Non-standard Boundary Conditions Research Articles

Optimal Control of Distributed Parameter Systems with Application to Transient Thermoelectric Cooling

We give a solution of optimal control problem for distributed parameter systems described by nonlinear partial differential equations with nonstandard boundary conditions. The variatio ...

Time Origin and Universe Uniform Expanding

A new cosmological hypothesis is considered which states that our Universe is a black hole in some external world. Because of that it expands linearly depending on so-called Parametric Time. For such the model we use the typical Einstein-Friedmann’s equation system, but non-standard boundary conditions should be applied. A number of results are established that are indicative of the model efficacy.

Surface effects in anti-plane deformations of a micropolar elastic solid: integral equation methods

The theory of linear micropolar elasticity is used in conjunction with a new representation of micropolar surface mechanics to develop a comprehensive model for the deformations of a linearly micropolar elastic solid subjected to anti-plane shear loading. The proposed model represents the surface effect as a thin micropolar film of separate elasticity, perfectly bonded to the bulk. This model captures not only the micro-mechanical behavior of the bulk which is known to be considerable in many real materials but also the contribution of the surface effect which has been experimentally well observed for bodies with significant size-dependency and large surface area to volume ratios. The contribution of the surface mechanics to the ensuing boundary-value problem gives rise to a highly nonstandard boundary condition not accommodated by classical studies in this area. Nevertheless, the corresponding interior and exterior mixed boundary-value problems are formulated and reduced to systems of singular integro-differential equations using a representation of solutions in the form of modified single-layer potentials. Analysis of these systems demonstrates that the classical Noether theorems reduce to Fredholms theorems leading to results on well-posedness of the corresponding mathematical model.

A discrete duality finite volume discretization of the vorticity‐velocity‐pressure stokes problem on almost arbitrary two‐dimensional grids

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity‐velocity‐pressure formulation of the two‐dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015

A Study on Numerical Solutions for Rotating Flows of a Third Grade Fluid with Partial Slip Conditions

In this paper rotating flow of a third grade fluid past a porous plate with partial slip is studied .The nonlinear boundary value problem is solved using finite difference method. The variations of velocity components for various values of partial parameter �� 1are discussed the results are reported for conclusion. Keyword: third grade fluid, partial slip parameter, finite difference method, non - Newtonian fluid. I. Introduction Materials such as polymer solutions or melts, drilling mud, clastomers, certain oils and greases and many other emulsions are classified as non-Newtonian fluids. Due to complexity of fluids, there are many models describing the properties, but not all of non-Newtonian fluids. These models, however, cannot predict all the behaviours of non-Newtonian fluids, for example, normal stress differences, shear thinning or shear thickening, shear relaxation, elastic and memory effects etc. The pioneers of the field such as Coulomb, Navier, Girad, Poisson, Stokes, St. Venant and others recognised that boundary conditions are constitutive equations that should be determined by the material on either side of the boundary. The usual prescription of Dirichlet and Neumann conditions are often unsuitable for a realistic physical problem, for example the flow of polymers that stick-slip on the boundary. Recently non-standard boundary conditions have been considered from a rigorous mathematical perspective by Rao and Rajagopal. The earliest work S.Asgar,M.Mudassar Gulzar, M.Ayub (2006) are derived an analytical study of the rotating flow of a third grade fluid past a porous plate with partial slip effects. It serves as a flow model for the study of polymers. The analytic solution has been determined using homotopy analysis method (HAM). The work is extended to examine the effects of partial slip on the rotating flow past a uniformly porous plate. The fluid is incompressible and third grade .The numerical solution has been obtained using Finite difference method and finally focused on the effects of suction, blowing, third grade and slip parameter. II. A Study on rotating flow of a third grade fluid Consider the Cartesian coordinate system rotating uniformly with an angular velocity ٠about the z- axis ,taken positive in the vertically upward direction and the plate coinciding with the plane ��= 0 . The fluid flowing past a porous plate is third grade and incompressible .All the material parameters of the fluid are assumed to be constants . For the rotating frame, the equation of momentum

An ill-posed parabolic evolution system for dispersive deoxygenation–reaeration in water

We consider an inverse problem that arises in the management of water resources and pertains to the analysis of surface water pollution by organic matter. Most physically relevant models used by engineers derive from various additions and corrections to enhance the earlier deoxygenation–reaeration model proposed by Streeter and Phelps in 1925, the unknowns being the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) concentrations. The one we deal with includes Taylor’s dispersion to account for the heterogeneity of the contamination in all space directions. The system we obtain is then composed of two reaction-dispersion equations. The particularity is that both Neumann and Dirichlet boundary conditions are available on the DO tracer while the BOD density is free of any conditions. In fact, for real-life concerns, measurements on the DO are easy to obtain and to save. On the contrary, collecting data on the BOD is a sensitive task and turns out to be a lengthy process. The global model pursues the reconstruction of the BOD density, and especially of its flux along the boundary. Not only is this problem plainly worth studying for its own interest but it could also be a mandatory step in other applications such as the identification of the location of pollution sources. The non-standard boundary conditions generate two difficulties in mathematical and computational grounds. They set up a severe coupling between both equations and they are the cause of the ill-posed data reconstruction problem. Existence and stability fail. Identifiability is therefore the only positive result one can search for; it is the central purpose of the paper. Finally, we have performed some computational experiments to assess the capability of the mixed finite element in missing data recovery.

Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type

The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

This paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi–Douglas–Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart–Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška–Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

High order numerical simulation of the transmission and scattering of waves using the method of difference potentials

The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems. It offers several key advantages, such as the capability of handling boundaries/interfaces that are not aligned with the discretization grid, variable coefficients, and nonstandard boundary conditions. In doing so, the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid.Previously, we have applied the method of difference potentials to solving several variable coefficient interior Helmholtz problems with fourth and sixth order accuracy. We have employed compact finite difference schemes as a core discretization methodology. Those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself. Numerical experiments corroborate the high order accuracy of our method for variable coefficients, regular grids, and non-conforming boundaries.In the current paper, we extend the previously developed methodology to exterior problems. We present a complete theoretical analysis of the algorithm, as well as the results of a series of numerical simulations. Specifically, we study the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions. We also solve the transmission/scattering problems, in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside. In all the cases, our methodology guarantees high order accuracy for regular grids and non-conforming boundaries and interfaces.

A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate the resulting numerical capabilities by solving a range of nonstandard boundary value problems for the Helmholtz equation. These include problems with variable coefficient Robin boundary conditions (including discontinuous coefficients) and problems with mixed (Dirichlet/Neumann) boundary conditions. In all our simulations, we use a Cartesian grid and a circular boundary curve. For those test cases where the overall solution was smooth, our methodology has consistently demonstrated the design fourth-order rate of grid convergence, whereas when the regularity of the solution was not sufficient, the convergence slowed down, as expected. We also show that every additional boundary condition requires only an incremental additional expense.

Method for electric field and potential calculations in Hall plates

Electrostatic field problems occurring in Hall plates are difficult to solve, mainly because of a non-standard boundary condition defining an oblique angle of the electric field w.r.t. an isolating boundary. A new approach for solving Hall-related field problems is presented. Compared to prior approaches, the technique leads more easily to closed-form expressions for the electric field, and allows obtaining voltage-related Hall characteristics in numerically well conditioned forms.

Failure-Oriented Multi-scale Variational Formulation: Micro-structures with nucleation and evolution of softening bands

This contribution presents the theoretical foundations of a Failure-Oriented Multi-scale variational Formulation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phenomena. The multi-scale model considers two coupled mechanical problems at different physical length scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linked to a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homogenization of the micro-mechanical problem.At the macroscopic level, the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled.The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical information transference from the macro-to-micro scales during the complete loading history, even in those points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-models consistently coupled:(i)a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response.(ii)A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated, for modeling the macro-crack evolution.When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a change in the rule that defines how the increments of generalized macro-strains are inserted into the micro-scale and (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures are obtained. Then, as a consequence of the Hill–Mandel Variational Principle and the proposed kinematical assumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuum part of the body, as well as, for the traction acting on the macro-discontinuity surface.The assumed macro-to-micro mechanism of kinematical coupling defines a specific admissible RVE-displacement space, which is obtained by incorporating additional boundary conditions, Non-Standard Boundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Boundary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we call through the paper “objectivity” of the homogenized constitutive response.Several numerical examples are presented showing the objectivity of the formulation, as well as, the capabilities of the new multi-scale approach to model material failure problems.

Lp-THEORY FOR VECTOR POTENTIALS AND SOBOLEV'S INEQUALITIES FOR VECTOR FIELDS: APPLICATION TO THE STOKES EQUATIONS WITH PRESSURE BOUNDARY CONDITIONS

In a three-dimensional bounded possibly multiply connected domain, we give gradient and higher-order estimates of vector fields via div and curl in Lp-theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with non-standard boundary conditions of the form u × n = g × n and π = π0 on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf–Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u⋅n and u × n on Γ.

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben'kii and can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve. Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils--one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation. We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.

Optimization of piezoelectric patch positioning for passive sound radiation control of plates

The purpose of this article is the optimization of piezoelectric patch positioning for reducing the radiated sound power of thin plates. To this end, an aluminium plate equipped with a set of piezoelectric patches connected to a passive circuit is considered. The difficulties in designing a smart structure are not only related to the conception of the electric circuit used as controller, but also to the choice of how the circuit itself is coupled with the structure. The selection of the number of transducers to be used, and their positioning, is a crucial step in the designing process. In this work the entire design process of a smart structure is proposed, with the goal of obtaining the best efficiency in terms of reduction of radiated sound power, while using a lightweight optimization procedure in terms of computational costs. To this end, classical instruments of vibrations mechanics are used together with acoustic concepts, such as the modal radiation efficiency. The introduction of this acoustic characterization of the structure in the optimization process is done by using a new utility function, the acoustic controllability, which will be used for the optimization of transducer positioning. The proposed optimization process is applied to the case study of an aluminium plate with nonstandard boundary conditions, and experimental results confirm the validity of this novel procedure.

Spherical perturbations of hairy black holes in designer gravity theories

We study the spectrum of the scalar l = 0 quasi-normal frequencies of anti-de Sitter hairy black holes in four- and five-dimensional designer gravity theories of the Einstein-scalar type, arising as consistent truncations of gauged supergravity. In the dual field theory, such hairy black holes represent thermal states in which the operator corresponding to the bulk scalar field is condensed, due to the multi-trace deformation associated with non-standard boundary conditions. We show that, in a particular class of models, the effective potential describing the vacua of the deformed dual theory can be identified, at large values of the condensate, with the deformation plus the conformal coupling of the condensate to the curvature of the boundary geometry. In this limit, we show that the least damped quasi-normal frequency of the corresponding hairy black holes can be accurately predicted by the curvature of the effective potential describing the field theory at finite entropy.

Periodic Planar Multilayered Substrates Analysis Using Wave Concept Iterative Process

Due to the practical importance and difficulties associated with their closed form solutions, the experimental and computational study of periodic planar multilayered structures, such as FSS in multilayered configuration and Multilayered Planar antennas array, are in complementary progress. During the past two decades, the widespread use of such methods has allowed a broad range of important scattering problems involving non-standard shapes, boundary conditions and material composition to be solved. In this sense, an efficient iterative technique based on the concept of wave is presented for computing periodic substrates in multilayered configuration. This paper presents an extensible approach of the iterative method to study multilayered substrates (n layers in which n = 2, 20) with spatial periodicity in multi- layer configuration. Our new approach is performed in order to study 3 dimensional structures by the method called Wave Concept Iterative Process (WCIP). This method is adapted in its original form to study 2 dimensional structures. The third dimension is modulated by transmission line as an approximation for every mode in spectral domain. The utility of the new WCIP appears because of its fast convergence and little consumption in memory.

On Modeling and Simulation of Electrokinetic Phenomena in Two-Phase Flow with General Mass Densities

We formulate phase-field models for electrowetting with electrolyte solutions for general mass densities taking, in particular, mass density contrast into account. Based on Onsager's variational principle and volume averaging for gross velocities, we obtain a nonlinear coupling of hydrodynamic equations with the Nernst--Planck equations of ion transport, the Cahn--Hilliard equation for the phase-field, and an elliptic transmission problem for the electrostatic potential. Nonstandard boundary conditions for the Cahn--Hilliard equation allow for contact angle hysteresis. We comment on sharp interface limits and present energy estimates in the diffuse- and in the sharp interface settings. Finally, two- and three-dimensional numerical simulations underline the capability of diffuse interface approaches to model droplet transport, coalescence, and splitting.

Spectral modeling of vibrating plates with general shape and general boundary conditions

The analysis and design of lightweight plate structures require efficient computational tools, because exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. This paper deals with vibration analysis of Kirchhoff plates of general shape with non-standard boundary conditions, adopting a Rayleigh-Ritz approach. Three different coordinate mappings are considered, using different kinds of functions: 1) trigonometric and polynomial interpolation functions for mapping the shape of the plate, 2) trigonometric and polynomial interpolation functions for mapping a constraint domain of general shape, 3) products of linearly independent eigenfunctions evaluated from a standard beam in flexural vibration for describing the transverse displacement field of the plate. Flexural free vibration analysis of different shaped plates is then performed using the same approach: skew, trapezoid and triangular plates, plates with parabolic curved edges, sectors of circular plates, circular and elliptic annular plates. Purely elastic plates are considered, but the method may also be applied to the analysis of viscoelastic plates. The results are compared with those available in the literature and using standard finite element analysis.