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Abstract The element free Galerkin (EFG) method, which is based on the moving least square (MLS) approximation, requires only nodal data and no element connectivity. These features make the method more flexible than the conventional FEM. Nevertheless, direct imposition of the essential boundary conditions in the EFG method is always difficult because the shape functions obtained from the MLS approximation do not have the Kronocker-delta property. A new method named "the complementary integral method" is proposed here to overcome this difficulty. The presented method is more consistent with the variational basis of the EFG method. Several numerical examples are used to illustrate the implementation and performance of the method. The numerical examples including the Poisson's equation and 2D static and dynamic elasticity problems show that the method converges fast with reasonably accurate result for both the unknown variables and its derivatives.

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In this paper, the Meshless Local Petrov-Galerkin (MLPG) method is used to analyze the fracture of an isotropic FGM plate. The stress intensity factor of Mode I and Mode II are determined under the influence of various non-homogeneity ratios, crack length and material gradation angle. Both the moving least square (MLS) and the direct method have been applied to estimate the shape function and to impose the essential boundary conditions. The enriched weight function method is used to simulate the displacement and stress field around the crack tip. Normalized stress intensity factors (NDSIF) are calculated using the path independent integral, J*, which is formulated for the non-homogeneous material. The Edge-Cracked FGM plate is considered here and analyzed under the uniform load and uniform fixed grip conditions. To validate results, at first, homogeneous and FGM plate with material gradation along crack length was analyzed and compared with exact solution. Results showed good agreement between MLPG and exact solution.

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Dynamic response of fully-clamped laminated plate subjected to small mass and low-velocity impact studied in this paper by using the suitable Algebraic Polynomials and Galerkin method. The first-order deformation theory as well as the displacement filed is used to solve the governing equations of the composite plate analytically. The interaction between the impactor and the target are considered in the impact analysis. This interaction is modeled with the help of a two degrees-of-freedom system, consisting of springs-masses. The results indicated that some of parameters like mass and velocity of the impactor in a constant impact energy level, mass of the plate (target), increasing the length-to-width ratio of the plate (a/b ratio) and orientation of composite fibers of plate are important factors affecting the impact process and the design of structures.

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A two dimensional generalized plane strain micromechanical model is developed to study electro-elastic behavior of piezoelectric fiber reinforced composites (PFRC) with transverse polarization. A small repeating area of the composite, representing a quarter of fiber surrounded by matrix is considered as representative volume element (RVE). The composite system consists of long parallel piezoelectric fibers with transversely isotropic properties and perfectly bounded to the isotropic matrix in a square array arrangement. In addition, the constituents are assumed to have both linear elastic and electrical behavior, whereas, the matrix is piezoelectrically passive. The element free Galerkin method is employed to obtain solution for the governing system of partial differential of equations. In this method, the Moving Least Square shape functions are used to approximate the field variable at arbitrary point. Comparison of the presented results with other techniques available in the literature reveals good agreement. It is demonstrated that the piezoelectric coefficient “e31” in the transverse polarization is considerably improved in comparison with corresponding coefficient of pure piezoelectric material. Furthermore, as a result, it is found that fibers with elliptical cross section may enhance the amount of electrical sensitivity of PFRC several times than circular fibers in a specific direction.

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In this paper, the buckling of rectangular plates subjected to non-uniform in-plane loading is investigated. At first the equilibrium equations of plate based on the first order shear deformation theory have been extracted. The kinematic relations have been assumed based on the von-Karman model and the Hook’s law has been considered as the constitutive equations. The adjacent equilibrium method has been used for deriving the stability equations. The equilibrium equations which are related to the prebuckling stress distribution, have been solved using the differential equations theory. To determine the buckling load of a simply supported plate, the Galerkin method has been used for solving the stability equations which are a system of differential equations with variable coefficients. In this paper, four types of in-plane loading, including the uniform, parabolic, cosine and triangular loading, have been considered and the effects of the plate aspect ratio and thickness on the buckling load has been investigated and the results have been compared with the finite element method and the classical plate theory. The comparison of the results show that for all loading cases, the buckling load computed by the classical plate theory is higher than the value obtained based on first order shear deformation theory.

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In this study, the static deflection and natural frequency of an electrostatically excited patch-coated microcantilever beam are analyzed. The proposed model is considered as the main element of many microsensors and microswitches. Firstly, the nonlinear motion equation is extracted by means of Hamilton principle, assuming shortening effect. Secondly, differential equations, governing the static deflection and free vibration equation around the stability point, are solved using Galerkin method and the three mode shapes of a uniform microbeam are employed as the comparison function. By assuming that the volume of deposited layer is constant, the variation of natural frequency and static deflection are examined in three different cases. In any cases, it is presumed that the second layer is initially deposited on the entire length of microbeam. In the first case, one end of coated layer is considered fix at the clamped side of microcantilever, and then its length is decreased from other side, where its thickness is increased. In the second case, one end of coated layer is perceived fix at the free side of microcantilever, and then its length is decreased from other side, where its thickness is escalated. In the third case, the length of second layer is decreased from both of left and right ends, where its thickness is expanded. In addition, the effect due to the change of the second layer position is considered on mechanical behavior of the system.

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In this article, a numerical solution of incompressible two-phase flow in isothermal condition, based on wetting pressure-wetting saturation formulation (Pw,Sw) using high order primal discontinuous Galerkin (DG) methods is considered which can capture the shock fronts of two-phase flow in heterogeneous porous media. In this presented model, the velocity field is reconstructed by a H(div) post-process in lowest order of Raviart-Thomas space (RT0). Also in this study, the scaled penalty and weighted average (harmonic average) formulation significantly improve the especial discretization formulation of governing equations which cause to reduce the instabilities in heterogamous media. The modified MLP slope limiter is used to remove the non-physical saturation values at end of each time step. In this study, the slope limiter should be considered as one of the main novelties due to the impressive effects in results stabilization. The proposed model is verified by pseudo 1D Buckley-Leverett and Mcwhorter problems. Two test cases, a problem for modeling the secondary recovery of petroleum reservoirs and other one a problem for detecting immiscible contamination are used to show the abilities of shock capturing two phases interface in porous media.

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The objective of this investigation is to present a semi-analytical method for studying the buckling of the moderately thick composite conical shells under axial compressive load. In order to derive the equilibrium equations of the conical shell, first order shear deformation shell theory is used. The equilibrium equations are derived by applying the principle of minimum potential energy to the energy function that they are in the type of partial differential equations. In the following, the partial differential equations are transformed to algebraic type by using Galerkin and differential quadrature methods and then the standard eigenvalue equation is formed and critical buckling load is calculated. Also, to validate the results obtained in this study, comparisons are made with outcomes of previous literatures and the results of Abaqus finite element software. Analyzing the results, shows the convergence speed and good accuracy of differential quadrature method and desired precision of Galerkin method in calculating the critical buckling load. Finally, the effect of cone angle, fiber orientation, boundary conditions, ratios of thickness to radius and length to radius of the critical buckling load are studied.

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In this paper, the dynamic behavior of atomic force microscope (AFM) based on non-classical strain gradient theory was analyzed. For this aim atomic force microscope micro-beam with attached tip has been modeled as a lumped mass. Micro-beam has stimulated via a piezoelectric element attached to the end of clamped and non-linear partial differential equation of the system has extracted based on Euler-Bernoulli theory and to be converted into ordinary differential equation by using Galerkin and separation method. The classic continuum theory because of lack of consideration size effect that has been observed in many experimental studies, has little accuracy in predicting the mechanical behavior of Nano devices. In this study, the stability region of micro-beam are determined analytically and validated by comparison with numerical results. Difference between presented analysis in dynamic behavior of micro-beam by classic and non-classic theories has been shown with variety of diagrams. It is clear that consideration the size effect changes the dynamical behavior of the problem completely and it is possible while classical theory predicts stable behavior for microscope the size effect is caused bi-stability. The results in this paper are very useful for the design and analysis of atomic force microscope.

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Dynamics analysis of the rotational axially moving pipe conveying fluid under simply supported condition investigated in this research. The pipe assumed as Euler Bernoulli beam. The gyroscopic force and mass eccentricity were considered in the research. Equations of motion are derived using Hamilton’s principle, resulting in two partial differential equations for the transverse motions. The non-dimensional equations were discretized via Galerkin’ method and were solved using Rung Kutta method (order 15s). The frequency response curve obtained in terms of non-dimensional rotational speed. The bifurcation diagrams are represented in the case that the non-dimensional fluid speed, non-dimensional axial speed and non-dimensional rotational speed were respectively varied and the dynamical behavior is numerically identified based on the Poincare' portrait. Numerical simulations indicated that the system response increases by increasing non-dimensional axial speed of the pipe, non-dimensional fluid speed and non-dimensional rotational speed of the pipe and then decreases after passing critical area. The system is unstable at critical point associated with non-dimensional axial speed. Poincare portrait indicates periodic motion in transverse vibrations of the pipe at some points of control parameters. Phase portrait and FFT (Fast Fourier Transform) diagrams were used for validation of the results.

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The present paper aims to evaluate a class of discontinuous Galerkin methods for modeling of coupled flow and mass transport equations in porous medium. Various combinations of primal discontinuous Galerkin methods were used for discretization of the coupled nonlinear system of flow and mass transport equations in a saturated porous medium and a fully implicit backward Euler scheme was applied for temporal discretization. The primal DGs have been developed successfully for density-dependent flows by applying both Cauchy and Dirichlet boundary conditions to the mass transport equation. To avoid the errors arising from non-compatible selection of DG methods for flow and mass transport equations, only compatible combinations were applied. To linearize the resulting nonlinear systems, Picard iterative technique was applied and a slope limiter was used to eliminate the nonphysical oscillations appeared in solution. For the purpose of consistent velocity approximation, Frolkovic-Knabner method was used. Three benchmark problems were simulated for validation and verification of the numerical code, which the results from the simulations show a good accuracy and low numerical dispersion for the model. Finally, to highlight the significance of consistent velocity approximation, a hydrostatic test problem was prepared.

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The complex behavior of the aquifer system is studied by solving a set of governing equations using either analytical or numerical methods. Numerical techniques like finite difference method (FDM) is being used to solve differential equation in some simple cases. Recently Meshless methods are developed in engineering fields. They are used for solving differential equations in both simple and complex cases. As this methods needs no meshing or re-meshing on the domain the shortages of meshing disappeared. Less studies already performed in groundwater flow modeling with meshless method. In this study Meshless local Petrov-Galerkin with moving least squares approximation function and spline weight function is used to model groundwater flow in Birjand unconfined aquifer in steady condition. The computed surface of groundwater with meshless local Petrov-Galerkin method is compared with the results observation. The results are found satisfactory. The relative mean error and root mean square error of computed groundwater surface from Meshless Local Petrov-Galerkin are 0.0002 and 0.483 respectively.

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In this paper, an implicit finite element-discontinuous Galerkin method for compressible viscous and inviscid flow is developed using Newton-Krylov algorithm with the objective of increasing the accuracy and convergence rate. For inviscid flows, an artificial viscosity is implemented in sharp gradient flow regions especially at high-order cases, increasing the accuracy of the solution. Moreover, for viscous flows, the accuracy is improved by using compact discontinuous Galerkin discretization method for elliptical terms. To reduce the computing CPU time and increase the convergence rate, an iterative Krylov type preconditioned linear solver is applied. For preconditioning, restarting, Block-Jacobi and block incomplete-LU factorization are employed for solving the linear system of the Jacobian matrix. The Jacobian matrix is constructed via finite difference perturbation technique. In this context, the performance of preconditioning matrix for three types of flow regimes of inviscid subsonic, inviscid transonic and viscous laminar subsonic are studied. In addition to complete the discussions, multigrid smoother with special conditions is applied for all preconditioning matrices. To improve the solver performance for higher order discretization, a lower order solution may be used as higher orders initial condition. Therefore, a middle phase is needed to transfer calculations from low to high order discretized domain and then the final Newton phase is continued. In addition, local time stepping is implemented to improve the rate of convergence. Consequently, the presented numerical method can be used as an efficient algorithm for high-order Discontinuous Galerkin flow simulation, especially for transonic inviscid and laminar viscous flows.

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In the present research, higher resonance frequencies are employed to improve the performance of the atomic force microscopy in the non-contact mode. Conventional models already used in the literature to study AFM microcantilever dynamics such as point-mass approach are not only incapable of modeling higher vibrational modes but also fail to predict microcantilever complicated dynamics with a sufficient accuracy. In this paper, the Hamilton’s extended principle is used to obtain equations governing the nonlinear oscillations of the AFM probe. Euler-Bernoulli beam assumptions and small deflection theory are assumed. The resulting partial differential equation is often converted to a set of ordinary differential equations and then this set is solved either numerically or based on perturbation methods. In the present research, however, the partial differential equation is attacked directly by a special perturbation technique. The accuracy of the present method is then verified by a combination of the Galerkin discretization scheme and a Rung - Kutta numerical solution. Finally, different behaviors of the AFM probe including static behavior, linear mode shapes and frequency response curves are investigated through several numerical simulations. It is found out that higher vibrational modes have smaller frequency shift. It is also found out that higher modes are faster in gathering surface information and also more sensitive to the excitation.

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In this study first the meshless local Petrov-Galerkin (MLPG) method by Radial Basis Function (RBF) has been explained entirely. In this way the governing channel flow expression that is based on the Laplace equation is expanded. In MLPG method, the problem domain is represented by a set of arbitrarily distributed nodes and Quadrature radial basis function is used for field function approximation and local integration is used to calculate the integrals. In the following, MLPG method is verified by exact solution in a numerical example. The Results show that MLPG method presented high accuracy and capability for solving the governing equation of the problem. Finally the velocity field is approximated in middle of nodes by RBF (MatLab code was adopted) in the uniform flow in a sloped channel problem. The MLPG results are compared with the isogeometric analysis (IA) method in the tutorial numerical example of Fluid flow modeling in channel, the velocity contours is detected, and their accuracy is demonstrated by means of several examples. The results showed good conformity compared to available analytical solution. The obtain results explain that Application of meshless method in Fluid flow modeling in channel show the applicability and efficiency of the meshless local Petrov-Galerkin method by Radial Basis Function method.

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Semi-supported steel shear walls (SSSW) are a new lateral resisting system whose plates do not have any direct connection to the main columns of structure. Instead, they are connected to secondary columns which do not carry the gravity loads. The applied lateral loads may create overturning moment on the middle storeys. The ultimate shear capacity of the SSSWs in presence of the overturning moment has been reasonably determined with an analytical procedure. It was finalized with some applicable interaction curves between the ultimate shear capacity and the overturning moment which can be used for analysis and design of this system. In addition, some experimental studies have been conducted to find an insight for the cyclic behavior of this system. As the elastic buckling of wall plate always occurs at the low levels of lateral loads, the system stays in a relatively large region of elastic post-buckling. In this region, the geometrical nonlinearity with linear material behavior appear in the wall plate. Thus, the storey shear force has a linear variation versus the lateral displacement until the first point of wall plate is yielded. Perhaps solution of the Von-karman plate equations is the best approach to find an analytical vision for the elastic stiffness of the SSSWs. These equations are described with two coupled nonlinear fourth order differential equations. The mentioned equations have been widely solved for many plates which are under combinations of different in-plane and out of plane loads and various boundary conditions and imperfections. In this study, the Galerkin method was employed in a semi analytical procedure to solve the Von-karman plate equations for the wall plate of SSSW system in a middle storey. This solution leads to achieve the displacement field of the SSSWs at the different levels of lateral loads until the first point of the wall plate is yielded. Thus, the linear variations of the in-plane displacement versus the lateral load will be obtained. Since the ultimate capacity has been previously measured, then an ideal elasto-plastic curve can be obtained for this system. The wall plate is supposed as a thin plate whose parallel edges have two different boundary conditions: two simply supported and two stiffened free edges where the wall plate is connected to the storeys beams and the secondary columns respectively. A sine monomial is considered as the deflection function which is satisfied the boundary conditions. Then, an algorithm is analytically developed to find the out of plane deflection of plate and the two-dimensional elasticity is used to determine the in-plane displacement of plate. The obtained results are compared with those of FE analysis and the suggested algorithm can be programmed in usual computers. The results show that some parameters such as the wall plate dimensions, the geometric properties of secondary columns (i.e. cross sectional area, moments of inertia), the storey shear force and yield stress of wall plate effect on the end point of elastic post-buckling. But, the slope of this region is independent from the variation of overturning moment and section of secondary columns.

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 While studying large-scale systems, non-local damping definition can beneficially model contact shear and long-range forces resulting from adjacent and non-adjacent elements in the set of interconnected dampers, damping patches and foundations which are modeled as non-local domains. If two or three dimensional systems are considered one dimensional (e.g. analyzing three dimensional beams based on Euler-Bernoulli, Rayleigh or Timoshenko Theories which simplifies the behavior of structures), the concept of non-local damping models arises to improve the accuracy of numerical results. Even though defining dissipative forces which are dependent on more parameters and quantities helpfully boost the validity of results compared to experimental cases in labs and three dimensional numerical analysis done by software, many researchers have widely employed viscous damping model to demonstrate damped behavior of the structures in the sake of simplicity. Actually viscous damping does not model accurately the dissipative behavior of real systems and practical structures often demonstrate some kind of viscoelasticity while vibration.  In the recent study, external damping force at any point in the domain is influenced by the past history of velocity and long-range interactions through convolution integral over proper kernel functions. As a consequence of applying Laplace transformation and using Galerkin method, the integro-partial differential equation of Rayleigh beam as a distributed parameter system turns to an ordinary differential equation governing a discrete system with finite degrees of freedom. Galerkin method is established based on error minimization of assumed mode method and despite Rayleigh and Rayleigh-Ritz methods can suitably analyze nonconservative systems including damped beams. Corresponding undamped mode shapes of Rayleigh beam which satisfy essential boundary conditions are chosen as the best admissible functions to expand the trial response of equation of motion. In order to get  continuous functions and finite weighted residual integral, the equation is presented in the weak form. Afterwards stiffness, damping and two types of mass matrices are determined with respect to generalized coordinates. To get scaled mode shapes the mass change method is considered to evaluate scaling factor, then the results are mass normalized. By equating dynamic stiffness matrix to zero and solving the resulting algebraic equation, complex and real eigenvalues are obtained which are respectively elastic and non-viscous modes in stable systems. It’s noteworthy to announce that contrasted with the viscously damped systems the degree of algebraic equation in the case of viscoelastic non-locally damped Rayleigh beam would generally be more than 2N (which N stands for total degrees of freedom). Accordingly eigenvectors are found based on Gaussian elimination method and null space of the dynamic stiffness matrix. Additionally, Neumann series expansion is developed to determine eigenvectors of vibrating systems effected by inertia. Finally, numerical results of Rayleigh and Euler-Bernoulli beams are compared. Very thin Rayleigh beams are verified and show great accuracy but when the thickness increases and inertial effect becomes bold, results of Rayleigh and Euler-Bernoulli beams are not matched anymore. Furthermore it’s understood from the graphs and tables that when the characteristic parameter of nonlocality and relaxation constant increase and tend to infinity, the nonlocal and non-viscous effects decline.