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The dynamic analysis of prestressed, bending-torsion coupled beams is revisited. The axially loaded beam is assumed to be slender, isotropic, homogeneous, and linearly elastic, exhibiting coupled flexural-torsional displacement caused by the end moment. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the vibration and stability of such beams are explored. Using the closed-form solutions of the uncoupled portions of the governing equations as the basis functions of approximation space, the dynamic, frequency-dependent, interpolation functions are developed, which are then used in conjunction with the weighted residual method to develop the Dynamic Finite Element (DFE) of the system. Having implemented the DFE in a MATLAB-based code, the resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies of illustrative beam examples, subjected to various boundary and load conditions. The proposed method is validated against limited available experimental and analytical data, those obtained from an in-house conventional Finite Element Method (FEM) code and FEM-based commercial software (ANSYS). In comparison with FEM, the DFE exhibits higher convergence rates and in the absence of end moment it produces exact results. Buckling analysis is also carried out to determine the critical end moment and compressive force for various load combinations.

Free vibration analysis of beams with single delamination undergoing bending-torsion coupling is made, using traditional finite element technique. The Galerkin weighted residual method is applied to convert the coupled differential equations of motion into to a discrete problem, where, in addition to the conventional mass and stiffness matrices, a delamination stiffness matrix, representing the extra stiffening effects at the delamination tips, is introduced. The linear eigenvalue problem resulting from the discretization along the length of the beam is solved to determine the frequencies and modes of free vibration. Both “free mode” and “constrained mode” delamination models are considered in formulation, and it is shown that the continuity (both kinematic and force) conditions at the beam span-wise locations corresponding to the extremities of the delaminated region, in particular, play a great role in “free mode” model formulation. Current trends in the literature are examined, and insight into different types of modeling techniques and constraint types are introduced. In addition, the data previously available in the literature and those obtained from a finite element-based commercial software are utilized to validate the presented modeling scheme and to verify the correctness of natural frequencies of the systems analyzed here. The paper ends with general discussions and conclusions on the presented theories and modeling approaches.

This paper presents a generalized formulation to describe nonuniform torsion in composite bars with variable cross section including higher-order warping modes. In that, the boundary-element subregion-by-subregion (BE SBS) technique is applied to determine the warping modes for cross sections constituted of any number of different materials. The distribution of the nonuniform warping along the axis is taken into account by multiplying the warping modes by derivatives of the associated twist angle functions. A new process based on the weighted residual method is proposed to solve the resulting global equilibrium equations. One also makes generic comments on the solution of the local coupled boundary-value problems (BVPs), which encompasses preconditioned Krylov solvers (embedded in the BE SBS technique), discontinuous boundary elements, and efficient (low-order) quadratures for singular and nearly-singular integrals. Efficiency and robustness of the technique are verified by comparing the present results with highly accurate 3D responses.

Free vibration of elastically constrained rectangular single-layered $$\hbox {MoS}_{\mathrm {2}}$$ is investigated by using a nonlocal Kirchhoff plate model with an initial stress. The variationally consistent elastically constrained boundary conditions are obtained by using the weighted residual method, while the governing equations of the nonlocal Kirchhoff plate model are known. A modified Fourier series method is applied to study the vibrational behaviors of elastically constrained nonlocal Kirchhoff plate models. The convergence and reliability of the modified Fourier series method is verified via comparison with the finite element method. A comprehensive parametric study is performed to show the influences of the boundary elastic constant, nonlocal parameter and initial stress on the vibrational behaviors of single-layered $$\hbox {MoS}_{\mathrm {2}}$$ . The results should be good for the design of nanoresonators .

An improved explicit time integration method is proposed for linear and nonlinear dynamics. Its calculation procedure is obtained with cubic B-spline interpolation approximation and weighted residual method. In the formulation, a momentum corrector is used to improve actual computation accuracy, especially for some special discontinuous loads. Analytical solutions of the local truncation errors, algorithmic damping and period elongation have been deduced to obtain the influence of algorithmic parameters on these basis algorithmic properties. The proposed method possesses at least second-order accuracy and can achieve at most third-order accuracy for no physical damping case. With free algorithmic parameters, the proposed method has controllable stability and numerical dissipation. Some demonstrative numerical examples are tested to confirm high efficiency of the proposed method for a variety of dynamic problems such as, dynamic response analysis of linear systems under various representative applied loads, finite element analysis (FEA) for dynamic response of engineering structures, and nonlinear dynamic analysis for strong nonlinear system.

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.

In this work, we present a new framework to couple peridynamics least squares minimization with finite element method (PDLSM-FEM) for three-dimensional (3D) problems based on the weighted residual method (WRM). In PDLSM-FEM, the problem domain is divided into a PD domain and a FEM domain, and both domains are discretized into uniform or non-uniform elements. Within this framework, the coupling between the PD domain and the FEM domain is straightforward, and the formulation can be converted conveniently between PDLSM, FEM, and PDLSM-FEM. To directly and implicitly solve for displacements, we present an algorithm to assemble the global stiffness matrix in a compressed sparse row (CSR) format by the message passing interface (MPI) technique. The system equations are solved by the parallel direct sparse solver (PDSS) of the Intel Math Kernel Library (MKL). Three static examples are performed and the results show the proposed PDLSM-FEM can substantially reduce computational cost in terms of model efficiency and memory usage. Comparison between PDLSM-FEM results and FEM results using ANSYS demonstrates the good accuracy of the proposed PDLSM-FEM.

As a major limitation on the process efficiency of the manufacturing industry, milling chatter can be effectively alleviated by the optimal parameter from the stability lobe diagrams. Numerical algorithms based on the equidistant discretization points are usually applied to solve delay differential equations that occurred in the milling dynamics. However, for the presence of the Runge phenomenon, the computational accuracy does not continuously improve with the increase of the approximation order. This paper proposes a novel Chebyshev-Gauss pseudospectral method and implements it in the angle domain for fast and accurate milling stability prediction. The highlights of the proposed method are that the state term in the forced vibration interval of the milling model is approximated with the barycentric Lagrange interpolation with nonuniform Chebyshev-Gauss points, the corresponding derivative term is calculated by an improved Chebyshev-Gauss differential matrix, and then the weighted residual technique with the Clenshaw-Curtis quadrature rule is applied for the first time to minimize the error function. By taking advantage of the above obtained algebraic equations and analytical solution of the free vibration interval, the Floquet transition matrix is constructed, and its critical eigenvalue is utilized for determining the milling stability. Finally, two benchmark milling examples demonstrate that the proposed method achieves the most stable and fastest spectral convergence rate, and brings a great improvement on the computational efficiency and accuracy, compared with the representative methods. Meanwhile, experimental verification in the low-speed and high-speed domains further indicates the applicability of the proposed method.

This article explores progressive solitary solution and shock solution for the dust–ion-acoustic waves (DIAWs) propagating in a collisional unmagnetized dusty plasma containing dust grains with negative dust charge, positive ions, neutral particles and Maxwellian electrons. In an earlier literature, Gao et al. in Braz. J. Phys. (2019) derived nonplanar KdV–Burgers (NKDVB) equation from the basic field equation utilizing reductive perturbation method (RPT). Further, assuming that conservation law holds in the system they obtained an approximate solitary wave solution, although the presence of Burgers term causes increase of the effect from viscosity and opposes the conservation law in the system. In that case, shock is wave usually generated due to the effect of strong anomalous dissipation. Being aware of the facts, simplified Hirota bilinear method is employed and shock-type wave solution is obtained. Again, if dissipation is weak, the balance between the dispersion and nonlinearity may lead to form solitary solution. Considering the case of weak dissipation, solitary wave solution is directly explored (without assuming conservation law) by applying a weighted residual method. Some 2D and 3D graphs are depicted to analyze the propagating behavior of nonplanar waves in different parametric spaces.

A boundary element method formulation is developed and validated through the solution of problems governed by the diffusion-wave equation, for which the order of the time derivative, say α, ranges in the interval (1, 2). This fractional time derivative is defined as an integro-differential operator in the Caputo sense. For α = 2, one recovers the classical wave equation. From the weighted residual method point of view, with the Laplace equation fundamental solution playing the role of the weighting function, a D-BEM type formulation is generated, in which the integrand of the domain integral contains the fractional time derivative. The Houbolt scheme is adopted to represent the second order time derivative of the variable of interest, say u, which appears in the Caputo operator. In all the examples, the proposed formulation provided accurate results for α as small as 1.05. Indeed, the analyses were carried out for α = 1.05, 1.2, 1.5, 1.8, 2.0, with the aim of verifying the influence of the order of the time derivative in the results.

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable.

In current investigation, a novel implementation of intelligent numerical computing solver based on multi-layer perceptron (MLP) feed-forward back-propagation artificial neural networks (ANN) with the Levenberg–Marquard algorithm is provided to interpret heat generation/absorption and radiation phenomenon in unsteady electrically conducting Williamson liquid flow along porous stretching surface. Heat phenomenon is investigated by taking convective boundary condition along with both velocity and thermal slip phenomena. The original nonlinear coupled PDEs representing the fluidic model are transformed to an analogous nonlinear ODEs system via incorporating appropriate transformations. A data set for proposed MLP-ANN is generated for various scenarios of fluidic model by variation of involved pertinent parameters via Galerkin weighted residual method (GWRM). In order to predict the (MLP) values, a multi-layer perceptron (MLP) artificial neural network (ANN) has been developed. There are 10 neurons in hidden layer of feed forward (FF) back propagation (BP) network model. The predictive performance of ANN model has been analyzed by comparing the results obtained from the ANN model using Levenberg-Marquard algorithm as the training algorithm with the target values. When the obtained Mean Square Error (MSE), Coefficient of Determination (R) and error rate values have been analyzed, it has been concluded that the ANN model can predict SFC and NN values with high accuracy. According to the findings of current analysis, ANN approach is accurate, effective and conveniently applicable for simulating the slip flow of Williamson fluid towards the stretching plate with heat generation/absorption. The obtained results showed that ANNs are an ideal tool that can be used to predict Skin Friction Coefficients and Nusselt Number values.

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

This research work performs the first time exploring and addressing the flexomagnetic property in a shear deformable piezomagnetic structure. The strain gradient reveals flexomagneticity in a magnetization phenomenon of structures regardless of their atomic lattice is symmetrical or asymmetrical. It is assumed that a synchronous converse magnetization couples both piezomagnetic and flexomagnetic features into the material structure. The mathematical modeling begins with the Timoshenko beam model to find the governing equations and non-classical boundary conditions based on shear deformations. Flexomagneticity evolves at a small scale and dominant at micro/nanosize structures. Meanwhile, the well-known Eringen’s-type model of nonlocal strain gradient elasticity is integrated with the mathematical process to fulfill the scaling behavior. From the viewpoint of the solution, the displacement of the physical model after deformation is carried out as the analytical solution of the Galerkin weighted residual method (GWRM), helping us obtain the numerical outcomes on the basis of the simple end conditions. The best of our achievements display that considering shear deformation is essential for nanobeams with larger values of strain gradient parameter and small amounts of the nonlocal coefficient. Furthermore, we showed that the flexomagnetic (FM) effect brings about more noticeable shear deformations’ influence.

In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates.

This article presents the approximate solution of non-linear dynamic energy model of multiple effect evaporator (MEE) using Fourier series and metaheuristics. The dynamic model of MEE involves first-order simultaneous ordinary differential equations (SODEs). Prior to solving the dynamic model, the non-linear steady-state model is solved to obtain the optimum steady-state process parameters. These process parameters serve as the initial conditions (constraints) for the SODEs. The SODEs are exemplified as an optimization problem by the weighted residual function to produce their approximate solutions. The optimization task is to find the best estimates of unknown coefficients in the Fourier series expansion using two preeminent metaheuristic approaches: Particle swarm optimization and harmony search. Besides, the influence of the number of approximation terms in Fourier series expansion on the accuracy of the approximate solutions has been investigated. The solution of the dynamic model assists in the investigation of open-loop dynamics of the MEE. Moreover, the acquired results may assist in designing suitable controllers to ensure energy-efficient performance of MEE and to monitor the product quality. The optimization results reveal that both the metaheuristic approaches offer minimum violation of the constraints and, therefore, validate their efficiency in solving such complex non-linear energy models.

This study focuses on the Bubnov-Galerkin finite element model used to obtain the stresses and pressure fields set up at various cross-sections of a blank during the metal forming process. Four Lagrange quadratic elements were assembled to represent the various blanks. The governing equation adopted is a one-dimensional differential equation describing the pressures and stresses exerted on the forming process. In conducting the analysis, the various blanks are divided into a finite number of elements and further applying the Bubnov-Galerkin-weighted residual method to obtain the weighted integral form; the finite element model is obtained in a matrix form, from the weighted residual boundary conditions which are now applied to obtain the pressure distribution across the cross-section of the various blanks. Finite element results are obtained for a value of the coefficient of friction, die angle, length, and blank radius. Solution of the finite element method was compared with the exact solution, along with an experimental test known as press-fit analysis using Ansys Workbench, a program-controlled mesh of polar hexahedral elements which was utilized to further validate the result. Furthermore, the use of a computer-aided software assists in visualizing the solution of the tube sinking problem. The results were all presented in both tabular and graphical modes. This study shows that there are potentials for using this approach in engineering practices to ensure that the strength of materials be considered before usage in design and fabrication. These results are expected to improve advance knowledge in manufacturing processes.

For the discretization of the integral fractional Laplacian $(-��)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.

In this paper, a variational approach is developed to solve the exterior acoustic-structure interaction problem. The Reissner-Naghdi's thin shell theory and Helmholtz equation are adopted to formulate the energy functionals for structural and acoustic domains, respectively. With introduced Lagrangian multipliers, a domain partitioning technique for both structural and acoustic domains is developed. Meanwhile, a least-square weighted residual method is adopted on the common interfaces of adjacent segments to guarantee the numerical stability of the present method. Taking the Sommerfeld radiation condition into account, the local absorbing boundary condition is introduced at the boundary of the acoustic domain. The robust performance of the present method at the interfaces and boundaries allows the use of any closed orthogonal polynomials to expand the structural displacements and acoustic pressure. Free and forced vibration responses of a submerged spherical shell are compared with available exact solutions or FEM results to validate the present method. Additionally, the advantages and stability of the local absorbing boundary condition are carefully examined. The effects of the radius of the acoustic domain and the structural damping are also analyzed to present the power of proposed method to address the acoustic-structure interaction problem.

We present a fully coupled level set and volume of fluid method for free surface flow simulations on unstructured grids in two- and three-dimensions. The evolution of the level set function and the volume fraction are updated at each time step. The level set advection equation is solved by a least squares weighted residual method while the volume fraction advection is solved using an unsplit Eulerian-Lagrangian scheme. The reconstruction of the interface in the volume of fluid method is performed using the volume fraction information together with the normal vector obtained from the level set function. The reconstructed interface is then used to reinitialize the level set function. Thanks to the fully coupling of the two methods, the interface reconstruction is carried out efficiently and the mass conservation is preserved exactly. The proposed method is validated against several benchmarks and is coupled with the incompressible Navier-Stokes solver to solve two-phase flow problems. Numerical results show that the method is capable of resolving complex interface changes efficiently and accurately.

System health monitoring as the basis of prognostics and health management (PHM) aims to explore health indices/features for PHM to do condition monitoring, perform abnormal detection, and provide degradation trends for prognostic models. Kurtosis and negative entropy are two classic and popular indices to measure the sparsity of impulsive transients, and they are prone to be affected by impulsive noise. The purpose of this article is twofold. Theoretically, new propositions and their proofs are proposed to illustrate how kurtosis and negative entropy can help to characterize impulsive transients and why they are affected by impulsive noise. Next, a weighted residual regression-based index is proposed to relieve the sensitivity of kurtosis and entropy to impulsive noise and to provide monotonic trends for gear and bearing degradation assessment. Theoretical results show that kurtosis and negative entropy are changed with the length of nonimpulsive regions of impulsive transients. Experimental results demonstrate that the proposed method has better incipient fault detection ability and monotonic trending ability than kurtosis and negative entropy for bearing and gear health monitoring and degradation assessment.

Solutions of 3D thermal analysis in inhomogeneous media with variable thermal source are divided into homogeneous and particular solutions by a novel boundary-type element-free method, namely virtual boundary element-free Galerkin method (VBEFGM). The homogeneous solution can be obtained by the virtual boundary element-free method (VBEFM). The virtual source function of the homogeneous solution and the unknown coefficient of the particular solution are constructed by the radial basis function interpolation (RBFI). And VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source is real meshfree and does not require the background integral element. Its equation is obtained the Galerkin method of the weighted residual methods. The demonstration of ANSYS Parametric Design Language (APDL) can greatly facilitate the programming for the extraction of the coordinate information and the outer normal cosines on nodes. The calculation scheme and the specific weighted coefficients are deducted in detail for 3D thermal analysis in inhomogeneous media with variable thermal source by VBEFGM. The numerical calculational steps and flow chart are given. Three numerical examples are computed. The results are compared with other numerical methods and the exact solutions. The stability and the accuracy of the proposed method is proved by VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source.

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.

Differential equations can often be seen in many fields of scientific research and engineering. Typically, finding the analytical (exact) solution is expensive task in terms of computational effort and may not an attainable task for some complex tasks. To effectively handle a wide variety of linear and nonlinear differential equations, this paper presents an approximate methodology based on hybrid Harris hawks–Nelder–Mead optimization algorithm with the aim to achieve accurate and reliable solution. The proposed methodology is introduced on basis of Fourier series expansion and Harris hawks–Nelder–Mead optimization algorithm. In this sense, the differential equation is represented as an optimization model by the means of the weighted residual function (cost function) that needed to be minimized, where the boundary and initial conditions of the differential equation are considered as the constraints of the optimization model. The practicality and efficiency of the proposed algorithm are demonstrated through six differential equations with different nature as well as four mechanical engineering differential equations. The comparison against different algorithms, by using the generational distance metric and Wilcoxon sign rank test, showed the effectiveness of the proposed algorithm.

A simple boundary method is developed for the solution of isotropic thick plates with in-plane arbitrarily variable material properties or thickness. Equilibrated basis functions which have proved to be effective in a variety of problems, are adopted for the bending problem of thick plates. The bases are created through a weighted residual approach over a fictitious rectangular domain so as to approximately satisfy the partial differential equation of the problem. This omits the necessity of the bases to analytically satisfy the equilibrium, thus simplifying the application of the method. Boundary conditions are applied to the approximate solution through a collocation technique, which considerably reduces its computational expenses. Mindlin’s first order and Levinson’s third-order shear deformation theories are adopted for the formulation. To accommodate more complicated geometries, a simple domain decomposition approach is also developed.

This paper presents numerical solution of second order singularly perturbed self-adjoint boundary value problems using weighted residual method of Galerkin type. First, for the given problem, the residue was computed using appropriate approximated basis function which satisfies all the boundary conditions. Then, using the chosen weighting function, integrating the weighted residue over the domain and the given differential equation is transformed to linear systems of algebraic equations. Further, these algebraic equations were solved using Galerkin method. To validate the applicability of the proposed method, two model examples have been considered and solved for different values of perturbation parameter and with different order of basis function. Additionally, convergence of error bounds has been established for the method. As it can be observed from the numerical results, the present method approximates the exact solution very well. Moreover, the present method gives better accuracy when the order of basis function is increased and it also improves the result of the methods existing in the literature. Keywords: Singularly perturbed problems, Self-adjoint problem, Galerkin method, Boundary value problems.

Static and dynamic analyses for the pre- and post-buckling behavior of an axially loaded nanobeam modeled with a general form of Eringen’s nonlocal theory are conducted and discussed. The general nonlocal theory is a modified form of Eringen’s nonlocal elasticity theory that, contrary to Eringen’s, considers separate attenuation functions to account for the long-range interactions for the two different material moduli and thus introduces an additional nonlocal parameter. The nanobeam is modeled considering Euler–Bernoulli beam theory and the von Karman geometric nonlinearity for midplane stretching in end-constrained beams. Three sets of boundary conditions are studied, namely clamped–clamped, clamped-hinged, and hinged–hinged. The governing equations are derived by virtue of the Hamilton’s principle. Introducing the general nonlocal theory into a beam system increases the order of the resultant transverse equation of motion from fourth to sixth, thus requiring two additional higher-order boundary conditions. These higher-order boundary conditions are derived using a weighted residual approach and are thus variationally consistent. Through this effort, a focus is placed on the effects that these higher-order boundary conditions have on the static and dynamic responses of the system. Specifically, the nonlocal parameters, Poisson ratio, boundary conditions, and axial load are varied. The static critical buckling loads, bifurcation diagrams, and static post-buckling configurations are examined and presented. Additionally, through a linear eigenvalue analysis, the natural frequencies and mode shapes in the pre- and post-buckling regimes are examined. It is shown that the system is very sensitive to the selected nonlocal parameters and higher-order boundary conditions for both the static and dynamic responses. It is the hope that this expanded model can be used by other researchers to accurately model nanobeams composed of materials outside the limits of applicability of Eringen’s nonlocal theory.

The Conventional Finite Element Method for the analysis of plate bending based on a the principle of stationarity of total potential energy which is well known as Stiffness Formulation , using a variety of conformal and no-conformal elements was widely applicable in early of numerical solutions for such problems , later in order to improve the efficiency of such solutions , a more general and flexible formulation called Mixed Formulation based on variational principles which can be regarded as an extension of the Stiffness principle become important alternative. The explicit stiffness matrix for plate bending was given in the most if it`s not in all literatures so the reader can easy follow the solving procedure for such problems and verifying any published results , however such like-matrix (Augmented Matrix) in case of mixed formulation not given , at least for simple elements in the literatures as well as concerned researches . This quietly leads to some difficulties concerning the verification as well as understanding the published results. The main objective of this paper is to introduce this matrix in abbreviated size followed by a applicability through a some detailed examples for some bending models. The derived matrix will be helpful subject of research work in addition to , reveals a very good feeling with understanding and verification the published results in plus to comparing with the analytical solutions of different plate bending problems as the reader can do that.

Natural convection in an inclined cavity filled with saturated porous media has been investigated. A time-dependent solution using a spectral weighted residual method is determined. Preparing the g...

The main aim of this paper is to examine the efficiency and the effect of various single-term weighted residual methods, including Bubnov–Galerkin, least squares, and point collocation methods on the free vibration behavior of viscoelastic plates. The refined classical plate theory and Kelvin–Voigt/standard solid viscoelastic models are adopted for kinematic and constitutive relations of foam plates, respectively. The spatial domain is discretized using weighted residual methods with shape functions of bending component of transverse deflections. The resulted algebraic eigenvalue problems with frequency-dependent coefficients are solved via an iterative numerical algorithm. For elastic plates, the present results based on four different methods are compared with their counterparts which are obtained based on 3-dimensional, classical, first- and higher-order shear deformation theories under Navier, Levy, and fully clamped boundary conditions. For simply supported viscoelastic plates, frequencies are compared with exact solutions, and acceptable accuracy is observed. Then, parametric studies are undertaken to assess the vibration mode, bulk/shear ratio, thickness ratio, material model, and boundary condition. From these results, it is revealed that bulk–shear ratio is a key factor while material model makes no significant difference on the real part of the vibration frequencies of fully free plates.

PurposeThis paper is concerned with new formulations of local meshfree and finite element numerical methods, for the solution of two-dimensional problems in linear elasticity.Design/methodology/approachIn the local domain, assigned to each node of a discretization, the work theorem establishes an energy relationship between a statically admissible stress field and an independent kinematically admissible strain field. This relationship, derived as a weighted residual weak form, is expressed as an integral local form. Based on the independence of the stress and strain fields, this local form of the work theorem is kinematically formulated with a simple rigid-body displacement to be applied by local meshfree and finite element numerical methods. The main feature of this paper is the use of a linearly integrated local form that implements a quite simple algorithm with no further integration required.FindingsThe reduced integration, performed by this linearly integrated formulation, plays a key role in the behavior of local numerical methods, since it implies a reduction of the nodal stiffness which, in turn, leads to an increase of the solution accuracy and, which is most important, presents no instabilities, unlike nodal integration methods without stabilization. As a consequence of using such a convenient linearly integrated local form, the derived meshfree and finite element numerical methods become fast and accurate, which is a feature of paramount importance, as far as computational efficiency of numerical methods is concerned. Three benchmark problems were analyzed with these techniques, in order to assess the accuracy and efficiency of the new integrated local formulations of meshfree and finite element numerical methods. The results obtained in this work are in perfect agreement with those of the available analytical solutions and, furthermore, outperform the computational efficiency of other methods. Thus, the accuracy and efficiency of the local numerical methods presented in this paper make this a very reliable and robust formulation.Originality/valuePresentation of a new local mesh-free numerical method. The method, linearly integrated along the boundary of the local domain, implements an algorithm with no further integration required. The method is absolutely reliable, with remarkably-accurate results. The method is quite robust, with extremely-fast computations.

In this paper, a modified variational method is developed to study the free and forced vibration of coupled straight–curved beam systems with an arbitrary number of eccentric discontinuities (EDs). Based on the generalized shell theory, the kinetic and potential functional of the curved beam with arbitrary subtended angles is formulated. Since the shear and inertial (or radial–tangential–rotational coupling) effects are included for the curved beam, the longitudinal vibration is also introduced to the energy functional for a straight Timoshenko beam. Using corresponding coordinate transformations, the Lagrange multiplier method and least-square weighted residual method are employed to impose the continuity constraints on the internal interfaces and boundaries among the straight and curved beams. The proposed method allows a flexible choice of the admissible functions and can be used for various combinations of the straight and curved beams to model corresponding engineering structures. Concentrated forces, uniformly distributed loads and space-dependent loads are considered to demonstrate great efficiency and accuracy of the present approach for the forced as well as the free vibration of the coupled system. Most of the present results are compared with those from finite element program ANSYS, and good agreement is observed. Influences of the EDs on the dynamic responses of the coupled system are also examined.

In this paper, we demonstrate that with a set of bases weakly satisfying the governing equation of an engineering problem on fictitious subdomains, known here as equilibrated basis functions (EqBFs), a variety of structural problems may be solved with ease of domain decomposition/discretization. Such a feature enables us to select the sub-domains freely (in contrast to the finite element method for instance). The EqBFs are constructed using Chebyshev polynomials and through a weighted residual integration over fictitious rectangular subdomains. Through some numerical examples, it is shown that the method performs very well even in comparison with efficient existing methods.

The effects of different parameters on the nonlinear dynamic characteristics of macrofiber composite (MFC) microsheet with graphene (GP) skin under non-uniform thermal field are investigated. Firstly, the physical parameters of the MFC–GP structure are calculated by the mixing rule, and the constitutive equations of the structure are set up by employing the Eringen theory. The nonlinear dynamic equations of the microsheet are obtained by using Hamilton’s principle. Then, the heat conduction equations of the microsheet are considered, adopting Green and Naghdi’s generalized thermoelasticity theory. According to the Galerkin weighted residual method, the thermoelasticity coupling equations of the structure are obtained. Meanwhile, the influence of the positive piezoelectric effect of GP and MFC on the vibration response of the structure is also investigated. The nonlinear dynamic governing equations including displacement, coupled thermoelasticity, and electricity field are discretized by the Galerkin method. The effects of non-local parameter, volume fraction of GP, and thermal and electricity coupling coefficients on structural dynamic behavior are discussed in the numerical simulation.

Subsurface scatters are sometimes masked by reflectors in seismic migration images, because the diffractions are much weaker in energy than the reflections. We propose a novel imaging method, named reflection-damped plane-wave least-squares reverse time migration (RD_PLSRTM), to enhance the scatters in the migration image. We formulate seismic imaging as an inverse problem that minimizes a weighted residual between the modeled and observed seismic data. In the proposed approach, we use the plane-wave destruction filter to separate the diffractions from the reflections in the data residual. A reflection-damped weighting matrix is then used to govern the fitting of the diffractions and the reflections, and therefore emphasize the updates of the scatters. The inverse problem is finally solved by using an iteratively reweighted least-squares (IRLS) algorithm. The proposed method provides a generalized formulation that could be reduced to conventional PLSRTM and PLSRTM of diffractions (PLSRTM_D) by using specific damping factors. We conduct imaging tests on synthetic and field data that prove the superiority of the proposed method over PLSRTM in imaging deep scatters and subsalt scatters. Compared with PLSRTM_D, it could produce high-quality images of not only the scatters but also the reflectors.

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduct...

In this paper a novel method is presented to solve problems with weak singularities in two-dimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other mesh-based or mesh-less approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar co-ordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the well-known literature or solved by well-established techniques, will demonstrate the capability of the method in problems related to composite materials.

In this study, a novel method for improving the simulation of wave propagation in Peridynamic (PD) media is investigated. Initially, the dispersion properties of the nonlocal Bond-Based Peridynamic model are computed for 1-D and 2-D uniform grids. The optimization problem, developed through inverse analysis, is set up by comparing exact and numerical dispersion and minimizing the error. Various weighted residual techniques, i.e., point collocation, sub-domain collocation, least square approximation and the Galerkin method, are adopted and the modification of the wave dispersion is then proposed. It is found that the proposed methods are able to significantly improve the description of wave dispersion phenomena in both 1-D and 2-D PD models.

International audience; A numerical scheme grounded on the Boundary Element Method expressed in the Frequency Domain is proposed to perform Nonsmooth Modal Analysis of one-dimensional bar systems. The latter aims at finding continuous families of periodic orbits of mechanical components featuring unilateral contact constraints. The proposed formulation does not assume a semi-discretization in space of the governing Partial Differential Equations, as achieved in the Finite Element Method, and so mitigates a few associated numerical difficulties, such as chattering at the contact interface, or the questionable approximation of internal resonance conditions. The nonsmooth Signorini condition stemming from the unilateral contact constraint is enforced in a weighted residual sense via the Harmonic Balance Method. Periodic responses are investigated in the form of energy-frequency backbone curves along with the associated displacement fields. It is found that for the one-bar systems, the results compare well with existing works and the proposed methodology stands as a viable option in the field of interest. The two-bar system, for which no known results are reported in the literature, exhibits very rich nonsmooth modal dynamics with entangled nonsmooth modal motions combining hardening and softening effects via the intricate interaction of various, possibly subharmonic and internally resonant nonsmooth modes of the two bars.Two-bar system nonsmooth mode of vibration corresponding to Figure 14(b) Two-bar system nonsmooth mode of vibration corresponding to Figure 14(g) Two-bar system nonsmooth mode of vibration corresponding to Figure 18(b) Two-bar system nonsmooth mode of vibration corresponding to Figure 18(d) For all animations above, the two bars share a common colormap, from dark blue for negative displacements to yellow for positive displacements, in agreement with the axes x [left bar] and y [right bar] defined in figure 1(c), x positive to the right and y positive to the left.

An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

Accurate determination of the response of structures under dynamic loads such as earthquake loads plays an important role in the safe and economical design of structures. The purpose of this paper is to utilize a novel solution method based on the use of exponential basis functions for dynamic analysis of Bernoulli beam subjected to different types of base excitations. This method was firstly introduced for solving scalar wave propagation problems, named as stepwise time-weighted residual method. The proposed method considers the solution as a series of exponential basis functions with unknown constant coefficients; and the problem is solved in time without the need for spatial discretization of the beam and by using an appropriate recursive relation to correct the coefficients of the exponential bases. In order to apply the earthquake excitation, first by using the central finite difference relation, the earthquake acceleration history is converted to displacement history. Moreover, the displacement history is applied to the beam as a time-varying boundary condition. In this study, the capabilities of the proposed method in solving several sample problems of vibration of single and multi-span beams under various stimuli such as earthquake acceleration variations are compared with the results of other existing methods.

In this study, a non-probabilistic approach-based Navier’s method (NM) and Galerkin weighted residual method (GWRM) in terms of double parametric form have been proposed to investigate the buckling behavior of Euler–Bernoulli nonlocal beam under the framework of Eringen’s nonlocal elasticity theory, considering the structural parameters as imprecise or uncertain. The uncertainties in Young’s modulus and diameter of the beam are modeled in terms of triangular fuzzy numbers. The critical buckling loads are calculated for hinged–hinged, clamped–hinged, and clamped–clamped boundary conditions, and these results are compared with the deterministic model in special cases, demonstrating robust agreement. Further, a random sampling technique-based method, namely Monte Carlo simulation technique (MCST), has been implemented to compute the critical buckling loads of uncertain systems. Also, the critical buckling loads obtained from the uncertain model in terms of lower bound and upper bound by the non-probabilistic methods, viz. NM and GWRM, are again verified with the MCST with their time periods, demonstrating the efficacy, accuracy, and effectiveness of the proposed uncertain model. A comparative study is also carried out among the non-probabilistic methods and MCST to demonstrate the effectiveness of methods with respect to time. Additionally, a parametric study has been performed to display the propagation of uncertainties into the nonlocal system in the form of critical buckling loads.

In this paper, we present a weighted residual Galerkin method to solve linear functional differential equations. We consider the problem with variable coefficients under initial conditions. Assuming the exact solution of the problem has a Taylor series expansion convergent in the relevant domain, we seek a solution of the given problem in the form of a polynomial having degree $N$ of our choice. Substituting this polynomial with unknown coefficients in the given equation yields an expression linear in these coefficients. We then proceed as in the weighted residual method and take inner product of this expression with monomials up to degree $N$, resulting in $N+1$ linear algebraic equations. Appropriately incorporating the initial conditions and solving the resulting linear system, we obtain the approximate solution to the given problem. Additionally, we present a way of estimating the absolute error of the obtained approximation, which is then used to improve the original approximation through a method called residual correction. We also show that the upper bound for the error of the proposed method depends on the Taylor truncation error of the exact solution. The proposed scheme and the residual correction technique are illustrated in several example problems.

In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we ...

A new dynamic model of the variable-length cable used in a vertically tethered underwater drill rig is presented in this paper, which also includes the dynamics of its fixed-length counterpart. First, the differential equation of the variable-length cable is derived through the theorem of linear momentum, and its finite element formulation is obtained by the Galerkin weighted residual method correspondingly. Second, the implicit time integration formulation is derived by using the Newmark-β algorithm. And then the proposed model is validated by comparing with the experimental data of the fixed-length underwater steel-armored umbilical cable of a work-class Remotely Operated Vehicle (ROV), and the variable-length underwater cables made of nylon and rubber, respectively, and the calculated results show good agreement with the experimental data. Third, the variable-length cable dynamics of payout and reel-in of the underwater drill rig is analyzed under three kinds of sea conditions, i.e., still water, sinusoidal heave, and the actual heave, respectively. Some conclusions are summarized in the end.

The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.

To improve the efficiency for managing power systems with constant time delay, an improved frequency-weighted model reduction is proposed. First, the Pade approximant allows to represent the time delay in the state-space form and build a linear model of the closed-loop power system containing a wide-area damping controller. Then, a weight function is introduced to solve the equilibrium transformation matrix and balance the original system. Finally, the residual order reduction is applied to the model of the power system considering time delay. The proposed method is evaluated in the two-area four-machine test system and New England system, and the effectiveness of the proposed reduction method is verified.