNMs defined with respect to an appropriately extended nozzle. We also discuss the stability of our model with respect to small perturbations.

A Cartesian grid-based two-dimensional plus time method for simulating ship bow waves

Numerical modeling of ship bow waves is still hard work, partly due to their multiscale features. Direct three-dimensional (3D) computational fluid dynamics simulation could be an appropriate choice to investigate the problem. However, limited by computational resources, small scale phenomena such as spraying and wave breaking that could be observed during the ship wave generation process are usually simplified or not fully distinguished in a 3D simulation. In order to accurately capture the small scales flow field information with the available computational resource, a new Cartesian grid-based two-dimensional plus time (2D+t) method is developed in this paper, which is suitable for 3D slender ships. With this method, a 3D steady ship wave-making problem is transformed into a 2D unsteady wave-making problem of a deformable body. The boundary velocity of the deformable body is obtained with a novel interpolation algorithm, which is then enforced on the background Cartesian grid by a newly proposed immersed boundary method. The pressure boundary condition on the surface of the deformable body is explicitly considered in the solution of the pressure Poisson equation. Moreover, an extra open boundary condition is applied to the upper boundary of the computational domain to achieve a better conservation. The proposed model is validated with selected cases, showing that the model is capable of simulating both non-wave-breaking and wave-breaking problems.

Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions, I

Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions, I

DYNAMIC CHARACTERISTICS OF MIXTURE UNIFIED GRADIENT ELASTIC NANOBEAMS

The mixture unified gradient theory of elasticity is invoked for the rigorous analysis of the dynamic characteristics of elastic nanobeams. A consistent variational framework is established and the boundary-value problem of dynamic equilibrium enriched with proper form of the extra non-standard boundary conditions is detected. As a well-established privilege of the stationary variational theorems, the constitutive laws of the resultant fields cast as differential relations. The wave dispersion response of elastic nano-sized beams is analytically addressed and the closed form solution of the phase velocity is determined. The free vibrations of the mixture unified gradient elastic beam is, furthermore, analytically studied. The dynamic characteristics of elastic nanobeams is numerically evaluated, graphically illustrated, and commented upon. The efficacy of the established augmented elasticity theory in realizing the softening and stiffening responses of nano-sized beams is evinced. New numerical benchmark is detected for dynamic analysis of elastic nanobeams. The established mixture unified gradient elasticity model provides a practical approach to tackle dynamics of nano-structures in pioneering MEMS/NEMS.

Flow Analysis on Boundary Layer of Porous Horizontal Circular Cylinder Filled by Viscoelastic-Micropolar Fluid

This study emphasis on the analysis of boundary layer flow of viscoelastic fluid with microrotation moving over a porous horizontal circular cylinder. The model of the problem is based on Navier Stokes equations which involved continuity, momentum and micro inertia equations. The mentioned equations are first undergo Boussinesq and boundary layer approximation before transforming to non-dimensional form which in partial differential equations system. Since the boundary layer equations of viscoelastic fluid are an order higher than Newtonian (viscous) fluid, the adherence boundary conditions are insufficient to govern the solutions entirely. Hence, the augmentation of an extra boundary conditions is necessary to perform the computation. The computation is done by adopting the established procedures called Keller box method. The results are computed for velocity and microrotation distribution as well as skin friction coefficient. It is worth to mentioned at the special case, the present model can be deduced to the established model where the porosity, microinertia and magnetic term excluded. The output computed will be served as a reference to study the complex fluid especially when the fluid exhibit both viscous and elastic characteristics with microrotation effect.

Entropy analysis of unsteady magnetohydrodynamic thin liquid film flow of Maxwell nanofluids with variable fluid properties

The flow and heat transfer phenomena of a viscous fluid in a thin film over an expanding flat surface possesses optimistic applications in a large number of technological processes. Heat and mass transfer rate within the thin film bears an unequivocal factors on the performance of coating process and the chemical nature of the product. This article concerns with the study of unsteady boundary layer magnetohydrodynamic thin film flow of Maxwell nanofluid past over an exponentially stretching surface subject to slip effect on velocity and suction/injection at the boundary. The heat transfer mechanism is influenced by the thermal radiation, temperature dependent thermal conductivity, viscous dissipation of energy and a non-uniform heat source/sink. We have analyzed the entropy generation of the system by paying due attention to the variable fluid properties and film thickness. The system of arising time dependent coupled nonlinear partial differential equations governing the fluid flow are converted into a system of ordinary differential equation by utilizing suitable self-similar transformations. The reduced two-point boundary value problem is then solved with the aid of well known Runge–Kutta–Fehlberg integration technique based shooting method. The mathematical model suggests an extra kinematic boundary condition which is utilized to compute the thickness of the thin film by employing the Newton–Raphson method. The investigation highlights that the arising controlling parameters such as magnetic parameter, radiation parameter, viscosity parameter, thermal conductivity parameter have strong effect on velocity field, temperature field, Nusselt number, and the thickness of the thin film. Moreover, the Bejan number, a measure of thermal irreversibility, is highly affected by the variation of the magnetic field strength, thermal radiation, dissipation of energy, thermal conductivity parameter, viscosity parameter and heat source/sink.

Low-order continualization of an anisotropic membrane lattice with next-nearest interactions. Enhanced prediction of its dynamic behaviour

In this paper, a novel anisotropic membrane lattice with nearest and next-nearest interactions (long-range forces) has been continualized through different standard and non-standard continualization procedures, which enables the development of new non-classical continuum models capable of accurately capturing the scale effects, present in the matter due to its discrete nature. The performance of these continuum models is assessed by means of both dispersion and natural frequencies analyses, where the discrete model is considered as a reference. In addition, the appearance of certain physical inconsistencies in some of the developed models is analysed, concluding that these only appear for those developed with continualizations based on Taylor expansion. Interestingly, the non-standard models suitably capture the dispersive behaviour of the discrete one, without both physical inconsistencies and higher-order spatial derivatives, thus avoiding the need for extra boundary conditions when finite solids are involved.

New low-order continuum models for the dynamics of a Timoshenko beam lattice with next-nearest interactions

In this paper, the dynamic behaviour of a novel Timoshenko beam lattice with long-range interactions, accounting for both bending and shear deformations, is investigated. Several new non-classical continuum models are developed with the aim of capturing its dispersive behaviour with a lower computational cost. For this, innovative continualization procedures are used, comparing them with techniques commonly used in lattices continualization, as well as with advanced ones. Moreover, low-order continuum governing equations are pursued, thus avoiding the need for extra boundary conditions, whose physical meaning is unclear. A comprehensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been performed here for the first time for this lattice and the corresponding continuum models. The capability of these continuum models in capturing the behaviour of the lattice is assessed by conducting both dispersion and natural frequency analyses, for the latter providing an original method to treat the edges for the three possible boundary conditions in Timoshenko beam lattices. The influence of long-range interactions is analysed, and the way shear effect affects the shape vibration modes of the discrete model is interestingly illustrated, finally concluding that some of the new developed continuum models accurately capture the behaviour of the lattice.

Low-order non-classical continuum models for the improved prediction of an anisotropic membrane lattice’s dynamics

In this work, different standard and non-standard continualization methods are applied to an anisotropic membrane lattice to obtain non-classical continuum models, capable of accurately capturing its dynamic behaviour. Their performance is evaluated by comparing the continuum and discrete models through their dispersion relations, as well as their natural frequencies for the fixed-edge configuration. Moreover, the occurrence of certain physical inconsistencies in some of these models is examined. Interestingly, the novel Enriched Kinetic Energy model shows the best performance, presenting neither physical inconsistencies nor higher-order derivatives in its governing equation, thus not needing extra boundary conditions when dealing with finite solids.

Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data.

Boundary conditions for isolated asymptotically anti-de Sitter spacetimes

We revisit the propagation of classical scalar fields in a spacetime, which is asymptotically anti-de Sitter. The lack of global hyperbolicity of the underlying background gives rise to an ambiguity in the dynamical evolution of solutions of the wave equation, requiring the prescription of extra boundary conditions at the conformal infinity to be fixed. We show that the only boundary conditions that are compatible with the hypothesis that the system is isolated, as defined by the (improved) energy–momentum tensor, are of Dirichlet and Neumann types.

CONTINUOUS-FLOW EXTRACTION OF BIOCOMPOUNDS IN FIXED BED: INFLUENCE OF SWAPPING FROM DIRICHLET TO DANCKWERTS CONDITION AT INLET IN PHENOMENOLOGICAL MODELS

Phenomenological models have increasingly become vital to bioprocess engineering. In continuous-flow biocompounds extraction models, diffusion requires an extra boundary condition at exit (usually null Neumann condition) while either Dirichlet or Danckwerts condition can be imposed at inlet. By taking an extant case study and with the help of an in-house lattice-Boltzmann simulator, this work numerically examines prospective effects of interchanging aforesaid inlet conditions. Trial simulations were performed for scenarios ranging from convective-dominant to diffusive-dominant. Extraction yields numerically simulated under each inlet condition were compared with experimental data. Expected shape of extraction yield curves was simulated whenever process parameters were properly provided and differences due to switching inlet conditions became evident only in diffusion-dominant extraction scenarios. At diffusivities of order 10-6 m2 s-1, numerical results suggest that Danckwerts boundary condition should be preferred at bed inlet.

Dynamic analysis and non-standard continualization of a Timoshenko beam lattice

In this paper, a Timoshenko beam lattice, made up of a chain of masses and straight segments, is proposed, considering bending and shear deformation by means of linear rotational and transverse springs, respectively. Different standard and non-standard continualization methods are applied to it, highlighting here for the first time the suitability of taking the coupled discrete governing equations as a starting point for deriving new continuum models. Several novel low order non-classical continuum models are obtained, with the aim of reliably capturing size-effects and reflecting the dispersive behaviour of the discrete system. Low order governing equations prevents the need for extra boundary conditions when finite (bounded) solids are treated. An extensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been carried out, examining its influence for the discrete and non-standard continuum models. The natural frequencies of a finite solid with two different boundary conditions are obtained through an edge treatment applied here for the first time to this kind of lattices, thus making it possible to solve the clamped-free edges configuration. The reliability of these approaches is evaluated by comparing their dynamic behaviours with that of the discrete system (taken as a reference), through both dispersion and vibration analyses, some of the new proposed continuum models successfully capturing the behaviour of the discrete one, even for high wavenumbers. Moreover, the appearance of physical inconsistencies is examined.

Low order nonstandard continualization of a beam lattice with next-nearest interactions: Enhanced prediction of the dynamic behavior

In this work, a one-dimensional beam lattice composed of masses and rotational springs with nearest and next-nearest interactions is proposed, applying to it several nonstandard continalization procedures. The reliability of nonclassical continuum models to capture the dynamic behavior of the lattice – considered as a reference –, is evaluated through dispersion and natural frequency analyses. A detailed boundary conditions treatment is presented and the existence of physical inconsistencies in the new continuum models is examined. The novel enriched kinetic energy model proposed shows the best performance, its governing equation being of low order, thus, avoiding the use of extra boundary conditions.

Time-varying materials in the presence of dispersion: plane-wave propagation in a Lorentzian medium with temporal discontinuity

We study the problem of a temporal discontinuity in the permittivity of an unbounded medium with Lorentzian dispersion. More specifically, we tackle the situation in which a monochromatic plane wave forward-traveling in a (generally lossy) Lorentzian-like medium scatters from the temporal interface that results from an instantaneous and homogeneous abrupt temporal change in its plasma frequency (while keeping its resonance frequency constant). In order to achieve momentum preservation across the temporal discontinuity, we show how, unlike in the well-known problem of a nondispersive discontinuity, the second-order nature of the dielectric function now gives rise to two shifted frequencies. As a consequence, whereas in the nondispersive scenario the continuity of the electric displacement D and the magnetic induction B suffices to find the amplitude of the new forward and backward wave, we now need two extra temporal boundary conditions. That is, two forward and two backward plane waves are now instantaneously generated in response to a forward-only plane wave. We also include a transmission-line equivalent with lumped circuit elements that describes the dispersive time-discontinuous scenario under consideration.

Hyperbolic Solutions to Bernoulli’s Free Boundary Problem

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.

Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending

AbstractThe Eringen's fully nonlocal elasticity model is known to lead to ill‐posed boundary‐value problems and to suffer some boundary effects arising from particle interactions impeded by the body's boundary surface. An enhanced model is derived from the original fully nonlocal one by the addition of a regularizing non‐homogeneous local phase which accounts for boundary effects and which leads to a Fredholm integral equation of the second kind, hence to well‐posed boundary‐value problems, without paradoxes, nor other drawbacks. The enhanced integral model applied to a beam in bending proves to be equivalent to a sixth order differential equation with variable coefficients, with extra nonlocality boundary conditions here also derived. Both the integral approach and the differential one lead to a same unique solution of the small‐scale beam problem. An efficient numerical algorithm is presented in which the sixth order differential equation with variable coefficients is reduced to one of the second order, which is addressed by a finite difference method. The proposed theory is applied to a set of engineering beam problems, for each of which the inherent size effects are reported and graphically illustrated. The influence of the length scale parameter upon the beam's response is highlighted by means of a function representing the normalized maximum deflection of the beam as a function of the length scale parameter. It is shown that the enhanced model always predictssoftening size effectsno matter the boundary and loading conditions, and that the related response function generally exhibits awaved pattern with positive slopes first, then negative, as the length scale parameter increases, with a limit asymptotic behavior like an atomic lattice model. A comparison with other theories is also presented together with possible future developments.