Different Types Of Boundary Conditions Research Articles

Nonlinear vibration analysis of functionally graded material tubes with conveying fluid resting on elastic foundation by a new tubular beam model

In this paper, nonlinear free vibration analysis of shear deformable functionally graded material (FGM) tubes conveying fluid with immovable support conditions resting on a Pasternak-type foundation is presented. Based on Hamilton’s principle, the equations of motion and boundary conditions are obtained by using a new high-order shear deformation tubular beam model with von Kármán nonlinearity and the exact expression of curvature, in which contributions of fluid velocity to the kinetic energy and body forces are also considered. The differential quadrature method (DQM) is employed to determine the nonlinear frequencies and amplitude–frequency responses of fluid-conveying FGM tubes with different boundary conditions. A detailed parametric study is conducted to analyze the influences of different types of boundary conditions, geometric and physical properties. The numerical results reveal that the geometrical and physical properties, including elastic foundation, boundary conditions and flow velocity in the fluid-conveying pipes are crucial factors on their dynamical behavior.

Size Effect on the Axisymmetric Vibrational Response of Functionally Graded Circular Nano-Plate Based on the Nonlocal Stress-Driven Method

In this work, the axisymmetric-vibrational behavior of a size-dependent circular nano-plate with functionally graded material with different types of boundary conditions was investigated. The analysis was performed based on the Stress-driven model (SDM) and Strain-gradient theory (SGT) in conjunction with classical plate theory. The governing equations of motion and their corresponding equations for boundary conditions were obtained based on Hamilton’s principle and solved using the generalized differential quadrature rule. Results show that this method is applicable to the vibrational analysis of such structures with a fast convergence rate; as N approaches 6 for the first mode, and 10 for the second as well as the third and fourth modes, regardless of the type of boundary condition. In both models, the influences of various parameters such as size-effect parameter Lc, material heterogeneity index n, and types of boundary conditions were obtained on the first four modes and compared with each other. Results indicate that the natural frequencies in these modes increase with an increase in the heterogeneity index n, and size-effect parameter Lc. Additionally, these parameters appear to have a stiffening effect on the nano-plate vibrational behavior. However, for a nano-plate resting on a knife or simply supported edge, in the first mode, the SDM shows a more stiffening effect on the plate behavior as compared with the SGT. Nonetheless, for the clamped and free edge boundary conditions, both models predicted the same behavior. The SGT showed a higher-stiffening effect only in the fourth mode, for all types of considered boundary conditions.

Advanced nonlinear buckling analysis of a compressed single layer graphene sheet using the molecular mechanics method

The standard molecular mechanics (MM) method with the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909) and the molecular structural mechanics (MSM) method with Bernoulli–Euler beam elements are used to study the quasi-static nonlinear buckling and post-buckling behavior of a compressed nearly square single layer graphene sheet (SLGS) for different types of boundary conditions. The novelty of this study is the finding that well-calibrated parameter sets provide similar values of buckling forces/modes and similar post-buckling deformations for stable equilibrium configurations of SLGSs in simulations using the standard MM and MSM methods. In addition, the effect of accounting for non-bonded van der Waals (vdW) forces on the advanced post-buckling deformation modes of compressed SLGSs was studied for the first time. It is shown that the column-like post-buckling deformation modes of compressed SLGSs obtained without the inclusion of vdW interatomic forces are similar to the well-known ones for the planar Euler elastica, and the advanced post-buckling modes obtained with the inclusion of interatomic vdW forces are qualitatively different from the corresponding modes for the planar Euler elastica. In addition, the study shows the theoretical possibility of the existence of post-buckling out-of-plane equilibrium configurations whose stability is provided by attractive vdW forces.

Fourier transform approach to nonperiodic boundary value problems in porous conductive media

AbstractIn this article, we develop an extension of the Fourier transform solution method in order to solve conduction equation with nonperiodic boundary conditions (BC). The periodic Lippmann–Schwinger equation for porous materials is extended to the case of non‐periodicity with relevant source terms on the boundary. The method is formulated in Fourier space based on the temperature as unknown, using the exact periodic Green function and form factors to describe the boundaries. Different types of BC: flux, temperature, mixed and combined with periodicity can be treated by the method. Numerical simulations show that the method does not encounter convergence issues due to the infinite contrast and yields accurate results for both local fields and effective conductivity.

Analytical solutions for extended Graetz problem in infinite domains via integral transforms

A methodology for obtaining analytical solutions for the extended Graetz problem including the effects of axial conduction, viscous dissipation heating, and additional volumetric heating effects in an infinite domain has been developed. The domain extends infinitely in the directions parallel to the flow, being divided into a preparation upstream region, in which the walls are either thermally insulated or isothermal, and a downstream region, in which the walls are either uniformly heated or isothermal, at temperature that is different than that of the upstream walls; regardless of the selected configuration, there is always a step change in the wall heating condition at the origin — where both regions meet. The solution methodology is based on the Generalized Integral Transform Technique, in which eigenfunction expansions in terms of orthogonal bases are employed. Simple Sturm–Liouville eigenfunction bases in terms of Helmholtz problems are utilized for maintaining the calculation of eigenfunctions-related quantities as simple as possible, thus minimizing the amount of necessary computational work. A thorough error analysis is performed for verification purposes, demonstrating that larger truncation orders are required for cases with smaller Péclet number values, and with wall boundary conditions of different types in each region. It was also shown that positions near the origin or occurring near discontinuities in the Nusselt number, also require more terms in the eigenseries expansion. An interesting convergence phenomenon was observed for cases with the same type of wall conditions in both regions: as the Péclet number is large enough or at positions sufficiently far from the origin, the Nusselt number error decreases following a negative power of the truncation order; also, although smaller Péclet numbers were shown to have a worse convergence rate, they shift to the power-like form after a minimum truncation order is reached. The presented methodology was also verified through comparisons with literature studies. Finally, illustrative results were provided showing that the boundary condition in the upstream preparation region can have a prominent effect on the thermal developing Nusselt behavior, especially for lower Péclet number values.

Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a ‘discretize‐then‐project’ approach

AbstractA novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This ‘discretize‐then‐project’ approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier‐Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence‐free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence‐free in the former method. For both methods, accurate results have been obtained for test cases with different types of boundary conditions: a lid‐driven cavity and an open‐cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence‐free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method.

Analysis of thermal injuries using classical Fourier and DPL models for multi-layer of skin under different boundary conditions

In this paper, the temperature distribution in the multi-layer of the skin is studied when the skin surface is subjected to most generalized boundary condition. Our skin model consists of three layers known as the epidermis, dermis, and subcutaneous layers. All layers of skin are assumed to be connected with point of interface condition and taking the barrier in between each of the two layers by symmetric flux condition and analyzing each layer separately. The classical Fourier and non-Fourier (DPL) models are extended to analyze the behavior of heat transfer in the multi-layer of the skin. The Laplace transform technique is used to derive analytical solutions for the multi-layer of skin models. The effects of the variability of different parameters such as relaxation time, layer thickness, and different types of boundary conditions on the behavior of temperature distribution in the multi-layer of skin are analyzed and discussed in detail. All the effects are shown graphically. It has been observed that during temperature distribution in the multi-layer of skin, the measurement of skin damage is less on the DPL model ([Formula: see text]) in comparison to the classical Fourier model.

Influence of relativistic rotation on the confinement-deconfinement transition in gluodynamics

In this paper we consider the influence of relativistic rotation on the confinement/deconfinement transition in gluodynamics within lattice simulation. We perform the simulation in the reference frame which rotates with the system under investigation, where rotation is reduced to external gravitational field. To study the confinement/deconfinement transition the Polyakov loop and its susceptibility are calculated for various lattice parameters and the values of angular velocities which are characteristic for heavy-ion collision experiments. Different types of boundary conditions (open, periodic, Dirichlet) are imposed in directions, orthogonal to rotation axis. Our data for the critical temperature are well described by a simple quadratic function $T_c(\Omega)/T_c(0) = 1 + C_2 \Omega^2$ with $C_2>0$ for all boundary conditions and all lattice parameters used in the simulations. From this we conclude that the critical temperature of the confinement/deconfinement transition in gluodynamics increases with increasing angular velocity. This conclusion does not depend on the boundary conditions used in our study and we believe that this is universal property of gluodynamics.

FDwave3D: a MATLAB solver for the 3D anisotropic wave equation using the finite-difference method

Seismic modeling plays an important role in geophysics and seismology for estimating the response of seismic sources in a given medium. In this work, we present a MATLAB-based package, FDwave3D, for synthetic wavefield and seismogram modeling in 3D anisotropic media. The seismic simulation is carried out using the finite-difference method over the staggered grid, and it is applicable to both active and passive surveys. The code package allows the incorporation of arbitrary source mechanisms and offers spatial derivative operators of accuracy up to tenth-order along with different types of boundary conditions. First, the methodological aspects of finite-difference method are briefly introduced. Then, the code has been tested and verified against the analytical solutions obtained for the homogeneous model. Further, the numerical examples of layered and overthrust models are presented to demonstrate its reliability.

A Novel Peridynamic Mindlin Plate Formulation Without Limitation on Material Constants

In this study, a new peridynamic Mindlin plate formulation is introduced by utilising Euler-Lagrange equations. The classical strain energy density of a material point is converted to its corresponding peridynamic form by using Taylor’s expansions. The formulation is suitable for thick plates by considering the transverse shear deformation. Material constants do not have any limitation in the current formulation. Different types of boundary conditions are considered in numerical examples including simply supported, clamped and mixed (clamped-supported). To verify the current formulation, peridynamic solutions of the transverse displacements and rotations are compared against solutions obtained from finite element analysis.

Admissible Extremals in the Problems of Variational Calculus with Constraints in the Form of a System of Linear Inhomogeneous Differential Equations of the First Order with Rectangular Matrices

We establish necessary and sufficient conditions for the existence of smooth admissible extremals in the problems of variational calculus with constraints in the form of a system of linear inhomogeneous differential equations of the first order with rectangular matrices and different types of boundary conditions.

The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade

We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain \(\varOmega \). The corresponding Stokes–type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the curves \(\varGamma _{\hspace {-1.1pt} 0}\) and \(\varGamma _{\hspace {-1.1pt} 1}\), the Dirichlet boundary conditions on \(\varGamma _{\hspace {-1.1pt} \mathrm {in}}\) and \(\varGamma _{\hspace {-1.1pt} p}\) and an artificial “do nothing”–type boundary condition on \(\varGamma _{\hspace {-1.1pt} \mathrm {out}}\). (See Fig. 1.) We explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain \(\varOmega \) is not smooth and different types of boundary conditions meet in the corners of \(\varOmega \), the considered problem has a strong solution with the so called maximum regularity property.

Heat transfer from a ferrofluid during generalized Couette flow through parallel plates in the presence of an orthogonal magnetic field

In the present work, we report heat transfer during the unidirectional laminar flow of ferrofluids in a parallel plate channel in the presence of a homogeneously applied magnetic field perpendicular to the flow direction. We have considered generalized Couette flow which results from the combination of shear stress, due to the relative motion between the parallel plates, and applied pressure gradient. For a comprehensive investigation, four different thermal boundary conditions have been considered. The application of magnetic field modifies the viscosity of the colloidal ferrofluid and thereby causes variations in the local flow field. Further, it also modifies the thermal conductivity of the fluid by changing particle orientation. As a consequence, the temperature distribution between the parallel plate channel is uniquely modified. The complex interplay between the hydrodynamics and the magnetics results in implicitly coupled governing differential equations, which have been solved numerically. We report that the local temperature in the domain could be reduced by increasing the applied field strength irrespective of the imposed thermal boundary conditions. However, the influence of the applied field finally gets saturated. We also discuss the effect of Brinkman number, loading of magnetic particles and pressure gradient on temperature distribution. Finally, analytical solutions, which presents closed-form expressions of the temperature profiles for four different types of boundary conditions, have been proposed.

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.

Computational modelling of light propagation in a turbid medium illuminated by a LED array–diaphragm system

Light sources generated by LASER and LED are progressively used in the medical field. For many applications there is the need to use light sources with a well-defined size. This can be achieved when a large LED array is associated to a thin plane mask pierced with a circular hole. In this paper three tissue illumination modes were studied. The one (reference mode) allows to illuminate the tissue with a collimated light beam of finite radius, while the two others generate an illumination that depends on the relative position of the diaphragm and LED array with respect to the tissue boundary. Finite Element Method (FEM) and Monte Carlo (MC) were used to compute the fluence rate (axial and radial profiles) inside the tissue as function of source radius, realistic source shapes, tissue optical properties, and boundary conditions. Different types of boundary conditions (i.e. Robin boundary conditions (RBC), and altered with the consideration of the optical features of the interface tissue-diaphragm: -reflecting,-absorbing,-diffusing,-and partially absorbing-diffusing) were evaluated in order to compare the three above mentioned tissue illumination modes. FEM provided accurate results in cases of flat top source profiles and for simple tissue-diaphragm interfaces (absorbing and reflecting) as well as for tissue-air boundary conditions (RBC). MC simulations showed good agreements with experiments when a LED array–diaphragm system (partially absorbing-diffusing) is positioned near the tissue (2.5 mm), while FEM underestimated the fluence rate by a large error within tissue-depths of 5–10 mean-free-paths. These investigations emphasize the need to accurately model all optical events occurring in complex tissue-illumination modes. Monte Carlo simulations further confirmed that a critical beam radius Rc (Rc≥4δ) should be used to retrieve the light penetration depth δ within the tissue, in the same conditions as those of the 1-D geometry.

An Efficient and Convergent Finite Element Scheme for Cahn--Hilliard Equations with Dynamic Boundary Conditions

The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical treatment of a recent model which introduces an additional Cahn--Hilliard type equation on the boundary as closure for the Cahn--Hilliard equation in the domain [C. Liu, H. Wu, Arch. Ration. Mech. An., 2019]. By identifying a mapping between the phase-field parameter and the chemical potential inside of the domain, we are able to postulate an efficient, unconditionally energy stable finite element scheme. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. This serves also as an additional pathway to establish existence of weak solutions. Furthermore, we present simulations underlining the practicality of the proposed scheme and investigate its experimental order of convergence.

Nonlinear vibration and stability of FG nanotubes conveying fluid via nonlocal strain gradient theory

In this work, a model of a functionally graded (FG) nanotube conveying fluid embedded in an elastic medium is developed based on the nonlocal strain gradient theory (NSGT) in conjunction with Euler-Bernoulli beam theory (EBT). The main objective of this research is to investigate the nonlinear vibration and stability analysis of fluid-conveying nanotubes. The governing equations of motion are derived by means of Hamiltonian principle. The analytical expressions of nonlinear frequencies and critical flow velocities for two different types of boundary conditions including pinned-pinned (P-P) and clamped-clamped (C-C) conditions are obtained by employing Galerkin method as well as Hamiltonian Approach (HA). Comparison of the obtained results with the published works show the acceptable accuracy of the current solutions. The effects of the power-law index, the nonlocal and material length scale parameters and the elastic medium on the stability and nonlinear responses of FG nanotubes are thoroughly investigated and discussed.

Crack Detection in Plate‐Like Structures Using Modal Strain Energy Method considering Various Boundary Conditions

Among vibration‐based damage detection methods, one of the effective approaches for damage localization is the modal strain energy method. In this paper, the modal strain energy method is developed for damage detection in plate‐like structures with various boundary conditions. Firstly, the theory of the modal strain energy method is briefly outlined. In order to overcome the limitation of measuring points, a central difference method is newly employed to compute the partial differential terms in the modal strain energy formula. Finite element analysis is conducted on an aluminum thin plate to obtain the mode shapes before and after the occurrence of damage. Feasibility of the proposed method is verified by investigating plates with different types of boundary conditions. A damage index is presented to identify the location and extent of crack in the plate‐like structures. The analytical results show that the proposed method accurately identifies the crack in the plate structure with various types of boundary conditions by using appropriate mode shapes and damage threshold.

A semi-analytical study on effects of geometric imperfection and curved fiber paths on nonlinear response of compression-loaded laminates

Variable stiffness composite plate can adjust the stiffness properties of the plate in order to satisfy the design requirements of studied problem. In this paper, the effects of different types of boundary conditions, lay-ups, sizes and shapes of geometric initial imperfection on the nonlinear behavior of VAT plates are investigated; these conditions influence the behavior of the plate completely. Moreover, the first-order shear deformation plate theory and Von-Karman assumptions are applied to analyze the effects of fiber lay-up sequences of VAT-laminate under different boundary conditions. Ritz method is also exploited by using Legendre polynomials to approximate the unknown displacement fields of the problem. To yield more accurate results, potential energy integrals are numerically calculated by employing Gauss-Lobatto formulas. Lastly, the system of nonlinear equations is solved by the well-known Newton-Raphson technique. In order to validate the results, finite element analysis is also implemented by the commercial finite element package ABAQUS.

Effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam in HDQ analysis

The aim of this investigation is to study the effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam. The harmonic differential quadrature (HDQ) method is used to compute the natural frequency of Timoshenko beam. In this regards, three different types of meshing are used namely uniform meshing, normalized meshing and non-uniform. The results obtained using these meshing are compared with results obtained in previous published work under different types of boundary conditions. It is found that non-uniform meshing gives better convergence as well as good results as compare to results using uniform and normalized meshing under all boundary conditions.