Non-standard Boundary Conditions Research Articles

Spectral element discretization of the time-dependent Stokes problem with nonstandard boundary conditions

This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem.

A second-gradient elasticity model and isogeometric analysis for the pantographic ortho-block

We present a continuum model for a three-dimensional pantographic structure that we call pantographic ortho-block. The model is based on the embedding of four families of elastic fibers in a three-dimensional domain. Similar to pantographic sheets, we find an energy density depending on the first and second-gradient of displacement resulting in a three-dimensional incomplete second-gradient material. For certain kinematic assumptions concerning perfect joints, we find two distinct zero-energy modes. Finally, we perform simulations based on isogeometric analysis to showcase standard and non-standard boundary conditions.

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.

Nonlinear flexure mechanics of mixture unified gradient nanobeams

The mixture unified gradient theory of elasticity is invoked for the nanoscopic study of the nonlinear flexure mechanics of nanobeams. A mixed variational framework is conceived based on ad hoc functional space of kinetic test fields, wherein a complete set of governing equations, classical and non-standard boundary conditions, and the constitutive relations are integrated into a solitary functional. The size-effect phenomenon associated with the nonlinear flexure of nanobeams are efficiently realized as the stress gradient theory, the strain gradient theory, and the classical elasticity theory are consistently unified within the framework of the mixture unified gradient elasticity. A viable numerical approach is introduced based on the conceived mixed variational framework while the autonomous series solution of the kinematic and kinetic field variables is implemented. As the kinetic field variables are directly determined rather than utilizing post-computations, the established consistent variational theory provides an advantageous means for efficacious approximate approaches. The nonlinear flexural response of the mixture unified gradient beams with kinematics constraints of practical interest in nano-mechanics is detected and thoroughly compared with the linear counterparts in terms of the characteristic parameters. The detected nonlinear flexural characteristics of nano-sized beams can pave the way ahead in the nonlinear mechanics of nano-structures.

Nonlocal gradient integral models with a bi-Helmholtz averaging kernel for functionally graded beams

In this paper, we develop the bi-Helmholtz averaging kernel-based variationally nonlocal strain and stress gradient (NStrainG and NStressG) integral models to study the size-dependent bending of functionally graded (FG) beams. The integral constitutive relation is transformed into a higher-order differential form while four non-standard constitutive boundary conditions are derived in detail. The Laplace transform technique is utilized to derive closed-form solutions. Interestingly, we find that, unlike the Helmholtz kernel case, the local stress contribution must be included in the bi-Helmholtz kernel-based NSrainG model to avoid the conflict between equilibrium and constitutive requirements. In the numerical simulations, a series of examples are presented to study the size-effects on the bending deformation of the beams with various boundary types. Numerical results show that the bi-Helmholtz kernel-based nonlocal gradient theories are capable of describing both softening and stiffening size-effects in small structures by suitably tuning the nonlocal and gradient relevant parameters involved in nonlocal gradient models. Moreover, the effect of FG parameter is also presented.

Stress-driven local/nonlocal mixture model for buckling and free vibration of FG sandwich Timoshenko beams resting on a nonlocal elastic foundation

We study the buckling and free vibration of functionally graded (FG) sandwich Timoshenko beams resting on an elastic foundation. In contrast to the majority of the literature on this subject, the behaviors of both the beam and elastic foundation are considered as nonlocal by applying the stress-driven strategy equipped with a bi-Helmholtz kernel. We find that in the presence of a nonlocal elastic foundation, the local/nonlocal mixture should be adopted in order to obtain a well-posed formulation of the problem at hand. The equations of motion and the standard boundary conditions are obtained by invoking Hamilton's principle. Each integro-differential constitutive law is transformed into an equivalently differential form equipped with four non-standard constitutive boundary conditions. The generalized differential quadrature method (GDQM) is then utilized to solve the corresponding eigenvalue problems numerically. Several comparative studies are conducted to validate the effectiveness of our solution. The numerical simulation results present the size-effect on the critical buckling loads and natural frequency of the beams with various boundary types, providing a new benchmark for further study of modeling small-scale beam structures using the bi-Helmholtz kernel-based stress-driven mixture model. Moreover, the influence of considering the size-dependency of the elastic foundation is also investigated.

Bootstrapping boundary-localized interactions II. Minimal models at the boundary

We provide evidence for the existence of non-trivial unitary conformal boundary conditions for a three-dimensional free scalar field, which can be obtained via a coupling to the m’th unitary diagonal minimal model. For large m we can demonstrate the existence of the fixed point perturbatively, and for smaller values we use the numerical conformal bootstrap to obtain a sharp kink that smoothly matches onto the perturbative predictions. The wider numerical analysis also yields universal bounds for the spectrum of any other boundary condition for the free scalar field. A second kink in these bounds hints at a second class of non-standard boundary conditions, as yet unidentified.

Unsteady non-Newtonian fluid flow with heat transfer and Tresca\u2019s friction boundary conditions

We consider an unsteady non-isothermal flow problem for a general class of non-Newtonian fluids. More precisely the stress tensor follows a power law of parameter p, namely sigma = 2 mu ( theta , upsilon , | D(upsilon ) |) |D( upsilon ) |^{p-2} D(upsilon ) - pi mathrm{Id} where θ is the temperature, π is the pressure, υ is the velocity, and D(upsilon ) is the strain rate tensor of the fluid. The problem is then described by a non-stationary p-Laplacian Stokes system coupled to an L^{1}-parabolic equation describing thermal effects in the fluid. We also assume that the velocity field satisfies non-standard threshold slip-adhesion boundary conditions reminiscent of Tresca’s friction law for solids. First, we consider an approximate problem (P_{delta }), where the L^{1} coupling term in the heat equation is replaced by a bounded one depending on a small parameter 0 < delta ll 1, and we establish the existence of a solution to (P_{delta }) by using a fixed point technique. Then we prove the convergence of the approximate solutions to a solution to our original fluid flow/heat transfer problem as δ tends to zero.

On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

Analysis of an electroless plating problem

Abstract Electroless plating in microfluidic channels is a novel technology at the micrometer scale. As the microchannel depth varies with the flow of the chemicals, care must be taken for the channel not to run dry. Owing to the deposited chemical species the physical domain of the flow changes with time, leading to a free boundary problem. As the motion of the free boundary is small it is modelled by a transpiration approximation. With this simplification, the mathematical model consists of a Navier–Stokes flow and an equation for the concentration of the plating chemical coupled by nonstandard and nonlinear boundary conditions. Existence and uniqueness are proved for the concentration equation, assuming that the flow is given. Numerical analysis is carried out and justifies the proposed numerical schemes and the nonlinear algorithms. The numerical study is performed, in the two-dimensional case, with the finite element method and an implicit Euler time scheme for the coupled problem with the Navier–Stokes flow and the nonlinear concentration equation.

On the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams based on the higher-order nonlocal gradient elasticity

Dispersion characteristics of flexural waves in a functionally graded (FG) porous nanobeam are analytically examined. Kinematics of the nano-sized beam is assumed to be consistent with the Timoshenko-Ehrenfest beam model. Material behavior of the FG porous beam is considered to vary symmetrically along the beam thickness while appropriate symmetric porosity models are applied to account for two diverse distributions of porosity with variable volume of voids. For the first time, size-dependent response of the symmetric FG porous nanobeam is realized within the framework of the higher-order nonlocal gradient elasticity theory. The integro-differential constitutive laws of the stress resultant fields are established and reinstated with the equivalent differential relations equipped with non-standard boundary conditions. The closed-form solution of the phase velocity of flexural waves is analytically determined. The ensuing numerical results of the flexural wave dispersion detect new benchmarks for numerical analysis and can be effectively exploited in design and optimization of composite nano-structural elements of advanced nano-electro-mechanical systems.

Analytical Solutions of Model Problems for Large-Deformation Micromorphic Approach to Gradient Plasticity

The objective of this work is to present analytical solutions for several 2D model problems to demonstrate the unique plastic fields generated by the implementation of micromorphic approach for gradient plasticity. The approach is presented for finite deformations and several macroscopic and nonstandard microscopic boundary conditions are applied to a gliding plate to illustrate the capability to predict the size effects and inhomogeneous plastic fields promoted by the gradient terms. The constitutive behavior of the material undergoing plastic deformation is analyzed for softening, hardening and perfect plastic response and corresponding solutions are provided. The analytical solutions are also shown to match with the numerical results obtained by implementing a user element subroutine (UEL) to the commercial finite element software Abaqus/Standard.

Dynamics of nonlocal strain gradient nanobeams with longitudinal magnetic field

Flexural dynamics of carbon nanotubes under the longitudinal magnetic field have been investigated in the paper. Carbon nanotube has been assumed as a hollow beam with circular cross section. Nonlocal strain gradient model has been employed in the modeling of nanobeam. Transverse component of Lorentz force, which occurs with longitudinal magnetic field, has been considered as an external force on nanobeam. Stress and strain energy functionals have been defined for the nonlocal strain gradient model using Hamilton principle. Higher‐order governing differential equation of motion for the present problem has been solved with Differential Quadrature Method. Lorentz force and nonlocal strain gradient parameters have been considered in the standard and nonstandard boundary conditions in the present model formulation. Length scale and magnetic field parameters effect on the flexural vibration response of nanobeam have been investigated. Mode shapes of nonlocal strain gradient nanobeam have been depicted on various cases. Softening nonlocal strain gradient model gives physically consistent results and should be used in the analysis. Magnetic field effect shifts the mode shapes of clamped‐free nanobeam. Present study could give useful results for designing of magnetically actuated nanomotors.

Elastostatics of Bernoulli-Euler Beams Resting on Displacement-Driven Nonlocal Foundation.

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

Non-standard boundary conditions for the linearized Korteweg-de Vries equation

<p style='text-indent:20px;'>This paper aims to solve numerically the linearized Korteweg-de Vries equation. We begin by deriving suitable boundary conditions then approximate them using finite difference method. The methodology of derivation, used in this paper, yields to Non-Standard Boundary Conditions (NSBC) that perfectly absorb wave reflections at the boundary. In addition, these NSBC are exact and local in time and space for non necessarily supported initial data and source terms. We finish with numerical examples that show the absorbing quality of these boundary conditions. Further comparisons are made using standard boundary conditions like, Dirichlet, Neumann and a variant of absorbing boundary conditions called discrete artificial ones.

Dynamics of nonlocal thick nano-bars

The thick bar model, accounting for the lateral deformation, shear stiffness, and lateral inertia effect, is the most comprehensive structural theory to study the axial deformation of carbon nanotubes. Physically motivated definition of the axial force field and associated higher order boundary conditions are determined applying a consistent variational framework. The effects of long-range interactions are suitably realized in the framework of the nonlocal integral elasticity. The integral convolutions of the nonlocal constitutive law are determined and suitably resorted with the equivalent nonlocal differential model equipped with non-standard boundary conditions. Preceding contributions on the elastodynamic analysis of the elastic thick bar are, therefore, amended by properly taking into account the higher order and non-standard boundary conditions. The established size-dependent thick bar model is demonstrated to be exempt from the inherent drawbacks of the nonlocal differential formulation and leads to well-posed elastodynamic problems. The wave desperation response and free vibrational behavior of elastic thick bars with kinematic constraints of nano-mechanics interest are rigorously investigated by making recourse to a viable solution approach. New numerical benchmarks are detected for the elastodynamic response of nonlocal thick nano-bars. A consistent approach for nanoscopic study of the field quantities in the nonlocal mechanics is proposed that is capable of properly confirming the smaller-is-softer phenomenon.

Imposing nonlocal boundary conditions in Galerkin-type methods based on non-interpolatory functions

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.

On Solvability of Boundary Value Problems for Elastic Micropolar Shells with Rigid Inclusions

In the framework of the linear theory of micropolar shells, existence and uniqueness theorems for weak solutions of boundary value problems describing small deformations of elastic micropolar shells connected to a system of absolutely rigid bodies are proved. The definition of a weak solution is based on the principle of virial movements. A feature of this problem is non-standard boundary conditions at the interface between the shell and solids.

Trefftz collocation method for two‐dimensional strain gradient elasticity

AbstractIndirect Trefftz method is proposed for solving two‐dimensional boundary value problems of the strain gradient elasticity theory (SGET). A system of trial functions satisfying the fourth‐order equilibrium equations of SGET are developed based on the generalized Papkovich‐Neuber potentials. The classical part of the displacement solution is represented through the T‐complete system of functions satisfying the Laplace equation. The gradient part of the solution is represented through the system of heuristic functions satisfying the Helmholtz equation. The least squares collocation method is used to enforce the boundary conditions. Numerical examples are presented for the square domain under non‐uniform tensile and bending loads. It is shown, that the advantage of the presented method is that it allows to directly control the accuracy of the fulfillment of all nonstandard boundary conditions, that are prescribed in SGET on the surfaces and edges of the body.