Search results for: mixed boundary conditions

Authors: Alexandra Leukauf, Alexander Schirrer, Emir Talic

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Numerical computation of wave propagation in a large domain usually requires significant computational effort. Hence, the considered domain must be truncated to a smaller domain of interest. In addition, special boundary conditions, which absorb the outward travelling waves, need to be implemented in order to describe the system domains correctly. In this work, the linear one dimensional wave equation is approximated by utilizing the Fourier Galerkin approach. Furthermore, the artificial boundaries are realized with absorbing boundary conditions. Within this work, a systematic work flow for setting up the wave problem, including the absorbing boundary conditions, is proposed. As a result, a convenient modal system description with an effective absorbing boundary formulation is established. Moreover, the truncated model shows high accuracy compared to the global domain.

Keywords: absorbing boundary conditions, boundary control, Fourier Galerkin approach, modal approach, wave equation

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Authors: M. Y. Malika, Farzana, Abdul Rehman

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The study deals with the steady, 3D MHD boundary layer flow of a non-Newtonian Maxwell fluid flow due to a horizontal surface stretched exponentially in two lateral directions. The temperature at the boundary is assumed to be distributed exponentially and possesses convective boundary conditions. The governing nonlinear system of partial differential equations along with associated boundary conditions is simplified using a suitable transformation and the obtained set of ordinary differential equations is solved through numerical techniques. The effects of important involved parameters associated with fluid flow and heat flux are shown through graphs.

Keywords: boundary layer flow, exponentially stretched surface, Maxwell fluid, numerical solution

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Authors: Zhenjie Liu

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This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method. The theorems obtained are very general. For the different time scale, the problem may be the corresponding continuous or discrete boundary value problem.

Keywords: mixed monotone operator, boundary value problem, time scale, green's function, positive solution, singularity

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Authors: K. N. S. Kasi Viswanadham

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Coupled system of second order linear and nonlinear boundary value problems occur in various fields of Science and Engineering. In the formulation of the problem, any one of 81 possible types of boundary conditions may occur. These 81 possible boundary conditions are written as a combination of four boundary conditions. To solve a coupled system of boundary value problem with these converted boundary conditions, a collocation method with cubic B-splines as basis functions has been developed. In the collocation method, the mesh points of the space variable domain have been selected as the collocation points. The basis functions have been redefined into a new set of basis functions which in number match with the number of mesh points in the space variable domain. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Several linear and nonlinear boundary value problems are presented to test the efficiency of the proposed method and found that numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

Keywords: collocation method, coupled system, cubic b-splines, mesh points

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Authors: Suleiman Bashir Adamu, Lawan Sani Taura

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We study the role of boundary conditions via -symmetric quantum mechanics, where denotes parity operator and denotes time reversal operator. Using the one-dimensional Schrödinger Hamiltonian for a free particle in an infinite square well, we introduce symmetric boundary conditions. We find solutions of the 1D Klein-Gordon equation for a free particle in an infinite square well with Hermitian boundary and symmetry boundary conditions, where in both cases the energy eigenvalues and eigenfunction, respectively, are obtained.

Keywords: Eigenvalues, Eigenfunction, Hamiltonian, Klein- Gordon equation, PT-symmetric quantum mechanics

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Authors: Manana Chumburidze, David Lekveishvili

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We have considered the harmonic oscillations and general dynamic (pseudo oscillations) systems of theory generalized Green-Lindsay of couple-stress thermo-elasticity for isotropic, homogeneous elastic media. Approximate solution of some mixed boundary value problems for finite domain, bounded by the some closed surface are constructed.

Keywords: the couple-stress thermoelasticity, boundary value problems, dynamic problems, approximate solution

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Authors: Anuar Ishak

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The mixed convection stagnation point flow toward a vertical plate is investigated. The external flow impinges normal to the heated plate and the surface temperature is assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the mixed convection parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.

Keywords: dual solutions, heat transfer, mixed convection, stability analysis

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Authors: Arturo Ayala-Hernandez, Humberto Hijar

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We propose a new alternative method for imposing fluid-solid boundary conditions in simulations of Multiparticle Collision Dynamics. Our method is based on the introduction of an explicit potential force acting between the fluid particles and a surface representing a solid boundary. We show that our method can be used in simulations of plane Poiseuille flows. Important quantities characterizing the flow and the fluid-solid interaction like the slip coefficient at the solid boundary and the effective viscosity of the fluid, are measured in terms of the set of independent parameters defining the numerical implementation. We find that our method can be used to simulate the correct hydrodynamic flow within a wide range of values of these parameters.

Keywords: Multiparticle Collision Dynamics, fluid-solid, boundary conditions, molecular dynamics

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Authors: M. Eskandarighadi, C. R. McGann

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It is observed from past experiences of earthquakes that local site conditions can significantly affect the strong ground motion characteristicssuch as frequency content, amplitude, and duration of seismic waves. The most common method for investigating site response is one-dimensional seismic site response analysis. The infinite horizontal length of the model and the homogeneous characteristic of the soil are crucial assumptions of this method. One boundary condition that can be used in the sides is tying the sides horizontally for vertical 1D wave propagation. However, 1D analysis cannot account for the 2D nature of wave propagation in the condition where the soil profile is not fully horizontal or has heterogeneity within layers. Therefore, 2D seismic site response analysis can be used to take all of these limitations into account for a better understanding of local site conditions. Different types of boundary conditions can be appliedin 2D site response models, such as tied boundary condition, massive columns, and free-field boundary condition. The tied boundary condition has been used in 1D analysis, which is useful for 1D wave propagation. Employing two massive columns at the sides is another approach for capturing the 2D nature of wave propagation. Free-field boundary condition can simulate the free-field motion that would exist far from the domain of interest. The goal for free-field boundary condition is to minimize the unwanted reflection from sides. This research focuses on the comparison between these methods with examples and discusses the details and limitations of each of these boundary conditions.

Keywords: boundary condition, free-field, massive columns, opensees, site response analysis, wave propagation

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Authors: Judith Stein, Elfriede Friedmann

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The injection of a drug into the vitreous body for the treatment of retinal diseases like wet aged-related macular degeneration (AMD) is the most common medical intervention worldwide. We develop mathematical models for drug transport in the vitreous body of a human eye to analyse the impact of different rheological models of the vitreous on drug distribution. In addition to the convection diffusion equation characterizing the drug spreading, we use porous media modeling for the healthy vitreous with a dense collagen network and include the steady permeating flow of the aqueous humor described by Darcy's law driven by a pressure drop. Additionally, the vitreous body in a healthy human eye behaves like a viscoelastic gel through the collagen fibers suspended in the network of hyaluronic acid and acts as a drug depot for the treatment of retinal diseases. In a completely liquefied vitreous, we couple the drug diffusion with the classical Navier-Stokes flow equations. We prove the global existence and uniqueness of the weak solution of the developed initial-boundary value problem describing the drug distribution in the healthy vitreous considering the permeating aqueous humor flow in the realistic three-dimensional setting. In particular, for the drug diffusion equation, results from the literature are extended from homogeneous Dirichlet boundary conditions to our mixed boundary conditions that describe the eye with the Galerkin's method using Cauchy-Schwarz inequality and trace theorem. Because there is only a small effective drug concentration range and higher concentrations may be toxic, the ability to model the drug transport could improve the therapy by considering patient individual differences and give a better understanding of the physiological and pathological processes in the vitreous.

Keywords: coupled PDE systems, drug diffusion, mixed boundary conditions, vitreous body

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Authors: M. Jamil Amir, Rabia Iqbal, M. Yaseen

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In this paper, we solve some differential equations analytically by using differential transform method. For this purpose, we consider four models of Laplace equation with two Dirichlet and two Neumann boundary conditions and K(2,2) equation and obtain the corresponding exact solutions. The obtained results show the simplicity of the method and massive reduction in calculations when one compares it with other iterative methods, available in literature. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

Keywords: differential transform method, laplace equation, Dirichlet boundary conditions, Neumann boundary conditions

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Authors: Khaled Moaddy

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In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: standard finite difference schemes, non-standard schemes, Laplace equation, Dirichlet boundary conditions

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Authors: Rezwana Rahman

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Various simulations have been conducted to understand the particle's macroscopic behavior in the solid-gas multiphase flow in rotating drums in the past. In these studies, the particle-wall no-slip boundary condition was usually adopted. However, the non-slip boundary condition is rarely encountered in real systems. A little effort has been made to investigate the particle behavior at slip boundary conditions. The paper represents a study of the gas-solid flow in a horizontal rotating drum at a slip boundary wall condition. Two different sizes of particles with the same density have been considered. The Eulerian–Eulerian multiphase model with the kinetic theory of granular flow was used in the simulations. The granular pressure at the rolling flow regime with specularity coefficient 1 was examined and compared with that obtained based on the no-slip boundary condition. The results reveal that the profiles of granular pressure distribution on the transverse plane of the drum are similar for both boundary conditions. But, overall, compared with those for the no-slip boundary condition, the values of granular pressure for specularity coefficient 1 are larger for the larger particle and smaller for the smaller particle.

Keywords: boundary condition, eulerian–eulerian, multiphase, specularity coefficient, transverse plane

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Authors: Pradeepa Teegala, Ramreddy Chitteti

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This article analyzes the mixed convection flow of chemically reacting micropolar fluid over a truncated cone embedded in non-Darcy porous medium with convective boundary condition. In addition, heat generation/absorption and Joule heating effects are taken into consideration. The similarity solution does not exist for this complex fluid flow problem, and hence non-similarity transformations are used to convert the governing fluid flow equations along with related boundary conditions into a set of nondimensional partial differential equations. Many authors have been applied the spectral quasi-linearization method to solve the ordinary differential equations, but here the resulting nonlinear partial differential equations are solved for non-similarity solution by using a recently developed method called the spectral quasi-linearization method (SQLM). Comparison with previously published work on special cases of the problem is performed and found to be in excellent agreement. The effect of pertinent parameters namely, Biot number, mixed convection parameter, heat generation/absorption, Joule heating, Forchheimer number, chemical reaction, micropolar and magnetic field on physical quantities of the flow are displayed through graphs and the salient features are explored in detail. Further, the results are analyzed by comparing with two special cases, namely, vertical plate and full cone wherever possible.

Keywords: chemical reaction, convective boundary condition, joule heating, micropolar fluid, mixed convection, spectral quasi-linearization method

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Authors: M. S. Ahmed, F. A. Mohammad

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This paper focuses on the dynamic behavior of reinforced concrete (RC) slabs. Therefore, the theoretical modal analysis was performed using two different types of boundary conditions. Modal analysis method is the most important dynamic analyses. The analysis would be modal case when there is no external force on the structure. By using this method in this paper, the effects of freely and simply supported boundary conditions on the frequencies and mode shapes of RC square slabs are studied. ANSYS software was employed to derive the finite element model to determine the natural frequencies and mode shapes of the slabs. Then, the obtained results through numerical analysis (finite element analysis) would be compared with an exact solution. The main goal of the research study is to predict how the boundary conditions change the behavior of the slab structures prior to performing experimental modal analysis. Based on the results, it is concluded that simply support boundary condition has obvious influence to increase the natural frequencies and change the shape of mode when it is compared with freely supported boundary condition of slabs. This means that such support conditions have direct influence on the dynamic behavior of the slabs. Thus, it is suggested to use free-free boundary condition in experimental modal analysis to precisely reflect the properties of the structure. By using free-free boundary conditions, the influence of poorly defined supports is interrupted.

Keywords: natural frequencies, mode shapes, modal analysis, ANSYS software, RC slabs

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Authors: Yongxu Fu, Shaolong Wan

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We study the band degeneracy, defectiveness, as well as exceptional points of non-Hermitian systems and materials analytically. We elaborate on the energy bands, the band degeneracy, and the defectiveness of eigenstates under open boundary conditions based on developing a general theory of one-dimensional (1D) non-Hermitian systems. We research the presence of the exceptional points in a generalized non-Hermitian Su-Schrieffer-Heeger model under open boundary conditions. Beyond our general theory, there exist infernal points in 1D non-Hermitian systems, where the energy spectra under open boundary conditions converge on some discrete energy values. We study two 1D non-Hermitian models with the existence of infernal points. We generalize the infernal points to the infernal knots in four-dimensional non-Hermitian systems.

Keywords: non-hermitian, degeneracy, defectiveness, exceptional points, infernal points

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Authors: Cagri Mollamahmutoglu, Aykut Levent, Ali Mercan

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Micro-beams are one of the most common components of Nano-Electromechanical Systems (NEMS) and Micro Electromechanical Systems (MEMS). For this reason, static bending, buckling, and free vibration analysis of micro-beams have been the subject of many studies. In addition, micro-beams restrained with elastic type foundations have been of particular interest. In the analysis of microstructures, closed-form solutions are proposed when available, but most of the time solutions are based on numerical methods due to the complex nature of the resulting differential equations. Thus, a robust and efficient solution method has great importance. In this study, a mixed finite element formulation is obtained for a functionally graded Timoshenko micro-beam resting on two-parameter elastic foundation. In the formulation modified couple stress theory is utilized for the micro-scale effects. The equation of motion and boundary conditions are derived according to Hamilton’s principle. A functional, derived through a scientific procedure based on Gateaux Differential, is proposed for the bending and buckling analysis which is equivalent to the governing equations and boundary conditions. Most important advantage of the formulation is that the mixed finite element formulation allows usage of C₀ type continuous shape functions. Thus shear-locking is avoided in a built-in manner. Also, element matrices are sparsely populated and can be easily calculated with closed-form integration. In this framework results concerning the effects of micro-scale length parameter, power-law parameter, aspect ratio and coefficients of partially or fully continuous elastic foundation over the static bending, buckling, and free vibration response of FG-micro-beam under various boundary conditions are presented and compared with existing literature. Performance characteristics of the presented formulation were evaluated concerning other numerical methods such as generalized differential quadrature method (GDQM). It is found that with less computational burden similar convergence characteristics were obtained. Moreover, formulation also includes a direct calculation of the micro-scale related contributions to the structural response as well.

Keywords: micro-beam, functionally graded materials, two-paramater elastic foundation, mixed finite element method

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Authors: Benedict Barnes, Anthony Y. Aidoo

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A Divergence Regularization Method (DRM) is used to regularize the ill-posed Helmholtz equation where the boundary deflection is inhomogeneous in a Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes the inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation. This ensures the existence, as well as, uniqueness of solution for the equation. The DRM restores all the three conditions of well-posedness in the sense of Hadamard.

Keywords: divergence regularization method, Helmholtz equation, ill-posed inhomogeneous Cauchy boundary conditions

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Authors: M. Ashaque Meah, Md. Fazlul Karim, M. Shah Noor, Nazmun Nahar Papri, M. Khalid Hossen, M. Ismoen

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Tsunami and inundation modelling due to far field tsunami propagation in a limited area is a very challenging numerical task because it involves many aspects such as the formation of various types of waves and the irregularities of coastal boundaries. To compute the effect of far field tsunami and extent of inland inundation due to far field tsunami along the coastal belts of west coast of Malaysia and Southern Thailand, a formulated boundary condition and a moving boundary condition are simultaneously used. In this study, a boundary fitted curvilinear grid system is used in order to incorporate the coastal and island boundaries accurately as the boundaries of the model domain are curvilinear in nature and the bending is high. The tsunami response of the event 26 December 2004 along the west open boundary of the model domain is computed to simulate the effect of far field tsunami. Based on the data of the tsunami source at the west open boundary of the model domain, a boundary condition is formulated and applied to simulate the tsunami response along the coastal and island boundaries. During the simulation process, a moving boundary condition is initiated instead of fixed vertical seaside wall. The extent of inland inundation and tsunami propagation pattern are computed. Some comparisons are carried out to test the validation of the simultaneous use of the two boundary conditions. All simulations show excellent agreement with the data of observation.

Keywords: open boundary condition, moving boundary condition, boundary-fitted curvilinear grids, far-field tsunami, shallow water equations, tsunami source, Indonesian tsunami of 2004

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Authors: M. Eskandarighadi, C. R. McGann

Abstract:

It is observed from past experiences of earthquakes that local site conditions can significantly affect the strong ground motion characteristics experience at the site. One-dimensional seismic site response analysis is the most common approach for investigating site response. This approach assumes that soil is homogeneous and infinitely extended in the horizontal direction. Therefore, tying side boundaries together is one way to model this behavior, as the wave passage is assumed to be only vertical. However, 1D analysis cannot capture the 2D nature of wave propagation, soil heterogeneity, and 2D soil profile with features such as inclined layer boundaries. In contrast, 2D seismic site response modeling can consider all of the mentioned factors to better understand local site effects on strong ground motions. 2D wave propagation and considering that the soil profile on the two sides of the model may not be identical clarifies the importance of a boundary condition on each side that can minimize the unwanted reflections from the edges of the model and input appropriate loading conditions. Ideally, the model size should be sufficiently large to minimize the wave reflection, however, due to computational limitations, increasing the model size is impractical in some cases. Another approach is to employ free-field boundary conditions that take into account the free-field motion that would exist far from the model domain and apply this to the sides of the model. This research focuses on implementing free-field boundary conditions in OpenSees for 2D site response analysisComparisons are made between 1D models and 2D models with various boundary conditions, and details and limitations of the developed free-field boundary modeling approach are discussed.

Keywords: boundary condition, free-field, opensees, site response analysis, wave propagation

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Authors: Ajay R, Rammohan B, Sridhar K S S, Gurusharan N

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The objective of the work is to develop a numerical method to solve the inverse mode shape problem by determining the cross-sectional area of a structure for the desired mode shape via the vibration response study of the humerus bone, which is in the form of a cantilever beam with anisotropic material properties. The humerus bone is the long bone in the arm that connects the shoulder to the elbow. The mode shape is assumed to be a higher-order polynomial satisfying a prescribed set of boundary conditions to converge the numerical algorithm. The natural frequency and the mode shapes are calculated for different boundary conditions to find the cross-sectional area of humerus bone from Eigenmode shape with the aid of the inverse mode shape algorithm. The cross-sectional area of humerus bone validates the mode shapes of specific boundary conditions. The numerical method to solve the inverse mode shape problem is validated in the biomedical application by finding the cross-sectional area of a humerus bone in the human arm.

Keywords: Cross-sectional area, Humerus bone, Inverse mode shape problem, Mode shape

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Authors: Shuang Luo, Er-Xiang Song

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Many problems in geotechnical engineering, such as foundation deformation, groundwater seepage, seismic wave propagation and geothermal transfer problems, may involve analysis in the ground which can be seen as extending to infinity. To that end, consideration has to be given regarding how to deal with the unbounded domain to be analyzed by using numerical methods, such as finite element method (FEM), finite difference method (FDM) or finite volume method (FVM). A simple artificial boundary approach derived from the analytical solutions for transient radial seepage problems, is introduced. It should be noted, however, that the analytical solutions used to derive the artificial boundary are particular solutions under certain boundary conditions, such as constant hydraulic head at the origin or constant pumping rate of the well. When dealing with unbounded domains with unsteady boundary conditions, a more sophisticated artificial boundary approach to deal with the infinity of the domain is presented. By applying Laplace transforms and introducing some specially defined auxiliary variables, the global artificial boundary conditions (ABCs) are simplified to local ones so that the computational efficiency is enhanced significantly. The introduced two local ABCs are implemented in a finite element computer program so that various seepage problems can be calculated. The two approaches are first verified by the computation of a one-dimensional radial flow problem, and then tentatively applied to more general two-dimensional cylindrical problems and plane problems. Numerical calculations show that the local ABCs can not only give good results for one-dimensional axisymmetric transient flow, but also applicable for more general problems, such as axisymmetric two-dimensional cylindrical problems, and even more general planar two-dimensional flow problems for well doublet and well groups. An important advantage of the latter local boundary is its applicability for seepage under rapidly changing unsteady boundary conditions, and even the computational results on the truncated boundary are usually quite satisfactory. In this aspect, it is superior over the former local boundary. Simulation of relatively long operational time demonstrates to certain extents the numerical stability of the local boundary. The solutions of the two local ABCs are compared with each other and with those obtained by using large element mesh, which proves the satisfactory performance and obvious superiority over the large mesh model.

Keywords: transient seepage, unbounded domain, artificial boundary condition, numerical simulation

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Authors: Tayeb Bensattalah, Mohamed Zidour, Mohamed Ait Amar Meziane, Tahar Hassaine Daouadji, Abdelouahed Tounsi

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In this paper, the Differential Transform Method (DTM) is employed to predict and to analysis the non-local critical buckling loads of carbon nanotubes with various end conditions and the non-local Timoshenko beam described by single differential equation. The equation differential of buckling of the nanobeams is derived via a non-local theory and the solution for non-local critical buckling loads is finding by the DTM. The DTM is introduced briefly. It can easily be applied to linear or nonlinear problems and it reduces the size of computational work. Influence of boundary conditions, the chirality of carbon nanotube and aspect ratio on non-local critical buckling loads are studied and discussed. Effects of nonlocal parameter, ratios L/d, the chirality of single-walled carbon nanotube, as well as the boundary conditions on buckling of CNT are investigated.

Keywords: boundary conditions, buckling, non-local, differential transform method

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Authors: Sadia Siddiqa, Z. Khan, M. A. Hossain

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In this article, the effect of thermal radiation on mixed convection boundary layer flow of a viscous fluid along a highly heated vertical flat plate is considered with varying density and volumetric expansion coefficient. The density of the fluid is assumed to vary exponentially with temperature, however; volumetric expansion coefficient depends linearly on temperature. Boundary layer equations are transformed into convenient form by introducing primitive variable formulations. Solutions of transformed system of equations are obtained numerically through implicit finite difference method along with Gaussian elimination technique. Results are discussed in view of various parameters, like thermal radiation parameter, volumetric expansion parameter and density variation parameter on the wall shear stress and heat transfer rate. It is concluded from the present investigation that increase in volumetric expansion parameter decreases wall shear stress and enhances heat transfer rate.

Keywords: thermal radiation, mixed convection, variable density, variable volumetric expansion coefficient

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Authors: Chinh Phuong Do

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Dynamic analysis of slope under seismic condition requires the elimination of spurious reflection at the bounded domain. This paper studies the performances of wave transmitting boundaries, including the standard viscous boundary and the viscoelastic boundary to the material point method (MPM) framework. First, analytical derivations of these non-reflecting conditions particularly to the implicit MPM are presented. Then, a number of benchmark and geotechnical examples will be shown. Overall, the results agree well with analytical solutions, indicating the ability to accurately simulate the radiation at the bounded domain.

Keywords: dynamic analysis, implicit, MPM, non-reflecting boundary

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Authors: Mohamed Houas, Maamar Benbachir

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In this paper, under weak assumptions, we study the existence and uniqueness of solutions for a nonlinear fractional boundary value problem. New existence and uniqueness results are established using Banach contraction principle. Other existence results are obtained using scheafer and krasnoselskii's fixed point theorem. At the end, some illustrative examples are presented.

Keywords: caputo derivative, boundary value problem, fixed point theorem, local conditions

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Authors: Theddeus T. Akano, Omotayo A. Fakinlede

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The modelling of physical phenomena, such as the earth’s free oscillations, the vibration of strings, the interaction of atomic particles, or the steady state flow in a bar give rise to Sturm-Liouville (SL) eigenvalue problems. The boundary applications of some systems like the convection-diffusion equation, electromagnetic and heat transfer problems requires the combination of Dirichlet and Neumann boundary conditions. Hence, the incorporation of Robin boundary condition in the analyses of Sturm-Liouville problem. This paper deals with the computation of the eigenvalues and eigenfunction of generalized Sturm-Liouville problems with Robin boundary condition using the finite element method. Numerical solutions of classical Sturm–Liouville problems are presented. The results show an agreement with the exact solution. High results precision is achieved with higher number of elements.

Keywords: Sturm-Liouville problem, Robin boundary condition, finite element method, eigenvalue problems

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Authors: Korosh Khorshidi, Mohammad Khodadadi

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In this paper, exact close form solution for out of plate free flexural vibration of moderately thick rectangular nanoplates are presented based on nonlocal modified trigonometric shear deformation theory, with assumptions of the Levy's type boundary conditions, for the first time. The aim of this study is to evaluate the effect of small-scale parameters on the frequency parameters of the moderately thick rectangular nano-plates. To describe the effects of small-scale parameters on vibrations of rectangular nanoplates, the Eringen theory is used. The Levy's type boundary conditions are combination of six different boundary conditions; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free. Governing equations of motion and boundary conditions of the plate are derived by using the Hamilton’s principle. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Numerical results are presented to illustrate the effectiveness of the proposed method compared to other methods reported in the literature. Finally, the effect of boundary conditions, aspect ratios, small scale parameter and thickness ratios on nondimensional natural frequency parameters and frequency ratios are examined and discussed in detail.

Keywords: exact solution, nonlocal modified sinusoidal shear deformation theory, out of plane vibration, moderately thick rectangular plate

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Authors: S. Zenhari, M. R. Hematiyan, A. Khosravifard, M. R. Feizi

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The boundary element method (BEM) and the method of fundamental solutions (MFS) are well-known fundamental solution-based methods for solving a variety of problems. Both methods are boundary-type techniques and can provide accurate results. In comparison to the finite element method (FEM), which is a domain-type method, the BEM and the MFS need less manual effort to solve a problem. The aim of this study is to compare the accuracy and reliability of the BEM and the MFS. This comparison is made for 2D potential and elasticity problems with different boundary and loading conditions. In the comparisons, both convex and concave domains are considered. Both linear and quadratic elements are employed for boundary element analysis of the examples. The discretization of the problem domain in the BEM, i.e., converting the boundary of the problem into boundary elements, is relatively simple; however, in the MFS, obtaining appropriate locations of collocation and source points needs more attention to obtain reliable solutions. The results obtained from the presented examples show that both methods lead to accurate solutions for convex domains, whereas the BEM is more suitable than the MFS for concave domains.

Keywords: boundary element method, method of fundamental solutions, elasticity, potential problem, convex domain, concave domain

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Authors: Dara Singh, Keikhosrow Firouzbakhsh, Mohammad Taghi Ahmadian

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In this study, a validated 3D finite volume model of human eye is developed to study the fluid flow and heat transfer in the human eye at steady state conditions. For this purpose, discretized bio-heat transfer equation coupled with Boussinesq equation is analyzed with different anatomical, environmental, and physiological conditions. It is demonstrated that the fluid circulation is formed as a result of thermal gradients in various regions of eye. It is also shown that posterior region of the human eye is less affected by the ambient conditions compared to the anterior segment which is sensitive to the ambient conditions and also to the way the gravitational field is defined compared to the geometry of the eye making the circulations and the thermal field complicated in transient states. The effect of variation in material and boundary conditions guides us to the conclusion that thermal field of a healthy and non-healthy eye can be distinguished via computer simulations.

Keywords: bio-heat, boussinesq, conduction, convection, eye

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