Non-standard Boundary Research Articles

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.

The consistency of the nonlocal strain gradient integral model in size-dependent bending analysis of beam structures

Despite the extensive applications of the nonlocal strain gradient theory to predict the size-effect of beam-like structures, the inconsistencies of its differential form have been uncovered recently. For bounded beam structures, there are additional constitutive boundary conditions should be taken into account to ensure the strict equivalence between the differential constitutive relation and the originally integral ones. Such an improvement would lead the problems to have more mandatory boundary conditions than those required, and therefore, there must be a trade-off between these boundary conditions. A strategy by omitting the higher-order (non-standard) boundary conditions has been presented recently and from which consistently size-effects can be obtained. However, it was pointed out that this treatment no longer fully meets the requirements of the energy-variational principle. In this paper, an alternative methodology by directly using the originally integral model is proposed for the first time. The advantages of such a strategy are pertinently evidenced by investigating Euler-Bernoulli curved beams made of functionally graded materials. The Laplace transform technique is applied to solve the integral constitutive equations and differential governing equations, which is an original contribution of this paper. In contrast to the differential forms, the originally integral model can satisfy energy conservation well. Meanwhile, adopting specific higher-order boundary conditions, numerical results show consistently stiffening and softening effects for all boundary edges and loading conditions.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method

In this work, an initial-boundary value problem with a non-classic condition for the one-dimensional wave equation is presented and the reduced differential transform method is applied to ascertain the solution of the problem. We will investigate a new kind of non-local boundary value problems in which are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The advantage of this method is its simplicity in using, it solves the problem directly and straightforward without using perturbation, linearization, Adomian’s polynomial or any other transformation and gives the solution in the form of convergent power series with simply determinable components. Also, the convergence of the method is proved and seven examples are tested to shows the competency of our study.

A multiphase model of growth factor-regulated atherosclerotic cap formation.

Atherosclerosis is characterised by the growth of fatty plaques in the inner artery wall. In mature plaques, vascular smooth muscle cells (SMCs) are recruited from adjacent tissue to deposit a collagenous cap over the fatty plaque core. This cap isolates the thrombogenic plaque content from the bloodstream and prevents the clotting cascade that leads to myocardial infarction or stroke. Despite the protective role of the cap, the mechanisms that regulate cap formation and maintenance are not well understood. It remains unclear why some caps become stable, while others become vulnerable to rupture. We develop a multiphase PDE model with non-standard boundary conditions to investigate collagen cap formation by SMCs in response to diffusible growth factor signals from the endothelium. Platelet-derived growth factor stimulates SMC migration, proliferation and collagen degradation, while transforming growth factor (TGF)-[Formula: see text] stimulates SMC collagen synthesis and inhibits collagen degradation. The model SMCs respond haptotactically to gradients in the collagen phase and have reduced rates of migration and proliferation in dense collagenous tissue. The model, which is parameterised using in vivo and in vitro experimental data, reproduces several observations from plaque growth in mice. Numerical and analytical results demonstrate that a stable cap can be formed by a relatively small SMC population and emphasise the critical role of TGF-[Formula: see text] in effective cap formation. These findings provide unique insight into the mechanisms that may lead to plaque destabilisation and rupture. This work represents an important step towards the development of a comprehensive in silico plaque model.

Non-standard contact conditions in generalized continua: microblock contact model for a Cosserat body

Generalized continuum theories involve non-standard boundary conditions that are associated with the additional kinematic variables introduced in those theories, e.g., higher gradients of the displacement field or additional kinematic degrees of freedom. Accordingly, formulation of a contact problem for such a continuum necessarily requires that adequate contact conditions are formulated for the additional kinematic variables and/or for the respective generalized tractions. In this paper, we address several related open problems, namely, how to enhance the classic contact conditions to include the effects of the additional kinematic variables, how to link the enhanced contact model to the underlying microstructure of the solid, and how to do it in a consistent manner. As a first step towards a new class of contact models for generalized continua, a microblock contact model is derived for a Cosserat solid based on simple micromechanical considerations. To illustrate the non-trivial effects introduced by the non-standard boundary conditions, the problem of compression of an infinite strip with nonaligned microblocks is considered, and the analytical solution is derived for the corresponding boundary layers. A Hertz-like contact problem is also solved numerically with the focus on non-standard features of the solution and on the related size effects.

The spectral implementation of the nonstationary Stokes problem with nonstandard boundary conditions

This paper deals with the implementation of the spectral discretization of the vorticity–velocity–pressure formulation of the nonstationary Stokes problem. We use the implicit Euler scheme for the discretization in time. We propose a global method for the resolution of the linear system resulting from the discrete problem. This method is implemented, and some numerical results are presented which confirm the good convergence for the three unknowns.

Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and H > 0. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold H) it is either absorbed, with probability α, or reflected upwards (downwards), with probability 1 − α, for 0 < α < 1. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through linear boundaries. Our analysis includes also some comparisons between suitable stopping times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac’s scaling conditions.

Spectral discretization of the time‐dependent vorticity–velocity–pressure formulation of the Stokes problem

In this paper, we study the time‐dependent vorticity–velocity–pressure formulation of Stokes problem in two‐ and three‐dimensional domains provided with nonstandard boundary conditions, related to the normal component of the velocity and the tangential components of the vorticity. This problem is discretized by implicit Euler's scheme in time and spectral method in space. We prove an optimal error estimate for the three unknowns.

Non-isothermal energy–momentum time integrations with drilling degrees of freedom of composites with viscoelastic fiber bundles and curvature–twist stiffness

3D fiber-reinforced composites demand for a special simulation technique, because they consist of fiber bundles. Therefore, in the corresponding representative volume element of these metamaterials, secondary effects as a micro inertia and a curvature–twist stiffness have to bear in mind. The latter increases the strength-to-weight ratio of thin-walled lightweight structures due to a separate twisting and bending stiffness of the fiber bundles. In this paper, these secondary effects are introduced in a continuum formulation by means of independent drilling degrees of freedom. The resulting non-isothermal constrained micropolar continuum is derived by a principle of virtual power, which simultaneously generates in the discrete setting a mixed B-bar method and a Galerkin-based energy–momentum scheme of higher order. This work also takes into account viscoelastic material behavior in the fiber bundles, which arises from a mixture of organic and inorganic fibers. Here, the viscous evolution equation is solved elementwise by using a mixed field as viscous internal variable. Representative numerical examples demonstrate the inelastic material behavior, the effect of micro inertia on the physical properties of the continuum as well as on its space and time discretization and, finally, the twisting and bending stiffness of the fiber bundles. Further, non-standard boundary conditions are applied in the dynamic simulations performed by higher order energy–momentum schemes.